\input{"preamble.tex"}

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\title{
\rule{\linewidth}{1pt} \\
\textbf{
    Sheaf Cohomology
  }
    \\ {\normalsize Lectures by Valery Alexeev. University of Georgia,
Spring 2022} \\
  \rule{\linewidth}{2pt}
}
\titlehead{
    \begin{center}
  \includegraphics[width=\linewidth,height=0.45\textheight,keepaspectratio]{figures/cover.png}
  \end{center}
       \begin{minipage}{.35\linewidth}
    \begin{flushleft}
      \vspace{2em}
      {\fontsize{6pt}{2pt} \textit{Notes: These are notes live-tex'd
from a graduate course in Sheaf Cohomology taught by Valery Alexeev at
the University of Georgia in Spring 2022. As such, any errors or
inaccuracies are almost certainly my own. } } \\
    \end{flushleft}
    \end{minipage}
    \hfill
    \begin{minipage}{.65\linewidth}
    \end{minipage}
  }







\begin{document}

\date{}
\maketitle
\begin{flushleft}
\textit{D. Zack Garza} \\
\textit{University of Georgia} \\
  \textit{\href{mailto: dzackgarza@gmail.com}{dzackgarza@gmail.com}} \\
{\tiny \textit{Last updated:} 2022-05-29 }
\end{flushleft}


\newpage

% Note: addsec only in KomaScript
\addsec{Table of Contents}
\tableofcontents
\newpage

\hypertarget{intro-motivations-monday-january-10}{%
\section{Intro, Motivations (Monday, January
10)}\label{intro-motivations-monday-january-10}}

\begin{remark}

Topic: cohomology of sheaves and derived categories. The plan:

\begin{itemize}
\tightlist
\item
  Sheaves (see ELC notes)
\item
  Derived functors and coherent sheaves (see ELC notes)
\item
  Derived categories (Gelfand-Manin, Tohoku)
\end{itemize}

References:

\begin{itemize}
\tightlist
\item
  Valery's notes (see ELC)
\item
  Gelfand-Manin, \emph{Methods of Homological Algebra}.
\end{itemize}

\end{remark}

\begin{remark}

Compare (genus \(g\)) Riemann surfaces in the classical topology to
(genus \(g\), projective) algebraic curves over \({\mathbb{C}}\) in the
Zariski topology. Recall that
\begin{align*}
H^*(\Sigma_g; {\mathbb{Z}}) =
\begin{cases}
{\mathbb{Z}}& *=0, 2 
\\
{\mathbb{Z}}^{2g} & *=1
\\
0 & \text{else}.
\end{cases}
\end{align*}
Note that this is a linear invariant in the sense that the constituents
are free abelian groups, and we can extract a numerical invariant. For
surfaces up to homeomorphism, this distinguishes them completely.

For algebraic curves, note that the topology is very different: the only
closed sets are finite. In this topology,
\begin{align*}
H^*(X; {\mathbb{Z}})=
\begin{cases}
{\mathbb{Z}}& *=0 
\\
0 & \text{else},
\end{cases}
\end{align*}
which doesn't see the genus at all. In fact all such curves are
homeomorphic in this topology, witnessed by picking any bijection and
noting that it sends closed sets to closed sets. The linear replacement:
\(H^*(X; {\mathcal{O}}_X)\) for \({\mathcal{O}}_X\) the structure sheaf,
which yields
\begin{align*}
H^*(X; {\mathcal{O}}_X)
= 
\begin{cases}
{\mathbb{C}}& *=0 
\\
{\mathbb{C}}^g & *=1
\\
0 & \text{else}.
\end{cases}
\end{align*}
These surfaces can be parameterized by the moduli space
\({ \mathcal{M}_{g} }\), which is dimension \(3g-3\) for \(g \geq 2\).

\end{remark}

\begin{remark}

The POV in classical topology is to fix the coefficients:
\({\mathbb{Z}}, {\mathbb{R}}, {\mathbb{C}}, {\mathbb{Z}}/n\), or \(R\) a
general ring. A minor variation is to consider a local system
\({\mathcal{L}}\), which are locally constant but may have nontrivial
monodromy around loops. For example, one might have \({\mathbb{R}}\)
locally, but traversing a loop induces an automorphism
\(f\in \mathop{\mathrm{Aut}}({\mathbb{R}}) = {\mathbb{R}}^{\times}\). In
this setting, we have a functor \(F({-}) = H({-}; R)\). For sheaf
cohomology, instead fix \(X\) and take \(G({-}) = H(X; {-})\). In
general, one can take sheaves of abelian groups,
\({\mathcal{O}}_X{\hbox{-}}\)modules, quasicoherent sheaves, or coherent
sheaves:
\begin{align*}
{\mathsf{Sh}}(X, {\mathsf{Ab}}{\mathsf{Grp}}) \hookrightarrow{\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}\hookrightarrow{\mathsf{QCoh}}(X) \hookrightarrow{\mathsf{Coh}}(X)
.\end{align*}

\end{remark}

\begin{remark}

We'll be looking at three kinds of topologies:

\begin{itemize}
\item
  The order topology: start with a poset and define the open sets to be
  the \emph{decreasing/lower sets}, i.e.~subsets \(U_{x_0}\) that
  contain every element below a point \(x_0\). In other words, if
  \(x\in U\) and \(y\leq x\), then \(y\in U\).
\item
  The Zariski topology: let \(R\) be a DVR, so
  \(\operatorname{Spec}R = \left\{{ \left\langle{0}\right\rangle, {\mathfrak{m}}}\right\}\).
  E.g. for \(R \coloneqq{\mathbb{C}} { \left[ {t} \right] }\),
  \({\mathfrak{m}}= \left\langle{t}\right\rangle\), and the open sets
  are
  \(\left\{{\left\langle{0}\right\rangle}\right\}, \operatorname{Spec}R\),
  corresponding to the poset
  \({\operatorname{pt}}\to{\operatorname{pt}}\).
\item
  The classical topology, usually paracompact and Hausdorff.
\end{itemize}

One can define sheaves in all three cases, which have different
properties. For posets, e.g.~one can take \(C^0({-}, R)\) for
\(R = {\mathbb{R}}, {\mathbb{C}}, { {\mathbb{Z}}_{\widehat{p}} }\).

\end{remark}

\begin{remark}

Some computational tools:

\begin{itemize}
\tightlist
\item
  Vanishing theorems
\item
  Riemann-Roch
\end{itemize}

\end{remark}

\hypertarget{topological-notions-wednesday-january-12}{%
\section{Topological Notions (Wednesday, January
12)}\label{topological-notions-wednesday-january-12}}

\begin{remark}

Some topological notions to recall:

\begin{itemize}
\tightlist
\item
  \(T_0\), Kolmogorov spaces: distinct points don't have the exact same
  neighborhoods, i.e.~there exists a neighborhood of \(x\) not
  containing \(y\) \textbf{or} a neighborhood of \(y\) not containing
  \(x\).
\item
  \(T_1\), Frechet spaces: points are separated, so replace ``or'' with
  ``and'' above.
\item
  \(T_2\), Hausdorff spaces: points are separated by disjoint
  neighborhoods.
\item
  Alexandrov spaces: arbitrary intersections of opens are open.
\item
  Metrizability
\item
  Paracompactness
\end{itemize}

\end{remark}

\begin{remark}

Recall that a topology \(\tau\) on \(X\) satisfies

\begin{itemize}
\tightlist
\item
  \(\emptyset, X\in \tau\)
\item
  \(A,B\in \tau \implies A \cap B \in \tau\)
\item
  \(\displaystyle\bigcup_{j\in J} A_j \in \tau\) if \(A_j\in \tau\) for
  all \(j\).
\end{itemize}

Equivalently one can specify the closed sets and require closure under
finite unions and arbitrary intersections.

\end{remark}

\begin{example}[of topologies]

Running examples:

\begin{itemize}
\tightlist
\item
  Any subset \(S \subseteq {\mathbb{R}}^n\) is Hausdorff and
  paracompact.
\item
  Order topologies on posets
\item
  Zariski topologies on varieties over
  \(k= \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\),
  e.g.~\(\operatorname{mSpec}A\) for
  \(A\in {\mathsf{Alg}}^{\mathrm{fg}}_{/ {k}}\) or affine schemes
  \(\operatorname{Spec}A\).
\item
  The discrete/initial topology \(\tau = 2^X\).
\item
  The indiscrete topology \(\tau = \left\{{\emptyset, X}\right\}\).
\end{itemize}

\end{example}

\begin{remark}

Recall the separation axioms:

\begin{itemize}
\item
  \(T_0\): points can be topologically distinguished. Note that the
  indiscrete topology s not \(T_0\) if \({\sharp}X\geq 2\).
\item
  \(T_1\): points can be separated by (not necessarily disjoint)
  neighborhoods. Equivalently, points are closed.
\item
  \(T_2\)/Hausdorff: points can be separated by disjoint neighborhoods.
\item
  \(T_{3.5}\)/Tychonoff:?
\item
  \(T_6\):?
\end{itemize}

\end{remark}

\begin{exercise}[?]

Show that points are closed in \(X\) iff \(X\) is \(T_1\).

\end{exercise}

\begin{definition}[Paracompactness]

A space \(X\) is \textbf{paracompact} iff every open cover
\({\mathcal{U}}\rightrightarrows X\) admits a \emph{locally} finite
refinement \({\mathcal{V}}\rightrightarrows X\), i.e.~any \(x\in X\) is
in only finitely many \(V_i\).

\end{definition}

\begin{exercise}[Euclidean space is paracompact]

Show that any \(S \subseteq {\mathbb{R}}^n\) is paracompact, and indeed
any metric space is paracompact.

\end{exercise}

\begin{solution}

Let \({\mathcal{U}}\rightrightarrows X \coloneqq{\mathbb{R}}^d\) be an
open cover and define a proposed locally open refinement in the
following way:

\begin{itemize}
\tightlist
\item
  Write
  \({\mathcal{U}}\coloneqq\left\{{U_\alpha {~\mathrel{\Big\vert}~}\alpha\in A}\right\}\)
  for some index set.
\item
  Use that
  \(W_n \coloneqq{ \operatorname{cl}} _{X}({\mathbb{B}}_n(\mathbf{0}))\)
  is compact, and since \({\mathcal{U}}\rightrightarrows W_n\) there is
  a finite subcover
  \({\mathcal{V}}_n \coloneqq\left\{{U_{n, 1}, \cdots, U_{n, m}}\right\}\rightrightarrows{ \operatorname{cl}} _X({\mathbb{B}}^n(\mathbf{0}))^c\).
\item
  Show that
  \({\mathcal{V}}\coloneqq\left\{{{\mathcal{V}}}\right\}_{n\in {\mathbb{Z}}_{\geq 0}}\)
  is an open refinement of \({\mathcal{U}}\).

  \begin{itemize}
  \tightlist
  \item
    Why: it is a subcollection, and every \(x\in X\) is in a ball of
    radius \(R\approx N\coloneqq\ceil{{\left\lVert {x} \right\rVert}}\).
    So \(x\in {\mathbb{B}}_N(0)\), thus \(x\in U_{N, k}\) for some
    \(k\).
  \end{itemize}
\item
  Show that \({\mathcal{V}}\) is locally finite.

  \begin{itemize}
  \tightlist
  \item
    Why: each \({\mathcal{V}}_n\) misses the \({\mathbb{B}}_{k<n}(0)\),
    so each \(x\not\in \displaystyle\bigcup_{k\geq N} {\mathcal{V}}_n\)
    if \(N\) is defined as above. So \(x\) is in only finitely many
    \({\mathcal{V}}_n\).
  \end{itemize}
\end{itemize}

\end{solution}

\begin{fact}

Paracompact spaces admit a POU -- for
\({\mathcal{U}}\rightrightarrows X\), a collection \(A\) of function
\(f_\alpha: X\to {\mathbb{R}}\) where for all
\(\alpha \in \mathop{\mathrm{supp}}f_\alpha = { \operatorname{cl}} \qty{\left\{{f\neq 0}\right\}}\),
for all \(x\in X\), there exists a \(V\ni x\) such that for only
finitely many
\(\alpha, { \left.{{f_\alpha}} \right|_{{V}} } \not\equiv 0\), and
\(\sum_{\alpha\in A} f_{ \alpha}(x) = 1\).

\end{fact}

\begin{remark}

Recall the order topology: for \((P, \leq )\) a poset, so

\begin{itemize}
\tightlist
\item
  \(x\leq y, y\leq x\implies x=y\),
\item
  \(x\leq y\leq z\implies x\leq z\)
\item
  \(x\leq x\)
\end{itemize}

Define

\begin{itemize}
\tightlist
\item
  Open sets to be increasing sets, so
  \(x\in U, x\leq y \implies y\in U\),
\item
  Closed sets to be decreasing sets, so
  \(x\in U, x\geq y \implies y\in U\)
\end{itemize}

Note that this is a free choice!

\end{remark}

\begin{exercise}[?]

Show that the order topology is closed under arbitrary unions \emph{and}
intersections of opens.

\end{exercise}

\begin{exercise}[?]

Show that the order topology is not \(T_1\) by showing
\({ \operatorname{cl}} _P\qty{\left\{{x}\right\}} = Z_{\leq}(x) \coloneqq\left\{{y\in P{~\mathrel{\Big\vert}~}y\leq x}\right\}\).

\end{exercise}

\begin{fact}

For \(k\) an infinite field, \({\mathbb{A}}^1_{/ {k}}\) is the cofinite
topology and thus not Hausdorff.

\end{fact}

\hypertarget{friday-january-14}{%
\section{Friday, January 14}\label{friday-january-14}}

\hypertarget{posets-zariski-topologies}{%
\subsection{Posets, Zariski
Topologies}\label{posets-zariski-topologies}}

\begin{remark}

Recall the definition of a \textbf{poset}.

\end{remark}

\begin{example}[?]

Given a polytope, one can take its face poset
\(\mathrm{FP} (P) = \left\{{F \leq P}\right\}\) where \(F_1 \leq F_2\)
iff \(F_1 \subseteq F_2\) for the faces \(F_i\). More generally, one can
take a complex of polytopes, i.e.~a collection of polytopes that only
intersect at faces. An example of a complex is the fan of a toric
variety.

Similarly, one can take \textbf{cones}
\(\sum c_i \mathbf{v}_i \subseteq {\mathbb{R}}^d\) for some positive
coefficients.

\end{example}

\begin{remark}

Conversely, given a poset \(I\), one can associate a simplicial complex
\({ {\left\lvert {I} \right\rvert} }\), the geometric realization. Any
chain \(i_{n_1} < \cdots i_{n_k}\) is sent to a face and glued.

\end{remark}

\begin{example}[?]

Consider a polytope \(P\), taking the face poset \(\mathrm{FP} (P)\),
and its geometric realization
\({ {\left\lvert { \mathrm{FP} (P)} \right\rvert} }\). A square has

\begin{itemize}
\tightlist
\item
  \({\sharp}P_2 = 1\)
\item
  \({\sharp}P_1 = 4\)
\item
  \({\sharp}P_0 = 1\)
\end{itemize}

\includegraphics{figures/2022-01-14_10-37-55.png}

Note that one can take the geometric realization of a category by using
the nerve to first produce a poset.

\end{example}

\begin{remark}

With the right choices, there exists a continuous map
\({ {\left\lvert {I} \right\rvert} } \to I\) where \(I\) is given the
order topology. Pulling back sheaves on the latter yields constructible
sheaves on convex objects, which are locally constant on the interior
components.

\end{remark}

\begin{remark}

A first version of the Zariski topology: let
\(k = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu \in \mathsf{Field}\)
and let \(R\in{\mathsf{Alg}}^{\mathrm{fg}}_{/ {k}}\) be of the form
\(R = k[x_1, \cdots, x_{n}]/\left\langle{f_a}\right\rangle\). We can
consider
\(X\coloneqq\operatorname{mSpec}R \subseteq {\mathbb{A}}^n_{/ {k}}\) as
the points \(\mathbf{x}\in k{ {}^{ \scriptscriptstyle\times^{n} } }\)
such that \(f_a(\mathbf{x}) = 0\). Recall Noether's theorem -- the
\(f_a\) can be replaced with a finite collection. The closed subsets are
of the form \(V(g_b)\). Note that this is \(T_1\) since points are
closed: given \(\mathbf{p} = {\left[ {p_0,\cdots, p_n} \right]}\), take
\(f(\mathbf{p}) = \prod_{i\leq n} (x-p_i)\) so that
\(V(f) = \left\{{\mathbf{p}}\right\}\). These points biject with maximal
ideals in \(R\).

\end{remark}

\begin{remark}

An improved version of the Zariski topology:
\(X = \operatorname{Spec}R\), including prime ideals. The points are as
before, and additionally for every irreducible subvariety
\(Z \subseteq X\), there is a generic point \(\eta_Z\). This adds new
points which can't be described in coordinates.

\end{remark}

\begin{remark}

Note that this generalizes to arbitrary (associative, commutative)
rings. For rings that aren't finitely generated, one loses the
coordinate interpretation. These generally won't embed into
\({\mathbb{A}}^n_{/ {k}}\) for any \(n\), but can be embedded into (say)
\({\mathbb{A}}^1_{/ {R}}\). Use that a closed embedding
\(X\hookrightarrow Y\) corresponds precisely to a surjection of
associated rings \(R_Y \twoheadrightarrow R_X\).

\end{remark}

\hypertarget{sheaves}{%
\subsection{Sheaves}\label{sheaves}}

\begin{example}[?]

Let \(U \subseteq \Omega \subseteq {\mathbb{C}}\) and consider
\(C^0(\Omega, {\mathbb{C}}) \coloneqq\mathop{\mathrm{Hom}}_{\mathsf{Top}}(\Omega, {\mathbb{C}})\)
-- this forms a sheaf of abelian groups,
\({\mathbb{C}}{\hbox{-}}\)algebras, rings, sets, etc.

\includegraphics{figures/2022-01-14_11-04-15.png}

We'll refer to this as \({\mathcal{O}}^\text{cts}_X\).

\end{example}

\begin{remark}

Some properties:

\begin{itemize}
\item
  For every \(\iota: V \subseteq U \implies\) there is a restriction ma
  \begin{align*}
  {\mathcal{F}}(\iota): {\mathcal{F}}(U) &\to {\mathcal{F}}(V) \\
  f &\mapsto { \left.{{f}} \right|_{{V}} }
  .\end{align*}
\item
  \({\mathcal{F}}({ \mathscr \emptyset^{\scriptscriptstyle \downarrow} }) = { \mathscr{1}_{\scriptscriptstyle \uparrow} }\),
  so e.g.~for rings
  \({ \mathscr{1}_{\scriptscriptstyle \uparrow} }= \left\{{0}\right\}\)
  is the zero ring.
\item
  (Sheaf 1, uniqueness): if \({\mathcal{U}}\rightrightarrows U\) and
  \(s_1, s_2 \in {\mathcal{F}}({\mathcal{U}})\), then
  \({ \left.{{s_1}} \right|_{{U_i}} } = { \left.{{s_2}} \right|_{{U_i}} } \implies s_1 = s_2\).
\item
  (Sheaf 2, existence): if \(s_i\in {\mathcal{F}}(U_{ij})\) and
  \({ \left.{{s_1}} \right|_{{U_{ij}}} } = { \left.{{s_2}} \right|_{{U_{ij}}} }\),
  then there is a global section \(s\in {\mathcal{F}}(U_1 \cup U_2)\).
\end{itemize}

\end{remark}

\begin{example}[?]

Other examples of sheaves:

\begin{itemize}
\tightlist
\item
  \({\mathcal{O}}^\text{cts}_X\). One can check the sheaf properties
  directly.
\item
  \({\mathcal{O}}^\text{hol}_X = {\mathcal{O}}^{\mathrm{an}}_X\) the
  holomorphic (complex differentiable) and thus analytic (locally equal
  to a convergent power series) functions on \(X\).
\item
  Given a fixed continuous map \(f: Y\to X\), setting
  \({\mathcal{F}}(U) = \left\{{s: U\to Y}\right\}\) the set of
  continuous sections of \(f\).
\end{itemize}

\end{example}

\hypertarget{wednesday-january-19}{%
\section{Wednesday, January 19}\label{wednesday-january-19}}

\begin{example}[of sheaves]

Some examples of sheaves:

\begin{itemize}
\tightlist
\item
  For \(X \subseteq {\mathbb{C}}\) open, consider
  \({\operatorname{pr}}_1: X\times {\mathbb{C}}\to X\) and consider the
  space of continuous sections
  \({\mathcal{O}}_X^\text{cts}(U) \coloneqq\mathop{\mathrm{Hom}}_{\mathsf{Top}}(U, U\times {\mathbb{C}})\).
\end{itemize}

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Spring/SheafCohomology/sections/figures}{2022-01-19_10-26.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

\begin{itemize}
\item
  Analytic functions \({\mathcal{O}}_X^{\mathrm{an}}\)
\item
  \({\mathcal{O}}_X^\text{cts}\) where \({\mathbb{C}}\) is given the
  discrete topology instead of the Euclidean topology. The opens in
  \(U\times {\mathbb{C}}\) are of the form \(U\times V\) for
  \(V \subseteq {\mathbb{C}}\) any set at all:
\end{itemize}

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Spring/SheafCohomology/sections/figures}{2022-01-19_10-29.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

\begin{itemize}
\tightlist
\item
  Constant sheaves \(\underline{{\mathbb{C}}}(U)\) defined as the
  locally constant continuous \({\mathbb{C}}{\hbox{-}}\)valued functions
  on \(U\).
\end{itemize}

\end{example}

\begin{remark}

Recall the sheaf properties:

\begin{itemize}
\tightlist
\item
  \(U\to F(U)\) and
  \(\iota_{U, V} \mapsto \mathop{\mathrm{Res}}{F(V),F(U)}\).
\item
  \({ \mathscr \emptyset^{\scriptscriptstyle \downarrow} }\mapsto F({ \mathscr \emptyset^{\scriptscriptstyle \downarrow} }) = { \mathscr{1}_{\scriptscriptstyle \uparrow} }\).
\item
  Sheaf conditions:

  \begin{itemize}
  \tightlist
  \item
    Unique gluing: \({\mathcal{U}}\rightrightarrows X\) with
    \(\mathop{\mathrm{Res}}_{X, U_i} s = \mathop{\mathrm{Res}}_{X, U_i} t \implies s=t\in F(X)\)
  \item
    Existence of gluing: \(\left\{{s_i\in F(U_i)}\right\}\) with
    \(\mathop{\mathrm{Res}}_{U_i, U_{ij}} s_i = \mathop{\mathrm{Res}}_{U_j, U_{ij}} s_j\)
    implies \(\exists ! s\in F(X)\) with
    \(\mathop{\mathrm{Res}}_{X, U_i} s = s_i\) for all \(i\).
  \end{itemize}
\end{itemize}

\end{remark}

\begin{example}[?]

Recall that a \textbf{basis} of a topology is a collection \(B_i\) where
every \(U \in \tau_X\) can be written as
\(\displaystyle\bigcup_{i\in I} B_i\) for some index set \(I = I(X)\).
Some examples:

\begin{itemize}
\tightlist
\item
  For \(X\in {\mathsf{Alg}}{\mathsf{Var}}_{/ {k}}\), the distinguished
  opens \(D(f) = \left\{{f\neq 0}\right\}\) and
  \(Z(f) = \left\{{f=0}\right\}\).
\item
  For
  \(X =\operatorname{Spec}R \in {\mathsf{Aff}}{\mathsf{Sch}}_{/ {k}}\),
  take
  \(D(f) = \left\{{{\mathfrak{p}}\in \operatorname{Spec}R {~\mathrel{\Big\vert}~}f\neq 0 \in R/{\mathfrak{p}}}\right\} = \left\{{{\mathfrak{p}}\in \operatorname{Spec}R {~\mathrel{\Big\vert}~}f\not\in {\mathfrak{p}}}\right\}\)

  \begin{itemize}
  \tightlist
  \item
    Note that
    \({\mathcal{O}}_{\operatorname{Spec}R}(D(f)) = R \left[ { \scriptstyle { {f}^{-1}} } \right]\).
  \end{itemize}
\end{itemize}

\end{example}

\begin{exercise}[?]

Formulate the sheaf condition with a basis instead of arbitrary opens.

\begin{quote}
Hint: keep all of the same conditions, but since intersections may not
be basic opens, write \(B_\alpha \cap B_\beta = \cup_k B_k\).
\end{quote}

\end{exercise}

\begin{remark}

Some upcoming standard notions:

\begin{itemize}
\tightlist
\item
  Stalks \(F_x\)
\item
  Sheafification \(F \mapsto F^+\)
\end{itemize}

A less standard topic:

\begin{itemize}
\tightlist
\item
  The espace etale or ``flat space'' of \(F\).
\end{itemize}

\end{remark}

\begin{definition}[Stalks]

Recall that
\begin{align*}
F_x = \colim_{U\ni x} F(U) = \left\{{(U, s\in F(U))}\right\} / \sim && (U, s) \sim (V, t) \iff \exists W \supseteq U, V,\, \mathop{\mathrm{Res}}_{U, W}s = \mathop{\mathrm{Res}}_{V, W} t
.\end{align*}

Example:
\({\mathcal{O}}_{X, p}^{\mathrm{an}}= \left\{{f(z) \coloneqq\sum c_k (z-p)^k {~\mathrel{\Big\vert}~}f\text{ has a positive radius of convergence} }\right\}\).
Note that \({\mathcal{O}}_{X, p}^\text{cts}\) doesn't have such a nice
description, since continuous functions can be distinct while agreeing
on a small neighborhood. Similarly,
\(\underline{{\mathbb{C}}}_p = {\mathbb{C}}\), since locally constant is
actually constant on a small enough neighborhood.

\end{definition}

\begin{remark}

Recall that morphisms of (pre)sheaves are natural transformations of
functors. There is a forgetful functor
\(\mathop{\mathrm{Forget}}: {\mathsf{Sh}}(X) \to \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)\),
which has a left adjoint
\(({-})^+: \underset{ \mathsf{pre} } {\mathsf{Sh} }(X) \to {\mathsf{Sh}}(X)\).
There is a description of \(F^+(U)\) as collections of local compatible
sections of \(F\) modulo equivalence -- compatibility here means that if
\({\mathcal{U}}\rightrightarrows X\), then writing \(U_{ij} = \cup V_k\)
we have
\(\mathop{\mathrm{Res}}_{X, V_k}s_i = \mathop{\mathrm{Res}}_{X, V_k}s_j\)
for all \(i, j\).

\end{remark}

\hypertarget{friday-january-21}{%
\section{Friday, January 21}\label{friday-january-21}}

\begin{remark}

Last time: definitions of presheaves and sheaves. There is an adjunction
\begin{align*}
\adjunction{({-})^+}{\mathop{\mathrm{Forget}}}{ \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)}{{\mathsf{Sh}}(X)}
.\end{align*}

Recall that constant sheaves for \(A\in \mathsf{D}\) are defined as
\(\underline{A}({-}) \coloneqq\mathop{\mathrm{Hom}}_{\mathsf{Top}}({-}, A)\)
where \(A\) is equipped with the discrete topology.

\end{remark}

\begin{exercise}[?]

What is \(\Gamma(\underline{A}, X)\) for
\(X\coloneqq\left\{{1/n}\right\}_{n\in {\mathbb{Z}}_{\geq 0}} \subseteq {\mathbb{R}}\)?
So \(A(U) \neq A^{{\sharp}\pi_0 U}\) in general, since there may not be
a notion of connected components for an arbitrary topological space.

\end{exercise}

\begin{exercise}[?]

Is it true that for any \(X\in {\mathsf{Top}}\) there is a unique
decomposition \(X = {\textstyle\coprod}_{i\in I} U_i\) into connected
components?

\begin{quote}
Hint: form a poset of such decompositions ordered by refinement and
apply Zorn's lemma.
\end{quote}

\end{exercise}

\begin{example}[?]

Consider the following poset with a prescribed topology, and applying
some functor \(F\):

\includegraphics{figures/2022-01-21_10-49-22.png}

For this to be a sheaf, this forces

\begin{itemize}
\tightlist
\item
  \(F(\emptyset) = { \mathscr{1}_{\scriptscriptstyle \uparrow} }\)
\item
  \(F_{12} \cong F_1 \oplus F_2\) by the universal property of
  \(\oplus\) if this is to be a sheaf.
\item
  \(F_3\) can be anything mapping to \(F_{12}\).
\end{itemize}

What are the stalks?

\begin{itemize}
\tightlist
\item
  \(F_x = F(X)\) for \(x=3\), since \(X\) is the smallest open set
  containing \(3\).
\item
  \(F_{x_i} = F_i\) for \(x_i = 1, 2\).
\end{itemize}

\end{example}

\begin{example}[?]

Consider now a poset in the order topology:

\includegraphics{figures/2022-01-21_10-57-13.png}

Now \(F\) is a sheaf iff
\(F_{124}\cong F_1 { \underset{\scriptscriptstyle {F_4} }{\times} }F_2\)
is the fiber product.

\end{example}

\begin{definition}[Sheaf space]

A map \(\pi: Y\to X \in {\mathsf{Top}}\) is a \textbf{sheaf space} if it
is a local homeomorphism, so every \(y\in Y\) admits a neighborhood
\(U_y\ni y\) where
\({ \left.{{\pi}} \right|_{{U_y}} }: U_y\to \pi(U_y)\) is a
homeomorphism onto its image.

\end{definition}

\begin{example}[?]

Some examples:

\begin{itemize}
\tightlist
\item
  \(X\times A\to X\) for \(A\) discrete.
\end{itemize}

\includegraphics{figures/2022-01-21_11-04-47.png}

\end{example}

\begin{example}[?]

One possibility: ``jumping''. Take
\(Y \coloneqq X\displaystyle\coprod_{X\setminus\left\{{0}\right\}} X\)
for \(X\subseteq {\mathbb{R}}\), which is a version of the line with two
zeros. Then \(Y\to X\) is a sheaf space, since it is a local
homeomorphism.

The other possibility is ``skipping'':

\includegraphics{figures/2022-01-21_11-10-26.png}

\end{example}

\begin{remark}

These two definitions of sheaf coincide: for new to old, given
\(Y \xrightarrow{\pi} X\) apply
\(\mathrm{ContSec}_\pi \subseteq \mathop{\mathrm{Hom}}_{\mathsf{Top}}(X, Y)\).
In the other direction, define
\(Y\coloneqq\displaystyle\coprod_{x\in X} F_x\) and prove it is a local
homeomorphism.

\end{remark}

\begin{remark}

Next time: direct/inverse image, shriek functors, sheaves of modules.

\end{remark}

\hypertarget{monday-january-24}{%
\section{Monday, January 24}\label{monday-january-24}}

\begin{remark}

Recall the definitions of presheaves and sheaves, and sheafification as
an adjoint to
\(\mathop{\mathrm{Forget}}: {\mathsf{Sh}}(X)\to \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)\).
For \(F\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)\) we concretely
construct its sheafification \(F^+\) using the sheaf space
\(\pi: Y\coloneqq\displaystyle\coprod_{x\in X} F_x \to X\).

What are the sections of \(\pi\)? For a basic open
\(U \subseteq X \ni x\), the fiber is
\(\pi^{-1}(x) = F_x \coloneqq\colim_{V\ni x} F(V)\), which receives a
map \(\mathop{\mathrm{Res}}_{U, x}: F(U) \to F_x\). Writing
\(s\in F(U)\), define \(s_x \coloneqq\mathop{\mathrm{Res}}_{U, x}(s)\),
and set
\(W_{s, U} \coloneqq\left\{{s_x {~\mathrel{\Big\vert}~}x\in U}\right\}\)
to be \(\pi^{-1}(U)\). Then define \(F^+\) to be the continuous sections
of \(Y \xrightarrow{\pi} X\). What does such a section look like? For
\(t:U\to \pi^{-1}(U)\) and \(x\in U\), the vertical fiber is \(F_x\).
For a basic open \(V\ni X\) in the base, there is a basic open
\(W_{s, V}\) in \(Y\) for \(s\in F(V)\):

\includegraphics{figures/2022-01-24_10-39-51.png}

There are maps \(s_{ij}: U_{ij}\to \pi^{-1}(U_{ij})\), but note that
\(\mathop{\mathrm{Res}}(U_i, U_{ij}) s_i\) does not necessarily equal
\(\mathop{\mathrm{Res}}(U_j,U_{ij}) s_j\) in \(F(U_{ij})\) -- instead,
there is an open cover \(U_{ij} = \displaystyle\bigcup V_{\alpha}\) with
\(\mathop{\mathrm{Res}}(U_i, V_\alpha) s_i = \mathop{\mathrm{Res}}(U_j, V_\alpha) s_j\)
for each \(\alpha\).

\begin{quote}
Todo
\end{quote}

\end{remark}

\begin{remark}

For \(f\in {\mathsf{Top}}(X, Y)\) we have the following constructions:

\begin{itemize}
\tightlist
\item
  The direct image \(f_*: {\mathsf{Sh}}(X) \to {\mathsf{Sh}}(Y)\), which
  is easy with the sheaf definition, and
\item
  The inverse image \(f^{-1}: {\mathsf{Sh}}(Y)\to {\mathsf{Sh}}(X)\)
  which is easier with the sheaf space definition.
\end{itemize}

Recall the definition of a morphism of sheaves as a natural
transformation.

For sheaves of abelian groups and \(\phi: F\to G\) a morphism of
sheaves, there are notions of
\(\ker \phi, \operatorname{coker}\phi, \operatorname{im}\phi\), and
extension of a sheaf by zero.

To show these exist as presheaves, one only has to show existence of the
following blue morphisms of abelian groups:

\begin{center}
\begin{tikzcd}
    &&&&& {} \\
    0 && {\ker \phi_U} && {F(U)} && {G(U)} && {\operatorname{coker}\phi_U} && 0 \\
    &&&&& {\operatorname{im}\phi_U} \\
    0 && {\ker \phi_V} && {F(V)} && {G(V)} && {\operatorname{coker}\phi_V} && 0 \\
    &&&&& {\operatorname{im}\phi_V}
    \arrow[from=2-5, to=2-7]
    \arrow[from=2-5, to=4-5]
    \arrow[from=4-5, to=4-7]
    \arrow[from=2-7, to=4-7]
    \arrow[hook, from=2-3, to=2-5]
    \arrow[two heads, from=2-7, to=2-9]
    \arrow[hook, from=4-3, to=4-5]
    \arrow[two heads, from=4-7, to=4-9]
    \arrow[two heads, from=2-5, to=3-6]
    \arrow[hook, from=3-6, to=2-7]
    \arrow[two heads, from=4-5, to=5-6]
    \arrow[hook, from=5-6, to=4-7]
    \arrow[from=2-1, to=2-3]
    \arrow[from=4-1, to=4-3]
    \arrow[from=2-9, to=2-11]
    \arrow[from=4-9, to=4-11]
    \arrow["\exists"{description, pos=0.3}, color={rgb,255:red,92;green,92;blue,214}, dashed, from=2-3, to=4-3]
    \arrow["\exists"{description, pos=0.3}, color={rgb,255:red,92;green,92;blue,214}, dashed, from=3-6, to=5-6]
    \arrow["\exists"{description, pos=0.3}, color={rgb,255:red,92;green,92;blue,214}, dashed, from=2-9, to=4-9]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

Write \((\operatorname{coker}\phi)^-\) and \((\operatorname{im}\phi)^-\)
for these presheaves.

\end{remark}

\begin{proposition}[?]

\(\ker \phi\) is a sheaf.

\end{proposition}

\begin{proof}[?]

Axiom 1: use that \(F\) is a sheaf and \(\ker \phi_U \subseteq F(U)\)
can be viewed as an inclusion. Axiom 2: write
\(s_i\in \ker \qty{F(U_i) \xrightarrow{\phi_{U_i}} F(U_j) }\), then
there exists a unique \(s\in F(U)\). Then check that
\(s\in \ker\qty{F(U) \to G(U)}\) by noting that if \(s\mapsto t\) then
\({ \left.{{t}} \right|_{{U_i}} } = 0\) for all \(i\), making
\(t\equiv 0\) by the sheaf property of \(G\).

\end{proof}

\begin{definition}[Cokernel and image sheaves]

Define
\begin{align*}
\operatorname{coker}\phi &\coloneqq( (\operatorname{coker}\phi)^-)^+ \\
\operatorname{im}\phi &\coloneqq( (\operatorname{im}\phi)^-)^+ 
.\end{align*}

\end{definition}

\begin{example}[of necessity of sheafifying]

Take \(X = {\mathbb{C}}\) and consider
\(\exp: \mathop{\mathrm{Hol}}(X) \to G\) the sheaf of nowhere zero
holomorphic functions. Then on
\(U_i \in {\mathbb{C}}\setminus\left\{{0}\right\}\), take \(z\in G\).
Then \(z = \exp(f_i)\) in each \(U_i\) with
\(f_i \in \mathop{\mathrm{Hol}}(X)\), so \(f_i = \log(z)\) locally and
\(z = \exp(\log z)\), but there is no global
\(f\in \mathop{\mathrm{Hol}}(X)\) with \(\exp(f) = z\). So
\(z\in \ker \phi_i(\mathop{\mathrm{Hol}}(U_i) \to G(U_i))\) but
\(z\not \in \ker \exp\). For the same reason, \(z = 0\) in
\(\operatorname{coker}\phi_i\) since it's locally in the image. but
\(z\neq 0 \in \operatorname{coker}\exp\) since it's not globally in the
image.

\end{example}

\hypertarget{wednesday-january-26}{%
\section{Wednesday, January 26}\label{wednesday-january-26}}

\begin{remark}

Recall last time: presheaf vs sheaf properties, images, kernel,
cokernel. We can state the uniqueness sheaf axiom as the following: if
\(s\in F(U)\) with \({ \left.{{s}} \right|_{{U_i}} } = 0\) for
\({\mathcal{U}}\rightrightarrows U\), then \(s = 0\) in \(F(U)\).

\begin{itemize}
\tightlist
\item
  \({\mathcal{F}}\coloneqq(\operatorname{im}\phi)^-\) satisfies
  uniqueness.
\item
  \({\mathcal{G}}\coloneqq(\operatorname{coker}\phi)^-\) satisfies
  existence.
\item
  \({\mathcal{F}}\) fails existence \(\iff {\mathcal{G}}\) fails
  uniqueness
\item
  \({\mathcal{F}}\) fails uniqueness iff \({\mathcal{G}}\) fails
  existence.
\end{itemize}

The presheaf image and cokernel can sometimes fail to be a sheaf: use
\(\mathop{\mathrm{Hol}}(X) \xrightarrow{\exp} \mathop{\mathrm{Hol}}(X)^{\times}\).
The kernel presheaf \((\ker \phi)^-\) is already a sheaf.

\end{remark}

\begin{exercise}[?]

Show the following:

\begin{itemize}
\tightlist
\item
  A sheaf \({\mathcal{F}}\) is the zero sheaf iff \({\mathcal{F}}_p =0\)
  for all \(p\).
\item
  \(\ker(\phi)_p = \ker(\phi_p)\), which is
  \(\ker({\mathcal{F}}_p \xrightarrow{\phi_p} {\mathcal{G}}_p)\) the
  kernel of the induced map.
\item
  \(\operatorname{coker}(\phi)_p = \operatorname{coker}(\phi_p) \coloneqq\operatorname{coker}({\mathcal{F}}_p \xrightarrow{\phi_p} {\mathcal{G}}_p)\).
\item
  \(\phi: {\mathcal{F}}\to {\mathcal{G}}\) is injective iff
  \(\phi_p: {\mathcal{F}}_p \to {\mathcal{G}}_p\) is injective for all
  \(p\).
\item
  \(\phi: {\mathcal{F}}\to {\mathcal{G}}\) is surjective iff
  \(\phi_p: {\mathcal{F}}_p \to {\mathcal{G}}_p\) is surjective for all
  \(p\).
\end{itemize}

\end{exercise}

\begin{remark}

Hints:

\begin{itemize}
\tightlist
\item
  Suppose \(s\neq 0\) in \(F(U)\), does there exist a \(p\) with
  \(s_p = 0\)?
\item
  Use that \(s_p \in (\ker \phi)_p\) can be regarded as
  \(s\in \ker(F(V) \to G(V))\) mod equivalence.
\end{itemize}

\end{remark}

\begin{definition}[?]

If there exists an injective morphism
\(\phi:{\mathcal{F}}\to {\mathcal{G}}\), we regard
\({\mathcal{F}}\leq {\mathcal{G}}\) as a \textbf{subsheaf} and define
the \textbf{quotient sheaf}
\({\mathcal{F}}/{\mathcal{G}}\coloneqq\operatorname{coker}({\mathcal{F}}\xrightarrow{\phi} {\mathcal{G}})\).

\end{definition}

\begin{exercise}[?]

Show by example that \(({\mathcal{F}}/{\mathcal{G}})^-\) need not be a
sheaf.

\end{exercise}

\begin{remark}

Note that for \(\phi: {\mathcal{F}}\to {\mathcal{G}}\), the image
\(\operatorname{im}\phi\) is a secondary notion in additive categories,
and can instead be defined as either

\begin{itemize}
\tightlist
\item
  \(\operatorname{coker}(\ker \phi \to {\mathcal{F}})\)
\item
  \(\ker({\mathcal{G}}\to \operatorname{coker}\phi)\)
\end{itemize}

These need not coincide in general.

\end{remark}

\begin{remark}

Defining the direct image: easier using the sheaf axioms. For
\(f\in {\mathsf{Top}}(X, Y)\), define
\(f_*: {\mathsf{Sh}}(X) \to {\mathsf{Sh}}(Y)\) by
\begin{align*}
f_* {\mathcal{F}}(U) \coloneqq{\mathcal{F}}(f^{-1}(U)), \qquad \in {\mathsf{Sh}}(Y) \text{ for } {\mathcal{F}}\in {\mathsf{Sh}}(X)
.\end{align*}

\end{remark}

\begin{remark}

For the preimage: easier to use the espace étalé. As a special case,
consider \(\iota: S \hookrightarrow Y\) where \(S\) is a subspace of
\(Y\) (with the subspace topology). Then for
\({\mathcal{F}}\in {\mathsf{Sh}}(Y)\), we can now define sections not
only on open subsets \(U\) but arbitrary subsets \(S\) as
\begin{align*}
{\mathcal{F}}(S) \coloneqq(\iota^{-1}{\mathcal{F}})(S)
.\end{align*}

\end{remark}

\hypertarget{friday-january-28}{%
\section{Friday, January 28}\label{friday-january-28}}

\begin{remark}

Last time:

\begin{itemize}
\tightlist
\item
  Morphisms of sheaves \(\phi\),
\item
  \(\ker \phi\) (already a sheaf),
\item
  \((\operatorname{im}\phi)^-, (\operatorname{coker}\phi)^-\) (need to
  sheafify),

  \begin{itemize}
  \tightlist
  \item
    All defined to commute with taking stalks:
    \((\ker \phi)_p = \ker(\phi_p)\), etc
  \end{itemize}
\item
  \((\operatorname{im}\phi)^-\) may fail the existence axioms for
  sheaves, using
  \(\exp: {\mathcal{O}}^{\mathrm{an}}\to ({\mathcal{O}}^{\mathrm{an}})^{\times}\)
  for \(X\) a complex analytic space,
\item
  \((\operatorname{coker}\phi)^-\) may fail the uniqueness axioms for
  sheaves,
\item
  \((\operatorname{im}\phi)^-\) satisfies existence
  \(\iff (\operatorname{coker}\phi)^-\) satisfies uniqueness,
\item
  For \({\mathcal{F}}\hookrightarrow{\mathcal{G}}\) injective, the
  presheaf quotient \(({\mathcal{G}}/{\mathcal{F}})^-\) may fail to be a
  sheaf.
\end{itemize}

\end{remark}

\begin{example}[of the last claim]

For \(X\in {\mathsf{Alg}}{\mathsf{Var}}_{/ {k}}\) for
\(k= { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu }\),
let \({\mathcal{O}}_X\) be its regular algebraic functions. Take
\(X = {\mathbb{P}}^1\) and
\(U \coloneqq{\mathbb{A}}^1\setminus\left\{{{\operatorname{pt}}}\right\} \subseteq {\mathbb{A}}^1 \subseteq {\mathbb{P}}^1\setminus\left\{{a_1,\cdots, a_k}\right\}\).
Then
\({\mathcal{O}}_X(U) = k[x] \left[ { \scriptstyle { {f}^{-1}} } \right]\)
for \(f(x) \coloneqq\prod (x-a_k)\), \({\mathcal{O}}_X(X) = k\),
\(K_X(U) = k(x)\), and
\(K_X^{\times}(U) = k(x)\setminus\left\{{0}\right\}\) if
\(U \neq \emptyset\). Define \textbf{Cartier divisors} as global
sections of the sheaf
\(\mathop{\mathrm{Cart}}\operatorname{Div}\coloneqq K_X^{\times}/{\mathcal{O}}_X^{\times}\).
Recall that Weil divisors are finite sums of codimension 1 subvarieties,
and these notions coincide for nonsingular varieties.

For \(p\in {\mathbb{A}}^1 \subseteq {\mathbb{P}}^1\), we have
\begin{align*}
(K_X^{\times}/{\mathcal{O}}_X^{\times})_p 
= {K_{X, p} \over {\mathcal{O}}_{X, p}} 
= {k(x) \over \left\{{f/g {~\mathrel{\Big\vert}~}f(p)\neq 0 , g(p) \neq 0}\right\}}
\cong {\mathbb{Z}}
,\end{align*}
using that any element in the quotient can be written as
\(h(x) = (x-p)^n g(x)\) for some \(g\in {\mathcal{O}}_{X, p}^{\times}\).
Here \(\mathop{\mathrm{Cart}}\operatorname{Div}(X) = \sum n_p P\) are
all finite sums with \(n_p\in {\mathbb{Z}}\). The claim is that sheaf
existence fails for this quotient -- there are local sections that do
not glue. Here

\begin{itemize}
\tightlist
\item
  \(K^{\times}({\mathbb{P}}^1) = k(x)^{\times}\)
\item
  \({\mathcal{O}}^{\times}({\mathbb{P}}^1) = k^{\times}\)
\item
  \(K^{\times}({\mathbb{P}}^1)/{\mathcal{O}}^{\times}({\mathbb{P}}^1) = {k(x)^{\times}\over k^{\times}}\)
\end{itemize}

For any \(s\) in the quotient, we can associated
\((s)_0 - (s)_\infty = \sum n_p P\), but not every Cartier divisor is of
this form -- these are the \emph{principal} divisors. This form a group
\({\operatorname{Pic}}(X) = \mathop{\mathrm{Cart}}\operatorname{Div}(X) / \mathop{\mathrm{Prin}}\mathop{\mathrm{Cart}}\operatorname{Div}(X)\),
which may not be trivial. This proof generalizes to locally Noetherian
schemes, not necessarily reducible, with no embedded components.

\end{example}

\begin{remark}

Note that \({\operatorname{Pic}}(X)\) is also the group of invertible
sheaves on \(X\), and for irreducible algebraic varieties these
coincide. Use the SES
\(0\to {\mathcal{O}}^{\times}\to K^{\times}\to K^{\times}/{\mathcal{O}}^{\times}\to 0\)
to obtain
\begin{align*}
1 \to H^0({\mathcal{O}}^{\times}) \to H^0(K^{\times}) \to \mathop{\mathrm{Prin}}\mathop{\mathrm{Cart}}\operatorname{Div}(X) \to H^1({\mathcal{O}}^{\times})\cong \text{invertible sheaves}/\sim \to 0,
,\end{align*}
where \(H^1(K^{\times})\) vanishes since it's a constant sheaf on an
irreducible scheme in the Zariski topology.

\end{remark}

\begin{proposition}[?]

\((\operatorname{im}\phi)^-\) satisfies existence
\(\iff (\operatorname{coker}\phi)^-\) satisfies uniqueness.

\end{proposition}

\begin{proof}[?]

\(\implies\): Let \(s\in \operatorname{coker}(F(U) \to G(U))\) and write
\(U = \cup U_i\). We want to show that \(s_{U_i}\) implies
\(s\in \operatorname{coker}(F(U_i) \to G(U_i))\) for all \(i\). Note
that \(s = 0\) in \(\operatorname{coker}(F(U) \to G(U))\) iff
\(s\in \operatorname{im}(F(U) \to G(U))\)

\end{proof}

\hypertarget{monday-january-31}{%
\section{Monday, January 31}\label{monday-january-31}}

\begin{remark}

Direct image sheaf: for
\({\mathcal{F}}\in {\mathsf{Sh}}(X), {\mathcal{G}}\in {\mathsf{Sh}}(Y), f\in {\mathsf{Top}}(X, Y)\),
the \(f_* \in [{\mathsf{Sh}}(X), {\mathsf{Sh}}(Y)]\) is defined by
\(f_* {\mathcal{F}}(U) \coloneqq{\mathcal{F}}(f^{-1}U)\). The inverse
image functor \(f^{-1}\in [{\mathsf{Sh}}(Y), {\mathsf{Sh}}(X)]\) is
slightly more complicated. An easy case: if
\(\iota: S \hookrightarrow Y\) is a subspace, then it is just
restriction: \((\iota^{-1}G)(S) \coloneqq G(S)\).

Idea for sheaf space: there are strictly horizontal neighborhoods as the
homeomorphic preimages of small opens in the base. So for
\(\text{Ét}_{{\mathcal{G}}} \xrightarrow{\pi} Y\) the sheaf space of
\({\mathcal{G}}\), define the inverse image as
\begin{align*}
\text{Ét}_{\iota^{-1}{\mathcal{G}}} \coloneqq\pi^{-1}(S) \subseteq \text{Ét}_{{\mathcal{G}}}
,\end{align*}
and define a basis of sections in the following way: for
\(s\in {\mathcal{G}}(U)\), set
\(t(U) \coloneqq s(U) \cap\pi^{-1}(S) \in \text{Ét}_{{\mathcal{G}}}\) to
be sections of \(\text{Ét}_{\iota^{-1}{\mathcal{G}}}\). Declare these to
be a basis of opens, i.e.~take the subspace topology for
\(\pi^{-1}(S) \subseteq \text{Ét}_{{\mathcal{G}}}\) in the sheaf
topology on the total space. More generally, for
\(f\in {\mathsf{Top}}(X, Y)\), set
\begin{align*}
\text{Ét}_{f^{-1}{\mathcal{G}}} \coloneqq\text{Ét}_{\mathcal{G}}{ \underset{\scriptscriptstyle {Y} }{\times} } X
.\end{align*}

The fibers are identical:

\begin{center}
\begin{tikzcd}
    \textcolor{rgb,255:red,92;green,92;blue,214}{(\pi^*)^{-1}(x) } && \textcolor{rgb,255:red,92;green,92;blue,214}{\pi^{-1}(x)} \\
    & {\text{Ét}_{f^{-1}{\mathcal{G}}}} && {\text{Ét}_{{\mathcal{G}}}} \\
    \\
    & X && Y \\
    \textcolor{rgb,255:red,92;green,92;blue,214}{x} && y
    \arrow["\pi", from=2-4, to=4-4]
    \arrow["f", from=4-2, to=4-4]
    \arrow["{\pi^*}"', from=2-2, to=4-2]
    \arrow["{f^*}", from=2-2, to=2-4]
    \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=4-4]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-3, to=5-3]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, maps to, from=5-1, to=5-3]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, Rightarrow, dashed, no head, from=1-1, to=1-3]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-1, to=5-1]
    \arrow[from=1-3, to=2-4]
    \arrow[from=1-1, to=2-2]
    \arrow[from=5-1, to=4-2]
    \arrow[from=5-3, to=4-4]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

The topology on \(\text{Ét}_{f^{-1}{\mathcal{G}}}\) is the coarsest
topology for which \(\pi^*\) and \(f^*\) are continuous. This is
generated by \(\qty{ f^{-1}(s)(f^{-1}U)} \cap(\pi^*)^{-1}(W)\) for
\(W \subseteq X\) open. Define
\(f^{-1}(s) \in f^{-1}{\mathcal{G}}(f^{-1}(U)) \coloneqq(f^{-1}U){ \underset{\scriptscriptstyle {U} }{\times} } s(U)\).
This makes the pullback continuous both vertically and horizontally.

\end{remark}

\begin{corollary}[?]

\begin{align*}
(f^{-1}{\mathcal{G}})_y = {\mathcal{G}}_{f(y)}
.\end{align*}

\end{corollary}

\begin{definition}[Inverse image sheaf]

\begin{align*}
f^{-1}{\mathcal{G}}\coloneqq\qty{V\mapsto \colim_{U, V \subseteq f^{-1}(U)} {\mathcal{G}}(U)}^+
.\end{align*}

\end{definition}

\begin{remark}

How to prove this coincides with the previous definition:

\begin{itemize}
\tightlist
\item
  Show the stalks are isomorphic,
\item
  Show that there is a map of presheaves
  \((f^{-1}{\mathcal{G}}) \to f^{-1}{\mathcal{G}}\),
\item
  Show that the map induces an isomorphism on stalks, and lift using the
  universal property of sheafification.
\end{itemize}

\end{remark}

\begin{exercise}[?]

Try to prove this by commuting limits.

\end{exercise}

\begin{remark}

Recall that
\(K^{\times}/{\mathcal{O}}^{\times}\cong \bigoplus _{x\in X} (\iota_*)_* \iota_*^{-1}\underline{{\mathbb{Z}}}\)
which had stalks \({\mathbb{Z}}\) but was not constant -- check that the
local sections differ.

\end{remark}

\begin{question}

For \(S\hookrightarrow Y\), does every section of \({\mathcal{G}}\) over
\(S\) extend to \(Y\)?

\end{question}

\hypertarget{wednesday-february-02}{%
\section{Wednesday, February 02}\label{wednesday-february-02}}

\begin{remark}

Extending by zero: for \(i: U \hookrightarrow X\) an open subspace and
\({\mathcal{F}}\in {\mathsf{Sh}}(U)\), define
\(i_!{\mathcal{F}}\in {\mathsf{Sh}}(X)\). If the target category has a
zero object, define this in the sheaf space by extending the zero
section:

\includegraphics{figures/2022-02-02_10-28-52.png}

Thus
\(\text{Ét}_{i_! {\mathcal{F}}} = \text{Ét}_{{\mathcal{F}}} {\textstyle\coprod}\left\{{s_0}\right\}\)
for \(s_0\) the zero section.

\end{remark}

\begin{proposition}[?]

Define a presheaf are given by
\begin{align*}
(i_! {\mathcal{F}})^-(V) = 
\begin{cases}
F(V) & V \subseteq U 
\\
0 & \text{else}.
\end{cases}
.\end{align*}

Sheafifying produces an equivalent sheaf,
i.e.~\((i_! {\mathcal{F}})^{-+} \cong i_! {\mathcal{F}}\).

\end{proposition}

\begin{proof}[?]

Idea: produce a map \((i_! {\mathcal{F}})^- \to i_! {\mathcal{F}}\) and
show it is an isomorphism on stalks. What are the stalks? By the sheaf
space definition,
\begin{align*}
(i_! {\mathcal{F}})_p
=
\begin{cases}
{\mathcal{F}}_p & p\in U 
\\
0 & \text{else}.
\end{cases}
.\end{align*}
On the other hand,
\((i_! {\mathcal{F}})_p^- = \colim_{V\ni p} {\mathcal{F}}(V)\), but this
limit can be taken over the system of open sets \(V \subseteq U\), so it
yields \({\mathcal{F}}_p\).

\end{proof}

\begin{remark}

Consider \(X = U {\textstyle\coprod}Z\) with \(U\) open and \(Z\)
closed. Let \(U \xhookrightarrow{i} X\) and \(Z\xhookrightarrow{j} X\),
and consider \(i_* { \left.{{{\mathcal{F}}}} \right|_{{U}} }\) and
\(j_* { \left.{{{\mathcal{F}}}} \right|_{{U}} }\). There is a SES
\begin{align*}
0 \to i_! { \left.{{{\mathcal{F}}}} \right|_{{U}} } \to {\mathcal{F}}\to j_* { \left.{{{\mathcal{F}}}} \right|_{{Z}} }\to 0
.\end{align*}

\end{remark}

\begin{example}[?]

The sheaf \(i_!{ \left.{{{\mathcal{F}}}} \right|_{{U}} }\) is a subsheaf
of \({\mathcal{F}}\), and
\(j_*{ \left.{{{\mathcal{F}}}} \right|_{{Z}} }\) is a quotient.

\includegraphics{figures/2022-02-02_10-51-43.png}

Here
\(\text{Ét}_{\underline{{\mathbb{Z}}}} = \displaystyle\coprod_{n\in {\mathbb{Z}}} X\),
and
\(\text{Ét}_{j_* { \left.{{\underline{{\mathbb{Z}}}}} \right|_{{Z}} }} X\)
glued along \(X\setminus Z\). So
\(i_! { \left.{{{\mathcal{F}}}} \right|_{{U}} } \hookrightarrow{\mathcal{F}}\).
It's important that \(Z\) is closed here to get a surjection, since then
any point in its complement has a neighborhood \(V\) missing \(Z\)
entirely and \((i_! {\mathcal{F}})^-(V) = 0\). Checking the stalks:

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.10}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.16}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.36}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.36}}@{}}
\toprule
& \({\mathcal{F}}\) & \(i_! { \left.{{{\mathcal{F}}}} \right|_{{U}} }\)
& \(j_* { \left.{{{\mathcal{F}}}} \right|_{{V}} }\) \\
\midrule
\endhead
\(p\in U\) & \({\mathcal{F}}_p\) & \({\mathcal{F}}_p\) & 0 \\
\(p\in Z\) & \({\mathcal{F}}_p\) & 0 & \({\mathcal{F}}_p\) \\
\bottomrule
\end{longtable}

\end{example}

\begin{example}[?]

Let \(X\in {\mathsf{Alg}}{\mathsf{Var}}_{/ {k}}\),
e.g.~\(X={\mathbb{P}}^1\), let \(Z \subseteq X\) be closed, and let
\({\mathcal{F}}\coloneqq{\mathcal{O}}_X\). There is a SES
\(0\to I_Z \to {\mathcal{O}}_X\to {\mathcal{O}}_Z \to 0\).

\end{example}

\begin{remark}

Note that we have adjunctions
\begin{align*}
\adjunction{f^{-1}}{f_*}{{\mathsf{Sh}}X}{{\mathsf{Sh}}Y} \\
\adjunction{i_!}{{ \left.{{{-}}} \right|_{{U}} }}{{\mathsf{Sh}}?}{{\mathsf{Sh}}?} \\
\adjunction{j_*}{{ \left.{{{-}}} \right|_{{Z}} } }{{\mathsf{Sh}}?}{{\mathsf{Sh}}?}
.\end{align*}

\end{remark}

\hypertarget{friday-february-04}{%
\section{Friday, February 04}\label{friday-february-04}}

\begin{remark}

Last time: extension by zero, inverse image, pushforward on closed sets
and adjunctions.
\begin{align*}
f\in \mathop{\mathrm{Hom}}_{{\mathsf{Top}}}(X, Y) \leadsto \mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(X)}(f^{-1}{\mathcal{G}}, {\mathcal{F}}) \cong \mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(Y)}({\mathcal{G}}, f_* {\mathcal{F}})
.\end{align*}

\end{remark}

\begin{warnings}

Pushing forward open sets is not generally a good idea! Take
\(X = {\mathbb{R}}^{\mathrm{zar}}\),
\(Z = \left\{{{\operatorname{pt}}}\right\}, U = X\setminus Z\). Then
\((i_* \underline{{\mathbb{Z}}_U})_p = {\mathbb{Z}}{ {}^{ \scriptscriptstyle\oplus^{2} } }\)
if \(p= {\operatorname{pt}}\), since any neighborhood of \(p\) pulls
back to two connected components.

\end{warnings}

\begin{remark}

Consider \(U \xhookrightarrow{i} X\) with \(U\) open and
\(Z \xhookrightarrow{j} X\) with \(Z\) closed, then for
\({\mathcal{F}}\in {\mathsf{Sh}}(X), {\mathcal{H}}\in {\mathsf{Sh}}(U), {\mathcal{G}}\in {\mathsf{Sh}}(Z)\),
\begin{align*}
\mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(Z)}( { \left.{{{\mathcal{F}}}} \right|_{{Z}} }, {\mathcal{G}}) & { \, \xrightarrow{\sim}\, }\mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(X)}({\mathcal{F}}, j_* {\mathcal{G}}) \\
\mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(U)}({\mathcal{H}}, { \left.{{{\mathcal{F}}}} \right|_{{U}} } ) & { \, \xrightarrow{\sim}\, }\mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(X)}(i_! {\mathcal{H}}, {\mathcal{F}})
.\end{align*}

\end{remark}

\begin{remark}

We'll consider
\((X, {\mathcal{O}}_X) \in \mathsf{Loc}\mathsf{RingSp}_{/ {\mathsf{CRing}}}\)
with sheaves of reduced commutative rings -- note that noncommutative
rings are also important, e.g.~\(\operatorname{GL}_n\) or
\({\mathfrak{gl}}_n\).

\end{remark}

\begin{example}[?]

Common examples of locally ringed spaces:

\begin{itemize}
\tightlist
\item
  \((X, \underline{R})\) any space with a constant sheaf.
\item
  \((X, {\mathcal{F}})\) for
  \({\mathcal{F}}\coloneqq{\mathcal{O}}_X^\text{cts}\coloneqq\mathop{\mathrm{Hom}}_{\mathsf{Top}}({-}, {\mathbb{R}})\).
\item
  \((X, {\mathcal{O}}_X^{\mathrm{zar}})\) for
  \(X\in {\mathsf{Aff}}{\mathsf{Alg}}{\mathsf{Var}}_{/ {k}}\) and
  \({\mathcal{O}}_X^{\mathrm{zar}}\) the sheaf of Zariski-regular
  functions. In this case, for
  \(k= { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu }\),
  these are of the form
  \(\operatorname{mSpec}R \subseteq {\mathbb{A}}^n_{/ {k}}\) for
  \(R\coloneqq k[x_1, \cdots, x_{n}]/\left\langle{f}\right\rangle\).
  Recall distinguished opens are \(D(g) = \left\{{g\neq 0}\right\}\) for
  \(g\in k[x_1, \cdots, x_{n}]\), and sections are
  \({\mathcal{O}}_X(D(g)) = R \left[ { \scriptstyle { {g}^{-1}} } \right]\)
  are functions \(\rho: X\to k\) of the form \(\rho = h/g^k\) for some
  regular function \(h\). It's a theorem that these assemble to a sheaf.
\end{itemize}

\end{example}

\begin{remark}

Define algebraic varieties as locally ringed spaces
\((X, {\mathcal{O}}_X)\) that

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(X\) is covered by finitely affine algebraic varieties, so
  \(X = \cup U_i\) with \((U_i, {\mathcal{O}}_{U_i})\) affine algebraic,
  and
\item
  \(X\) is separated,
  i.e.~\(X \xrightarrow{\Delta_X} X{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{2} } }\)
  is closed.
\end{enumerate}

Note that affine and even quasiprojective schemes are automatically
separated. We require the separated condition here to rule out things
like \({\mathbb{A}}^1\) with two origins,
i.e.~\(X \coloneqq{\mathbb{A}}^1{ \displaystyle\coprod_{{\mathbb{A}}^1\setminus\left\{{0}\right\}} }{\mathbb{A}}^1\).

\end{remark}

\begin{example}[?]

Affine schemes: for \(R\in \mathsf{CRing}\), take
\(X\coloneqq\operatorname{Spec}R\) with a basis \(D(g)\) and define a
presheaf by
\({\mathcal{O}}_X(D(g)) = R \left[ { \scriptstyle { {g}^{-1}} } \right]\).
It's a theorem that this yields a sheaf.

\end{example}

\begin{definition}[$\OO_X\dash$modules]

For \((X, {\mathcal{O}}_X) \in \mathsf{Loc}\mathsf{RingSp}\),
\({\mathcal{F}}\) is a \textbf{sheaf of
\({\mathcal{O}}_X{\hbox{-}}\)modules} iff every section \(F(U)\) is an
\({\mathcal{O}}_X(U){\hbox{-}}\)module and restriction is compatible
with the module structures in the sense that
\({ \left.{{(rm)}} \right|_{{V}} } = { \left.{{r}} \right|_{{V}} } { \left.{{m}} \right|_{{V}} }\):

\begin{center}
\begin{tikzcd}
    m\in & {{\mathcal{F}}(U)} && {{\mathcal{O}}_X(U)} & {\ni r} \\
    \\
    & {{\mathcal{F}}(V)} && {{\mathcal{O}}_X(V)}
    \arrow[from=1-2, to=3-2]
    \arrow[from=1-2, to=1-4]
    \arrow[from=3-2, to=3-4]
    \arrow[from=1-4, to=3-4]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMSwwLCJcXG1jZihVKSJdLFsxLDIsIlxcbWNmKFYpIl0sWzMsMCwiXFxPT19YKFUpIl0sWzMsMiwiXFxPT19YKFYpIl0sWzAsMCwibVxcaW4iXSxbNCwwLCJcXG5pIHIiXSxbMCwxXSxbMCwyXSxbMSwzXSxbMiwzXV0=}{Link
to Diagram}
\end{quote}

\end{definition}

\begin{example}[?]

Any constant sheaf \(\underline{M}\) for
\(M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\).

\end{example}

\begin{definition}[Quasicoherent and coherent sheaves]

An \({\mathcal{O}}_X{\hbox{-}}\)module is

\begin{itemize}
\tightlist
\item
  \textbf{Quasicoherent} if locally \({\mathcal{F}}\cong \underline{M}\)
  (there exists an open cover \(X = \cup U_i\) with
  \({ \left.{{{\mathcal{F}}}} \right|_{{U_i}} } \cong \underline{M_{U_i}}\)),
\item
  \textbf{Coherent} iff \({\mathcal{F}}\) is quasicoherent and
  \(M \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\mathrm{fg}}\) and \(X\)
  is locally Noetherian.
\end{itemize}

\end{definition}

\begin{example}[?]

Of an \({\mathcal{O}}_X{\hbox{-}}\)module for a constant sheaf:
\(M = R/p\) for \({\mathcal{O}}_X = \underline{R}\).

\end{example}

\begin{example}[?]

For complex analytic varieties, take
\((X, {\mathcal{O}}_X^{\mathrm{an}})\) so \({\mathcal{O}}_X(U)\) are
locally meromorphic functions regular on \(U\), i.e.~whose denominator
does not vanish on \(U\). This is the setting where Cartan, Serre, etc
defined original notions of coherence, and e.g.~Serre vanishing, and
scheme theory is developed by analogy to this situation. Here,
\({\mathcal{F}}\) is a \textbf{coherent} sheaf iff \({\mathcal{F}}\) is
a sheaf of \({\mathcal{O}}_X^{\mathrm{an}}{\hbox{-}}\)modules and admits
a presentation
\begin{align*}
{\mathcal{O}}_X^{\mathrm{an}}{ {}^{ \scriptscriptstyle\oplus^{m} }  } \to {\mathcal{O}}_X^{\mathrm{an}}{ {}^{ \scriptscriptstyle\oplus^{n} }  } \to {\mathcal{F}}\to 0
.\end{align*}

\end{example}

\begin{remark}

Next time: locally free, invertible, tensor, and hom.

\end{remark}

\hypertarget{monday-february-07}{%
\section{Monday, February 07}\label{monday-february-07}}

\begin{remark}

Examples of sheaves:

\begin{itemize}
\tightlist
\item
  \({\mathcal{O}}_X^\text{cts}\) for \(X\in {\mathsf{Top}}\), where
  \({\mathcal{O}}_X^\text{cts}({-}) = {\mathsf{Top}}({-}, {\mathbb{R}})\)
\item
  \({\mathcal{O}}_X^{{\mathsf{sm}}}({-}) = C^\infty({-}, {\mathbb{R}})\).
\item
  \({\mathcal{O}}_X^{\text{hol}}({-}) = \mathop{\mathrm{Hol}}({-}, {\mathbb{C}})\)
\item
  \({\mathcal{O}}_X^{{\mathrm{an}}}({-}) \subseteq {\mathsf{Top}}({-}, {\mathbb{R}})\)
  the sheaf of analytic functions, those locally equal to power series.
\item
  For \(X\in {\mathsf{Alg}}{\mathsf{Var}}_{/ {k}}\),
  \({\mathcal{O}}_X({-}) = {\mathsf{Top}}(({-})^{\mathrm{zar}}, k)\) the
  Zariski-regular \(k{\hbox{-}}\)valued functions.
\end{itemize}

In all cases, \({\mathcal{O}}_X\) can be regarded as sheaves of
\emph{regular} sections to
\(X\times {\mathbb{A}}^1_{/ {k}} \xrightarrow{\pi} X\). Note that this
doesn't necessarily coincide with sections of the espace etale, since
e.g.~the fibers are \({\mathbb{A}}^1\) and not necessarily the stalks.
For \({\mathcal{O}}{ {}^{ \scriptscriptstyle\oplus^{d} } }\), one
instead takes \(X\times {\mathbb{A}}^d_{/ {k}} \to X\).

\end{remark}

\begin{definition}[Locally free and invertible sheaves]

A sheaf \({\mathcal{F}}\in {\mathsf{Sh}}(X)\) is \textbf{locally free}
iff there exists an open cover \({\mathcal{U}}\rightrightarrows X\) with
\({ \left.{{{\mathcal{F}}}} \right|_{{U_i}} } \cong {\mathcal{O}}_{U_i}{ {}^{ \scriptscriptstyle\oplus^{n} } }\).
The quantity \(n\) is the \textbf{rank} of \({\mathcal{F}}\). If
\(\operatorname{rank}{\mathcal{F}}= 1\), then \({\mathcal{F}}\) is
\textbf{invertible}.

A \textbf{vector bundle} over \(X\) is \(V \xrightarrow{\pi} X\) with
\(\pi^{-1}(U_i) \cong U_i \times {\mathbb{A}}^r\). For \(r=1\), this is
a \textbf{line bundle}.

\end{definition}

\begin{remark}

Maps between bundles are linear in the second coordinate.

Note that there is a correspondence between vector bundles and locally
free sheaves. Consider the rank 1 case, matching invertible sheaves and
line bundles. The necessary data:

\begin{itemize}
\tightlist
\item
  An open cover \({\mathcal{U}}\rightrightarrows X\), where
  \({\mathcal{U}}= \left\{{U_i}\right\}_{i\in I}\)
\item
  For all \(i, j\in I\), transition functions
  \(\phi_{ij}\in {\mathcal{O}}^{\times}(U_{ij}) = \mathop{\mathrm{Aut}}_{{\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}}(U_{ij})\).
\item
  A cocycle condition:
  \(\phi_{ii} = \operatorname{id}, \phi_{ij} = \phi_{ji}^{-1}\), and
  \(\phi_{ij} \phi_{jk} \phi_{ki} =\operatorname{id}\in {\mathcal{O}}^{\times}(U_{ijk})\)
\end{itemize}

Note that any morphism of sheaves
\({\mathcal{O}}_V \to {\mathcal{O}}_V\) induces a morphism of
\({\mathcal{O}}_V{\hbox{-}}\)modules on global sections
\begin{align*}
{\mathcal{O}}_V(V) & { \, \xrightarrow{\sim}\, }{\mathcal{O}}_V(V) \in {\mathsf{{\mathcal{O}}_V}{\hbox{-}}\mathsf{Mod}} \\
1 &\mapsto \phi
,\end{align*}
and this being an isomorphism manes \(\phi\) is invertible. Note that
these are not isomorphic as rings.

Write
\(Z_1({\mathcal{U}}; {\mathcal{O}}^{\times}) = \left\{{\phi_{ij} \in {\mathcal{O}}^{\times}(U_{ij}) {~\mathrel{\Big\vert}~}\cdots }\right\}\)
for those \(\phi_{ij}\) satisfying the conditions above, and
\(B_1({\mathcal{U}}; {\mathcal{O}}^{\times}) = \left\{{\phi_{ij} \in {\mathcal{O}}^{\times}(U_{ij}) {~\mathrel{\Big\vert}~}\phi_{ij} \sim \phi_{ij}{\psi_j \over \psi_i} }\right\}\)
for any
\({\psi_i \over \psi_j} \in \operatorname{GL}_1({\mathcal{O}}) \cong {\mathcal{O}}^{\times}\).
More generally, we let \(\phi_{ij} = \psi_j \phi_{ij} \psi_i^{-1}\) for
\(\psi_i, \psi_j \in \operatorname{GL}_n({\mathcal{O}})\).

\end{remark}

\begin{remark}

Recall that for a given space \(X\), the open covers of \(X\) form a
poset under refinement, where \({\mathcal{U}}\geq {\mathcal{V}}\) iff
for every \(U_i\in {\mathcal{U}}\) there is some
\(V_j \in {\mathcal{V}}\) with \(U_i \supseteq V_j\). This yields a
system of maps
\(Z^1({\mathcal{U}}; {\mathcal{O}}^{\times}) \to Z^1({\mathcal{V}}; {\mathcal{O}}^{\times})\)
compatible with transition maps, so we define
\begin{align*}
{\check{H}}^1(X; {\mathcal{O}}_X^{\times}) \coloneqq\colim_{{\mathcal{U}}\rightrightarrows X} {\check{H}}^1({\mathcal{U}}; {\mathcal{O}}^{\times})
.\end{align*}

\end{remark}

\begin{exercise}[?]

Compute
\({\check{H}}^1({\mathbb{P}}^1; {\mathcal{O}}_{{\mathbb{P}}^1}^{\times})\)
using an open cover by two sets.

\end{exercise}

\hypertarget{wednesday-february-09}{%
\section{Wednesday, February 09}\label{wednesday-february-09}}

\begin{remark}

Plan:

\begin{itemize}
\tightlist
\item
  \(\mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(X)}({-}, {-})\)
\item
  \(\mathop{\mathrm{Hom}}_{{\mathsf{ {\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}}({-}, {-})\)
\item
  \(({-}) \otimes_{{\mathcal{O}}_X}({-})\)
\end{itemize}

\end{remark}

\begin{remark}

For \((X, {\mathcal{O}}_X)\in \mathsf{Loc}\mathsf{RingSp}\) and
\({\mathcal{F}}, {\mathcal{G}}\in {\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\),
define
\(\mathop{\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathcal{F}}, {\mathcal{G}})\)
to be natural transformations which are
\({\mathcal{O}}_X{\hbox{-}}\)linear. This forms an abelian group under
pointwise operations, and more generally an
\({\mathcal{O}}_X{\hbox{-}}\)module since one can act on morphisms by
global sections. There is a sheaf version, the local hom
\(\underline{\mathop{\mathrm{Hom}}}_{{\mathcal{O}}_X}({\mathcal{F}}, {\mathcal{G}})(U) \coloneqq\mathop{\mathrm{Hom}}_{{\mathcal{O}}_U}({\mathcal{F}}_U, {\mathcal{G}}_U)\)
where we write
\({\mathcal{F}}_U \coloneqq{ \left.{{{\mathcal{F}}}} \right|_{{U}} }\).

\end{remark}

\begin{proposition}[?]

This forms a sheaf of \({\mathcal{O}}_X{\hbox{-}}\)modules.

\end{proposition}

\begin{proof}[?]

Let
\begin{align*}
f_i\in \mathop{\mathrm{Hom}}_{{\mathcal{O}}_{U_i}}({\mathcal{F}}_{U_i}, {\mathcal{G}}_{U_i}) \\
f_j\in \mathop{\mathrm{Hom}}_{{\mathcal{O}}_{U_j}}({\mathcal{F}}_{U_j}, {\mathcal{G}}_{U_j}) 
.\end{align*}
If
\({ \left.{{f_i}} \right|_{{U_{ij}}} } = { \left.{{f_j}} \right|_{{U_{ij}}} }\),
then the claim is that there exists a unique
\(F\in \mathop{\mathrm{Hom}}_{{\mathcal{O}}_{U_{ij}}}({\mathcal{F}}_{U_{ij}}, {\mathcal{G}}_{U_{ij}} )\).
For \(V \subseteq X\), decompose as \(V = \displaystyle\bigcup_i U_i\).

\end{proof}

\begin{proposition}[?]

If
\({\mathcal{F}}\in {\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}^{{\mathrm{lf}}, \operatorname{rank}=r}\)
and
\({\mathcal{G}}\in {\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}^{{\mathrm{lf}}, \operatorname{rank}=s}\)
then
\(\underline{\mathop{\mathrm{Hom}}}_{{\mathcal{O}}_X}({\mathcal{F}}, {\mathcal{G}})\in {\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}^{{\mathrm{lf}}, \operatorname{rank}= rs}\).

\end{proposition}

\begin{proof}[?]

Choose trivializations
\({\mathcal{F}}_{U_i} { \, \xrightarrow{\sim}\, }{\mathcal{O}}_{U_i}{ {}^{ \scriptscriptstyle\oplus^{r} } }\)
and
\({\mathcal{G}}_{U_i} { \, \xrightarrow{\sim}\, }{\mathcal{O}}_{U_i}{ {}^{ \scriptscriptstyle\oplus^{s} } }\).
The claim is that
\(\underline{\mathop{\mathrm{Hom}}}_{{\mathcal{O}}_U}({\mathcal{O}}_U, {\mathcal{O}}_U) = {\mathcal{O}}_U\)
for any \({\mathcal{O}}_U\). Given this,
\(\mathop{\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathcal{O}}_X{ {}^{ \scriptscriptstyle\oplus^{r} } }, {\mathcal{O}}_X{ {}^{ \scriptscriptstyle\oplus^{s} } }) \cong \operatorname{Mat}_{r\times s}({\mathcal{O}}_X)\)
split out as matrices. To prove this, just check on global sections that
\(\underline{\mathop{\mathrm{Hom}}}_{{\mathcal{O}}_X}({\mathcal{O}}_X, {\mathcal{O}}_X) \cong \mathop{\mathrm{Hom}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(R, R)\cong R\)
for \(R\coloneqq{{\Gamma}\qty{R} }\).

\end{proof}

\begin{remark}

Recall that
\({\mathsf{Sh}}(X)^{{\mathrm{lf}}, \operatorname{rank}=1} \cong {\mathsf{Bun}}_{\operatorname{GL}}^{\operatorname{rank}= 1}\),
i.e.~we identify rank 1 locally free sheaves with line bundles. We can
write
\(\mathop{\mathrm{Hom}}_{\mathcal{O}}({\mathcal{F}},{\mathcal{G}}) = \left\{{ {\phi_{ij} \over \psi_{ij} } {~\mathrel{\Big\vert}~}\phi_{ij} \in {\mathcal{O}}_X^{\times}(U_{ij}) \text{ satisfies the cocycle condition} }\right\}\).
What are the transition functions?

We also define
\(\mathop{\mathrm{Hom}}_{\mathcal{O}}({\mathcal{F}}, {\mathcal{O}}) \coloneqq{\mathcal{F}} {}^{ \vee }\),
and there is a relation to \({\operatorname{Pic}}(X)\).

\end{remark}

\begin{remark}

Note also that
\(\underline{\mathop{\mathrm{Hom}}}_{{\mathcal{O}}}({\mathcal{O}}, {\mathcal{F}}) \cong {\mathcal{F}}\),
so global sections coincide with homs. This will be useful later when
defining \(H^*\) in terms of derived functors.

\end{remark}

\begin{definition}[Tensor product]

Define the tensor product of
\({\mathcal{F}},{\mathcal{G}}\in {\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}\)
as the sheafification of
\begin{align*}
({\mathcal{F}}\otimes_{{\mathcal{O}}_X} {\mathcal{G}})^- \coloneqq U\mapsto  {\mathcal{F}}(U) \otimes_{{\mathcal{O}}_U} {\mathcal{G}}(U)
.\end{align*}

Note that there is a formula for stalks:
\begin{align*}
({\mathcal{F}}\otimes_{{\mathcal{O}}_X} {\mathcal{G}})_x = {\mathcal{F}}_x \otimes_{{\mathcal{O}}_x} {\mathcal{G}}_x
.\end{align*}
Moreover
\({\mathcal{F}}\otimes_{{\mathcal{O}}_X} {\mathcal{G}}\in {\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}^{{\mathrm{lf}}, \operatorname{rank}= rs}\).
This endows \({\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}\) with a
symmetric monoidal structure with duals, so

\begin{itemize}
\tightlist
\item
  \({\mathcal{F}}\otimes_{{\mathcal{O}}_X }{\mathcal{F}} {}^{ \vee }\cong {\mathcal{O}}_X\)
\item
  \({\mathcal{O}}_X \otimes_{{\mathcal{O}}_X} {\mathcal{F}}\cong {\mathcal{F}}\)
\end{itemize}

\end{definition}

\begin{remark}

Recall that \(f\in {\mathsf{Top}}(X, Y)\) for
\(X, Y\in {\mathsf{Aff}}{\mathsf{Sch}}\) induces
\(f^{-1}\in {\mathsf{Sh}}(X)(f^{-1}{\mathcal{O}}_Y, {\mathcal{O}}_X)\).
For varieties, this is just given by pullback of regular functions. More
generally, for \(X, Y\in \mathsf{Loc}\mathsf{RingSp}\), define the
\textbf{full pullback} \(f^*\) as
\begin{align*}
f^*{\mathcal{F}}= f^{-1}{\mathcal{F}}\otimes_{f^{-1}{\mathcal{O}}_Y} {\mathcal{O}}_X
.\end{align*}

\end{remark}

\begin{lemma}[?]

For the full pullback,
\begin{align*}
f^* {\mathcal{O}}_Y \cong {\mathcal{O}}_X
,\end{align*}
which is not true for \(f^{-1}\). This essentially follows from
\(R \otimes_R S \cong S\).

\end{lemma}

\begin{remark}

Consider \(f\in {\mathsf{Alg}_{/k} }(S, R)\) for
\(k=\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\) where
we only consider reduced algebra (no nonzero nilpotents). This induces
maps \(\tilde f: \operatorname{Spec}R\to \operatorname{Spec}S\) and
\(\tilde f': \operatorname{mSpec}R\to \operatorname{mSpec}S\). If
\({\mathcal{A}}\in {\mathsf{Sh}}(X; {{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Alg}})\),
there are induced maps \({\mathcal{O}}_X(U) \to {\mathcal{A}}(U)\) and
thus affine morphism \(\pi: \operatorname{Spec}{\mathcal{A}}(U) \to U\)
covering the affine open \(U\).

\end{remark}

\begin{example}[?]

\envlist

\begin{itemize}
\tightlist
\item
  \({\mathcal{A}}= {\mathcal{O}}_X[x_1,\cdots, x_n]\) yields a trivial
  vector bundle
  \(\operatorname{Spec}{\mathcal{A}}= X\times {\mathbb{A}}^n \to X\).
\item
  For
  \({\mathcal{F}}\in {\mathsf{Sh}}(X, {\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}^{{\mathrm{lf}}, \operatorname{rank}= n})\),
  set
  \begin{align*}
  {\mathcal{A}}= \operatorname{Sym}_{\mathcal{O}}^*({\mathcal{F}}) \coloneqq{\mathcal{O}}_X \oplus {\mathcal{F}}\oplus \operatorname{Sym}^2({\mathcal{F}}) \oplus \cdots
  ,\end{align*}
  which yields a nontrivial vector bundle
  \(\operatorname{Spec}{\mathcal{A}}\to X\).
\item
  For \({\mathcal{F}}\) rank 1,
  \({\mathcal{F}}{ {}^{ \scriptstyle\otimes_{{\mathcal{O}}_X}^{n} } } { \, \xrightarrow{\sim}\, }{\mathcal{O}}_X(-D)\),
  set
  \begin{align*}
  {\mathcal{A}}\coloneqq T_{{\mathcal{O}}}({\mathcal{A}}) \coloneqq{\mathcal{O}}\oplus {\mathcal{F}}\oplus {\mathcal{F}}{ {}^{ \scriptstyle\otimes_{{\mathcal{O}}_X}^{2} }  } \oplus \cdots
  ,\end{align*}
  then \(\operatorname{Spec}{\mathcal{A}}\to X\) is a cyclic Galois
  cover for \(G = \mu_n\).
\end{itemize}

\end{example}

\hypertarget{friday-february-11}{%
\section{Friday, February 11}\label{friday-february-11}}

\begin{remark}

Recall the definitions of:

\begin{itemize}
\tightlist
\item
  Cochain complexes,
\item
  Boundaries,
\item
  Cycles,
\item
  Homology as cycles mod boundaries,
\item
  Morphisms of chain complexes
\item
  Chain homotopies
\item
  Nullhomotopic morphisms
\item
  Homotopic morphisms of chain complexes
\item
  Short exact sequences of complexes:
\end{itemize}

\begin{center}
\begin{tikzcd}
    && 0 && 0 && 0 \\
    \\
    \cdots && {A^{n+1}} && {A^{n}} && {A^{n-1}} && \cdots \\
    \\
    \cdots && {C^{n}} && {B^{n}} && {B^{n-1}} && \cdots \\
    \\
    \cdots && {C^{n-1}} && {C^{n}} && {C^{n-1}} && \cdots \\
    \\
    && 0 && 0 && 0
    \arrow[from=3-1, to=3-3]
    \arrow[from=3-3, to=3-5]
    \arrow[from=3-5, to=3-7]
    \arrow[from=3-7, to=3-9]
    \arrow[from=5-1, to=5-3]
    \arrow[from=5-3, to=5-5]
    \arrow[from=5-5, to=5-7]
    \arrow[from=5-7, to=5-9]
    \arrow[hook, from=1-3, to=3-3]
    \arrow[hook, from=1-5, to=3-5]
    \arrow[hook, from=1-7, to=3-7]
    \arrow[hook, from=3-3, to=5-3]
    \arrow[hook, from=3-5, to=5-5]
    \arrow[hook, from=3-7, to=5-7]
    \arrow[two heads, from=5-3, to=7-3]
    \arrow[two heads, from=5-5, to=7-5]
    \arrow[two heads, from=5-7, to=7-7]
    \arrow[from=7-1, to=7-3]
    \arrow[from=7-3, to=7-5]
    \arrow[from=7-5, to=7-7]
    \arrow[from=7-7, to=7-9]
    \arrow[two heads, from=7-7, to=9-7]
    \arrow[two heads, from=7-5, to=9-5]
    \arrow[two heads, from=7-3, to=9-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

\begin{itemize}
\tightlist
\item
  Small categories

  \begin{itemize}
  \tightlist
  \item
    Sets of objects, sets of morphisms, a pairing
    \(\mathop{\mathrm{Mor}}(A, B)\times \mathop{\mathrm{Mor}}(B, C)\to \mathop{\mathrm{Mor}}(A, C)\).
  \end{itemize}
\item
  Universes
\end{itemize}

\end{remark}

\begin{exercise}[?]

Show that a morphism of chain complexes induces a morphism on homology.

\end{exercise}

\begin{exercise}[?]

Show that
\(f\simeq g \implies {{ {H}_{\scriptscriptstyle \bullet}} }(f) = {{ {H}_{\scriptscriptstyle \bullet}} }(g)\),
i.e.~homotopic chain morphisms induce equal maps on homology.

\begin{quote}
Hint: reduce to showing that \(f\) nullhomotopic implies
\({{ {H}_{\scriptscriptstyle \bullet}} }(f) = 0\).
\end{quote}

\end{exercise}

\begin{exercise}[Show a SES induces a LES in homology]

Show that a SES of complexes induces a LES in homology. Write a formula
for the connecting morphism, and do the check that everything is
well-defined! Use the grid diagram from above.

\end{exercise}

\begin{example}[?]

Examples of categories:

\begin{itemize}
\tightlist
\item
  \({\mathsf{Set}}\)
\item
  \(\mathsf{R}{\hbox{-}}\mathsf{Mod}\)
\item
  \(\mathsf{Mod}{\hbox{-}}\mathsf{R}\)
\item
  \({\mathsf{Top}}\)
\item
  \(\mathsf{CRing}\), assumed to be unital
\item
  \({\mathsf{Sch}}_{/ {k}}\)
\item
  \({\mathsf{Alg}}{\mathsf{Var}}_{/ {k}}\) for
  \(k= { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu }\)
\item
  \({\mathsf{Sh}}(X; {\mathbb{Z}{\hbox{-}}\mathsf{Mod}})\)
\item
  \({\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}\)
\item
  \({\mathsf{Top}}{\mathsf{Ab}}{\mathsf{Grp}}\)
\item
  \(G\curvearrowright{\mathsf{R}{\hbox{-}}\mathsf{Mod}}\)
\end{itemize}

Note that many of these are not abelian, since they are not even
additive, or e.g.~are not closed under kernels.

\end{example}

\hypertarget{monday-february-14}{%
\section{Monday, February 14}\label{monday-february-14}}

\begin{remark}

Recall the definitions of:

\begin{itemize}
\tightlist
\item
  Categories
\item
  Functors
\item
  Diagram/index categories

  \begin{itemize}
  \tightlist
  \item
    \(\bullet \to \bullet \leftarrow\bullet\)
  \item
    \(\bullet \leftarrow\bullet \to \bullet\)
  \item
    \(\bullet \rightleftharpoons\bullet\)
  \item
    \({\mathbb{N}}: \bullet \to \bullet \to \cdots\)
  \item
    \(\bullet \rightrightarrows\bullet\)
  \end{itemize}
\item
  Sets and posets as categories
\item
  Collections of objects \(\mathsf{C}\) as functors
  \(F\in [\mathsf{I}, \mathsf{C}]\) for \(\mathsf{I}\) an index category
\item
  Products and coproducts (via their universal properties). Useful
  mnemonic diagram:
\end{itemize}

\begin{center}
\begin{tikzcd}
    {\forall P} && {\prod A_i} \\
    \\
    & \cdots & {A_i} & \cdots \\
    \\
    && {\coprod A_i} && {\forall C}
    \arrow[from=3-2, to=3-3]
    \arrow[from=3-3, to=3-4]
    \arrow[from=1-3, to=3-3]
    \arrow[from=3-3, to=5-3]
    \arrow["{\forall \pi_P}", from=1-1, to=3-3]
    \arrow["\exists"', dashed, from=1-1, to=1-3]
    \arrow["{\forall \iota_C}", from=3-3, to=5-5]
    \arrow["\exists"', dashed, tail reversed, no head, from=5-5, to=5-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNyxbMSwyLCJcXGNkb3RzIl0sWzIsMiwiQV9pIl0sWzMsMiwiXFxjZG90cyJdLFsyLDAsIlxccHJvZCBBX2kiXSxbMiw0LCJcXGNvcHJvZCBBX2kiXSxbMCwwLCJcXGZvcmFsbCBQIl0sWzQsNCwiXFxmb3JhbGwgQyJdLFswLDFdLFsxLDJdLFszLDFdLFsxLDRdLFs1LDEsIlxcZm9yYWxsIFxccGlfUCJdLFs1LDMsIlxcZXhpc3RzIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzEsNiwiXFxmb3JhbGwgXFxpb3RhX0MiXSxbNiw0LCJcXGV4aXN0cyIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6ImFycm93aGVhZCJ9LCJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dXQ==}{Link
to Diagram}
\end{quote}

\begin{itemize}
\tightlist
\item
  Algebraic cats over sets (concrete categories) will be closed under
  products, i.e.~\(\prod A_i\) will admit the same algebraic structure
  by taking pointwise operations.
\item
  Examples of (co)products in common categories:

  \begin{itemize}
  \tightlist
  \item
    \({\mathsf{Set}}\): direct cartesian product and disjoint union.
  \item
    \({\mathsf{Ab}}{\mathsf{Grp}}\): direct cartesian product and direct
    sum \(\oplus\).
  \item
    \(\mathsf{Ring}\): \(\prod\) and \(\otimes_{\mathbb{Z}}\)
  \item
    \({\mathsf{Top}}\): \(\prod\) whose underlying set is the cartesian
    product with the product topology and \(\coprod\) as the disjoint
    union with the union of topologies Note the difference between the
    box and product topologies.
  \end{itemize}
\item
  A diagram in \(\mathsf{C}\) defined as a functor.
\item
  (co)filtered diagram categories \(\mathsf{I}\): for any pair \(i,j\),
  \({\sharp}\mathop{\mathrm{Mor}}_{\mathsf{I}}(i, j) \leq 1\) and there
  exists a \(k\) with \(i, j \to k\). Reverse arrows for cofiltered.

  \begin{itemize}
  \tightlist
  \item
    This allows for distinct but isomorphic objects, useful e.g.~in
    \({ \mathsf{Vect} }_{/ {k}}\) where abstractly
    \(V\cong V {}^{ \vee }\) but it's useful to distinguish.
  \end{itemize}
\item
  Limits (injective, cones that live above) and colimits (projective,
  cocones that live below):
\end{itemize}

\begin{center}
\begin{tikzcd}
    {\forall P} && {\cocolim A_i} \\
    \\
    & \cdots & {A_i} & \cdots \\
    \\
    && {\colim A_i} && {\forall C}
    \arrow[from=3-2, to=3-3]
    \arrow[from=3-3, to=3-4]
    \arrow[from=1-3, to=3-3]
    \arrow[from=3-3, to=5-3]
    \arrow["{\forall \pi_P}", from=1-1, to=3-3]
    \arrow["\exists"', dashed, from=1-1, to=1-3]
    \arrow["{\forall \iota_C}", from=3-3, to=5-5]
    \arrow["\exists"', dashed, tail reversed, no head, from=5-5, to=5-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNyxbMSwyLCJcXGNkb3RzIl0sWzIsMiwiQV9pIl0sWzMsMiwiXFxjZG90cyJdLFsyLDAsIlxcY29jb2xpbSBBX2kiXSxbMiw0LCJcXGNvbGltIEFfaSJdLFswLDAsIlxcZm9yYWxsIFAiXSxbNCw0LCJcXGZvcmFsbCBDIl0sWzAsMV0sWzEsMl0sWzMsMV0sWzEsNF0sWzUsMSwiXFxmb3JhbGwgXFxwaV9QIl0sWzUsMywiXFxleGlzdHMiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMSw2LCJcXGZvcmFsbCBcXGlvdGFfQyJdLFs2LDQsIlxcZXhpc3RzIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiYXJyb3doZWFkIn0sImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV1d}{Link
to Diagram}
\end{quote}

\begin{itemize}
\tightlist
\item
  Fiber products/pullbacks and pushouts
\item
  Equalizers/difference kernels \(K\) and coequalizers/difference
  cokernels \(C\) fitting into \(K \to A_1 \rightrightarrows A_2\to C\).
\item
  Computing cofiltered colimits in \({\mathsf{Ab}}{\mathsf{Grp}}\): for
  the cofiltered set
  \(\left\{{A_i, \phi_{ij}: A_i\to A_j}\right\}_{i, j}\), can construct
  as \(\colim A_i = {\textstyle\coprod}A_i/\sim\) here
  \(a_i \sim \phi_{ik}(a_i)\) for any \(k\) with \(i\to k\).

  \begin{itemize}
  \tightlist
  \item
    For filtered limits, one generally gets
    \(\cocolim A_i = \bigoplus A_i/\sim\) where
    \(a_i \sim \phi_{ik}(a_i)\)
  \end{itemize}
\item
  Example: \({\textstyle\coprod}A_i \in {\mathsf{Ab}}{\mathsf{Grp}}\) is
  not a cofiltered colimit, since the diagram category is discrete.

  \begin{itemize}
  \tightlist
  \item
    Claim: the underlying set is not \({\textstyle\coprod}A_i\).
  \end{itemize}
\item
  For fixed objects \(A\in \mathsf{C}\), the functors
  \(\mathop{\mathrm{Mor}}_{\mathsf{C}}(A, {-}): \mathsf{C}\to {\mathsf{Set}}\)
  and
  \(\mathop{\mathrm{Mor}}_{\mathsf{C}}({-}, A): \mathsf{C} \to { {{\mathsf{Set}}}^{\operatorname{op}}}\).

  \begin{itemize}
  \tightlist
  \item
    More generally the target can be
    \({\mathsf{Ab}}{\mathsf{Grp}}, \mathsf{CRing}\), etc.
  \end{itemize}
\end{itemize}

\end{remark}

\begin{remark}

Next time: additive and abelian categories, why
\({\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\) is an abelian
category.

\end{remark}

\hypertarget{wednesday-february-16}{%
\section{Wednesday, February 16}\label{wednesday-february-16}}

\begin{definition}[Equalizer and coequalizer]

The definition of equalizers and coequalizers:

\begin{center}
\begin{tikzcd}
    &&&&&& Y \\
    \\
    {K = \operatorname{eq}(f, g)} && A && B && {C = \operatorname{coeq}(f, g)} \\
    \\
    X
    \arrow[from=3-1, to=3-3]
    \arrow["f", shift left=2, from=3-3, to=3-5]
    \arrow[from=3-5, to=3-7]
    \arrow["{\exists !}"', dashed, from=3-7, to=1-7]
    \arrow[from=3-5, to=1-7]
    \arrow[from=5-1, to=3-3]
    \arrow["{\exists !}"', dashed, from=5-1, to=3-1]
    \arrow["g"', shift right=1, from=3-3, to=3-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMCwyLCJLID0gXFxlcShmLCBnKSJdLFsyLDIsIkEiXSxbNCwyLCJCIl0sWzYsMiwiQyA9IFxcY29lcShmLCBnKSJdLFs2LDAsIlkiXSxbMCw0LCJYIl0sWzAsMV0sWzEsMiwiZiIsMCx7Im9mZnNldCI6LTJ9XSxbMiwzXSxbMyw0LCJcXGV4aXN0cyAhIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzIsNF0sWzUsMV0sWzUsMCwiXFxleGlzdHMgISIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFsxLDIsImciLDIseyJvZmZzZXQiOjF9XV0=}{Link
to Diagram}
\end{quote}

\end{definition}

\begin{remark}

Notes:

\begin{itemize}
\tightlist
\item
  \(\ker f \to A \rightrightarrows^f_0 B\to \operatorname{coker}f\).
\item
  \(B\xhookrightarrow{h} X\) is injective iff
  \(A\rightrightarrows_g^f B \to X\)
\item
  \(X \xrightarrow[]{h} { \mathrel{\mkern-16mu}\rightarrow }\, A\) is
  surjective iff \(X \xrightarrow{h} A \rightrightarrows^f_g B\)
\item
  Iso = mono and epi
\end{itemize}

\end{remark}

\begin{exercise}[?]

Show that if \(\operatorname{eq}(f, g)\to A\) exists then
\(\operatorname{eq}(f, g) \hookrightarrow A\) is mono.

\end{exercise}

\begin{warnings}

Iso implies bijective on underlying sets, but not conversely.

Take the subcategory of \({\mathsf{Top}}{\mathsf{Ab}}{\mathsf{Grp}}\)
whose objects are \({\mathbb{R}}\) with various topologies, then take
\(\operatorname{id}: {\mathbb{R}}^{\mathrm{disc}}\to {\mathbb{R}}^{\mathrm{Euc}}\).
Note that
\(\ker \operatorname{id}= \operatorname{coker}\operatorname{id}= 0\) but
this is not an isomorphism. The issue: this is an additive category that
isn't abelian.

\end{warnings}

\begin{definition}[Additive categories]

For \(\mathsf{C} \in \mathsf{Cat}\),

\begin{itemize}
\tightlist
\item
  \(\mathop{\mathrm{Hom}}_{\mathsf{C}}(A, B) \in {\mathsf{Ab}}{\mathsf{Grp}}\)
\item
  Composition is distributive, so \(f(g+h) = fg+gh\) and
  \((g+h)f = gf + hf\).
\end{itemize}

\end{definition}

\begin{definition}[Abelian categories]

For \(\mathsf{C} \in \mathsf{Cat}\),

\begin{itemize}
\tightlist
\item
  Closed under all kernels and cokernels
\item
  Closed under products \(\prod A_i\)

  \begin{itemize}
  \tightlist
  \item
    Equivalently, closed under coproducts \(\bigoplus A_i\), and in fact
    \(A\times B = A \oplus B\) in \(\mathsf{C}\).
  \end{itemize}
\item
  There exists a zero object
  \(0 = { \mathscr \emptyset^{\scriptscriptstyle \downarrow} }= { \mathscr{1}_{\scriptscriptstyle \uparrow} }\)
  with
  \(\mathop{\mathrm{Hom}}(0, X) = \mathop{\mathrm{Hom}}(X, 0) = 0\).
\item
  Images are uniquely isomorphic to coimages:
\end{itemize}

\begin{center}
\begin{tikzcd}
    0 && {\ker f} && A && B && {\operatorname{coker}f} && 0 \\
    \\
    &&&& I && {I'} \\
    \\
    &&&& 0 && 0
    \arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=1-3]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-3, to=1-5]
    \arrow["f", from=1-5, to=1-7]
    \arrow[color={rgb,255:red,214;green,153;blue,92}, from=1-7, to=1-9]
    \arrow[color={rgb,255:red,214;green,153;blue,92}, from=1-9, to=1-11]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-5, to=3-5]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, from=3-5, to=5-5]
    \arrow[color={rgb,255:red,214;green,153;blue,92}, from=5-7, to=3-7]
    \arrow[color={rgb,255:red,214;green,153;blue,92}, from=3-7, to=1-7]
    \arrow["{\exists !}", dashed, from=3-5, to=3-7]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

\end{definition}

\begin{remark}

For \(\mathsf{C} = {\mathsf{Ab}}{\mathsf{Grp}}\),
\(\mathop{\mathrm{Hom}}_{\mathsf{C}}\) form abelian groups under
pointwise operations. For morphisms
\(\mathsf{C} = {\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\) and
\(f,g\in \mathsf{C}({\mathcal{F}}, {\mathcal{G}})\), writing
\(f = \left\{{f_U}\right\}, g = \left\{{g_U}\right\}\) in components,
one can set \(f+g = \left\{{f_U + g_U}\right\}\) to make
\(\mathop{\mathrm{Hom}}_{\mathsf{C}}\) an abelian group. Images will be
isomorphic to coimages in \(\mathsf{C}\) since the induced maps will be
isomorphisms on stalks, using that \({\mathsf{Ab}}{\mathsf{Grp}}\) is
abelian.

\end{remark}

\begin{remark}

If \({\mathcal{A}}\in {\mathsf{Ab}}\mathsf{Cat}\), then
\({\mathsf{Sh}}(X; {\mathcal{A}})\in {\mathsf{Ab}}\mathsf{Cat}\).

\end{remark}

\begin{exercise}[?]

Show that \(A_1\times A_2 = A_1 \oplus A_2\) in an abelian category
using the universal properties.

\end{exercise}

\begin{solution}

See course notes.

\end{solution}

\hypertarget{friday-february-18}{%
\section{Friday, February 18}\label{friday-february-18}}

\begin{remark}

Last time: abelian categories \(\mathsf{C}\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Existence of kernels, cokernels, and biproducts:
  \(\exists A\times B \iff \exists A \oplus B\).
\item
  Existence of isomorphisms
  \(\operatorname{coim}\phi \to \operatorname{im}\phi\) for all
  \(\phi\in \mathsf{C}(A, B)\)
\end{enumerate}

\end{remark}

\begin{corollary}[?]

For \(A\in {\mathsf{Ab}}\mathsf{Cat}\), every morphism has a mono-epi
factorization:

\begin{center}
\begin{tikzcd}
    A && B \\
    \\
    & I
    \arrow[dashed, two heads, from=1-1, to=3-2]
    \arrow[dashed, hook, no head, from=3-2, to=1-3]
    \arrow["f", from=1-1, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwwLCJBIl0sWzIsMCwiQiJdLFsxLDIsIkkiXSxbMCwyLCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifSwiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzIsMSwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifSwiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMCwxLCJmIl1d}{Link
to Diagram}
\end{quote}

\end{corollary}

\begin{remark}

The main technical tool: every SES induces a LES in cohomology. The
proof used for \(\mathsf{C} = {\mathsf{Ab}}{\mathsf{Grp}}\) works nearly
identically in an arbitrary abelian category using either

\begin{itemize}
\tightlist
\item
  \emph{generalized elements}, c/o MacLane, or
\item
  the full Freyd-Mitchell embedding.
\end{itemize}

MacLane's idea: define a functor
\begin{align*}
F: \mathsf{A} \to {\mathsf{Set}}_{{\operatorname{pt}}} \\
A &\mapsto \left\{{X\in \mathsf{A} {~\mathrel{\Big\vert}~}X\hookrightarrow A}\right\}/\sim
,\end{align*}
sending \(A\) to the set of its subobjects (equivalence classes of
monomorphisms), and on morphisms \(A \xrightarrow{f} B\) sending
\(X\hookrightarrow A\) to its image \(f(X)\hookrightarrow B\), so
\(F(f)(X) = \operatorname{im}_B(X)\). The point in the pointed set is
the subobject \(0_A \to A\). One then proves

\begin{itemize}
\tightlist
\item
  \(f = 0 \iff F(f) = 0\),
\item
  \(f\) is mono/epi \(\iff F(f)\) is mono/epi,
\item
  Thus \(F\) is exact.
\end{itemize}

So one can reduce checking exactness of \(f\) (where \(\mathsf{A}\) may
not have sets of elements) to checking exactness of \(F(f)\), where the
source/target are sets.

\end{remark}

\begin{theorem}[Freyd-Mitchell]

For \(\mathsf{A} \in {\mathsf{Ab}}\mathsf{Cat}\), there is a fully
faithful embedding
\(\mathsf{C} \xhookrightarrow{F} {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\)
for some ring \(R\). Here \emph{full} means that
\(\hom_{\mathsf{A}}(A, B) \cong \hom_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(FA, FB)\).

\end{theorem}

\begin{proof}[Idea]

\envlist

\begin{itemize}
\item
  Use the Yoneda/functor of points embedding, which is fully faithful:
  \begin{align*}
  \mathsf{A} &\to [\mathsf{A}, {\mathsf{Set}}] \\
  X &\mapsto h^X({-}) \coloneqq\mathop{\mathrm{Mor}}_{\mathsf{A}}(X, {-})
  .\end{align*}
\item
  Identify
  \([A, {\mathsf{Set}}] \simeq{\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) where
  \(R = \mathop{\mathrm{Mor}}_{\mathsf{A}}(I, I)\) for \(I\) an
  injective generator of this category, so every object comes from a
  subobject or quotient of \(I\). Then every
  \(M = \mathop{\mathrm{Mor}}_{\mathsf{A}}(I, M)\) becomes an
  \(R{\hbox{-}}\)module.
\end{itemize}

\end{proof}

\begin{observation}

Some observations about abelian categories:

\begin{itemize}
\tightlist
\item
  \({\mathsf{Ab}}\mathsf{Cat}\) is closed under
  \({ {({-})}^{\operatorname{op}}}\),
  i.e.~\(\mathsf{A} \in {\mathsf{Ab}}\mathsf{Cat}\iff { {\mathsf{A}}^{\operatorname{op}}} \in {\mathsf{Ab}}\mathsf{Cat}\)
\item
  \(\mathsf{A} \in {\mathsf{Ab}}\mathsf{Cat}\implies \mathsf{Ch}\mathsf{A} \in {\mathsf{Ab}}\mathsf{Cat}\).
\item
  If \(\mathsf{I}\) is any index category,
  \(\mathsf{A}^{\mathsf{I}} = [\mathsf{I}, \mathsf{A}] \in {\mathsf{Ab}}\mathsf{Cat}\).

  \begin{itemize}
  \tightlist
  \item
    E.g. \({\mathbb{Z}}\) with \(i\to j\iff i\leq j\) yields
    \(\mathsf{A}^{{\mathbb{Z}}}\) the category of sequences of elements
    of \(\mathsf{A}\),
    i.e.~\(\cdots \to A_{-1}\to A_0 \to A_1\to \cdots\).
  \item
    E.g. for \(I = \bullet \to \bullet \leftarrow\bullet\),
    \(\mathsf{A}^{\mathsf{I}}\) is the category of pushouts in
    \(\mathsf{A}\) whose morphisms are commuting diagrams:
  \end{itemize}
\end{itemize}

\begin{center}
\begin{tikzcd}
    {A_1} && {A_2} \\
    && {B_1} && {B_2} \\
    {A_3} \\
    && {B_3}
    \arrow[dashed, from=1-3, to=2-5]
    \arrow[dashed, from=1-1, to=2-3]
    \arrow[dashed, from=3-1, to=4-3]
    \arrow[from=2-3, to=2-5]
    \arrow[from=2-3, to=4-3]
    \arrow[from=1-1, to=3-1]
    \arrow[from=1-1, to=1-3]
    \arrow[from=4-3, to=2-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMCwwLCJBXzEiXSxbMiwwLCJBXzIiXSxbMCwyLCJBXzMiXSxbMiwxLCJCXzEiXSxbNCwxLCJCXzIiXSxbMiwzLCJCXzMiXSxbMSw0LCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMCwzLCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMiw1LCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMyw0XSxbMyw1XSxbMCwyXSxbMCwxXSxbNSwzXV0=}{Link
to Diagram}
\end{quote}

\end{observation}

\begin{remark}

Some additional axioms that hold in \({\mathsf{Ab}}{\mathsf{Grp}}\)
which we could ask \(\mathsf{A}\in {\mathsf{Ab}}\mathsf{Cat}\) to have:

\begin{itemize}
\item
  AB3: existence of arbitrary sums \(\bigoplus_i A_i\).
\item
  AB4: AB3 and if \(A_i\hookrightarrow B_i\) for all \(i\), then
  \(\bigoplus _i A_i \hookrightarrow\bigoplus _i B_i\) is again
  injective.
\item
  The dual of AB4, with products replaced by coproducts and injectives
  replaced by surjections.
\item
  AB5: AB3 and for all filtered system of subobjects \(A_i \subseteq A\)
  and a subobject \(B \subseteq A\),
  \begin{align*}
  (\sum A_i) \cap B \cong \sum (A_i \cap B)
  .\end{align*}
\item
  AB6: AB3 and for all filtered systems
  \(B_{i}^j \subseteq B^j \subseteq A\),
  \begin{align*}
  \cap_{j\in J} \qty{ \sum_{i\in I_j} B_i^j } = \sum_{i\in {\textstyle\coprod}I_j}\qty{\cap_{j\in J} B_i^j }
  .\end{align*}
\item
  AB: AB6 and AB4\({}^{ \vee }\), the dual conditions for AB4.
\end{itemize}

The categories \({\mathsf{Sh}}_X({\mathsf{Ab}}{\mathsf{Grp}})\) and
\({\mathsf{Sh}}_X({\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}})\)
satisfy AB5 and AB3\({}^{ \vee }\)

\end{remark}

\hypertarget{monday-february-21}{%
\section{Monday, February 21}\label{monday-february-21}}

\begin{remark}

Recall the definitions of \(\cocolim F\) and \(\colim F\) for
\(F \in [\mathsf{I}, \mathsf{C}] = \mathsf{C}^{\mathsf{I}}\) with
\(\mathsf{I}\) a small index category. Note that if
\(\mathsf{N} \coloneqq{ { {\mathsf{Open}}(X)}^{\operatorname{op}}}\),
the functor category
\(\mathsf{C}^{\mathsf{N}} = \underset{ \mathsf{pre} } {\mathsf{Sh} }(X; \mathsf{C})\)
consists of presheaves on \(X\).

\end{remark}

\begin{lemma}[?]

If any of the following exist in \(\mathsf{C}\):

\begin{itemize}
\tightlist
\item
  \(\prod A_i\)
\item
  \(\coprod A_i\)
\item
  \(\cocolim F\)
\item
  \(\colim F\)
\end{itemize}

Then the same is true in \(\mathsf{C}^{\mathsf{N}}\).

\end{lemma}

\begin{proof}[?]

\begin{center}
\begin{tikzcd}
    i &&&& {F_i(U)} &&& {F_i(V)} \\
    \\
    j &&&& {F_j(U)} &&& {F_j(V)} \\
    \\
    &&& {\colim_i F_i(U)} &&& {\colim F_i(V)} \\
    \\
    & {\forall\, G(U)} &&& {\forall \,G(V)}
    \arrow[from=1-1, to=3-1]
    \arrow[from=1-5, to=3-5]
    \arrow[from=1-8, to=3-8]
    \arrow[from=3-8, to=5-7]
    \arrow[from=3-5, to=5-4]
    \arrow["{\exists !}"{description}, dashed, from=5-4, to=7-2]
    \arrow["{\exists !}"{description}, dashed, from=5-7, to=7-5]
    \arrow[from=1-5, to=5-4]
    \arrow[from=1-5, to=7-2]
    \arrow[from=1-8, to=5-7]
    \arrow[from=1-8, to=7-5]
    \arrow[curve={height=-30pt}, from=3-5, to=7-2]
    \arrow[curve={height=-30pt}, from=3-8, to=7-5]
    \arrow["{\text{claim: } \exists}", color={rgb,255:red,214;green,92;blue,92}, squiggly, from=5-4, to=5-7]
    \arrow[from=1-5, to=1-8]
    \arrow[from=3-5, to=3-8]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMTAsWzAsMCwiaSJdLFswLDIsImoiXSxbNCwwLCJGX2koVSkiXSxbNCwyLCJGX2ooVSkiXSxbMyw0LCJcXGNvbGltX2kgRl9pKFUpIl0sWzcsMCwiRl9pKFYpIl0sWzcsMiwiRl9qKFYpIl0sWzYsNCwiXFxjb2xpbSBGX2koVikiXSxbMSw2LCJcXGZvcmFsbFxcLCBHKFUpIl0sWzQsNiwiXFxmb3JhbGwgXFwsRyhWKSJdLFswLDFdLFsyLDNdLFs1LDZdLFs2LDddLFszLDRdLFs0LDgsIlxcZXhpc3RzICEiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbNyw5LCJcXGV4aXN0cyAhIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzIsNF0sWzIsOF0sWzUsN10sWzUsOV0sWzMsOCwiIiwxLHsiY3VydmUiOi01fV0sWzYsOSwiIiwxLHsiY3VydmUiOi01fV0sWzQsNywiXFx0ZXh0e2NsYWltOiB9IFxcZXhpc3RzIiwwLHsiY29sb3VyIjpbMCw2MCw2MF0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6InNxdWlnZ2x5In19fSxbMCw2MCw2MCwxXV0sWzIsNV0sWzMsNl1d}{Link
to Diagram}
\end{quote}

\end{proof}

\begin{lemma}[?]

If \(\mathsf{C}\) has coproducts or colimits, then so does
\({\mathsf{Sh}}(X; \mathsf{C})\).

\end{lemma}

\begin{proof}[?]

Factor through the sheafification:

\begin{center}
\begin{tikzcd}
    {F_i} \\
    \\
    && {(\colim F)^-} &&&& G \\
    \\
    {F_j} &&&& \textcolor{rgb,255:red,92;green,92;blue,214}{(\colim F)^{-+}}
    \arrow["{\in  \underset{ \mathsf{pre} } {\mathsf{Sh} }(X, \mathsf{C})}"{description}, dashed, from=1-1, to=3-3]
    \arrow["{\in  \underset{ \mathsf{pre} } {\mathsf{Sh} }(X, \mathsf{C})}"{description}, dashed, from=5-1, to=3-3]
    \arrow[from=1-1, to=5-1]
    \arrow[from=1-1, to=3-7]
    \arrow[from=5-1, to=3-7]
    \arrow[dashed, from=3-3, to=3-7]
    \arrow["{\exists ! ({-})^+}", color={rgb,255:red,92;green,92;blue,214}, dotted, from=3-3, to=5-5]
    \arrow["{\exists!}", color={rgb,255:red,92;green,92;blue,214}, dotted, from=5-5, to=3-7]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

\end{proof}

\begin{remark}

In \({\mathsf{Ab}}{\mathsf{Grp}}\), we have
\(\prod,\coprod = \bigoplus, \colim, \cocolim\).

\begin{center}
\begin{tikzcd}
    &&&& {A_i} \\
    \\
    {\cocolim A_i} && {\prod A_i} &&&& {\coprod A_i = \bigoplus A_i} && {\colim A_i} \\
    {\left\{{(a_i) {~\mathrel{\Big\vert}~}f_{ij}(a_i) = a_j }\right\}} &&&&&&&& {\bigoplus A_i/ \left\langle{a_i - f_{ij}(a_i)}\right\rangle} \\
    &&&& {A_j}
    \arrow["{f_{ij}}", from=1-5, to=5-5]
    \arrow[squiggly, from=1-5, to=3-7]
    \arrow[squiggly, from=5-5, to=3-7]
    \arrow[squiggly, from=3-3, to=1-5]
    \arrow[squiggly, from=3-3, to=5-5]
    \arrow[from=3-1, to=1-5]
    \arrow[from=3-1, to=5-5]
    \arrow[hook, from=3-1, to=3-3]
    \arrow[from=1-5, to=3-9]
    \arrow[from=5-5, to=3-9]
    \arrow[two heads, from=3-7, to=3-9]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

Note that the inner diamond doesn't necessarily commute. The same
diagram holds in \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\).

\end{remark}

\begin{corollary}[?]

In \({\mathsf{Sh}}(X, {\mathsf{Ab}}{\mathsf{Grp}})\) and
\({\mathsf{Sh}}(X, {\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}})\),
both \(\oplus\) and \(\colim\) exist.

\end{corollary}

\begin{lemma}[?]

In \({\mathsf{Sh}}(X, {\mathsf{Ab}}{\mathsf{Grp}})\) and
\({\mathsf{Sh}}(X, {\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}})\),
both \(\prod\) and \(\cocolim\) exist.

\end{lemma}

\begin{proof}[?]

In
\(\underset{ \mathsf{pre} } {\mathsf{Sh} }(X,{\mathsf{Ab}}{\mathsf{Grp}})\),
there exist \(\prod, \cocolim\) where \((\prod F_i)(U) = \prod F_i(U)\),
but this already forms a sheaf. Check that if
\(U = \displaystyle\bigcup U_\alpha\), then a collection of sections
\(F_i(U_\alpha)\) agreeing on intersections is the same as an element of
the product.

\end{proof}

\begin{warnings}

Luckily we don't need to sheafify here, since the arrow for
sheafification goes the wrong way. However, the presheaf
\(U \mapsto \oplus_i F_i(U)\) is not necessarily a sheaf. Take
\(X = {\mathbb{Z}}\) with the discrete topology, then any global section
has infinitely many nonzero components. Note that
\((\oplus F_i)^{-+} \subseteq \prod F_i\) is the subsheaf of the product
where every local section has all but finitely many entries zero.

\end{warnings}

\begin{question}

\begin{align*}
\qty{\bigoplus F_i}^{-+}_p =_? \oplus (F_i)_p
,\end{align*}
i.e.~is the stalk given as
\(\left\{{ (a_i) \in (F_i)_p {~\mathrel{\Big\vert}~}\text{ all but finitely many entries are zero}}\right\}\).
Idea: each \(a_n\) might only lift to a disc of radius \(1/n\), which
intersect to \(\left\{{p}\right\}\). For example, take
\({\mathcal{F}}= C^\infty\) and take smooth compactly supported
functions on \([-1/n, 1/n]\) converging to \(\chi_{x=0}\).

\end{question}

\hypertarget{wednesday-february-23}{%
\section{Wednesday, February 23}\label{wednesday-february-23}}

\begin{remark}

Recall the definition of an additive category:

\begin{itemize}
\tightlist
\item
  \(\mathop{\mathrm{Mor}}_{\mathsf{C}}({-}, {-})\) are abelian groups,
\item
  Compositions distribute
\item
  A zero object
\item
  Finite products \(A\times B \iff\) finite coproducts
  \(A \oplus B \iff\) finite biproducts:
\end{itemize}

\begin{center}
\begin{tikzcd}
    A && {A \oplus B} && B
    \arrow["{i_1}", shift left=3, from=1-1, to=1-3]
    \arrow["{i_2}"', shift right=3, from=1-5, to=1-3]
    \arrow["{p_2}"', shift right=1, from=1-3, to=1-5]
    \arrow["{p_1}", shift left=1, from=1-3, to=1-1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwwLCJBIl0sWzIsMCwiQSBcXG9wbHVzIEIiXSxbNCwwLCJCIl0sWzAsMSwiaV8xIiwwLHsib2Zmc2V0IjotM31dLFsyLDEsImlfMiIsMix7Im9mZnNldCI6M31dLFsxLDIsInBfMiIsMix7Im9mZnNldCI6MX1dLFsxLDAsInBfMSIsMCx7Im9mZnNldCI6LTF9XV0=}{Link
to Diagram}
\end{quote}

where we require

\begin{itemize}
\item
  \(p_j i_j = \operatorname{id}\)
\item
  \(p_j i_k = 0\) for \(i\neq j\)
\item
  \(i_1 p_1 + i_2 p_2 = \operatorname{id}_{A \oplus B}\).
\item
  Abelian cats: additive, plus existence of kernels, cokernels, images.
\end{itemize}

\end{remark}

\begin{definition}[Additive Functors]

A functor \(F\in [\mathsf{A}, \mathsf{B}]\) is \textbf{additive} iff the
induced map
\(F_*: \mathop{\mathrm{Mor}}_{\mathsf{A}}(A, B) \to \mathop{\mathrm{Mor}}_{\mathsf{B}}(FA, FB) \in {\mathsf{Ab}}{\mathsf{Grp}}\)
is a morphism of groups.

\end{definition}

\begin{slogan}

Additive functors preserve

\begin{itemize}
\tightlist
\item
  polynomial identities in morphisms,
\item
  biproducts, so \(F(A \oplus B) \cong FA \oplus FB\),
\item
  complexes, so \(d_{n+1} d_n = 0\),
\item
  chain homotopy equivalences of complexes, which is a polynomial
  identity of the form \(ds + sd = h\).
\end{itemize}

\end{slogan}

\begin{example}[of additive functors]

\envlist

\begin{itemize}
\tightlist
\item
  For \(A\in \mathsf{A} \in {\mathsf{Add}}\mathsf{Cat}\), the functors
  \(\mathop{\mathrm{Mor}}_{\mathsf{A}}(A, {-}): \mathsf{A}\to {\mathsf{Ab}}{\mathsf{Grp}}\)
  and
  \(\mathop{\mathrm{Mor}}_{\mathsf{A}}({-}, A):\mathsf{A}\to { {{\mathsf{Ab}}{\mathsf{Grp}}}^{\operatorname{op}}}\).
\item
  For \(A\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\),
  \(F_A({-}) \coloneqq A\otimes_R({-}): {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\to{\mathsf{Ab}}{\mathsf{Grp}}\).

  \begin{itemize}
  \tightlist
  \item
    If \(R \in \mathsf{CRing}\), is commutative
    \(F_A({-}): {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\to{\mathsf{R}{\hbox{-}}\mathsf{Mod}}\).
  \end{itemize}
\item
  For \(\mathsf{I}\) and index category, recalling
  \(\mathsf{A}^{\mathsf{I}} = [\mathsf{I}, \mathsf{A}]\), the functors
  \(\cocolim \mathsf{A}^{\mathsf{I}} \to \mathsf{A}\) and
  \(\colim: \mathsf{A}^{\mathsf{I}} \to \mathsf{A}\) when they exist.
\item
  For \({\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\), the global
  sections functor
  \({{\Gamma}\qty{X; {-}} }: {\mathsf{Sh}}(X, {\mathsf{Ab}}{\mathsf{Grp}}) \to {\mathsf{Ab}}{\mathsf{Grp}}\).

  \begin{itemize}
  \tightlist
  \item
    For \(f\in {\mathsf{Top}}(X, Y)\), pushforward
    \(f_*: {\mathsf{Sh}}(X) \to {\mathsf{Sh}}(Y)\) (which includes
    inclusion of a point, i.e.~taking stalks at a point) and
    \(f^{-1}: {\mathsf{Sh}}(Y)\to {\mathsf{Sh}}(X)\) (which includes
    restriction).
  \end{itemize}
\item
  Local homs
  \(\mathop{\mathrm{Hom}}({\mathcal{F}}, {-}): {\mathsf{Sh}}(X; {\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}) \to {\mathsf{Sh}}(X; {\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}})\).
\item
  \({~\mathrel{\Big\vert}~}_x: {\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\to {\mathsf{Ab}}{\mathsf{Grp}}\)
  where \({\mathcal{F}}\mapsto {\mathcal{F}}_x\).
\end{itemize}

\end{example}

\begin{remark}

Recall the definition of exactness for chain complexes over abelian
categories: \(\operatorname{im}d^{n-1} = \ker d^n\). Note that one can
use epi-mono factorization to \textbf{splice}:

\begin{center}
\begin{tikzcd}
    \cdots && {C^{n-1}} && {C^n} && {C^{n+1}} && \cdots \\
    \\
    &&& {Z^{n-1}} && {Z^n} \\
    && 0 && 0 && 0
    \arrow["{d^{n-1}}", from=1-3, to=1-5]
    \arrow["{d^{n}}", from=1-5, to=1-7]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, two heads, from=1-3, to=3-4]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, hook, from=3-4, to=1-5]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, two heads, from=1-5, to=3-6]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, hook, from=3-6, to=1-7]
    \arrow["{d^{n-2}}", from=1-1, to=1-3]
    \arrow["{d^{n+1}}", from=1-7, to=1-9]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, from=4-3, to=3-4]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, from=3-4, to=4-5]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, from=4-5, to=3-6]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, from=3-6, to=4-7]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

This yields collections of SESs,
\begin{align*}
0\to Z^{n-1} \to C^n\to Z^{n}\to 0
.\end{align*}

Recall the definition of right/left/middle exactness: for
\(0\to A\to B\to C\to 0\) and covariant functors \(F\):

\begin{itemize}
\tightlist
\item
  Right exact: \(FA \to FB \to FC\to 0\),
\item
  Long exact: \(0\to FA \to FB \to FC\),
\item
  Middle exact: \(FA \to FB \to FC\).
\end{itemize}

For contravariant functors, e.g.~left exactness means
\(0\to FC\to FB \to FA\), so injectivity is preserved. Equivalently,
\(F: \mathsf{A}\to \mathsf{B}\) is left exact iff the covariant
\(F: { {\mathsf{A}}^{\operatorname{op}}} \to \mathsf{B}\) is left-exact.

\end{remark}

\begin{example}[?]

Of exactness:

\begin{itemize}
\tightlist
\item
  \({{\Gamma}\qty{{-}} }\) is left-exact,
\item
  \({~\mathrel{\Big\vert}~}_x\) is fully exact,
\item
  \(f_*\) is left exact,
\item
  \(f^{-1}\) is fully exact, since this preserves stalks,
\item
  \(A\otimes_R({-})\) is right exact,
\item
  \(\mathop{\mathrm{Hom}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(A, {-}), \mathop{\mathrm{Hom}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}({-}, A)\)
  are both left exact, which we'll prove.
\end{itemize}

\end{example}

\begin{proposition}[?]

\(\mathop{\mathrm{Hom}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(A, {-}), \mathop{\mathrm{Hom}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}({-}, A)\)
are both left exact.

\end{proposition}

\begin{proof}[?]

Use that kernels are monomorphisms:

\begin{center}
\begin{tikzcd}
    0 && {B'} && B && {B''} && 0 \\
    \\
    &&&& A \\
    {?} && {\mathop{\mathrm{Hom}}(A, B')} && {\mathop{\mathrm{Hom}}(A, B)} && {\mathop{\mathrm{Hom}}(A, B'')}
    \arrow["\exists", dashed, from=3-5, to=1-3]
    \arrow[from=1-1, to=1-3]
    \arrow["i", hook, from=1-3, to=1-5]
    \arrow[""{name=0, anchor=center, inner sep=0}, "p", from=1-5, to=1-7]
    \arrow[from=1-7, to=1-9]
    \arrow[from=4-1, to=4-3]
    \arrow[from=4-3, to=4-5]
    \arrow[""{name=1, anchor=center, inner sep=0}, from=4-5, to=4-7]
    \arrow["f"', from=3-5, to=1-5]
    \arrow["{\mathop{\mathrm{Hom}}(A, {-})}", shorten <=13pt, shorten >=13pt, Rightarrow, from=0, to=1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

Then show \(if=0 \implies f=0\), using that \(B'\to B\) is mono.
Similarly \(pf=0\implies f=ig\) for some \(g\).

\end{proof}

\begin{remark}

A nice proof that \({{\Gamma}\qty{{-}} }\) is left-exact: realize
\({{\Gamma}\qty{X; {-}} } \cong \mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(X)}(\underline{{\mathbb{Z}}}, {-})\),
which is left-exact for free. Use that the map
\(\underline{{\mathbb{Z}}} \to {\mathcal{F}}(X)\) is determined by
\(1\mapsto s\) and extend using \(n = n\cdot 1\).

\end{remark}

\hypertarget{friday-february-25}{%
\section{Friday, February 25}\label{friday-february-25}}

\hypertarget{adjoint-functors-exactness}{%
\subsection{Adjoint Functors,
Exactness}\label{adjoint-functors-exactness}}

\begin{remark}

Consider the setup:
\begin{align*}
\adjunction F G { \mathsf{A}}{ \mathsf{B} }
.\end{align*}
We say \(F\) is a \textbf{left adjoint} and \(G\) is a \textbf{right
adjoint}, so \(F\) \emph{has} a right adjoint and \(G\) \emph{has} a
left adjoint, if there are natural isomorphisms
\begin{align*}
[FA, B]_{\mathsf{B}}  { \, \xrightarrow{\sim}\, }[A, GB]_{\mathsf{A}}
,\end{align*}
i.e.~there is a natural isomorphism of functors
\([A, G({-})] { \, \xrightarrow{\sim}\, }[FA, ({-})]\). For a fixed
object \(B\), there is a natural transformation
\({\varepsilon}_B: FG\to \operatorname{id}_B\) which we call the
\textbf{counit} and \(\eta_A: \operatorname{id}_A\to GF\) called the
\textbf{unit}:

\begin{center}
\begin{tikzcd}
    A &&&& FA \\
    \\
    GB &&&& FGB && B
    \arrow["{\exists {\varepsilon}_B}", from=3-5, to=3-7]
    \arrow[""{name=0, anchor=center, inner sep=0}, from=1-1, to=3-1]
    \arrow[""{name=1, anchor=center, inner sep=0}, from=1-5, to=3-5]
    \arrow[from=1-5, to=3-7]
    \arrow["F", shorten <=26pt, shorten >=26pt, Rightarrow, from=0, to=1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCwwLCJBIl0sWzAsMiwiR0IiXSxbNCwwLCJGQSJdLFs0LDIsIkZHQiJdLFs2LDIsIkIiXSxbMyw0LCJcXGV4aXN0cyBcXGVwc19CIl0sWzAsMV0sWzIsM10sWzIsNF0sWzYsNywiRiIsMCx7InNob3J0ZW4iOnsic291cmNlIjoyMCwidGFyZ2V0IjoyMH19XV0=}{Link
to Diagram}
\end{quote}

\end{remark}

\begin{theorem}[?]

If \(\mathsf{A}, \mathsf{B} \in {\mathsf{Ab}}\mathsf{Cat}\), then

\begin{itemize}
\tightlist
\item
  If \(F\) is a right adjoint, \(F\) is left exact.
\item
  If \(G\) is a left adjoint, \(G\) is right exact.
\end{itemize}

\end{theorem}

\begin{proof}[?]

Note that the following lift exists iff \(\ker(A\to A'') = (A'\to A)\):

\begin{center}
\begin{tikzcd}
    0 && {A'} && A && {A''} && 0 \\
    \\
    &&&& X
    \arrow[from=3-5, to=1-5]
    \arrow["0"', from=3-5, to=1-7]
    \arrow[from=1-1, to=1-3]
    \arrow["i"', from=1-3, to=1-5]
    \arrow["p"', from=1-5, to=1-7]
    \arrow[from=1-7, to=1-9]
    \arrow["\exists", dashed, from=3-5, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMCwwLCIwIl0sWzIsMCwiQSciXSxbNCwwLCJBIl0sWzYsMCwiQScnIl0sWzgsMCwiMCJdLFs0LDIsIlgiXSxbNSwyXSxbNSwzLCIwIiwyXSxbMCwxXSxbMSwyLCJpIiwyXSxbMiwzLCJwIiwyXSxbMyw0XSxbNSwxLCJcXGV4aXN0cyIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==}{Link
to Diagram}
\end{quote}

Given \(0\to B'\to B\to B''\), we want to show
\(0\to GB'\to GB\to GB''\) is exact. Given \(A\to B''\) factoring
through zero, we can use adjointness to flip diagrams:

\begin{center}
\begin{tikzcd}
    0 && {GB'} && GB && {GB''} && 0 \\
    \\
    &&&& A \\
    \\
    0 && {B'} && B && {B''} && 0 \\
    \\
    &&&& FA
    \arrow[from=3-5, to=1-5]
    \arrow[""{name=0, anchor=center, inner sep=0}, "0"', from=3-5, to=1-7]
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-3, to=1-5]
    \arrow[from=1-5, to=1-7]
    \arrow[from=1-7, to=1-9]
    \arrow["{\therefore \exists}", dashed, from=3-5, to=1-3]
    \arrow[from=5-1, to=5-3]
    \arrow[from=5-3, to=5-5]
    \arrow[""{name=1, anchor=center, inner sep=0}, from=5-5, to=5-7]
    \arrow[from=5-7, to=5-9]
    \arrow[from=7-5, to=5-5]
    \arrow["0"', from=7-5, to=5-7]
    \arrow["\exists", dashed, from=7-5, to=5-3]
    \arrow["{F({-})}"', shorten <=13pt, shorten >=13pt, Rightarrow, from=1, to=0]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

\end{proof}

\begin{example}[?]

There is an adjunction between global sections and constant sheaves:
\begin{align*}
\adjunction{ {{\Gamma}\qty{X; {-}} } }{ \underline{({-}) }  }{{\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}}) }{{\mathsf{Ab}}{\mathsf{Grp}}}
.\end{align*}

One can define the map explicitly:
\begin{align*}
[A, {{\Gamma}\qty{X; {\mathcal{F}}} } ]_{{\mathsf{Ab}}{\mathsf{Grp}}} &\to [\underline{A}, {\mathcal{F}}]_{{\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})} \\
(a\mapsto s_a) &\mapsto (a_U \mapsto { \left.{{s_a}} \right|_{{U}} } )
.\end{align*}
It suffices to check this locally. Use that
\({{\Gamma}\qty{X; \underline{A}} }\) contains a copy of \(A\) to define
the reverse map, and check they are mutually inverse.

\end{example}

\begin{example}[?]

For \(f\in [X, Y]_{{\mathsf{Top}}}\), there is an induced adjunction
\begin{align*}
\adjunction{f_*}{f^{-1}}{{\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})}{{\mathsf{Sh}}(Y; {\mathsf{Ab}}{\mathsf{Grp}})}
.\end{align*}
Thus \(f_*\) is left exact.

\end{example}

\begin{exercise}[?]

Define the map
\begin{align*}
[{\mathcal{G}}, f_* {\mathcal{F}}]_{{\mathsf{Sh}}_Y} \to [f^{-1}{\mathcal{G}}, {\mathcal{F}}]_{{\mathsf{Sh}}_X}
.\end{align*}

\end{exercise}

\begin{remark}

Note that \(f_*\) is fully exact, as we knew before by checking on
stalks. Also note that \({~\mathrel{\Big\vert}~}_x\) for
\({\mathcal{F}}\in {\mathsf{Sh}}(X)\) is \(f^{-1}{\mathcal{F}}\) for
\(f:\left\{{x}\right\} \hookrightarrow X\).

\end{remark}

\begin{example}[?]

\begin{align*}
\adjunction{({-})^+}{\mathop{\mathrm{Forget}}}{ \underset{ \mathsf{pre} } {\mathsf{Sh} }(X; {\mathsf{Ab}}{\mathsf{Grp}})}{{\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})}
,\end{align*}
so sheafification is right exact and the forgetful functor is left
exact. In fact, \(({-})^+\) is fully exact since it preserves stalks.

\end{example}

\begin{example}[?]

For \(j\in [U, X]_{{\mathsf{Top}}}\) with \(U\) open in \(X\),
\begin{align*}
\adjunction{j_!}{j^{-1}}{{\mathsf{Sh}}(U; {\mathsf{Ab}}{\mathsf{Grp}}) }{{\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}}) }
.\end{align*}
In general there is a SES
\begin{align*}
0 \to j_! { \left.{{{\mathcal{F}}}} \right|_{{U}} } \to {\mathcal{F}}\to i_* { \left.{{{\mathcal{F}}}} \right|_{{X\setminus U}} } \to 0
.\end{align*}

\end{example}

\begin{example}[from algebra]

\begin{align*}
\adjunction{({-})\otimes_R ({-}) }{[{-}, {-}]_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}}{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}
,\end{align*}
so tensoring is right exact when an object is fixed. Note the
isomorphism
\begin{align*}
[A\otimes_R B]_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}} { \, \xrightarrow{\sim}\, }[A, [B,C]_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}]_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}
.\end{align*}

\end{example}

\hypertarget{monday-february-28}{%
\section{Monday, February 28}\label{monday-february-28}}

\hypertarget{tensors}{%
\subsection{Tensors}\label{tensors}}

\begin{remark}

Recall
\(\mathsf{R}{\hbox{-}}\mathsf{Mod} = \mathsf{Mod}{\hbox{-}}\mathsf{R} = ({R}, {R}){\hbox{-}}\mathsf{biMod}\)
for \(R \in \mathsf{CRing}\) associative, but for noncommutative rings
these may differ.

\begin{itemize}
\item
  The tensor product is a bifunctor
  \begin{align*}
  ({-})\otimes_R ({-}): \mathsf{Mod}{\hbox{-}}\mathsf{R} \times \mathsf{R}{\hbox{-}}\mathsf{Mod} &\to {\mathsf{Ab}}{\mathsf{Grp}}\\
  M_R \times _{R}N &\mapsto M_R \otimes_R {}_R N = { F(M\times N) \over (m_1 + m_2) \otimes n - m_1 \otimes n - m_2\otimes n, ma\otimes n - m\otimes an, \cdots}
  ,\end{align*}
  satisfying the usual universal property.
\item
  This generalizes:
  \begin{align*}
  ({-})\otimes_R ({-}): ({R}, {R}){\hbox{-}}\mathsf{biMod} \times \mathsf{R}{\hbox{-}}\mathsf{Mod} &\to \mathsf{R}{\hbox{-}}\mathsf{Mod}
  .\end{align*}
\item
  If \(\phi\in \mathsf{CRing}(R, S)\), then
  \begin{align*}
  ({-})\otimes_R S: \mathsf{Mod}{\hbox{-}}\mathsf{R} \to \mathsf{S}{\hbox{-}}\mathsf{Mod}
  .\end{align*}
\item
  This extends to algebras:
  \begin{align*}
  ({-})\otimes_R ({-}): \mathsf{Alg}{R}\times \mathsf{Alg}{R} \to \mathsf{Alg}{R} 
  ,\end{align*}
  with multiplication given by
  \((s_1\otimes s_2)\cdot(t_1\otimes t_2) \coloneqq(s_1t_1)\otimes(s_2t_2)\).
\item
  There is an adjunction:
  \begin{align*}
  [A\otimes_R C, B]_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}  { \, \xrightarrow{\sim}\, }[A, B^C]_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}
  .\end{align*}
\end{itemize}

\end{remark}

\begin{corollary}[?]

Since \(A\otimes_R ({-})\) is a left adjoint, it is right exact. Thus
presentations \(R^J\to R^I \to M \to 0\) yield presentations
\(M^J \to M^I \to M\otimes_R N \to 0\).

\end{corollary}

\begin{example}[?]

\begin{align*}
{\mathbb{C}}\otimes_RR {\mathbb{C}}\cong {\mathbb{C}}\oplus {\mathbb{C}}
,\end{align*}
writing
\({\mathbb{C}}= {\mathbb{R}}[x]/\left\langle{x^2+1}\right\rangle\), so
\begin{align*}
{\mathbb{C}}\otimes_{\mathbb{R}}{ {\mathbb{R}}[x] \over \left\langle{x^2+1}\right\rangle} \cong { {\mathbb{C}}[x] \over \left\langle{x^2+1}\right\rangle} \cong {{\mathbb{C}}[x] \over \left\langle{x-i}\right\rangle} \oplus {{\mathbb{C}}[x] \over \left\langle{x+i}\right\rangle}
.\end{align*}
Geometrically, this corresponds to
\(\colim(\operatorname{Spec}{\mathbb{C}}\to \operatorname{Spec}{\mathbb{R}}\leftarrow\operatorname{Spec}{\mathbb{C}})\cong X\coloneqq\operatorname{Spec}{\mathbb{C}}\otimes_{\mathbb{R}}{\mathbb{C}}\),
where point \(\left\langle{x^2+1}\right\rangle\) splits geometrically
and \(X\to \operatorname{Spec}{\mathbb{R}}\) is a 2-to-1 cover over this
point.

\begin{center}
\begin{tikzcd}
    {X{ \underset{\scriptscriptstyle {Y} }{\times} }\operatorname{Spec}k \cong \operatorname{Spec}(S\otimes_R k)} && X && {{\mathbb{A}}^{m+n}} &&&& {S=R=k[t_1,\cdots, t_n, x_1,\cdots, x_m]/\left\langle{q_j(t, x)}\right\rangle} && {k[t_1,\cdots, t_n, x_1,\cdots, x_m]} \\
    \\
    {\operatorname{Spec}k} && Y && {{\mathbb{A}}^n \ni {\left[ {a_1,\cdots, a_n} \right]}} &&&& {R=k[t_1,\cdots, t_n]/\left\langle{p_i(x)}\right\rangle} && {k[t_1,\cdots, t_n]} \\
    &&&&&& {} \\
    && {\operatorname{Spec}k} &&&&&& k
    \arrow[two heads, from=1-11, to=1-9]
    \arrow[two heads, from=3-11, to=3-9]
    \arrow[hook, from=3-3, to=3-5]
    \arrow[hook, from=1-3, to=1-5]
    \arrow[from=1-3, to=3-3]
    \arrow[from=1-5, to=3-5]
    \arrow[from=3-11, to=1-11]
    \arrow[from=3-9, to=1-9]
    \arrow[from=5-3, to=3-3]
    \arrow[from=5-3, to=3-5]
    \arrow["{t_i\mapsto a_i}", from=3-11, to=5-9]
    \arrow[from=3-1, to=3-3]
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-1, to=3-1]
    \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
    \arrow["{p_i(\mathbf{a}) = 0, \quad t_i\mapsto a_i}"', from=3-9, to=5-9]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

Conclusion:
\begin{align*}
S\otimes_R k = {k[x_1,\cdots, x_m] \over \left\langle{q_j(a, x) }\right\rangle }
.\end{align*}

In the previous example, the fiber over \(a\) is
\(\operatorname{Spec}k[x]/\left\langle{x^2-a}\right\rangle\) and the
covering map looks like the following:

\includegraphics{figures/2022-02-28_10-53-31.png}

\end{example}

\begin{question}

Is direct sum exact as a functor
\(\mathsf{A}{ {}^{ \scriptscriptstyle\times^{2} } }\to \mathsf{A}\)?
Regard
\(\mathsf{A}{ {}^{ \scriptscriptstyle\times^{2} } } = \mathsf{A}^{\mathsf{I}}\)
where \(I = \left\{{\bullet, \bullet}\right\}\) is the discrete 2-object
diagram category. The map \((A_1, A_2)\to A_1 \oplus A_2\) is exact by
just summing SESs.

\end{question}

\hypertarget{cohomology}{%
\subsection{Cohomology}\label{cohomology}}

\begin{remark}

Recall that one can compute \(H_*(S^2; {\mathbb{Z}})\) in several ways.

Method 1: triangulation.

\includegraphics{figures/2022-02-28_11-03-50.png}

This yields
\begin{align*}
0 \leftarrow{\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{4} }  } \leftarrow{\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{6} }  }\leftarrow{\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{4} }  } \leftarrow 0 \leadsto 0\leftarrow{\mathbb{Z}}\leftarrow 0 \leftarrow{\mathbb{Z}}\leftarrow 0
.\end{align*}

Method 2: cell complexes.

\includegraphics{figures/2022-02-28_11-06-45.png}

This directly yields
\begin{align*}
0\leftarrow{\mathbb{Z}}\leftarrow 0 \leftarrow{\mathbb{Z}}\leftarrow 0
.\end{align*}

\end{remark}

\begin{question}

Why are simplices \(\Delta_n\) or discs \(D^n\) the right things?

\end{question}

\begin{answer}

They are contractible, but more importantly do not themselves have
higher homology and are thus \emph{acyclic}.

\end{answer}

\begin{remark}

More generally, for
\(F\in {\mathsf{Ab}}\mathsf{Cat}(\mathsf{A},\mathsf{B})\), we'll want to
resolve by acyclic objects. Injectives and projectives will be universal
such objects, but are often hard to work with, so we'll work on finding
more economical acyclic resolutions. Next time: injectives/projectives
and derived functors.

\end{remark}

\hypertarget{wednesday-march-02}{%
\section{Wednesday, March 02}\label{wednesday-march-02}}

\begin{remark}

For \(F\in {\mathsf{Ab}}\mathsf{Cat}(\mathsf{A}, \mathsf{B})\) left
exact, assuming \(\mathsf{A}\) has enough injectives, there is a right
derived functor \({\mathbb{R}}F\) so that a SES
\(0\to A\to B\to C\to 0\) admits a LES with a connecting morphism
\(\delta\):
\begin{align*}
0\to {\mathbb{R}}FA\to {\mathbb{R}}F B\to {\mathbb{R}}F C \xrightarrow{\delta} \Sigma^1 {\mathbb{R}}F A \to \cdots
.\end{align*}
Note that \(\delta\) depends on the triple appearing in the SES.

\end{remark}

\begin{theorem}[Grothendieck]

\({\mathbb{R}}F\) and \(\delta\) are universal among
\(\delta{\hbox{-}}\)functors.

\end{theorem}

\begin{remark}

Injectives will be acyclic and homology will measure how things are
glued. Analogy: simplicial or cellular homology uses contractible
objects (with trivial homology) to measure how spaces are glued from
simplices or spheres.

\end{remark}

\begin{remark}

Recall the definitions of projective and injective objects, which
require existence (but not uniqueness) of certain lifts. In
\({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), free implies projective, so free
resolutions usually suffice and one can study generators, relations,
syzygies, etc.

We'll show that
\(\mathsf{A} \coloneqq{\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\)
has enough injectives, but usually won't have enough projectives. Recall
that this means that every \(A\in \mathsf{A}\) admits a monomorphism
\(A\hookrightarrow I\) for \(I\) an injective object. If there are
enough injectives, every object admits an injective resolution, and any
two such resolutions are homotopy equivalent.

\end{remark}

\begin{remark}

Recall that
\begin{align*}
 { {{\mathbb{R}}}^{\scriptscriptstyle \bullet}}  F(X) = { {H}_{\scriptscriptstyle \bullet}} ( F( X \leftleftarrows { {I}^{\scriptscriptstyle \bullet}}  ))
\end{align*}
and \({\mathbb{R}}^{i\geq 1} F(I) = 0\) if \(I\) is itself injective.

\end{remark}

\begin{remark}

Recall the Horseshoe lemma:

\begin{center}
\begin{tikzcd}
    0 && 0 && 0 \\
    A && {I^1_A} && {I^2_A} && \cdots \\
    B && \textcolor{rgb,255:red,92;green,92;blue,214}{\exists I^1_B \coloneqq I^1_A \oplus I^1_C} && \textcolor{rgb,255:red,92;green,92;blue,214}{\exists I^2_B \coloneqq I^2_A \oplus I^2_C} && \textcolor{rgb,255:red,92;green,92;blue,214}{\cdots} \\
    C && {I^1_C} && {I^2_C} && \cdots \\
    0 && 0 && 0
    \arrow[from=4-3, to=4-1]
    \arrow[from=4-5, to=4-3]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, from=3-7, to=3-5]
    \arrow[dashed, from=4-7, to=4-5]
    \arrow[dashed, from=2-7, to=2-5]
    \arrow[from=2-5, to=2-3]
    \arrow[from=2-3, to=2-1]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, from=3-5, to=3-3]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, from=3-3, to=3-1]
    \arrow[from=1-3, to=2-3]
    \arrow[from=2-3, to=3-3]
    \arrow[from=3-3, to=4-3]
    \arrow[from=4-3, to=5-3]
    \arrow[from=1-1, to=2-1]
    \arrow[from=2-1, to=3-1]
    \arrow[from=3-1, to=4-1]
    \arrow[from=4-1, to=5-1]
    \arrow[from=1-5, to=2-5]
    \arrow[from=2-5, to=3-5]
    \arrow[from=3-5, to=4-5]
    \arrow[from=4-5, to=5-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

Note that the complex in the middle is not the direct sum of the two
outer complexes, just the terms -- the differential \(d_B\) on
\({ {I}^{\scriptscriptstyle \bullet}} _B\) will be of the form
\begin{align*}
d_B = {
\begin{bmatrix}
  {d_A} & {*} 
\\
  {0} & {d_C}
\end{bmatrix}
}
.\end{align*}

\end{remark}

\begin{exercise}[?]

Prove this, using that additive functors preserve direct sums. Conclude
using that this construction yields a SES of complexes
\(0\to F { {I}^{\scriptscriptstyle \bullet}} _A \to F { {I}^{\scriptscriptstyle \bullet}} _B\to F { {I}^{\scriptscriptstyle \bullet}} _C\to 0\).

\end{exercise}

\begin{exercise}[?]

Prove that if \(I\) is injective then \(0\to I\to B\to C\to 0\) splits
by explicitly constructing a left and right splitting to show that \(B\)
satisfies the universal property of the biproduct. Show also that the
same conclusion holds for \(0\to A\to B\to P\to 0\) with \(P\)
projective.

\end{exercise}

\hypertarget{friday-march-04}{%
\section{Friday, March 04}\label{friday-march-04}}

\begin{remark}

Idea: regard \(A\) as a chain complex supported in degree zero and
\(A\underset{\leftleftarrows}{\eta} { {I}^{\scriptscriptstyle \bullet}}\)
an injective resolution, then the induced map
\(\eta^*: H^*(A)\to H^*( { {I}^{\scriptscriptstyle \bullet}} )\) is an
isomorphism, so \(A\) and \({ {I}^{\scriptscriptstyle \bullet}}\) are
quasi-isomorphic.

\end{remark}

\begin{exercise}[?]

Show that if
\(A\leftleftarrows { {I}^{\scriptscriptstyle \bullet}} , { {J}^{\scriptscriptstyle \bullet}}\),
then there exists a chain homotopy \(f: I \simeq J\).

\end{exercise}

\begin{remark}

Hints:

\begin{center}
\begin{tikzcd}
    A \\
    {X^0} && {I^0} & {X^1} & {I^1} & {X^2} & \bullet \\
    {Y_0} && {J^0} & {Y^1} & {J^1} & {Y^2} & \bullet \\
    B
    \arrow[Rightarrow, no head, from=1-1, to=2-1]
    \arrow[from=2-1, to=3-1]
    \arrow[Rightarrow, no head, from=3-1, to=4-1]
    \arrow[hook, from=2-1, to=2-3]
    \arrow[hook, from=3-1, to=3-3]
    \arrow[two heads, from=3-3, to=3-4]
    \arrow[two heads, from=2-3, to=2-4]
    \arrow[hook, from=2-4, to=2-5]
    \arrow[two heads, from=2-5, to=2-6]
    \arrow[hook, from=2-6, to=2-7]
    \arrow[hook, from=3-4, to=3-5]
    \arrow[two heads, from=3-5, to=3-6]
    \arrow[hook, from=3-6, to=3-7]
    \arrow[from=2-1, to=3-3]
    \arrow["{\exists \text{ since } J^0\text{ is injective}}", dashed, from=2-3, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

Given
\begin{align*}
f^{n-1} - g^{n-1} = h^n d^{n-1} - d^{n-2} h^{n-1}
,\end{align*}
construct \(h^{n+1} d_I^n\) such that
\begin{align*}
(f^n - g^n) d_I^n = ( h^{n+1} d_I^n + d_J^{n-1} h^n ) d_I^n
\end{align*}
and extend arbitrarily to \(h^{n+1}: I^{n+1} \to J^n\).

\end{remark}

\begin{exercise}[?]

Prove the Horseshoe lemma.

\end{exercise}

\hypertarget{monday-march-14}{%
\section{Monday, March 14}\label{monday-march-14}}

\begin{remark}

Call the definition of the derived functor \({\mathbb{R}}F\) for a
left-exact functor
\(F\in{\mathsf{Ab}}\mathsf{Cat}(\mathsf{C}, \mathsf{D})\) where
\(\mathsf{A}\) has enough injectives. These satisfy
\({\mathbb{R}}^0 F = F\), and for a SES \(0\to A\to B\to C\to 0\) there
is an induced LES
\({\mathbb{R}}F A\to {\mathbb{R}}F B \to {\mathbb{R}}F C \to {\mathbb{R}}F A[1]\)
which is functorial in the triple \((A,B,C)\). Next: Grothendieck's
universality theorem.

\end{remark}

\begin{definition}[$\delta\dash$functor]

A \(\delta{\hbox{-}}\)functor is a sequence of functors
\(\left\{{S^i: \mathsf{A}\to \mathsf{B}}\right\}_{i\geq 0}\) such that
for all SESs \(0\to A\to B\to C\to 0\) there is a (not necessarily
exact) complex:

\begin{center}
\begin{tikzcd}
    0 \\
    A && B && C \\
    \\
    {S^1A} && {S^1B} && {S^1 C} \\
    \\
    {S^2A} && {S^2B} && \cdots
    \arrow[from=1-1, to=2-1]
    \arrow[from=2-1, to=2-3]
    \arrow[from=2-3, to=2-5]
    \arrow[from=2-5, to=4-1]
    \arrow[from=4-1, to=4-3]
    \arrow[from=4-3, to=4-5]
    \arrow[from=4-5, to=6-1]
    \arrow[from=6-1, to=6-3]
    \arrow[from=6-3, to=6-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMTAsWzAsMSwiQSJdLFsyLDEsIkIiXSxbNCwxLCJDIl0sWzAsMywiU14xQSJdLFswLDUsIlNeMkEiXSxbMiwzLCJTXjFCIl0sWzQsMywiU14xIEMiXSxbMiw1LCJTXjJCIl0sWzQsNSwiXFxjZG90cyJdLFswLDAsIjAiXSxbOSwwXSxbMCwxXSxbMSwyXSxbMiwzXSxbMyw1XSxbNSw2XSxbNiw0XSxbNCw3XSxbNyw4XV0=}{Link
to Diagram}
\end{quote}

A \textbf{morphism} of \(\delta{\hbox{-}}\)functors is a collection
\(\left\{{f^i: S^i\to T^i}\right\}_{i\geq 0}\) such that for all such
SESs, there is a commutative diagram:

\begin{center}
\begin{tikzcd}
    {S^iA} && {S^iB} && {S^iC} && {S^{i+1}A} \\
    \\
    {T^iA} && {T^iB} && {T^iC} && {T^{i+1}A}
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-3, to=1-5]
    \arrow["{\phi_s^i}", color={rgb,255:red,92;green,92;blue,214}, from=1-5, to=1-7]
    \arrow[from=3-1, to=3-3]
    \arrow[from=3-3, to=3-5]
    \arrow["{\phi_t^i}", color={rgb,255:red,92;green,92;blue,214}, from=3-5, to=3-7]
    \arrow[from=1-1, to=3-1]
    \arrow[from=1-3, to=3-3]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-5, to=3-5]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-7, to=3-7]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsOCxbMCwwLCJTXmlBIl0sWzIsMCwiU15pQiJdLFs0LDAsIlNeaUMiXSxbMCwyLCJUXmlBIl0sWzIsMiwiVF5pQiJdLFs0LDIsIlReaUMiXSxbNiwwLCJTXntpKzF9QSJdLFs2LDIsIlRee2krMX1BIl0sWzAsMV0sWzEsMl0sWzIsNiwiXFxwaGlfc15pIiwwLHsiY29sb3VyIjpbMjQwLDYwLDYwXX0sWzI0MCw2MCw2MCwxXV0sWzMsNF0sWzQsNV0sWzUsNywiXFxwaGlfdF5pIiwwLHsiY29sb3VyIjpbMjQwLDYwLDYwXX0sWzI0MCw2MCw2MCwxXV0sWzAsM10sWzEsNF0sWzIsNSwiIiwxLHsiY29sb3VyIjpbMjQwLDYwLDYwXX1dLFs2LDcsIiIsMSx7ImNvbG91ciI6WzI0MCw2MCw2MF19XV0=}{Link
to Diagram}
\end{quote}

Note that the first 2 square are commutative by functoriality, and the
content here is that the map commutes with the connecting morphisms.

\end{definition}

\begin{definition}[Effaceable functors]

An additive functor \(G: \mathsf{A}\to \mathsf{B}\) is
\textbf{effaceable} iff for all \(A\in \mathsf{A}\) there is a
monomorphism \(A\xhookrightarrow{f} M\) such that
\(GA \xrightarrow{Gf} GM\) is the zero map.

\end{definition}

\begin{slogan}

Effaceable functors are those which erase some monomorphism.

\end{slogan}

\begin{definition}[Universal delta functors]

A delta functor \((S_i, \phi_S)\) is \emph{exact} iff the induced
complex is a LES, and is \textbf{universal} iff for any other delta
functor \((T_i, \phi_T)\) and any natural transformation
\(\eta: S^0\to T^0\), there is a unique morphism
\((S_i, \phi_S) \to (T_i, \phi_T)\) extending \(\eta\).

\end{definition}

\begin{theorem}[Grothendieck, Tohoku: exact fully effaceable functors are universal]

Suppose \(\left\{{S^i F, \phi}\right\}_{i\geq 0}\) is an exact delta
functor and that the \(S^i\) are effaceable for all \(i\). Then it is a
universal \(\delta\) functor.

\end{theorem}

\begin{corollary}[?]

When \(F\in {\mathsf{Ab}}\mathsf{Cat}(\mathsf{A}, \mathsf{B})\) where
\(\mathsf{A}\) has enough injectives, \(({\mathbb{R}}^i F, \phi)\) is
universal and there is a unique such delta functor with
\({\mathbb{R}}^0 F = F\).

\end{corollary}

\begin{proof}[of corollary]

Embed \(A\hookrightarrow I\) into an injective object, which is
\(F{\hbox{-}}\)acyclic, and thus
\({\mathbb{R}}^i F A \xrightarrow{0} {\mathbb{R}}^i F I = 0\).

\end{proof}

\begin{proof}[of theorem]

Proceed by induction. Let \(0\to A \to M \to Q \to 0\) be arbitrary, and
use a diagram chase to define a map \(f^i(\iota)\):

\begin{center}
\begin{tikzcd}
    {S^i M} && {S^i Q} && {S^{i+1}A} && {S^{i+1}M} \\
    \\
    {T^i M} && {T^i Q} && {T^{i+1}A} && {T^{i+1}B}
    \arrow[from=3-1, to=3-3]
    \arrow["{\phi_T^i}", from=3-3, to=3-5]
    \arrow[from=3-5, to=3-7]
    \arrow[from=1-1, to=1-3]
    \arrow["{\phi_S^i}", two heads, from=1-3, to=1-5]
    \arrow["0", from=1-5, to=1-7]
    \arrow[from=1-1, to=3-1]
    \arrow[from=1-3, to=3-3]
    \arrow["{\exists?}"', dashed, from=1-5, to=3-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsOCxbMCwwLCJTXmkgTSJdLFsyLDAsIlNeaSBRIl0sWzQsMCwiU157aSsxfUEiXSxbNiwwLCJTXntpKzF9TSJdLFs0LDIsIlRee2krMX1BIl0sWzYsMiwiVF57aSsxfUIiXSxbMiwyLCJUXmkgUSJdLFswLDIsIlReaSBNIl0sWzcsNl0sWzYsNCwiXFxwaGlfVF5pIl0sWzQsNV0sWzAsMV0sWzEsMiwiXFxwaGlfU15pIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzIsMywiMCJdLFswLDddLFsxLDZdLFsyLDQsIlxcZXhpc3RzPyIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==}{Link
to Diagram}
\end{quote}

One needs to show:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(f^i(\iota)\) does not depend on \(\iota\)
\item
  It is a ? for all \(A\to B\)
\item
  This map commutes with \(\phi_S, \phi_T\).
\end{enumerate}

\end{proof}

\hypertarget{wednesday-march-16}{%
\section{Wednesday, March 16}\label{wednesday-march-16}}

\hypertarget{grothendiecks-universal-theorem}{%
\subsection{Grothendieck's Universal
Theorem}\label{grothendiecks-universal-theorem}}

\begin{remark}

Setup from last time:
\(F\in {\mathsf{Add}}\mathsf{Cat}(\mathsf{A}, \mathsf{B})\) left-exact,
\(\left\{{(S^n, \phi_S^n)}\right\}_{n\geq 0}\) exact
\(\delta{\hbox{-}}\)functors where for \(n > 0\) the \(S^n\) are
effaceable. Then it is universal: for all \(\delta{\hbox{-}}\)functors
\(\left\{{(T^n, \phi_T^n)}\right\}_{n\geq 0}\) with a natural
transformation \(S^0 \to T^0\) there exist unique morphisms
\((S^n, \phi_S^n) \to (T^n, \phi_T^n)\), i.e.~natural transformations
\(S^n\to T^n\) commuting with the \(\phi^n\).

\end{remark}

\hypertarget{proof-of-universality}{%
\subsubsection{Proof of Universality}\label{proof-of-universality}}

\begin{remark}

Take an effacement \(0\to A \xhookrightarrow{i} M\) for \(S^{n+1}\) and
extend to a SES \(0\to A\to M\to Q\to 0\). We'll define the ladder of
morphisms inductively using the following commutative diagram:

\begin{center}
\begin{tikzcd}
    {S^nQ} && {S^{n+1}A} && {S^{n+1}M} \\
    \\
    {T^nQ} && {T^{n+1}A}
    \arrow["{\phi_S}", from=1-1, to=1-3]
    \arrow[from=1-3, to=1-5]
    \arrow["{\phi_T}", from=3-1, to=3-3]
    \arrow["{f^n}"', from=1-1, to=3-1]
    \arrow["{\exists f^{n+1} = f^{n+1}(A, i)}", dashed, from=1-3, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCwwLCJTXm5RIl0sWzIsMCwiU157bisxfUEiXSxbNCwwLCJTXntuKzF9TSJdLFswLDIsIlReblEiXSxbMiwyLCJUXntuKzF9QSJdLFswLDEsIlxccGhpX1MiXSxbMSwyXSxbMyw0LCJcXHBoaV9UIl0sWzAsMywiZl5uIiwyXSxbMSw0LCJcXGV4aXN0cyBmXntuKzF9ID0gZl57bisxfShBLCBpKSIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==}{Link
to Diagram}
\end{quote}

We need to show

\begin{itemize}
\tightlist
\item
  \(f^{n+1}(A, i)\) only depends on \(A\)
\item
  \(f^{n+1}\) is functorial in \(A\)
\item
  \(f^{n+1}\) commutes with \(\phi_S, \phi_T\).
\end{itemize}

\end{remark}

\begin{lemma}[?]

Assume that given two effacements of two delta functors, there exist
morphisms:

\begin{center}
\begin{tikzcd}
    0 && {A_1} && {M_1} \\
    \\
    0 && {A_2} && {M_2}
    \arrow[from=1-1, to=1-3, "{g}"]
    \arrow["{i_1}", from=1-3, to=1-5]
    \arrow[from=3-1, to=3-3]
    \arrow[from=1-3, to=3-3]
    \arrow["{i_2}", from=3-3, to=3-5]
    \arrow[from=1-5, to=3-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMCwwLCIwIl0sWzIsMCwiQV8xIl0sWzQsMCwiTV8xIl0sWzAsMiwiMCJdLFsyLDIsIkFfMiJdLFs0LDIsIk1fMiJdLFswLDFdLFsxLDIsImlfMSJdLFszLDRdLFsxLDRdLFs0LDUsImlfMiJdLFsyLDVdXQ==}{Link
to Diagram}
\end{quote}

Then there is a commuting square

\begin{center}
\begin{tikzcd}
    {S^{n+1}A_1} && {S^{n+1}A_2} \\
    \\
    {T^{n+1}A_1} && {T^{n+1}A_2}
    \arrow["{S^{n+1}(g)}", from=1-1, to=1-3]
    \arrow["{T^{n+1}(g)}", from=3-1, to=3-3]
    \arrow[from=1-3, to=3-3]
    \arrow[from=1-1, to=3-1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJTXntuKzF9QV8xIl0sWzIsMCwiU157bisxfUFfMiJdLFswLDIsIlRee24rMX1BXzEiXSxbMiwyLCJUXntuKzF9QV8yIl0sWzAsMSwiU157bisxfShnKSJdLFsyLDMsIlRee24rMX0oZykiXSxbMSwzXSxbMCwyXV0=}{Link
to Diagram}
\end{quote}

\end{lemma}

\begin{proof}[?]

There is a cube:

\begin{center}
\begin{tikzcd}
    &&& {S^nQ_1} && {S^nQ_2} \\
    \\
    {S^{n+1}A_1} && {S^{n+1}A_2} & {T^nQ_1} && {T^nQ_2} \\
    \\
    {T^{n+1}A_1} && {T^{n+1}A_2}
    \arrow["{S^{n+1}(g)}"{description}, color={rgb,255:red,214;green,92;blue,92}, from=3-1, to=3-3]
    \arrow["{T^{n+1}(g)}"{description}, color={rgb,255:red,92;green,92;blue,214}, from=5-1, to=5-3]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, from=3-3, to=5-3]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, from=3-1, to=5-1]
    \arrow[from=1-4, to=1-6]
    \arrow[from=3-4, to=3-6]
    \arrow["{f^{n+1}}"{description}, from=1-6, to=3-6]
    \arrow["{f^n}"{description}, from=1-4, to=3-4]
    \arrow["{\phi_S^n}"{description}, color={rgb,255:red,214;green,92;blue,92}, from=1-4, to=3-1]
    \arrow["{\phi_S^n}"{description}, from=1-6, to=3-3]
    \arrow["{\phi_T^n}"{description}, from=3-6, to=5-3]
    \arrow["{\phi_T^n}"{description}, from=3-4, to=5-1]
    \arrow[from=3-4, to=3-6]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

Here all faces but the front form commuting squares.

\begin{exercise}[?]

Show that one can move the red path to the blue through the other
commuting faces.

\end{exercise}

\end{proof}

\begin{corollary}[?]

\(f^{n+1}(A, i)\) only depends on \(A\). Take two effacements, and
assume there is a commuting diagram:

\begin{center}
\begin{tikzcd}
    0 && A && {M_1} \\
    \\
    0 && A && {M_2}
    \arrow[from=1-1, to=1-3]
    \arrow["{i_1}", from=1-3, to=1-5]
    \arrow[from=3-1, to=3-3]
    \arrow["{i_2}", from=3-3, to=3-5]
    \arrow["{\operatorname{id}_A}"{description}, Rightarrow, no head, from=1-3, to=3-3]
    \arrow["\exists"{description}, dashed, from=1-5, to=3-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMCwwLCIwIl0sWzIsMCwiQSJdLFs0LDAsIk1fMSJdLFs0LDIsIk1fMiJdLFsyLDIsIkEiXSxbMCwyLCIwIl0sWzAsMV0sWzEsMiwiaV8xIl0sWzUsNF0sWzQsMywiaV8yIl0sWzEsNCwiXFxpZF9BIiwxLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMiwzLCJcXGV4aXN0cyIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==}{Link
to Diagram}
\end{quote}

By the lemma:

\begin{center}
\begin{tikzcd}
    {S^{n+1}A} && {S^{n+1}A} \\
    \\
    {T^{n+1}A} && {T^{n+1}A}
    \arrow[Rightarrow, no head, from=1-1, to=1-3]
    \arrow["{f^n(i_1)}"', from=1-1, to=3-1]
    \arrow[Rightarrow, no head, from=3-1, to=3-3]
    \arrow["{f^{n}(i_2)}", from=1-3, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJTXntuKzF9QSJdLFsyLDAsIlNee24rMX1BIl0sWzAsMiwiVF57bisxfUEiXSxbMiwyLCJUXntuKzF9QSJdLFswLDEsIiIsMCx7ImxldmVsIjoyLCJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzAsMiwiZl5uKGlfMSkiLDJdLFsyLDMsIiIsMix7ImxldmVsIjoyLCJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzEsMywiZl57bn0oaV8yKSJdXQ==}{Link
to Diagram}
\end{quote}

\begin{center}
\begin{tikzcd}
    0 && A && {M_1} \\
    \\
    0 && A && {M_1\oplus M_2} \\
    \\
    0 && A && {M_2}
    \arrow[from=5-1, to=5-3]
    \arrow[from=5-3, to=5-5]
    \arrow[from=3-1, to=3-3]
    \arrow["{i_2}", from=3-3, to=3-5]
    \arrow[from=1-1, to=1-3]
    \arrow["{i_1 \oplus i_2}", from=1-3, to=1-5]
    \arrow[Rightarrow, no head, from=1-3, to=3-3]
    \arrow[Rightarrow, no head, from=3-3, to=5-3]
    \arrow["{(\operatorname{id}_{M_1}, 0)}"', from=3-5, to=1-5]
    \arrow["{(0, \operatorname{id}_{M_2})}", from=3-5, to=5-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsOSxbMCwwLCIwIl0sWzIsMCwiQSJdLFs0LDAsIk1fMSJdLFs0LDIsIk1fMVxcb3BsdXMgTV8yIl0sWzQsNCwiTV8yIl0sWzIsMiwiQSJdLFsyLDQsIkEiXSxbMCw0LCIwIl0sWzAsMiwiMCJdLFs3LDZdLFs2LDRdLFs4LDVdLFs1LDMsImlfMiJdLFswLDFdLFsxLDIsImlfMSJdLFsxLDUsIiIsMSx7ImxldmVsIjoyLCJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzUsNiwiIiwxLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMywyLCIoXFxpZF97TV8xfSwgMCkiLDJdLFszLDQsIigwLCBcXGlkX3tNXzJ9KSJdXQ==}{Link
to Diagram}
\end{quote}

\end{corollary}

\begin{quote}
See notes for finished proof.
\end{quote}

\hypertarget{friday-march-18}{%
\section{Friday, March 18}\label{friday-march-18}}

\begin{remark}

Given effacements:

\begin{center}
\begin{tikzcd}
    0 && {A_1} && {M_1} \\
    \\
    0 && {A_2} && {M_2}
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-3, to=1-5]
    \arrow[from=3-1, to=3-3]
    \arrow[from=3-3, to=3-5]
    \arrow["g", from=1-3, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMCwwLCIwIl0sWzIsMCwiQV8xIl0sWzQsMCwiTV8xIl0sWzQsMiwiTV8yIl0sWzIsMiwiQV8yIl0sWzAsMiwiMCJdLFswLDFdLFsxLDJdLFs1LDRdLFs0LDNdLFsxLDQsImciXV0=}{Link
to Diagram}
\end{quote}

There exists an effacement extending \(g\). Use

\begin{center}
\begin{tikzcd}
    0 && {A_1} && {M_1\oplus M_2} \\
    \\
    0 && {A_2} & {} & {M_2}
    \arrow[from=1-1, to=1-3]
    \arrow["{(i_1, gi_2)}", from=1-3, to=1-5]
    \arrow[from=3-1, to=3-3]
    \arrow["{i_2}", from=3-3, to=3-5]
    \arrow["g", from=1-3, to=3-3]
    \arrow["{(0, \operatorname{id})}", from=1-5, to=3-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNyxbMCwwLCIwIl0sWzIsMCwiQV8xIl0sWzQsMCwiTV8xXFxvcGx1cyBNXzIiXSxbNCwyLCJNXzIiXSxbMiwyLCJBXzIiXSxbMCwyLCIwIl0sWzMsMl0sWzAsMV0sWzEsMiwiKGlfMSwgZ2lfMikiXSxbNSw0XSxbNCwzLCJpXzIiXSxbMSw0LCJnIl0sWzIsMywiKDAsIFxcaWQpIl1d}{Link
to Diagram}
\end{quote}

There is a factorization:

\begin{center}
\begin{tikzcd}
    && {S^i A_2} \\
    \\
    {S^i A_1} &&&& {S^iM_2}
    \arrow["0", dashed, from=1-3, to=3-5]
    \arrow[dashed, from=3-1, to=1-3]
    \arrow[from=3-1, to=3-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwyLCJTXmkgQV8xIl0sWzQsMiwiU15pTV8yIl0sWzIsMCwiU15pIEFfMiJdLFsyLDEsIjAiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMCwyLCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMCwxXV0=}{Link
to Diagram}
\end{quote}

\begin{quote}
?? Concludes theorem from last time.z
\end{quote}

\end{remark}

\begin{remark}

Recall that \(\mathop{\mathrm{Hom}}(C, {-})\) is left exact covariant
and \(\mathop{\mathrm{Hom}}({-}, C)\) is left exact contravariant. For
left exact functors,

\begin{itemize}
\tightlist
\item
  Right derived functors are computed with injective resolutions.
\item
  \(\mathsf{C}\) needs enough injectives
\end{itemize}

For right exact functors,

\begin{itemize}
\tightlist
\item
  Left derived functors are computed with projective resolutions.
\item
  \(\mathsf{C}\) needs enough projectives
\end{itemize}

\end{remark}

\begin{remark}

Projective sheaves are locally free.

\end{remark}

\begin{exercise}[?]

Show:

\begin{itemize}
\tightlist
\item
  Injectives are closed under \(\prod\),
\item
  Projectives are closed under \(\bigoplus\).
\end{itemize}

\end{exercise}

\begin{proof}[?]

\begin{center}
\begin{tikzcd}
    0 && A && B \\
    \\
    &&& {I_i} \\
    \\
    &&& {\prod I_i}
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-3, to=1-5]
    \arrow[""{name=0, anchor=center, inner sep=0}, color={rgb,255:red,214;green,92;blue,92}, from=1-3, to=3-4]
    \arrow[from=3-4, to=5-4]
    \arrow[""{name=1, anchor=center, inner sep=0}, color={rgb,255:red,214;green,92;blue,92}, curve={height=30pt}, from=1-3, to=5-4]
    \arrow[""{name=2, anchor=center, inner sep=0}, "\exists"', color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-5, to=3-4]
    \arrow[""{name=3, anchor=center, inner sep=0}, "{\therefore \exists}", color={rgb,255:red,92;green,92;blue,214}, curve={height=-30pt}, dashed, from=1-5, to=5-4]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, shorten <=7pt, shorten >=7pt, Rightarrow, 2tail reversed, from=1, to=0]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, shorten <=7pt, shorten >=7pt, Rightarrow, dashed, 2tail reversed, from=2, to=3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

\end{proof}

\begin{exercise}[?]

Show that in \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), \(M\) is projective
\(\iff M\) is a direct summand of a free module iff \(M\) is locally
free.

\end{exercise}

\begin{solution}

Some hints:

\begin{center}
\begin{tikzcd}
    && P \\
    \\
    F && P && 0
    \arrow[two heads, from=3-1, to=3-3]
    \arrow[from=3-3, to=3-5]
    \arrow[Rightarrow, no head, from=1-3, to=3-3]
    \arrow[dashed, from=1-3, to=3-1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwyLCJGIl0sWzIsMiwiUCJdLFs0LDIsIjAiXSxbMiwwLCJQIl0sWzAsMSwiIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzEsMl0sWzMsMSwiIiwwLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMywwLCIiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=}{Link
to Diagram}
\end{quote}

\end{solution}

\begin{exercise}[?]

Show

\begin{itemize}
\item
  \({\mathbb{R}}\mathop{\mathrm{Hom}}_{{\mathbb{Z}{\hbox{-}}\mathsf{Mod}}}(C_n, M) = M[n] \oplus \Sigma^1(M/nM)\)
  using
  \(0\to {\mathbb{Z}}\xrightarrow{\times n} {\mathbb{Z}}\to {\mathbb{Z}}/n{\mathbb{Z}}\to 0\).

  \begin{itemize}
  \tightlist
  \item
    Conclude that divisible module has vanishing
    \(\operatorname{Ext} ^1(C_n, {-})\).
  \end{itemize}
\item
  If \(R\) is a PID, then \(M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) is
  injective \(\iff M\) is divisible.
\item
  For all rings \(R\), \(R\) is injective iff
\end{itemize}

\begin{center}
\begin{tikzcd}
    0 && N && R \\
    \\
    &&&& I
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-3, to=1-5]
    \arrow[from=1-3, to=3-5]
    \arrow[dashed, from=1-5, to=3-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCIwIl0sWzIsMCwiTiJdLFs0LDAsIlIiXSxbNCwyLCJJIl0sWzAsMV0sWzEsMl0sWzEsM10sWzIsMywiIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d}{Link
to Diagram}
\end{quote}

\end{exercise}

\hypertarget{monday-march-21}{%
\section{Monday, March 21}\label{monday-march-21}}

\begin{remark}

Recall free \(\implies\) projective and
\({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) has enough projectives and enough
injectives.

\end{remark}

\begin{exercise}[?]

Show \(I\) is injective iff

\begin{center}
\begin{tikzcd}
    0 && J && R \\
    \\
    &&&& I
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-3, to=1-5]
    \arrow["\exists", dashed, from=1-5, to=3-5]
    \arrow[from=1-3, to=3-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCIwIl0sWzIsMCwiSiJdLFs0LDAsIlIiXSxbNCwyLCJJIl0sWzAsMV0sWzEsMl0sWzIsMywiXFxleGlzdHMiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMSwzXV0=}{Link
to Diagram}
\end{quote}

Hint:

\includegraphics{figures/2022-03-21_10-35-47.png}

Extend to \(A' + Ra\) using \(1\mapsto a \mapsto i\in I\) under
\(R\to Ra\to I\). Take a poset of all \(B \subseteq A\) with
\(g:B\to I\) extending \(A'\to I\) and apply Zorn's lemma.

\end{exercise}

\begin{exercise}[?]

Show that for \(R\) a PID, \(M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\)
is injective iff divisible.

\end{exercise}

\begin{exercise}[?]

Show that \({\mathbb{Z}{\hbox{-}}\mathsf{Mod}}\) has enough innjectives.

\begin{quote}
Hint: write
\(A = \bigoplus {\mathbb{Z}}/K \hookrightarrow\bigoplus {\mathbb{Q}}/K\).
\end{quote}

\end{exercise}

\begin{remark}

On adjoint functors:
\begin{align*}
\adjunction F G {\mathsf{A}} {\mathsf{B}} \implies \mathsf{B}(FX, Y)  { \, \xrightarrow{\sim}\, }\mathsf{A}(X, GY)
.\end{align*}
Here \(F\) is a left adjoint hence right exact, and \(G\) is a right
adjoint and is left exact.

\end{remark}

\begin{exercise}[?]

Show that if \(F\) is left exact then \(G\) preserves in injectives, and
if \(F\) is right exact then \(G\) preserves projectives.

Hint:

\begin{center}
\begin{tikzcd}
    && 0 && {A'} && A \\
    \\
    0 && {FA'} && FA && GI \\
    \\
    &&&& I
    \arrow[from=1-3, to=1-5]
    \arrow[from=1-5, to=1-7]
    \arrow[from=3-1, to=3-3]
    \arrow[from=3-3, to=3-5]
    \arrow["\exists", dashed, from=3-5, to=5-5]
    \arrow[from=1-5, to=3-3]
    \arrow[from=1-7, to=3-5]
    \arrow[from=3-7, to=5-5]
    \arrow["{\therefore \exists}", dashed, from=1-7, to=3-7]
    \arrow[from=3-3, to=5-5]
    \arrow[from=1-5, to=3-7]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsOCxbMCwyLCIwIl0sWzIsMiwiRkEnIl0sWzQsMiwiRkEiXSxbNCw0LCJJIl0sWzIsMCwiMCJdLFs0LDAsIkEnIl0sWzYsMCwiQSJdLFs2LDIsIkdJIl0sWzQsNV0sWzUsNl0sWzAsMV0sWzEsMl0sWzIsMywiXFxleGlzdHMiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbNSwxXSxbNiwyXSxbNywzXSxbNiw3LCJcXHRoZXJlZm9yZSBcXGV4aXN0cyIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFsxLDNdLFs1LDddXQ==}{Link
to Diagram}
\end{quote}

\end{exercise}

\begin{remark}

For \(f\in \mathsf{CRing}(S\to R)\), there is an adjunction
\begin{align*}
\adjunction {M_R\mapsto M_S} { {\mathsf{S}{\hbox{-}}\mathsf{Mod}}(R, {-}) } {{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}{{\mathsf{S}{\hbox{-}}\mathsf{Mod}}}
\end{align*}
where
\({\mathsf{S}{\hbox{-}}\mathsf{Mod}}(R, N) \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\)
via the action \((rf)(x) \coloneqq f(rx)\), sometimes called the
\emph{induced \(R{\hbox{-}}\)module}. Note that
\({\mathsf{R}{\hbox{-}}\mathsf{Mod}}(R, N) { \, \xrightarrow{\sim}\, }N\)
by \(1_R \mapsto n\), and there is an iso
\begin{align*}
{\mathsf{S}{\hbox{-}}\mathsf{Mod}}(M_S, N) &\rightleftharpoons{\mathsf{R}{\hbox{-}}\mathsf{Mod}}(M_R, {\mathsf{S}{\hbox{-}}\mathsf{Mod}}(R, N)) \\
\qty{ m\mapsto \psi(m)(1) } &\mapsfrom \psi \\
\phi &\mapsto \qty{m \mapsto \psi(m)(i) \coloneqq\psi(im) \coloneqq\phi(im) }
.\end{align*}

\end{remark}

\begin{remark}

Proving \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) has enough injectives if
\({\mathsf{S}{\hbox{-}}\mathsf{Mod}}\) has enough injectives: use
\(M_R \cong {\mathsf{R}{\hbox{-}}\mathsf{Mod}}(R, M)\hookrightarrow{\mathsf{S}{\hbox{-}}\mathsf{Mod}}(R, M_S) \hookrightarrow{\mathsf{S}{\hbox{-}}\mathsf{Mod}}(R, I)\)
where \(M_S \hookrightarrow I\) embeds into some injective. Take \(R\)
arbitrary and \(S={\mathbb{Z}}\) to conclude any
\({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) has enough injectives.

\end{remark}

\begin{exercise}[?]

This is a theoretical tool and not particularly practical. Consider
\(S\to R \coloneqq{\mathbb{Q}}\to {\mathbb{C}}\) and
\(M = {\mathbb{C}}\). Then
\({\mathsf{{\mathbb{Q}}}{\hbox{-}}\mathsf{Mod}}({\mathbb{C}}, {\mathbb{C}}_{\mathbb{Q}}) = G{\mathbb{C}}_{\mathbb{Q}}\).

\end{exercise}

\begin{remark}

Any \(M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) admits a minimal
injective hull \(M\hookrightarrow I\).

\end{remark}

\begin{theorem}[?]

\({\mathsf{Sh}}(X \to {\mathsf{Ab}}{\mathsf{Grp}})\) and
\({\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}\) have enough
injectives.

\end{theorem}

\begin{proof}[?]

Take
\begin{align*}
{\mathcal{F}}\hookrightarrow\prod_{x\in X} (\iota_x)_* {\mathcal{F}}_x \hookrightarrow\prod_{x\in X} I_x
.\end{align*}
The claim is that the last term is an injective sheaf. Using that
products of injective are injective, it STS \(I_x\) is injective. For
\(\iota_x: \left\{{x}\right\} \hookrightarrow X\), use that modules on a
point are \({\mathbb{Z}{\hbox{-}}\mathsf{Mod}}\) and obtain an
adjunction
\begin{align*}
\adjunction {(\iota_x)_*} {(\iota_x)^{-1}} {{\mathbb{Z}{\hbox{-}}\mathsf{Mod}}}{{\mathsf{Sh}}(X\to {\mathsf{Ab}}{\mathsf{Grp}})}
.\end{align*}
Finally use that \({\mathbb{Z}{\hbox{-}}\mathsf{Mod}}\) has enough
injectives.

\end{proof}

\hypertarget{wednesday-march-23}{%
\section{Wednesday, March 23}\label{wednesday-march-23}}

\begin{remark}

Induced and coinduced modules:

\begin{center}
\begin{tikzcd}
    S \\
    && {{\mathsf{R}{\hbox{-}}\mathsf{Mod}}} && {{\mathsf{S}{\hbox{-}}\mathsf{Mod}}} \\
    R && {R\otimes_S N} && N \\
    && {\mathop{\mathrm{Hom}}_S(R, N)} && N
    \arrow["\mathop{\mathrm{Forget}}", from=2-3, to=2-5]
    \arrow[from=1-1, to=3-1]
    \arrow["\operatorname{Ind}", maps to, from=3-5, to=3-3]
    \arrow["\operatorname{coInd}", maps to, from=4-5, to=4-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsOCxbNCwxLCJcXG1vZHN7U30iXSxbMiwxLCJcXG1vZHN7Un0iXSxbMCwwLCJTIl0sWzAsMiwiUiJdLFs0LDIsIk4iXSxbMiwyLCJSXFx0ZW5zb3JfUyBOIl0sWzQsMywiTiJdLFsyLDMsIlxcSG9tX1MoUiwgTikiXSxbMSwwLCJcXEZvcmdldCJdLFsyLDNdLFs0LDUsIlxcaW5kIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dLFs2LDcsIlxcY29pbmQiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV1d}{Link
to Diagram}
\end{quote}

Note that coinduction sends injective to injectives, and induction sends
projectives to projectives. Recall that
\({\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\) and
\({\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}\) have enough
injectives, so left exact covariant functors \(F\) admit right-derived
functors \({\mathbb{R}}F\), and similarly right exact contravariant
functors \(F\) admit left-derived functors \({\mathbb{L}}F\).

\end{remark}

\begin{example}[?]

Important functors:

\begin{itemize}
\tightlist
\item
  Global sections
  \({{\Gamma}\qty{{-}} }: {\mathsf{Sh}}(X; \mathsf{C}) \to \mathsf{C}\)
  where \({\mathcal{F}}\mapsto {\mathcal{F}}(X)\), e.g.~for
  \(\mathsf{C} = {\mathsf{Ab}}{\mathsf{Grp}}\).
  \({\mathbb{R}}{{\Gamma}\qty{{\mathcal{F}}} } = H^i(X; {\mathcal{F}})\)
  is sheaf cohomology.
\item
  For \(f\in {\mathsf{Top}}(X, Y)\), the pushforwards
  \(f_*: {\mathsf{Sh}}(X; \mathsf{C}) \to {\mathsf{Sh}}(Y; \mathsf{C})\)
  where \({\mathcal{F}}\mapsto (U\mapsto {\mathcal{F}}(f^{-1}U))\).
  \({\mathbb{R}}f_* {\mathcal{F}}\) are \emph{derived pushforwards}.
\item
  Inverse image, which is exact.
\item
  \(({-})\otimes_{{\mathcal{O}}_X} {\mathcal{F}}\)
\item
  \(\mathop{\mathrm{Hom}}_{{\mathcal{O}}_X}({-}, {\mathcal{F}})\).
\end{itemize}

\end{example}

\begin{theorem}[?]

If \(F \in [\mathsf{A}, \mathsf{B}]\) is left exact covariant and
\(\mathsf{A}\) has enough injectives, then for every \(A\in \mathsf{A}\)
there exists an acyclic resolution
\(0\to A \leftleftarrows { {J}^{\scriptscriptstyle \bullet}}\) whose
homology computes \({\mathbb{R}}R\).

\end{theorem}

\begin{proof}[Sketch]

Why this homology computes the derived functors: let \(A = A^0\) and
take an injective resolution
\(A \leftleftarrows { {J}^{\scriptscriptstyle \bullet}}\). Break this
into SESs, letting \(Z_i\) denote images:

\begin{itemize}
\tightlist
\item
  \(0 \to Z^0 \to J^0\to Z^1\to 0\)
\item
  \(0 \to Z^1\to J^1 \to Z^2\to 0\)
\item
  \(\cdots\)
\end{itemize}

Note that
\(Z^n \leftleftarrows\Sigma^n { {J}^{\scriptscriptstyle \bullet}} = (J^n\to J^{n+1}\to\cdots)\)
is an injective resolution. Splice to obtain
\begin{align*}
0 \to FA \to FJ^0 \to FZ^1\to {\mathbb{R}}^1 FA \to 0, \qquad {\mathbb{R}}^n F Z^1  { \, \xrightarrow{\sim}\, }{\mathbb{R}}^{n+1} FA \\
0 \to \ker(FJ^0\to FJ^1) \to FJ^0 \to \ker(FJ^1\to FJ^2) \to {\mathbb{R}}^1 FA \to 0
.\end{align*}
Proceed by induction.

\end{proof}

\begin{remark}

Consider \(F = \mathsf{A}(A, {-})\) (covariant) or
\(\mathsf{A}({-}, A)\) (contravariant), so
\(F\in \mathsf{Cat}(\mathsf{A}, {\mathsf{Ab}}{\mathsf{Grp}})\). Note
that acyclic objects for \(F\) are exactly injectives: take
\(0\to A\to B\to C\to 0\) to obtain
\(0\to [C, I] \to [B, I]\to [A, I] \to \operatorname{Ext} ^1(C, I) = 0\)
by acyclicity of \(I\), meaning that \([B,I] \twoheadrightarrow[A, I]\)
and thus there exist lifts:

\begin{center}
\begin{tikzcd}
    0 & A & B & C & 0 \\
    \\
    & I
    \arrow[from=1-1, to=1-2]
    \arrow[from=1-2, to=1-3]
    \arrow[from=1-3, to=1-4]
    \arrow[from=1-4, to=1-5]
    \arrow[from=1-2, to=3-2]
    \arrow["\exists", dashed, from=1-3, to=3-2]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMCwwLCIwIl0sWzEsMCwiQSJdLFsyLDAsIkIiXSxbMywwLCJDIl0sWzQsMCwiMCJdLFsxLDIsIkkiXSxbMCwxXSxbMSwyXSxbMiwzXSxbMyw0XSxbMSw1XSxbMiw1LCJcXGV4aXN0cyIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==}{Link
to Diagram}
\end{quote}

\end{remark}

\begin{definition}[Flasque and soft sheaves]

A sheaf
\({\mathcal{F}}\in {\mathsf{Sh}}(X; {\mathbb{Z}{\hbox{-}}\mathsf{Mod}})\)
is \textbf{flasque} iff for all \(U \subseteq X\) open,
\(F(X) \twoheadrightarrow F(U)\). It is \textbf{soft} iff the same holds
for all \emph{closed} sets instead, and \textbf{fine} if
\({\mathcal{F}}\) has a partition of unity property.

\end{definition}

\begin{remark}

Note that fine \(\implies\) soft and flasque \(\implies\) soft. Fine
sheaves are best for paracompact Hausdorff spaces, and flasque are
better for e.g.~the order topology.

\end{remark}

\hypertarget{friday-march-25}{%
\section{Friday, March 25}\label{friday-march-25}}

\hypertarget{flasque-sheaves}{%
\subsection{Flasque Sheaves}\label{flasque-sheaves}}

\begin{remark}

Important classes of sheaves:

\begin{itemize}
\tightlist
\item
  Universal: flasque or flabby.
\item
  Classical topologies (Hausdorff, paracompact): fine \(\implies\) soft.
\item
  AG: quasicoherent sheaves on affine sets and covers.
\end{itemize}

\end{remark}

\begin{theorem}[Sufficient conditions for acyclicity]

Suppose \(\mathsf{A}\in {\mathsf{Ab}}\mathsf{Cat}\) has enough
injectives and \({\mathcal{F}}\in \mathsf{Cat}(\mathsf{A},\mathsf{B})\)
is left exact. Suppose
\({\mathcal{C}}\subseteq {\operatorname{Ob}}\mathsf{A}\) satisfies

\begin{itemize}
\tightlist
\item
  Any \(A\in \mathsf{A}\) admits an embedding \(A\hookrightarrow C\) for
  some \(C\in {\mathcal{C}}\).
\item
  If \(A_1 \bigoplus A_2 \in {\mathcal{C}}\) then
  \(A_1, A_2\in {\mathcal{C}}\).
\item
  Given a SES \(0\to A\to B\to C\to 0\) with \(A, B\in {\mathcal{C}}\),
  \(C\in {\mathcal{C}}\) and \(0\to FA\to FB\to FC\to 0\) is exact.
\end{itemize}

Then every \(C\in {\mathcal{C}}\) is \(F{\hbox{-}}\)acyclic.

\end{theorem}

\begin{exercise}[?]

Use this to show that flasque implies \(F{\hbox{-}}\)acyclic for
\(F({-}) \coloneqq{{\Gamma}\qty{{-}} }\).

\end{exercise}

\begin{solution}

Recall \(U \subseteq X\) open \(\implies F(X) \twoheadrightarrow F(U)\).

\begin{itemize}
\item
  Take an embedding \(0\to F\to \prod_{x\in X} (\iota_x)_* F_x\) where
  \(\iota_x: \left\{{x}\right\} \hookrightarrow X\). Use that for any
  group \(A\), \({\mathcal{G}}\coloneqq(\iota_x)_* A\) satisfies
  \({\mathcal{G}}(X) \twoheadrightarrow{\mathcal{G}}(S)\) for any
  \(S \subseteq X\) since \({\mathcal{G}}\) is flasque and soft and this
  is preserved under products.
\item
  Apply the lifting property to direct sums.
\item
  Use that restrictions of flasque sheaves to opens are again flasque to
  prove that there is a surjection:

  \begin{center}
  \begin{tikzcd}
    {B(X)} && {C(X)} \\
    \\
    {B(U)} && {C(U)}
    \arrow[two heads, from=1-1, to=1-3]
    \arrow[two heads, from=3-1, to=3-3]
    \arrow[two heads, from=1-1, to=3-1]
    \arrow["\therefore", dashed, two heads, from=1-3, to=3-3]
  \end{tikzcd}
  \end{center}
\end{itemize}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJCKFgpIl0sWzIsMCwiQyhYKSJdLFswLDIsIkIoVSkiXSxbMiwyLCJDKFUpIl0sWzAsMSwiIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzIsMywiIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzAsMiwiIiwxLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzEsMywiXFx0aGVyZWZvcmUiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifSwiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV1d}{Link
to Diagram}
\end{quote}

\end{solution}

\begin{proof}[of theorem]

Any injective is in \({\mathcal{C}}\) by assumption: since
\(J\hookrightarrow C\) splits for any injective \(J\), one has
\(C\cong J \oplus J'\), making \(J\) a direct summand and thus in
\({\mathcal{C}}\) by the 2nd property.

Since there are enough injectives, form \(0\to C\to I \to C''\to 0\).
Take the LES, using that \({\mathbb{R}}^{>0 } FI = 0\) to obtain

\begin{center}
\begin{tikzcd}
    0 & FC & FI & {FC''} & 0 \\
    0 & {{\mathbb{R}}^1 FC} & {{\mathbb{R}}^1 FI = 0} & {{\mathbb{R}}^1 FC''} \\
    & {{\mathbb{R}}^2 FC} & {{\mathbb{R}}^2 FI = 0} & {{\mathbb{R}}^2 FC''} \\
    & \cdots
    \arrow[from=2-1, to=2-2]
    \arrow[from=2-2, to=2-3]
    \arrow[from=2-3, to=2-4]
    \arrow["\cong"{description}, from=2-4, to=3-2]
    \arrow[from=3-2, to=3-3]
    \arrow[from=3-3, to=3-4]
    \arrow["\cong"{description}, from=3-4, to=4-2]
    \arrow[from=1-1, to=1-2]
    \arrow[from=1-2, to=1-3]
    \arrow[from=1-3, to=1-4]
    \arrow[from=1-4, to=1-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMTMsWzAsMCwiMCJdLFsxLDAsIkZDIl0sWzIsMCwiRkkiXSxbMywwLCJGQycnIl0sWzQsMCwiMCJdLFswLDEsIjAiXSxbMSwxLCJcXFJSXjEgRkMiXSxbMiwxLCJcXFJSXjEgRkkgPSAwIl0sWzMsMSwiXFxSUl4xIEZDJyciXSxbMiwyLCJcXFJSXjIgRkkgPSAwIl0sWzEsMiwiXFxSUl4yIEZDIl0sWzMsMiwiXFxSUl4yIEZDJyciXSxbMSwzLCJcXGNkb3RzIl0sWzUsNl0sWzYsN10sWzcsOF0sWzgsMTAsIlxcY29uZyIsMV0sWzEwLDldLFs5LDExXSxbMTEsMTIsIlxcY29uZyIsMV0sWzAsMV0sWzEsMl0sWzIsM10sWzMsNF1d}{Link
to Diagram}
\end{quote}

\end{proof}

\begin{remark}

There is a canonical flasque resolution:

\begin{center}
\begin{tikzcd}
    0 & {\mathcal{F}}& {S(F) \coloneqq\prod_{x\in X} (\iota_x)_* {\mathcal{F}}_X} \\
    \\
    0 & {\mathcal{G}}& {S({\mathcal{G}})}
    \arrow[from=1-1, to=1-2]
    \arrow[from=1-2, to=1-3]
    \arrow[from=3-1, to=3-2]
    \arrow[from=3-2, to=3-3]
    \arrow[from=1-2, to=3-2]
    \arrow[dashed, from=1-3, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMCwwLCIwIl0sWzEsMCwiXFxtY2YiXSxbMiwwLCJTKEYpIFxcZGEgXFxwcm9kX3t4XFxpbiBYfSAoXFxpb3RhX3gpXyogXFxtY2ZfWCJdLFswLDIsIjAiXSxbMSwyLCJcXG1jZyJdLFsyLDIsIlMoXFxtY2cpIl0sWzAsMV0sWzEsMl0sWzMsNF0sWzQsNV0sWzEsNF0sWzIsNSwiIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d}{Link
to Diagram}
\end{quote}

This is useful e.g.~for finite sets with the order topology, but less
useful if \({\left\lvert {X} \right\rvert}\) is infinite and there are
non-closed points.

\end{remark}

\begin{exercise}[?]

Show that if \(X\) is Hausdorff paracompact, flasque implies soft. As a
corollary, soft sheaves are acyclic for such spaces.

\end{exercise}

\begin{solution}

See notes.

\end{solution}

\hypertarget{fine-sheaves}{%
\subsection{Fine Sheaves}\label{fine-sheaves}}

\begin{remark}

Recall that a sheaf is fine iff it satisfies the POU property.

\begin{itemize}
\tightlist
\item
  Classically: there is an open cover
  \({\mathcal{U}}\rightrightarrows X\) and
  \(\phi_i: U_i \to {\mathbb{R}}\) with
  \(\mathop{\mathrm{supp}}\phi_i \subseteq U_i\) where
  \(\sum \phi_i = 1\) and locally there are only finitely many nonzero
  \(\phi\).
\item
  For sheaves: there is an open cover
  \({\mathcal{U}}\rightrightarrows X\) and
  \(\phi_i: {\mathcal{F}}\to {\mathcal{F}}\) with
  \(\mathop{\mathrm{supp}}\phi\) a closed set \(Z_i\) where
  \(\sum \phi_i = \operatorname{id}_{\mathcal{F}}\) and locally there
  are only finitely many \(i\) with \(\phi({\mathcal{F}})\neq 0\).
\end{itemize}

\includegraphics{figures/2022-03-25_11-02-23.png}

\end{remark}

\begin{example}[?]

Suppose \(X\) is Hausdorff paracompact, set
\({\mathcal{F}}\coloneqq{\mathcal{O}}_X^\text{cts}\). Thus
\({\mathcal{O}}_X\) has a POU property, as does any
\({\mathcal{O}}_X{\hbox{-}}\)module. Take a usual POU
\(\left\{{f_i}\right\}\) and define
\begin{align*}
\phi: {\mathcal{F}}&\to {\mathcal{F}}\\
s &\mapsto f_i s
.\end{align*}
So any
\({\mathcal{F}}\in {\mathsf{{\mathcal{O}}_X^\text{cts}}{\hbox{-}}\mathsf{Mod}}\)
is soft.

\end{example}

\begin{remark}

In this case, fine implies soft.

\end{remark}

\hypertarget{de-rham-and-dolbeaut-cohomology}{%
\subsection{de Rham and Dolbeaut
cohomology}\label{de-rham-and-dolbeaut-cohomology}}

\begin{remark}

Let \(X\) be a smooth manifold over \({\mathbb{R}}\). Note that
\(\underline{{\mathbb{R}}}\) is not fine and not soft, and not even an
\({\mathcal{O}}_X{\hbox{-}}\)module. However it admits a resolution
\(0 \leftarrow\underline{{\mathbb{R}}} \leftleftarrows { {\Omega_{X}}^{\scriptscriptstyle \bullet}}\)
where \(\Omega^0_{X} \coloneqq{\mathcal{O}}_X^{{\mathsf{sm}}}\), and
this resolution computes the sheaf cohomology
\({ {H}^{\scriptscriptstyle \bullet}} (X; \underline{{\mathbb{R}}})\).

Similarly,
\(0 \to \underline{{\mathbb{C}}} \leftleftarrows_{{ \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu}} \Omega^{0, \bullet}\)
where
\({ \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu}= \sum {\frac{\partial }{\partial \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu_i }\,} dz_i\).

\end{remark}

\hypertarget{computing-cohomology-monday-march-28}{%
\section{Computing Cohomology (Monday, March
28)}\label{computing-cohomology-monday-march-28}}

\begin{remark}

Upcoming topics related to
\({ {H}^{\scriptscriptstyle \bullet}} (X; {\mathcal{F}})\):

\begin{itemize}
\tightlist
\item
  General vanishing theorems
\item
  Čech cohomology
\item
  Riemann-Roch
\end{itemize}

\end{remark}

\hypertarget{vanishing-theorems}{%
\subsection{Vanishing Theorems}\label{vanishing-theorems}}

\begin{theorem}[Grothendieck]

If \(X\) is a Noetherian space, then
\(\tau_{\geq n+1} { {H}^{\scriptscriptstyle \bullet}} (X; {\mathcal{F}}) = 0\)
for \(n\coloneqq\dim X\).

\end{theorem}

\begin{remark}

\envlist

\begin{itemize}
\tightlist
\item
  Note that the theorem statement uses the Zariski topology, and so
  doesn't contradict that
  \({H}^{2d}_{{\mathrm{sing}}}(X; {\mathbb{Z}}) \neq 0\) for (say) \(X\)
  a compact complex manifold.

  \begin{itemize}
  \tightlist
  \item
    The theorem uses algebraic dimension
    \(d \coloneqq\dim_{\mathbb{C}}X\), which is generally twice the real
    dimension.
  \end{itemize}
\item
  Recall that \(X\) is Noetherian iff \(X\) satisfies the DCC on closed
  sets.
\item
  Algebraic varieties with the Zariski topology are Noetherian, since
  dimension strictly decreases on proper closed subsets.
\item
  Affine schemes over Noetherian rings are Noetherian, since closed
  subsets corresponds to radical ideals, which satisfy the ACC.
\item
  \(\dim X\) is defined as
  \(\sup \left\{{d {~\mathrel{\Big\vert}~}Z_0 \subsetneq Z_1 \subsetneq \cdots \subsetneq Z_d }\right\}\).
\item
  Noetherian spaces can have infinite dimension (see examples by Nagata)
\item
  Schemes are nonsingular if the completions of local rings are formal
  power series.
\item
  Smallest class of nice rings in AG: referred to as ``Japanese rings''
  in the literature, finitely generated rings over DVRs, plus
  localizations, completions, direct sums, etc.
\end{itemize}

\end{remark}

\begin{definition}[Quasicoherent sheaves]

A sheaf
\({\mathcal{F}}\in {\mathsf{Sh}}(X, {\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}})\)
is \textbf{quasicoherent} if for all
\(U = \operatorname{Spec}R \subseteq X\), the restrictions
\({ \left.{{{\mathcal{F}}}} \right|_{{U}} } \cong \tilde M\) for
\(M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\). Recall that
\({\mathcal{O}}_X(D(f)) = R{ \left[ { \scriptstyle \frac{1}{f} } \right] }\),
and we define
\(\tilde M(D(f)) \coloneqq M{ \left[ { \scriptstyle \frac{1}{f} } \right] }\),
so e.g.~\(\tilde R = {\mathcal{O}}_X\).

\end{definition}

\begin{theorem}[Serre]

A sheaf
\({\mathcal{F}}\in {\mathsf{Sh}}(X, {\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}})\)
is quasicoherent iff
\begin{align*}
{\mathcal{O}}_X { {}^{ \scriptscriptstyle\oplus^{J} }  } \to {\mathcal{O}}_X{ {}^{ \scriptscriptstyle\oplus^{I} }  } \to {\mathcal{F}}\to 0
.\end{align*}

\end{theorem}

\begin{remark}

Analogy:

\begin{itemize}
\tightlist
\item
  Quasicoherent: arbitrary modules \(M\)
\item
  Coherent: finitely presented modules \(M\).
\end{itemize}

\end{remark}

\begin{example}[Coherent sheaves]

Examples of coherent sheaves

\begin{itemize}
\tightlist
\item
  For \(X \subseteq {\mathbb{P}}^N\) projective (or quasiprojective,
  i.e.~open in a projective), the \textbf{twisting sheaves}
  \({\mathcal{O}}_X(d)\) whose local sections are
  \(p(\mathbf{x})/q(\mathbf{x})\) for \(p,q\) homogeneous where
  \(\deg p - \deg q = d\).
\item
  For any \(Z \subseteq X\) as above, the \textbf{ideal sheaf}
  \({\mathcal{I}}_Z \subseteq {\mathcal{O}}_Z\) and their twists
  \({\mathcal{I}}_Z(d) \coloneqq{\mathcal{I}}_Z \otimes_{{\mathcal{O}}_X} {\mathcal{O}}(d)\).
\item
  Tangent sheaves \({\mathbf{T}}_X\) and cotangent sheaves
  \({\mathbf{T}} {}^{ \vee }_X\), and their tensor powers,
  e.g.~\(\Omega^n_X\).
\end{itemize}

\end{example}

\begin{theorem}[Serre Vanishing 1]

\begin{align*}
{\mathcal{F}}\in {\mathsf{QCoh}}(X), \, X\in {\mathsf{Aff}}{\mathsf{Sch}}_{/ {k}}  \implies
\tau_{\geq 1}  { {H}^{\scriptscriptstyle \bullet}} (X; {\mathcal{F}}) = 0
.\end{align*}

\end{theorem}

\begin{theorem}[Serre Vanishing 2]

\begin{align*}
{\mathcal{F}}\in {\mathsf{Coh}}(X),\, X\in \mathop{\mathrm{Proj}}{\mathsf{Sch}}_{/ {k}}  \implies 
\tau_{\geq 1}  { {H}^{\scriptscriptstyle \bullet}} (X; {\mathcal{F}}(n) ) = 0 \text{ for some } n\gg 0 
.\end{align*}

\end{theorem}

\begin{remark}

Affine schemes correspond to general rings, and projective schemes
correspond to graded rings. In the second statement, coherence is used
as a kind of finiteness.

\end{remark}

\hypertarget{ux10dech-cohomology}{%
\subsection{Čech Cohomology}\label{ux10dech-cohomology}}

\begin{definition}[The Cech complex an differential]

For open covers, write \({\mathcal{U}}\rightrightarrows X\) iff
\(X = \cup_i U_i\). Define
\(U_{i_0, i_1,\cdots, i_p} \coloneqq U_{i_1} \cap U_{i_1} \cap\cdots \cap U_{i_p}\).
Define a complex
\begin{align*}
0 \to {\check{C}}^0({\mathcal{U}}; {\mathcal{F}}) = \bigoplus_{i_0\in I} {{\Gamma}\qty{{\mathcal{F}}; U_{i_0}} } \xrightarrow{{\partial}_1} \bigoplus _{i_1 < i_2} {{\Gamma}\qty{{\mathcal{F}}; U_{i_0, i_1}} } \xrightarrow{{\partial}_2}  \cdots
.\end{align*}
where we specify where elements land componentwise:
\begin{align*}
{ \left.{{{\partial}_i}} \right|_{{i_1 < \cdots < i_{p+1}}} }: \bigoplus _{i_0 < \cdots < i_p} {\mathcal{F}}(U_{i_0, \cdots , i_p}) \\
f &\mapsto \sum_{0\leq k \leq p+1} (-1)^k { \left.{{f}} \right|_{{i_0 < \cdots \widehat{k} < i_{p+1} }} }\mathrel{\Big|}_{U_{i_0, \cdots, i_{p+1}}}
.\end{align*}

\end{definition}

\begin{remark}

Why \({\partial}^2 = 0\): if \(k < \ell\), forget \(\ell\) first and
then \(k\) to get a sign \((-1)^\ell (-1)^k\), or forget \(k\) first
then \(\ell\) to get \((-1)^k (-1)^{\ell - 1}\) due to the shift. So
these contributions cancel.

\end{remark}

\begin{theorem}[?]

Suppose that for all inclusions
\(j_{i_0, \cdots, i_p}: U_{i_0, \cdots, i_p} \to X\), the pushforwards
of \({\mathcal{F}}\)
\begin{align*}
{ \left.{{ (j_{i_0, \cdots, i_p})_* {\mathcal{F}}}} \right|_{{U_{i_0, \cdots, i_p}}} }
\end{align*}
have vanishing cohomology in degrees \(p\geq 1\). Then
\begin{align*}
 { {H}^{\scriptscriptstyle \bullet}} (X; {\mathcal{F}})  { \, \xrightarrow{\sim}\, } { {{\check{H}}}^{\scriptscriptstyle \bullet}} ({\mathcal{U}}; {\mathcal{F}})
.\end{align*}

This is true for all affine schemes if
\({\mathcal{F}}\in {\mathsf{QCoh}}(X)\), e.g.~for algebraic varieties or
separated schemes.

\end{theorem}

\hypertarget{wednesday-march-30}{%
\section{Wednesday, March 30}\label{wednesday-march-30}}

\begin{remark}

Topics:

\begin{itemize}
\tightlist
\item
  General vanishing (Serre 1 and 2)
\item
  Čech cohomology
\item
  Riemann-Roch and Serre duality
\item
  Advanced vanishing (e.g.~Kodaira vanishing)
\end{itemize}

\end{remark}

\hypertarget{ux10dech-cohomology-1}{%
\subsection{Čech Cohomology}\label{ux10dech-cohomology-1}}

\begin{remark}

Setup: \(X\) and
\({\mathcal{F}}\in {\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\), an
open cover \({\mathcal{U}}\rightrightarrows X\). We defined the Čech
complex:
\begin{align*}
{\check{C}}^p({\mathcal{U}}; {\mathcal{F}}) = \bigoplus _{i_1 < \cdots < i_p} {\mathcal{F}}(U_{i_1, \cdots, i_p})
,\end{align*}
which had certain differentials.

\end{remark}

\begin{theorem}[?]

Suppose \(X\in {\mathsf{Alg}}{\mathsf{Var}}\) or \(X\in {\mathsf{Sch}}\)
is separated (e.g.~a quasiprojective scheme),
\(F\in {\mathsf{QCoh}}(X)\) an \({\mathcal{O}}_X{\hbox{-}}\)module, and
let \({\mathcal{U}}\rightrightarrows X\) be an affine open cover. Then
\begin{align*}
{\check{H}}({\mathcal{U}}; F) = {\mathbb{R}}\Gamma(X; F)
.\end{align*}

\end{theorem}

\begin{remark}

More generally, we can just assume that all intersections of affines are
affine, and instead there is a spectral sequence. This can fail if \(X\)
is not separated,
e.g.~\(X \coloneqq{\mathbb{A}}^2 { \displaystyle\coprod_{{\mathbb{A}}^2\setminus\left\{{0}\right\}} } {\mathbb{A}}^2\)
where the intersection \({\mathbb{A}}^2\setminus\left\{{0}\right\}\) is
not affine. Recall that \(X\) is separated iff
\(X \xhookrightarrow{\Delta_X} X{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{2} } }\)
is closed.

\end{remark}

\begin{example}[?]

Consider \(X={\mathbb{P}}^1\) and
\(F = \bigoplus _{d\in {\mathbb{Z}}} {\mathcal{O}}_X(d)\), we can
compute \({\check{H}}(X; {\mathcal{O}}(d))\) for all \(d\). Take a cover
\(U_i = \left\{{x_i\neq 0}\right\}\) where \(U_0\) has coordinate
\(x \coloneqq x_1/x_0\) and \(U_1\) has coordinate \(y= x_0/x_1\) which
intersect at \(U_{01} = \left\{{x,y\neq 0}\right\}\) and are glued by
\(y=1/x\). The Čech resolution is
\begin{align*}
0\to F(U_0) \oplus F(U_1) \xrightarrow{f} F(U_{01}) \to 0
,\end{align*}
so \(H^0 = \ker f\) and \(H^1 = \operatorname{coker}f\). Recall that
sections of \({\mathcal{O}}(d)\) are locally ratios of polynomials with
valuation \(d\). We have
\({ \left.{{{\mathcal{O}}_{{\mathbb{P}}^1}(d)}} \right|_{{{\mathbb{A}}_1}} } = x_0^d {\mathcal{O}}_{{\mathbb{P}}^1}\)
by rewriting \(p/q = x_0^d p'/q'\). We can thus write this sequence as
\begin{align*}
0 \to \bigoplus _{d\in {\mathbb{Z}}} x_0^d k { \left[ \scriptstyle {x= {x_1\over x_0}} \right] }  = 
\bigoplus_d \left\langle{\text{degree $d$ monomials in } x_0^{\pm 1}, x_1}\right\rangle \oplus  \bigoplus _{d} \left\langle{\text{degree $d$ monomials in } x_0, x_1^{\pm 1}}\right\rangle 
\to \bigoplus_d \left\langle{\text{degree $d$ monomials in } x_0^{\pm 1}, x_1^{\pm 1}}\right\rangle 
\to 0
.\end{align*}

\begin{claim}

\begin{align*}
H^0(X; F) = k[x_0, x_1], \qquad H^1 = {1\over x_0 x_1} k { \left[ \scriptstyle {{1\over x_0}, {1\over x_1}} \right] } 
.\end{align*}

\end{claim}

Being in the kernel means \(v_{x_0}(f)>0\) and \(v_{x_1}(f) > 0\), which
yields monomials \(x_0^n x_1^m\) where \(d=n+m\). For the cokernel, note
\((p, 1) \mapsto p-q\), what's missing? Monomials where both powers are
negative.

\end{example}

\begin{example}[?]

Similar computations work for \(X={\mathbb{P}}^n\) and yield
\begin{align*}
{H}^0\qty{X; \bigoplus _{d\in {\mathbb{Z}}} {\mathcal{O}}_{{\mathbb{P}}^n}(d) } = k[x_1, \cdots, x_{n}], \quad
{H}^n\qty{X; \bigoplus _{d\in {\mathbb{Z}}} {\mathcal{O}}_{{\mathbb{P}}^n}(d) } = {1\over \prod x_i} k { \left[ \scriptstyle {{1\over x_0}, \cdots, {1\over x_n}} \right] } 
.\end{align*}
Note that both sides are graded by degree. This can be done in affine
opens \(U_i = \left\{{x_i\neq 0}\right\} \cong {\mathbb{A}}^n\),
\({ \left.{{{\mathcal{O}}_X(d)}} \right|_{{=}} } x_i^d {\mathcal{O}}_X\),
and similarly
\begin{align*}
0 \to
\bigoplus_d \left\langle{\text{degree $d$ monomials in } x_0^{\pm 1}, x_1, \cdots, x_n}\right\rangle \oplus  
\bigoplus _d \left\langle{\text{degree $d$ monomials in} x_0, x_1^{\pm 1}, \cdots, x_n}\right\rangle 
\oplus \cdots
\to \cdots \to 0
.\end{align*}
The kernel is again spanned by monomials \(f\) with
\(v_{x_i}(f) \geq 0\) for all \(i\). Which monomials don't come from the
middle step? Those where \(v_{x_i}(f) < 0\) for all \(i\).

\end{example}

\begin{remark}

A combinatorial device to keep track of monomials: let
\(X={\mathbb{P}}^2\), and build simplices which track which monomials
are allowed to be negative.

See Hartshorne for a description of how to encode this as a simplicial
set:

\includegraphics{figures/2022-03-30_11-07-34.png}

\end{remark}

\begin{remark}

As a result, we can compute
\begin{align*}
\dim H^0({\mathbb{P}}^n; {\mathcal{O}}_{{\mathbb{P}}^n}(d)) = {n+d\choose n} = {n+d\choose d}
\end{align*}
by counting monomials using a stars and bars argument. Moreover
\begin{align*}
\dim H^n({\mathbb{P}}^n; {\mathcal{O}}(d)) = \dim H^0({\mathbb{P}}^n; {\mathcal{O}}(n-1-d)) = \dim H^0({\mathbb{P}}^n; {\mathcal{O}}(K) \otimes{\mathcal{O}}(d)^{-1})
\end{align*}
where the canonical class of \({\mathbb{P}}^n\) is given by
\({\mathcal{O}}(K_{{\mathbb{P}}^n}) = {\mathcal{O}}(-n-1)\).

\end{remark}

\hypertarget{friday-april-01}{%
\section{Friday, April 01}\label{friday-april-01}}

\begin{quote}
Reference for toric geometry: Fulton's Toric Varieties, Oda's
\emph{Convex bodies in algebraic geometry}.
\end{quote}

\begin{proposition}[?]

Claim from last time:
\begin{align*}
 { {H}^{\scriptscriptstyle \bullet}} ({\mathbb{P}}^n; {\mathcal{O}}(d)) \coloneqq{\mathbb{R}}\Gamma({\mathbb{P}}^n; {\mathcal{O}}_{{\mathbb{P}}^n}(d)) \cong {\check{H}}({\mathcal{U}}; {\mathcal{O}}_{{\mathbb{P}}^n}(d))
,\end{align*}
where this isomorphism is of graded vector spaces. We also saw
\begin{align*}
\bigoplus _{d\in {\mathbb{Z}}} H^0({\mathbb{P}}^n; {\mathcal{O}}(d)) \cong k[x_1, \cdots, x_{n}]= \bigoplus _{\mathbf{d} \geq 0} k \prod x_i^{d_i}
,\end{align*}
and in top degree,
\begin{align*}
\bigoplus _{d\in {\mathbb{Z}}} H^n({\mathbb{P}}^n; {\mathcal{O}}(d)) \cong \prod x_i^{-1}k { \left[ \scriptstyle {x_0^{-1}, \cdots, x_n^{-1}} \right] } 
,\end{align*}
with all intermediate degrees vanishing. There is a nondegenerate
pairing
\begin{align*}
H^0({\mathbb{P}}^n; {\mathcal{O}}(d)) \times H^n({\mathbb{P}}^n; {\mathcal{O}}(-n-1-d)) \to k\cdot \prod x_i^{-1}\cong k
\end{align*}
which is concretely realized by multiplying monomials and projecting
onto the span of \(\prod x_i^{-1}\) (so setting all other monomials to
zero). This is an instance of Serre duality, but this example is in fact
used in the proof.

\end{proposition}

\begin{proof}[?]

Compute \(\oplus_d {\check{H}}({\mathcal{U}}; {\mathcal{O}}(d))\) by
first writing
\({\mathbb{P}}^n = {\mathbb{A}}^n_{x_0\neq 0} \cup{\mathbb{A}}^n_{x_1\neq 0}\)
and look at global sections:
\begin{align*}
0 \to k[x_0^{\pm 1}, x_1, \cdots, x_n] \oplus 
k[x_0, x_1^{\pm 1}, x_2,\cdots, x_n] \oplus \cdots
\to k[x_0^{\pm 1}, x_1^{\pm 1}, x_2, \cdots ] \oplus \cdots
\to \cdots \to
\to k[x_0^{\pm 1}, x_1^{\pm 1}, \cdots, x_n^{\pm 1}] \to 0
,\end{align*}
where we choose 1 coordinate to invert at the 1st stage, 2 coordinate to
invert at the 2nd stage, and so on. Note that this is not only
\({\mathbb{Z}}{\hbox{-}}\)graded, but
\({\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{n+1} } }{\hbox{-}}\)
graded by monomials. The claim is that the contribution of a monomial
\(\prod x_i^{d_i}\) to cohomology will only depend on the pattern of
signs,
i.e.~\(I\coloneqq\left\{{k{~\mathrel{\Big\vert}~}d_k < 0}\right\} \subseteq [n]\).

\end{proof}

\begin{example}[?]

Consider \(I = \emptyset\), and the contribution of \(\prod x_i^{d_i}\)
with \(d_i \geq 0\) for all \(i\). Form a simplicial complex \(X\):

\includegraphics{figures/2022-04-01_10-53-14.png}

The cohomology computes
\({ {H}^{\scriptscriptstyle \bullet}} _{\Delta}(X; {\mathbb{Z}}) \cong {\mathbb{Z}}\)
since \(X\) is contractible.

\end{example}

\begin{example}[?]

For \(I = [n]\), so all \(d_i < 0\), one obtains just the faces of the
complex with the boundaries deleted.

\includegraphics{figures/2022-04-01_10-58-45.png}

This computes
\({ {H}^{\scriptscriptstyle \bullet}} _{\Delta}(X, \tilde X; {\mathbb{Z}}) \cong \tilde { {H}^{\scriptscriptstyle \bullet}} _{\Delta}(\tilde X)\)
by the LES of a pair:

\end{example}

\begin{remark}

Recall that this LES arises from
\begin{align*}
0 \to C^n(\tilde X) \to C^n(X)\to C^n(X, \tilde X)\to 0
.\end{align*}

\end{remark}

\begin{example}[?]

For \(I = \left\{{0}\right\}\), so \(I = \left\{{0}\right\}\) with
\(d_0 < 0\) and \(d_i \geq 0\) for \(i\geq 1\).

\includegraphics{figures/2022-04-01_11-02-23.png}

This computes
\({ {H}^{\scriptscriptstyle \bullet}} _{\Delta}(X, \tilde X; {\mathbb{Z}}) \cong \tilde { {H}^{\scriptscriptstyle \bullet}} _{\Delta}(\tilde X) = 0\).

\end{example}

\begin{remark}

When does this trick work? For any pair \((X, L)\) with
\(L\in {\operatorname{Pic}}X\) where the sections are
\({\mathbb{Z}}^{n+1}{\hbox{-}}\)graded where each graded piece is
dimension at most 1. These are referred to as
\textbf{multiplicity-free}. Examples: toric varieties:

\includegraphics{figures/2022-04-01_11-16-10.png}

\end{remark}

\hypertarget{monday-april-04}{%
\section{Monday, April 04}\label{monday-april-04}}

\hypertarget{riemann-roch-and-serre-duality}{%
\subsection{Riemann-Roch and Serre
Duality}\label{riemann-roch-and-serre-duality}}

\begin{remark}

Let \(X\in \mathop{\mathrm{Proj}}{\mathsf{Var}}_{/ {k}}\) and
\(F \in {\mathsf{Coh}}({\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}})\).
By Grothendieck, \({ {H}^{\scriptscriptstyle \bullet}} (X; F)\) is
supported in degrees \(0 \leq d \leq \dim X\) and
\(h^i = \dim_k H^d(X; F) < \infty\) for all \(d\).

\end{remark}

\begin{proposition}[Riemann-Roch]

If \(X\in {\mathsf{sm}}\mathop{\mathrm{Proj}}{\mathsf{Var}}_{/ {k}}\),
\begin{align*}
\chi(X; F) \coloneqq\sum_{0\leq i \leq \dim X} (-1)^i h^i(F) = \int_X {\mathrm{ch}}(F) \operatorname{Td} ({\mathbf{T}}_X)
.\end{align*}

\end{proposition}

\begin{remark}

What this formula means: for \(X\) smooth projective, there is a Chow
ring \(A^*(X) = \bigoplus _{0\leq i \leq \dim X} A^i(X)\) where \(A^i\)
is analogous to \(H^{2i}_{\mathrm{sing}}(X; {\mathbb{C}})\). These are
often different, but sometimes coincide (which can only happen if odd
cohomology vanishes). For curves, these differ, and
\(A^1(X) \cong {\operatorname{Pic}}(X)\) which breaks up as a discrete
part (degree) and continuous part (Jacobian). Define
\(A^i(X) \coloneqq{\mathbb{Z}}[C_i]\sim\) where \(C_i\) are codimension
\(i\) algebraic cycles (subvarieties) and we quotient by linear
equivalence. Recall that for divisors, \(D_1\sim D_2\) if \(D_1-D_2\) is
the divisor of zeros/poles of a rational functions. More generally, for
\(Z\) of codimension \(i\) and \(Z \xrightarrow{f} X\), consider
\(f_* D_1 \sim f_* D_2\) in order to define linear equivalence.

\end{remark}

\begin{example}[?]

Consider \(X_4 \subseteq {\mathbb{P}}^3\) a quartic, the easiest example
of a K3 surface. Then \(A^0[X] = {\mathbb{Z}}[X]\),
\(A^1(X) = {\operatorname{Pic}}(X)\), so what is \(A^2(X)\)? These are
linear equivalence classes of points, and any two points are equivalent
if they are equivalent in the image of a curve. It's a fact that K3s are
not covered by rational curves -- instead these form a countable
discrete set, with finitely many in each degree. There is a formula
which says that the generating function of curve counts is modular, and
\begin{align*}
\sum n_d x^d = {1\over x} {1\over \prod_{1\leq n\leq \infty} (1-x^n)^{24} } 
,\end{align*}
where \(n_d\) is the number of rational curves of degree \(2d\). So
\(A^2(X)\) is not obvious! A theorem of Mumford says that it's
torsionfree and infinitely generated. Note that \(n_d = p_{24}(d+1)\)
where \(p_\ell({-})\) is the numbered of \emph{colored} integer
partitions

\end{example}

\begin{remark}

The integration map:
\begin{align*}
\int_X: A^{\dim X}(X) &\to {\mathbb{Z}}\\
\sum n_i p_i &\to \sum n_i
.\end{align*}

There are two non-homogeneous polynomials \({\mathrm{ch}}(F)\) and
\(\operatorname{Td} ({\mathbf{T}}_X)\) in
\(A^*(X)\otimes_{\mathbb{Z}}{\mathbb{Q}}\), and the formula for
Riemann-Roch says to multiply and extract only the top-dimensional
component, i.e.~take
\(\deg({\mathrm{ch}}(F) \operatorname{Td} ({\mathbf{T}}_X))_{\dim X}\).
This is very computable!

\end{remark}

\begin{example}[?]

A Chern class: if \(F = {\mathcal{O}}_X(D)\), then
\begin{align*}
{\mathrm{ch}}(F) = e^D = \sum_{1\leq i\leq n} D^i/i!
\end{align*}
where
\begin{align*}
{\mathcal{O}}_X(D)(U) = \left\{{ f\in {\mathcal{O}}_X(U) {~\mathrel{\Big\vert}~}(f) + D \geq 0}\right\}
\end{align*}
and \(D^n = D \smile D \smile\cdots \smile D\) is the
\(n{\hbox{-}}\)fold self-intersection of \(D\). Note that
\(c_1(F) = D\).

\end{example}

\begin{remark}

The Chern character of \(F\) is additive on SESs,
i.e.~\(0\to A\to B\to C\to 0\) yields
\({\mathrm{ch}}(B) = {\mathrm{ch}}(A) + {\mathrm{ch}}(C)\).

\end{remark}

\begin{proposition}[RR for curves]

If \(X\) is a smooth projective curve,
\begin{align*}
h^0(X) - h^1(X) = \deg D - g(X) + 1
.\end{align*}
In this case, \({\mathrm{ch}}(F) = 1+D\) and
\(\operatorname{Td} ({\mathbf{T}}_X) = 1 + (1-g)[{\operatorname{pt}}]\)
where \([{\operatorname{pt}}]\) is a certain well-defined divisor in
\(A^1(X)\). One can rewrite this as
\(\operatorname{Td} _X = 1 + {1\over 2}c_1 = 1 - {1\over 2} K_X\) (the
canonical class, where \(\deg K_X = 2g-2\)). This uses that
\begin{align*}
c_1 = c_1({\mathbf{T}}_X) = -c_1(\Omega_X) = -K_X
.\end{align*}

\end{proposition}

\begin{example}[?]

For \(X\) a smooth surface,

\begin{itemize}
\tightlist
\item
  \({\mathrm{ch}}(F) = 1 + D + {D\over 2}\)
\item
  \(\operatorname{Td} ({\mathbf{T}}_X) = 1 - {1\over 2}c_1 + {1\over 12}(c_1^2 + c_2)\),
\end{itemize}

thus
\begin{align*}
\chi(X; {\mathcal{O}}_X(D)) = {D(D-2) \over K} + \chi(X; {\mathcal{O}}_X)
.\end{align*}

\end{example}

\begin{example}

If \(X\) is a K3 surface, then \(K_X = 0\) and
\(h^0({\mathcal{O}}_X) = h^1({\mathcal{O}}_X) = 0\), so
\(\chi(X; {\mathcal{O}}_X) = 2\) and
\begin{align*}
\chi(X; {\mathcal{O}}_X(D)) = {D^2\over 2} + 2
.\end{align*}

\end{example}

\begin{example}[?]

For \(X = {\mathbb{P}}^2\) with \(F = {\mathcal{O}}(d)\), note

\begin{itemize}
\tightlist
\item
  \(K_X = {\mathcal{O}}(-3)\)
\item
  \(h^0({\mathcal{O}}_X) = 1, h^1({\mathcal{O}}_X) = h^2({\mathcal{O}}_X) = 0\)
\end{itemize}

So
\begin{align*}
\chi(X; {\mathcal{O}}(d)) = {d(d+3) \over 2} +1 = {d+2\choose 2}
.\end{align*}
As a corollary, for \(d\geq 0\),
\begin{align*}
h^0({\mathcal{O}}_{{\mathbb{P}}^n}(d)) = {d+n\choose n}
.\end{align*}

\end{example}

\hypertarget{friday-april-08}{%
\section{Friday, April 08}\label{friday-april-08}}

\hypertarget{vanishing-theorems-1}{%
\subsection{Vanishing theorems}\label{vanishing-theorems-1}}

\begin{remark}

Setup:
\(X\in \mathop{\mathrm{Proj}}{\mathsf{Var}}_{/ {k}} , {\mathcal{F}}\in {\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\).
What is \(H^0(X; {\mathcal{F}})\)? Note that if
\begin{align*}
\chi(X; {\mathcal{F}}) \coloneqq\sum_k (-1)^k h^k(X; {\mathcal{F}})
,\end{align*}
if
\(\tau_{\geq 1} { {H}^{\scriptscriptstyle \bullet}} (X;{\mathcal{F}}) = 0\)
then this \(\chi(X;{\mathcal{F}}) = h^0(X; {\mathcal{F}})\). By Serre
duality,
\(h^n(X;{\mathcal{F}}) = h^0(\omega_X \otimes{\mathcal{F}}^{-1})\) which
holds if \(X\) is Gorenstein, e.g.~a locally complete intersection.

Recall that
\({\mathcal{O}}_X(D)(U) = \left\{{\phi {~\mathrel{\Big\vert}~}(\phi) + D \geq 0}\right\}\).
Note that if \({\mathcal{F}}= {\mathcal{O}}(D)\) then
\(h^0(X;{\mathcal{F}})\neq 0 \iff D\sim D'\) where \(D' > 0\) is
effective.

\end{remark}

\begin{remark}

If \(D\sim D'\) where \(-D' > 0\) is effective, then
\(h^0(X; {\mathcal{O}}(D)) = 0\). Note that if
\(D \subseteq X \subseteq {\mathbb{P}}^N\) is projective, take
\(H \subseteq {\mathbb{P}}^N\) and
\({\mathcal{O}}_{{\mathbb{P}}^N}(1) = {\mathcal{O}}_{{\mathbb{P}}^N}(H)\)
and intersect to obtain \(D \cdot H^{n-1} = \deg D\).

\end{remark}

\begin{example}[?]

If \(X\) is a smooth projective curve and
\({\mathcal{F}}= {\mathcal{O}}_X(D)\) is a line bundle. Riemann-Roch
yields
\begin{align*}
h^0(X;{\mathcal{F}}) - h^1(X;{\mathcal{F}}) = \deg D -g + 1
\end{align*}
and
\begin{align*}
\deg D = h^0(D) - h^0(K_X - D)
\implies \deg(K_X - D) = 2g-2 - \deg D
.\end{align*}

\end{example}

\begin{example}[?]

If \(X\) is a smooth projective curve,

\begin{itemize}
\tightlist
\item
  \({\mathcal{O}}_X(D)\) is ample \(\iff D > 0\) (some large multiple is
  a hyperplane section).
\item
  \({\mathcal{O}}_X(D)\) is very ample \(\impliedby \deg D \geq 2g-2+3\)
  (very ample: some multiple is ample).
\end{itemize}

There exists an embedding \(X\hookrightarrow{\mathbb{P}}^N\), and
\({\mathcal{O}}_X(D) = {\mathcal{O}}_X(1) = {\mathcal{O}}_{{\mathbb{P}}^N}(1)\mathrel{\Big|}_X\).
One can show \(h^0(D - {\operatorname{pt}}) < h^0(D)\).

\end{example}

\begin{example}[?]

An effective but not ample divisor: take two lines in
\({\mathbb{P}}^1\times {\mathbb{P}}^1\) which do not intersect.

\end{example}

\begin{theorem}[Kodaira]

Suppose
\(X\in{\mathsf{sm}}\mathop{\mathrm{Proj}}{\mathsf{Var}}_{/ {k}}\) where
\(k={\mathbb{C}}\) or \(\operatorname{ch}k = 0\) with
\(k= { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu }\)
and let \({\mathcal{F}}= \omega_X(L)\) with \(L\) ample. Then
\begin{align*}
\tau_{\geq 1}  { {h}^{\scriptscriptstyle \bullet}} (X;{\mathcal{F}}) = 0
.\end{align*}

\end{theorem}

\begin{remark}

A note on the proof: uses Deligne-Illusie and liftability from Witt
vectors. This liftability holds for all curves, all K3s, and some
Calabi-Yau threefolds.

\end{remark}

\begin{remark}

For curves, \(h^1(X; \omega_X(L)) = h^0(-L)\).

\end{remark}

\begin{theorem}[Kawamata-Viehweg vanishing (generalized Kodaira vanishing)]

Let
\(X \in {\mathsf{sm}}\mathop{\mathrm{Proj}}{\mathsf{Var}}_{/ {{\mathbb{C}}}}\)
with \(D = \cup_k D_K\) normal crossing union of smooth divisors and
write its formal boundary as \(\Delta \coloneqq\sum a_i D_i\) with
\(0 < a_i < 1\) and \(a_i \in {\mathbb{Q}}\). Suppose
\({\mathcal{F}}\equiv K_X + \Delta + A\) for \(A\) ample, then
\begin{align*}
\tau_{\geq 1} { {h}^{\scriptscriptstyle \bullet}} (X; {\mathcal{F}}) = 0
.\end{align*}

\end{theorem}

\begin{remark}

Say \(X\) has \textbf{klt singularities} (Kawamata log terminal) iff
there exists a projective morphism \(Y \xrightarrow{f} X\) with
\(Y\supseteq\cup_i D_i\) with each \(D_i\) snc, and
\(f^* K_X = K_Y + \Delta\). Generally \(Y\) is smooth and \(f\) is a
resolution.

\end{remark}

\begin{remark}

A note on the MMP: take \(X_0\) a variety, produce a variety \(X\) with
\(K_X\) nice, e.g.~\(-K_X > 0\) or \(K_X \geq 0\) numerically. At each
stage, contract a curve (the result is a \(-1\) curve) are perform a
\textbf{flip}. So if \(C \in X_0\), produce \(X_0 \to X_1\) with
\(CK_X < 0\).

\end{remark}

\hypertarget{monday-april-11}{%
\section{Monday, April 11}\label{monday-april-11}}

\hypertarget{spectral-sequences}{%
\subsection{Spectral sequences}\label{spectral-sequences}}

\begin{proposition}[Leray spectral sequence]

If \(f\in {\mathsf{Top}}(X, Y)\) and
\({\mathcal{F}}\in {\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\),
there is a spectral sequence
\begin{align*}
E_2^{p, q} = H^p(X; {\mathbb{R}}^{q} f_* {\mathcal{F}}) \Rightarrow H^{p+q}(X; {\mathcal{F}})
.\end{align*}

\end{proposition}

\begin{example}[?]

If \(0 \to A\leftleftarrows { {J}^{\scriptscriptstyle \bullet}}\) is an
injective resolution of a sheaf \(A\), then
\(E_1^{p, q} = H^p(J^q) \Rightarrow H^{p+q}(A)\). More generally, for
any functor \(F \in \mathsf{Cat}(\mathsf{A}, \mathsf{B})\),
\begin{align*}
E_1^{p, q} = {\mathbb{R}}^p F(J^q) \Rightarrow{\mathbb{R}}^{p+q} F(A)
.\end{align*}

So if \(J^q\) are \(F{\hbox{-}}\)acyclic, then
\(\tau_{\geq 1} { {{\mathbb{R}}}^{\scriptscriptstyle \bullet}} F(J^q) = 0\)
and thus \({\mathbb{R}}^n F(A)\) is the homology of the complex
\(F { {J}^{\scriptscriptstyle \bullet}}\).

\end{example}

\begin{proposition}[Grothendieck]

If

\begin{itemize}
\tightlist
\item
  \(\mathsf{A} \xrightarrow{F} \mathsf{B} \xrightarrow{G} \mathsf{C}\)
  are left-exact functors between abelian categories
\item
  \(\mathsf{A}, \mathsf{B}\) have enough injectives, and
\item
  \(F(I)\) for \(I\) injective in \(\mathsf{A}\) yields a
  \(G{\hbox{-}}\)acyclic object in \(\mathsf{B}\),
\end{itemize}

then there is a first-quadrant spectral sequence

\begin{align*}
E_2^{p, q} = {\mathbb{R}}^p G( {\mathbb{R}}^q G(A)) \Rightarrow{\mathbb{R}}^{p+q}(F\circ G)(A)
.\end{align*}

\end{proposition}

\begin{remark}

This recovers the Leray spectral sequence via
\({\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}}) \xrightarrow{f_*} {\mathsf{Sh}}(Y; {\mathsf{Ab}}{\mathsf{Grp}}) \xrightarrow{{{\Gamma}\qty{Y; {-}} }} {\mathsf{Ab}}{\mathsf{Grp}}\),
where the composition is \({{\Gamma}\qty{X; {-}} }\). Note that
injective sheaves are flasque, and pushforwards of flasque sheaves are
again flasque. Why flasque implies injective:

\begin{center}
\begin{tikzcd}
    0 && {j_!{ \left.{{ {\mathcal{O}}_X }} \right|_{{U}} }} && {{\mathcal{O}}_X} \\
    \\
    && I
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-3, to=1-5]
    \arrow[dashed, from=1-5, to=3-3]
    \arrow[from=1-3, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCIwIl0sWzIsMCwial8hXFxyb3sgXFxPT19YIH17VX0iXSxbNCwwLCJcXE9PX1giXSxbMiwyLCJJIl0sWzAsMV0sWzEsMl0sWzIsMywiIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzEsM11d}{Link
to Diagram}
\end{quote}

\end{remark}

\begin{remark}

Recall that cohomology vanishes above the dimension of a Noetherian
space. The analog for pushforward involves the relative dimension.

\end{remark}

\begin{remark}

General setup:

\begin{itemize}
\item
  \(d_r: E_r^{p, q} \to E_r^{p+r, q-r+1}\) (down and to the right) moves
  between diagonals.

  \begin{itemize}
  \tightlist
  \item
    For a fixed \(p, q\), all differentials out of \(E_{p, q}\) land on
    the same diagonal.
  \end{itemize}
\item
  \(E_{r+1} = H(E_r, d_r)\).
\item
  Letting \(E_n = \bigoplus_{p+q=n} E_\infty^{p, q}\), there is a
  descending filtration
  \({ {{\operatorname{Fil}}}_{\scriptscriptstyle \bullet}} E_n\) such
  that
  \({\mathsf{gr}\,}_p { {{\operatorname{Fil}}}_{\scriptscriptstyle \bullet}} E_n \coloneqq{\operatorname{Fil}}_p E_n/ {\operatorname{Fil}}_{p+1} E_n = E_\infty^{p, n-p}\).
\item
  Extension problem: \({\mathbb{Z}}\supseteq 2{\mathbb{Z}}\supseteq 0\)
  where \({\mathsf{gr}\,}_1 = C_2\) and
  \({\mathsf{gr}\,}_1 \cong {\mathbb{Z}}\), but another group and
  filtration may have the same associated graded,
  e.g.~\({\mathbb{Z}}\oplus C_2 \supseteq{\mathbb{Z}}\supseteq 0\).
\item
  Double complexes naturally arise by taking an injective resolution
  \(A \leftleftarrows { {J}^{\scriptscriptstyle \bullet}}\) and
  individually resolving the pieces by
  \(J^n \leftleftarrows C^{n, \bullet}\). Writing
  \({ \operatorname{Tot} }(C^{p, q})_n \coloneqq\oplus_{p+q=n} C^{p, q}\),
  there are maps
  \(A\to C^{0,0} \to (C^{0,1} \oplus C^{1, 0}) \to \cdots\) by summing
  horizontal and vertical differentials. Using the sign trick makes this
  a differential (multiply the vertical differentials in every even
  column by \(-1\)).
\item
  There are spectral sequences
  \begin{align*}
  E_1{p, q} &= H^p(C^{\bullet, q}, d_h) \Rightarrow H^{p+q}( { \operatorname{Tot} }( C^{\bullet, \bullet} ) ) \\
  E_1{p, q} &= H^q(C^{p, \bullet}, d_v) \Rightarrow H^{p+q}( { \operatorname{Tot} }( C^{\bullet, \bullet} ) )
  .\end{align*}

  \begin{itemize}
  \tightlist
  \item
    Why this is useful: resolve \(A\) by \(J\) which are not necessarily
    injective, and resolve each \(J^n\) by injectives, then
    \({ \operatorname{Tot} }\) is now an injective resolution.
  \end{itemize}
\item
\end{itemize}

\end{remark}

\hypertarget{wednesday-april-13}{%
\section{Wednesday, April 13}\label{wednesday-april-13}}

\hypertarget{spectral-sequences-continued}{%
\subsection{Spectral sequences
continued}\label{spectral-sequences-continued}}

\begin{remark}

Recall that for spectral sequences, the diagonal entries \(p+q=n\) are
the successive quotients in a filtration on
\(E^n\coloneqq{ \operatorname{Tot} }(E_\infty^{\bullet, \bullet})_n\).
Kodaira vanishing: for the original argument, go to characteristic \(p\)
and look at liftability.

\end{remark}

\begin{example}[Deligne-Illusie's proof of Kodaira vanishing]

We'll have some spectral sequence which we'll want to degenerate at
\(E_2\). It STS that \(d_r = 0\) for \(r\geq 1\), which in fact forces
\((E, d)\) to degenerate at \(E_1\). Strategy: find another spectral
sequence \((E', d')\) with the same \(E_1' \cong E_1\) and a
differential \(d\neq d'\) which converges to the same thing and more
patently stabilizes at \(E_1'\). It then follows that \(E\) stabilizes
at \(E_1\). Note the \(\dim_k E_r^{p, q} \leq \dim_{k} E_{r-1}^{p, q}\)
since we're taking kernels mod images.

\end{example}

\begin{lemma}[A 5-term sequence]

Suppose \(E_2^{p, q} \Rightarrow E^n\) for \(n=p+q\) is first quadrant.
Then

\begin{itemize}
\tightlist
\item
  \(E_2^{0, 0} = E^{\infty}_{0,0}\) and \(E_2^{1,0} = E_\infty^{1, 0}\).
\item
  \(E_3^{0, 1} = E_\infty^{0, 1}\) and \(E_3^{2, 0} = E_\infty^{2, 0}\)
\item
  There is a 5-term exact sequence
  \begin{align*}
  0\to E_2^{1,0} \to E^1\to E_2^{0, 1} \to E_2^{2, 0} \to E^2
  .\end{align*}
\end{itemize}

\end{lemma}

\begin{example}[?]

The Leray spectral sequence: for \(f\in {\mathsf{Top}}(X, Y)\) and
\({\mathcal{F}}\in {\mathsf{Sh}}(X; { \mathsf{Vect}_{/ {k}} })\),
\begin{align*}
E_2^{p, q} = H^p(Y; {\mathbb{R}}^q f_* {\mathcal{F}}) \Rightarrow H^{p+q}(X; {\mathcal{F}})
.\end{align*}
This yields
\begin{align*}
0 \to H^1(X; f_* {\mathcal{F}}) \to H^1(X;{\mathcal{F}}) \to H^0(X; {\mathbb{R}}^1 f_* {\mathcal{F}}) \to H^2(X; f_*{\mathcal{F}}) \to H^2(F)
.\end{align*}

Consider the filtration on \(E_\infty\):

\includegraphics{figures/2022-04-13_10-48-19.png}

This yields exact sequences

\begin{itemize}
\tightlist
\item
  \(0 \to E_\infty^{1, 0} \to E^1\to E_\infty^{0, 1} \to 0\)
\item
  \(0\to E_\infty^{2, 0} \to ? \to E_\infty^{1, 1} \to 0\)
\item
  \(0\to ? \to E^2\to E_\infty^{0, 2}\to 0\).
\end{itemize}

\end{example}

\begin{remark}

Recall the definition of a double complex:
\((C^{\bullet, \bullet}, d_h, d_v)\) where each row is a complex for
\(d_h\) and each column for \(d_v\), and each square skew-commutes. Note
that the sign trick does not change the cohomology. The totalized
complex is is \(({ \operatorname{Tot} }(C), {{\partial}})\) where
\(C^n \coloneqq\bigoplus _{p+q=n} C^{p, q} \xrightarrow{{{\partial}}} C^{n+1} \coloneqq\bigoplus_{p+q=n+1} C^{p, q}\)
and the differential is constructed from
\(C^{p, q} \xrightarrow{d_h \oplus d_v} C^{p+1, q} \oplus C^{p, q+1}\).
There is a descending filtration
\({ {{\operatorname{Fil}}}_{\scriptscriptstyle \bullet}} { \operatorname{Tot} }(C)\)
where
\({\operatorname{Fil}}_n { \operatorname{Tot} }(C) = \tau_{\geq n, \bullet} { \operatorname{Tot} }(C) = \bigoplus _{p\geq n} C^{p, q}\),
which is the double complex obtained by truncating all columns to the
left of column \(n\).

\end{remark}

\hypertarget{friday-april-15}{%
\section{Friday, April 15}\label{friday-april-15}}

\hypertarget{filtrations-and-gradings}{%
\subsection{Filtrations and Gradings}\label{filtrations-and-gradings}}

\begin{remark}

Given \({\operatorname{Fil}}A\) a descending filtration, define
\({\mathsf{gr}\,}_i A \coloneqq{\operatorname{Fil}}_i A/ {\operatorname{Fil}}_{i+1} A\).
Convention: everywhere we'll set \(p+q\coloneqq n, p = n-q\), etc.

This results in a collection of short exact sequences:
\begin{align*}
0 \to {\operatorname{Fil}}_{i+1} A\to {\operatorname{Fil}}_i A \to {\mathsf{gr}\,}_i A \to 0
.\end{align*}

\end{remark}

\begin{remark}

Our main example: a double complex
\({ {C}^{\scriptscriptstyle \bullet, \bullet}}\) with
\({ {A}^{\scriptscriptstyle \bullet}} \coloneqq { {{ \operatorname{Tot} }}^{\scriptscriptstyle \bullet}} { {C}^{\scriptscriptstyle \bullet, \bullet}}\)
with \(A^n \coloneqq\oplus _{p+q=n} C^{p ,q}\) and differentials
\({{\partial}}= (d_v, d_h)\) producing skew-commuting squares. The main
question is computing \(H^*(A)\).

Each \(A^n\) is a filtration \({\operatorname{Fil}}A^n\) where
\({{\partial}}{\operatorname{Fil}}^i A^n \subseteq {\operatorname{Fil}}^{i+1} A^n\).
The filtration is defined by
\begin{align*}
{\operatorname{Fil}}^{p_0} A^n = \bigoplus _{p+q=n, p\geq p_0} C^{p, q}
,\end{align*}
taking everything to the right of column \(p_0\). The claim is that this
induces a filtrations on \(Z^n(A), B^n(A), H^n(A)\) (cycles, boundaries,
and homology). One can restrict the differential on
\({ {A}^{\scriptscriptstyle \bullet}}\) to
\({\operatorname{Fil}} { {A}^{\scriptscriptstyle \bullet}}\); note that
cycles \(Z_n\mapsto 0\) and boundaries are the image and we're taking
cycles mod boundaries. Writing
\({\operatorname{Fil}}^p Z^n \coloneqq{\operatorname{Fil}}^p A^n \cap Z_n\)
and similarly for \(B^n, H^n\), one gets a filtration
\({\operatorname{Fil}}H({\operatorname{Fil}}^p A)\) on
\(H({\operatorname{Fil}}^p A)\). This yields
\begin{align*}
E_\infty^{p, q} = {\mathsf{gr}\,}_p H^n = {\operatorname{Fil}}^p H^n / {\operatorname{Fil}}^{p+1} H^n
.\end{align*}
If all of the SESs split, then
\(H^n = \bigoplus _{p+q=n} E^{p, q}_\infty\).

\end{remark}

\begin{remark}

Set

\begin{itemize}
\tightlist
\item
  \(E_0^{p, q} \coloneqq C^{p, q}\)
\item
  \(E_1^{p, q} = H^n( C^{p, \bullet}, d_v )\).
\item
  \(E_2^{p, q} = H^n( E_1^{p, q}, d_v) = H^*(\cdots\to H^{n-1}C^{p, \bullet} \to H^n(C^{p, \bullet}) \to H^{n+1} C^{p+1,\bullet}\to \cdots )\).
\end{itemize}

What are the cycles in \(E_0\)? To map to zero under the total
differential \({\partial}\), things emanating from column \(p\) must go
to zero, and for the columns \(p+k\), images under \(d_h^{p+k, \ell}\)
must cancel with images under \(d_h^{p+k+1, \ell-1}\). Define the
\emph{approximate homology}
\begin{align*}
{\operatorname{Fil}}^p H^{\approx}_{p \pm r} = {{\partial}^{-1}({\operatorname{Fil}}^{p+r} A^{n+1} ) \over {\partial}({\operatorname{Fil}}^{p-r+1} A^{n-1} ) }
.\end{align*}
Note that this \emph{increases} the number of allowed cycles and
\emph{decreases} the number of allowed boundaries. Then
\(E_r^{p, q} = {\mathsf{gr}\,}_p H^n_{p \pm r}\).

\end{remark}

\begin{remark}

Note that the statement is not the \(E_r\) is computed as
\(H^*(E_{r-1})\); instead there is a formula for \(E_r^{p, q}\) for all
\(r,p,q\) a priori, and it is a property that taking homology of pages
computes this.

\end{remark}

\begin{remark}

Claim: \({\mathsf{gr}\,}_p H^n_{p\pm 0} = C^{p, q}\). Check that
\begin{align*}
{\operatorname{Fil}}^{p_0} H^n_{p\pm 0} = {\bigoplus _{p+q=n, p\geq p_0} C^{p, q} \over d\qty{ \bigoplus _{p+q=n-1, p\geq p_0 + 1} C^{p, q} } }
.\end{align*}

\end{remark}

\hypertarget{monday-april-18}{%
\section{Monday, April 18}\label{monday-april-18}}

\hypertarget{spectral-sequences-1}{%
\subsection{Spectral Sequences}\label{spectral-sequences-1}}

\begin{remark}

A filtered complex:

\begin{center}
\begin{tikzcd}
    \cdots & {A^{n-1}} && {A^{n}} && {A^{n+1}} & \cdots & {} \\
    \\
    \cdots & {{\operatorname{Fil}}^p A^{n-1}} && {{\operatorname{Fil}}^p A^n} && {{\operatorname{Fil}}^p A^{n+1}} & \cdots \\
    \\
    \cdots & {{\operatorname{Fil}}^{p+1} A^{n-1}} && {{\operatorname{Fil}}^{p+1} A^n} && {{\operatorname{Fil}}^{p+1} A^{n+1}} & \cdots \\
    & \vdots && \vdots && \vdots
    \arrow[from=1-1, to=1-2]
    \arrow["{d^{n-1}}", from=1-2, to=1-4]
    \arrow["{d^n}", from=1-4, to=1-6]
    \arrow[from=1-6, to=1-7]
    \arrow[hook, from=5-2, to=3-2]
    \arrow[dotted, hook, from=3-2, to=1-2]
    \arrow[hook, from=5-4, to=3-4]
    \arrow[dotted, hook, from=3-4, to=1-4]
    \arrow[hook, from=5-6, to=3-6]
    \arrow[dotted, hook, from=3-6, to=1-6]
    \arrow["{d^{n-1}}", from=3-2, to=3-4]
    \arrow["{d^{n-1}}", from=5-2, to=5-4]
    \arrow["{d^n}", from=3-4, to=3-6]
    \arrow["{d^n}", from=5-4, to=5-6]
    \arrow[hook, from=6-2, to=5-2]
    \arrow[hook, from=6-4, to=5-4]
    \arrow[hook, from=6-6, to=5-6]
    \arrow[from=3-6, to=3-7]
    \arrow[from=5-6, to=5-7]
    \arrow[from=5-1, to=5-2]
    \arrow[from=3-1, to=3-2]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

This yields
\begin{align*}
H^n_{p\pm r} &= {A^n \cap d^{-1}({\operatorname{Fil}}^{p+r} A^{n+1} ) \over A^n \cap d({\operatorname{Fil}}^{p-r+1} A^{n+1} ) } \\ \\
H^n_{p \pm \infty} &= {A^n \cap d^{-1}(0) \over A^n \cap d(A^{n+1}) }
.\end{align*}

Notation: write
\begin{align*}
{}^n E^p_r \coloneqq E_{r}^{p, q} = {\mathsf{gr}\,}^p H^n_{p\pm r} = 
{
{\operatorname{Fil}}^p A^{n} \cap d^{-1}\qty{ {\operatorname{Fil}}^{p+r} A^{n+1} }
\over
{\operatorname{Fil}}^{p+1} A^n \cap d^{-1}({\operatorname{Fil}}^{p+r} A^{n+1} ) + {\operatorname{Fil}}^p A^{n} \cap d^{-1}\qty{{\operatorname{Fil}}^{p-r+1} A^{n-1} }
}
.\end{align*}

The main properties:

\begin{itemize}
\tightlist
\item
  \(d_r: E_r^{p, q} \to E_r^{p+r, q-r+1}\)
\item
  \(H( { {E_r}^{\scriptscriptstyle \bullet, \bullet}} , d_r) = { {E_{r+1}}^{\scriptscriptstyle \bullet, \bullet}}\).
\end{itemize}

Note that \({}^n E^p_r \xrightarrow{ d_r} {}^{n+1}E^{p+r}_r\), so
\begin{align*}
{
{\operatorname{Fil}}^p A^{n} \cap d^{-1}\qty{ {\operatorname{Fil}}^{p+r} A^{n+1} }
\over
{\operatorname{Fil}}^{p+1} A^n \cap d^{-1}({\operatorname{Fil}}^{p+r} A^{n+1} ) + {\operatorname{Fil}}^p A^{n} \cap d^{-1}\qty{{\operatorname{Fil}}^{p-r+1} A^{n-1} }
}
\xrightarrow{d_r} 
{
{\operatorname{Fil}}^{p+r} A^{n+1} \cap d^{-1}\qty{ {\operatorname{Fil}}^{p+2r} A^{n+2} }
\over
{\operatorname{Fil}}^{p+r+1} A^{n+1} \cap d^{-1}({\operatorname{Fil}}^{p+2r} A^{n+2} ) + \cdots 
}
,\end{align*}
and \(d_r^2 = 0\) since the first denominator above appears as the next
numerator.

\end{remark}

\hypertarget{applications}{%
\subsection{Applications}\label{applications}}

\begin{remark}

An application: consider a 2-step resolution
\(0 \to A \to J^0 \to J^1\), and take injective resolutions of each
\(J^i\) to form an \(E_0\):

\begin{center}
\begin{tikzcd}
    &&& \bullet & \vdots && \vdots \\
    \\
    &&&& {I^{0, 1}} && {I^{1,1}} && 0 \\
    \\
    &&&& {I^{0, 0}} && {I^{1, 0}} && 0 \\
    &&& \bullet & {} &&&&& \bullet \\
    0 && A && {J^0} && {J^1}
    \arrow[from=7-1, to=7-3]
    \arrow[from=7-3, to=7-5]
    \arrow[from=7-5, to=7-7]
    \arrow[from=5-5, to=5-7]
    \arrow[from=3-5, to=3-7]
    \arrow[from=7-5, to=5-5]
    \arrow[from=7-7, to=5-7]
    \arrow[from=5-5, to=3-5]
    \arrow[from=5-7, to=3-7]
    \arrow[dashed, no head, from=1-4, to=6-4]
    \arrow[dashed, no head, from=6-4, to=6-10]
    \arrow[from=5-7, to=5-9]
    \arrow[from=3-7, to=3-9]
    \arrow[from=3-5, to=1-5]
    \arrow[from=3-7, to=1-7]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

Then
\(0 \to A\to { {{ \operatorname{Tot} }}^{\scriptscriptstyle \bullet}} ( { {A}^{\scriptscriptstyle \bullet, \bullet}} )\)
is exact, i.e.~this is an injective resolution of \(A\). Take vertical
cohomology to get \(E_1\):

\begin{center}
\begin{tikzcd}
    \bullet & \vdots && \vdots \\
    \\
    & 0 && 0 && 0 \\
    \\
    & {J^0} && {J^1} && 0 \\
    \bullet & {} &&&&& \bullet
    \arrow[from=5-2, to=5-4]
    \arrow[from=3-2, to=3-4]
    \arrow[from=5-2, to=3-2]
    \arrow[from=5-4, to=3-4]
    \arrow[dashed, no head, from=1-1, to=6-1]
    \arrow[dashed, no head, from=6-1, to=6-7]
    \arrow[from=5-4, to=5-6]
    \arrow[from=3-4, to=3-6]
    \arrow[from=3-2, to=1-2]
    \arrow[from=3-4, to=1-4]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMTIsWzEsNV0sWzEsNCwiSl4wIl0sWzMsNCwiSl4xIl0sWzMsMiwiMCJdLFsxLDIsIjAiXSxbMSwwLCJcXHZkb3RzIl0sWzMsMCwiXFx2ZG90cyJdLFswLDAsIlxcYnVsbGV0Il0sWzAsNSwiXFxidWxsZXQiXSxbNiw1LCJcXGJ1bGxldCJdLFs1LDIsIjAiXSxbNSw0LCIwIl0sWzEsMl0sWzQsM10sWzEsNF0sWzIsM10sWzcsOCwiIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbOCw5LCIiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dLFsyLDExXSxbMywxMF0sWzQsNV0sWzMsNl1d}{Link
to Diagram}
\end{quote}

Since no functor has been applied, we obtain the follow \(E_2\) after
taking horizontal cohomology:

\begin{center}
\begin{tikzcd}
    \bullet & \vdots && \vdots \\
    \\
    & 0 && 0 && 0 \\
    \\
    & A && 0 && 0 \\
    \bullet & {} &&&&& \bullet
    \arrow[from=5-2, to=5-4]
    \arrow[from=3-2, to=3-4]
    \arrow[from=5-2, to=3-2]
    \arrow[from=5-4, to=3-4]
    \arrow[dashed, no head, from=1-1, to=6-1]
    \arrow[dashed, no head, from=6-1, to=6-7]
    \arrow[from=5-4, to=5-6]
    \arrow[from=3-4, to=3-6]
    \arrow[from=3-2, to=1-2]
    \arrow[from=3-4, to=1-4]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMTIsWzEsNV0sWzEsNCwiQSJdLFszLDQsIjAiXSxbMywyLCIwIl0sWzEsMiwiMCJdLFsxLDAsIlxcdmRvdHMiXSxbMywwLCJcXHZkb3RzIl0sWzAsMCwiXFxidWxsZXQiXSxbMCw1LCJcXGJ1bGxldCJdLFs2LDUsIlxcYnVsbGV0Il0sWzUsMiwiMCJdLFs1LDQsIjAiXSxbMSwyXSxbNCwzXSxbMSw0XSxbMiwzXSxbNyw4LCIiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dLFs4LDksIiIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzIsMTFdLFszLDEwXSxbNCw1XSxbMyw2XV0=}{Link
to Diagram}
\end{quote}

So \(H^n({ \operatorname{Tot} }I) = A[0]\).

\end{remark}

\begin{remark}

Let \(F\in \mathsf{Cat}(A, B)\) be additive left-exact, then
\({\mathbb{R}}^n FA = H^n(F{ \operatorname{Tot} } { {I}^{\scriptscriptstyle \bullet, \bullet}} )\)
for \(0\to A\to I\) a biresolution as above. Define \(E_0 = FI\), then
\(E_1^{p, q} = {\mathbb{R}}^q F J^p\).

\end{remark}

\begin{corollary}[?]

If \(J^p\) are \(F{\hbox{-}}\)acyclic, then \(E_1\) has the form

\begin{center}
\begin{tikzcd}
    \bullet & \vdots && \vdots \\
    \\
    & 0 && 0 && 0 \\
    \\
    & {FJ^0} && {FJ^1} && \cdots \\
    \bullet & {} &&&&& \bullet
    \arrow[from=5-2, to=5-4]
    \arrow[from=3-2, to=3-4]
    \arrow[from=5-2, to=3-2]
    \arrow[from=5-4, to=3-4]
    \arrow[dashed, no head, from=1-1, to=6-1]
    \arrow[dashed, no head, from=6-1, to=6-7]
    \arrow[from=5-4, to=5-6]
    \arrow[from=3-4, to=3-6]
    \arrow[from=3-2, to=1-2]
    \arrow[from=3-4, to=1-4]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMTIsWzEsNV0sWzEsNCwiRkpeMCJdLFszLDQsIkZKXjEiXSxbMywyLCIwIl0sWzEsMiwiMCJdLFsxLDAsIlxcdmRvdHMiXSxbMywwLCJcXHZkb3RzIl0sWzAsMCwiXFxidWxsZXQiXSxbMCw1LCJcXGJ1bGxldCJdLFs2LDUsIlxcYnVsbGV0Il0sWzUsMiwiMCJdLFs1LDQsIlxcY2RvdHMiXSxbMSwyXSxbNCwzXSxbMSw0XSxbMiwzXSxbNyw4LCIiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dLFs4LDksIiIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9LCJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzIsMTFdLFszLDEwXSxbNCw1XSxbMyw2XV0=}{Link
to Diagram}
\end{quote}

So \(E_2^{p, q} = H^q(FJ^p)\), i.e.~\({\mathbb{R}}F A\) can be compute
using the resolution
\(0\to A\to { {J}^{\scriptscriptstyle \bullet, \bullet}} \to \cdots\).
For example, for \(F({-}) = {{\Gamma}\qty{X; {-}} }\), we can resolve by
flasque, soft, or fine sheaves.

\end{corollary}

\begin{remark}

Using two spectral sequences for a single bicomplex: given
\({ {C}_{\scriptscriptstyle \bullet, \bullet}}\),
\begin{align*}
E_2^{p, q} = H_h^{p} H_v^q C^{p, { \scriptscriptstyle \bullet} }  &\Rightarrow H^n({ {{ \operatorname{Tot} }}_{\scriptscriptstyle \bullet}} { {C}_{\scriptscriptstyle \bullet, \bullet}} ) \\
E_2^{p, q} = H^q_v H^p_h C^{{ \scriptscriptstyle \bullet}, q}  &\Rightarrow H^n({ {{ \operatorname{Tot} }}_{\scriptscriptstyle \bullet}} { {C}_{\scriptscriptstyle \bullet, \bullet}} )
.\end{align*}

\end{remark}

\begin{remark}

Grothendieck spectral sequences: for
\(\mathsf{A} \xrightarrow{F} \mathsf{B} \xrightarrow{G} \mathsf{C}\),
form the composite \(\mathsf{A} \xrightarrow{GF} \mathsf{C}\) to obtain
\begin{align*}
E_2^{p, q} = {\mathbb{R}}^p G {\mathbb{R}}^q F A \Rightarrow{\mathbb{R}}^{p+q} GFA
,\end{align*}
provided \(F\) sends injectives to \(G{\hbox{-}}\)acyclics. This comes
from running the two spectral sequences above, where one collapses onto
a single row.

\end{remark}

\hypertarget{wednesday-april-20}{%
\section{Wednesday, April 20}\label{wednesday-april-20}}

\hypertarget{derived-categories}{%
\subsection{Derived Categories}\label{derived-categories}}

\begin{remark}

Recall how to construct derived functors. It is advantageous to embed
\(\mathsf{C} \hookrightarrow\mathsf{Ch}\mathsf{C}\) and resolve by nicer
objects. A complex contains strictly more information than homology:
e.g.~\(0\to {\mathbb{Z}}\xrightarrow{\cdot 2} {\mathbb{Z}}\to 0\) and
\(0 \to {\mathbb{Z}}\hookrightarrow{\mathbb{Z}}\oplus {{\mathbb{Z}}\over 2{\mathbb{Z}}}\to 0\)
have isomorphic homology but aren't isomorphic as complexes.

\end{remark}

\begin{definition}[Quasi-isomorphism]

A morphism \(f\in \mathsf{Ch}\mathsf{C}(A, B)\) is a
\textbf{quasi-isomorphism} iff the induced map
\(f^*\in \mathsf{Ch}\mathsf{C}( { {H}^{\scriptscriptstyle \bullet}} A, { {H}^{\scriptscriptstyle \bullet}} B)\)
is an isomorphism.

\end{definition}

\begin{definition}[The derived category]

There is a category \({\mathbb{D}}\mathsf{C}\) and a functor
\(\mathsf{Ch}\mathsf{C}\to {\mathbb{D}}\mathsf{C}\) with the following
universal property: if \(\mathsf{Ch}\mathsf{C} \to \mathsf{B}\) is any
functor sending quasi-isomorphisms to isomorphisms, there is a unique
functor \({\mathbb{D}}\mathsf{C} \to B\) factoring it. We call
\({\mathbb{D}}\mathsf{C}\) the \textbf{derived category} of
\(\mathsf{C}\).

\end{definition}

\begin{remark}

The basic morphisms in \({\mathbb{D}}\mathsf{C}\) are given by usual
chain maps \(f:A\to B\), and if \(f\) is a quasi-isomorphism we formally
add inverses \(X_f: B\to A\). A general morphism is a sequence of
morphisms \(\bullet \to \bullet \to \cdots \to \bullet\) where we
quotient by

\begin{itemize}
\tightlist
\item
  \(\bullet \xrightarrow{f} \bullet \xrightarrow{g} \bullet \sim \bullet \xrightarrow{g f} \bullet\)
\item
  \(A \xrightarrow{f} B \xrightarrow{X_f} A \sim A \xrightarrow{\operatorname{id}} A\)
\item
  \(B \xrightarrow{X_f} A \xrightarrow{f} B \sim B \xrightarrow{\operatorname{id}} B\).
\end{itemize}

One would like a calculus of fractions, so define:

\end{remark}

\begin{definition}[Localizing morphisms]

Given \(\mathsf{C}\in \mathsf{Cat}\), and subset
\(S \subseteq \mathop{\mathrm{Mor}}(\mathsf{ C})\) of morphisms is
\textbf{localizing} iff

\begin{itemize}
\tightlist
\item
  \(\operatorname{id}_A \in S\) for all objects \(A\)
\item
  \(S\) is closed under compositions
\item
  For every roof with \(f\) arbitrary and \(s\in S\), there exist
  arrows:
\end{itemize}

\begin{center}
\begin{tikzcd}
    && \bullet \\
    \\
    & \bullet && \bullet \\
    \\
    \bullet && \bullet && \bullet
    \arrow[from=3-2, to=5-1]
    \arrow["f"', from=3-2, to=5-3]
    \arrow["{s\in S}", from=3-4, to=5-3]
    \arrow[from=3-4, to=5-5]
    \arrow["\exists"{description}, dashed, from=1-3, to=3-2]
    \arrow["\exists"{description}, dashed, from=1-3, to=3-4]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMSwyLCJcXGJ1bGxldCJdLFswLDQsIlxcYnVsbGV0Il0sWzIsNCwiXFxidWxsZXQiXSxbMywyLCJcXGJ1bGxldCJdLFs0LDQsIlxcYnVsbGV0Il0sWzIsMCwiXFxidWxsZXQiXSxbMCwxXSxbMCwyLCJmIiwyXSxbMywyLCJzXFxpbiBTIl0sWzMsNF0sWzUsMCwiXFxleGlzdHMiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbNSwzLCJcXGV4aXN0cyIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==}{Link
to Diagram}
\end{quote}

As a corollary, arrows in
\(\mathsf{C}{ \left[ { \scriptstyle \frac{1}{S} } \right] }\) are roofs
modulo equivalence.

\end{definition}

\begin{remark}

The set \(S\) of quasi-isomorphisms in \(\mathsf{Ch}\mathsf{A}\) is
localizing.

Note that we can take

\begin{itemize}
\tightlist
\item
  \(\mathsf{Ch}\mathsf{C}:\) all complexes,
\item
  \(\mathsf{Ch}^+\mathsf{C}:\) complexes bounded from below,
\item
  \(\mathsf{Ch}^-\mathsf{C}:\) complexes from above,
\item
  \(\mathsf{Ch}^b\mathsf{C}:\) complexes from above and below.
\end{itemize}

These yield derived categories
\({\mathbb{D}}\mathsf{C}, {\mathbb{D}}^+\mathsf{C}, {\mathbb{D}}^-\mathsf{C}, {\mathbb{D}}^b\mathsf{C}\).
Note: frequently \({\mathbb{D}}\mathsf{C}\) actually means
\({\mathbb{D}}^+\mathsf{C}\) in the literature. When
\({\mathbb{D}}^b\mathsf{C}\) is used: if
\({\mathcal{F}}\in {\mathsf{Coh}}(X)\) and \(X\) is projective, which
corresponds to a graded module which (by Hilbert) has a finite
resolution.

One can similarly define homotopy categories
\({\mathsf{ho}}\mathsf{Ch}\mathsf{C}, {\mathsf{ho}}\mathsf{Ch}^+\mathsf{C}, {\mathsf{ho}}\mathsf{Ch}^-\mathsf{C}, {\mathsf{ho}}\mathsf{Ch}^b\mathsf{C}\)
with
\({\operatorname{Ob}}( {\mathsf{ho}}\mathsf{Ch}\mathsf{C} ) \coloneqq{\operatorname{Ob}}(\mathsf{Ch}\mathsf{C})\)
and
\(\mathop{\mathrm{Mor}}({\mathsf{ho}}\mathsf{Ch}\mathsf{C}) \coloneqq\mathop{\mathrm{Mor}}({\mathsf{ho}}\mathsf{Cat}{C})/\sim\)
where \(\sim\) denotes chain homotopy equivalence.

\end{remark}

\begin{theorem}[?]

\({\mathbb{D}}^+\mathsf{A} \cong {\mathsf{ho}}\mathsf{Ch}^+\mathsf{\mathsf{ I_A} }\)
where \(\mathsf{I_A}\) is the homotopy category of complexes of
injective objects in \(\mathsf{Ch}\mathsf{A}\).

\end{theorem}

\begin{remark}

Generally there is a functor
\({\mathsf{ho}}\mathsf{Ch}\mathsf{A} \hookrightarrow{\mathbb{D}}\mathsf{A}\)
since chain homotopy equivalences induce isomorphisms on homology (where
we apply the universal property of \({\mathbb{D}}\mathsf{A}\)) There is
also a functor
\({\mathbb{D}}\mathsf{A}\to {\mathsf{ho}}\mathsf{Ch}\mathsf{A}\) where
\(A\mapsto { \operatorname{Tot} }( { {I}^{\scriptscriptstyle \bullet, \bullet}} )\)
is a quasi-isomorphism.

\end{remark}

\hypertarget{friday-april-22}{%
\section{Friday, April 22}\label{friday-april-22}}

\begin{remark}

Recall that for \(S\subseteq \mathop{\mathrm{Mor}}(\mathsf{C})\), there
is a localized category
\(\mathsf{C} \left[ { \scriptstyle { {S}^{-1}} } \right]\) whose
morphisms are chains \(s_0^{-1}\circ f_0 \circ s_1^{-1}\circ \cdots\)
modulo an equivalence, and if \(S\) is \textbf{localizing} then

\begin{itemize}
\tightlist
\item
  Morphisms are single roofs (i.e.~we can collect the product fraction
  involving \(s_i, f_i\) into a single fraction).

  \begin{itemize}
  \tightlist
  \item
    Note that roofs can be multiplied, and roofs are equivalent when
    they admit a common roof:
  \end{itemize}
\end{itemize}

\begin{center}
\begin{tikzcd}
    && \bullet && \times && \bullet \\
    & \bullet && B && B && \bullet \\
    \\
    &&& \bullet \\
    {=} && \bullet && \bullet \\
    & \bullet && B && \bullet
    \arrow["{s_1}"', from=1-3, to=2-2]
    \arrow["{f_1}", from=1-3, to=2-4]
    \arrow["{s_2}", from=1-7, to=2-6]
    \arrow["{f_2}"', from=1-7, to=2-8]
    \arrow["{s_1}"', from=5-3, to=6-2]
    \arrow["{f_1}", from=5-3, to=6-4]
    \arrow["{s_2}"', from=5-5, to=6-4]
    \arrow["{f_2}", from=5-5, to=6-6]
    \arrow["\exists"{description}, dashed, from=4-4, to=5-3]
    \arrow["\exists"{description}, dashed, from=4-4, to=5-5]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=18pt}, from=4-4, to=6-2]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-18pt}, from=4-4, to=6-6]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMTQsWzEsMSwiXFxidWxsZXQiXSxbMiwwLCJcXGJ1bGxldCJdLFszLDEsIkIiXSxbNSwxLCJCIl0sWzYsMCwiXFxidWxsZXQiXSxbNywxLCJcXGJ1bGxldCJdLFs0LDAsIlxcdGltZXMiXSxbMSw1LCJcXGJ1bGxldCJdLFszLDUsIkIiXSxbMiw0LCJcXGJ1bGxldCJdLFs0LDQsIlxcYnVsbGV0Il0sWzUsNSwiXFxidWxsZXQiXSxbMywzLCJcXGJ1bGxldCJdLFswLDQsIj0iXSxbMSwwLCJzXzEiLDJdLFsxLDIsImZfMSJdLFs0LDMsInNfMiJdLFs0LDUsImZfMiIsMl0sWzksNywic18xIiwyXSxbOSw4LCJmXzEiXSxbMTAsOCwic18yIiwyXSxbMTAsMTEsImZfMiJdLFsxMiw5LCJcXGV4aXN0cyIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFsxMiwxMCwiXFxleGlzdHMiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMTIsNywiIiwwLHsiY3VydmUiOjMsImNvbG91ciI6WzAsNjAsNjBdfV0sWzEyLDExLCIiLDIseyJjdXJ2ZSI6LTMsImNvbG91ciI6WzAsNjAsNjBdfV1d}{Link
to Diagram}
\end{quote}

\begin{itemize}
\tightlist
\item
  Morphisms are equivalent when they admit a common roof:
\end{itemize}

\begin{center}
\begin{tikzcd}
    & \bullet \\
    \\
    \bullet && \bullet \\
    \\
    A && B
    \arrow[from=3-1, to=5-1]
    \arrow[from=3-1, to=5-3]
    \arrow[from=3-3, to=5-1]
    \arrow[from=3-3, to=5-3]
    \arrow[dashed, from=1-2, to=3-1]
    \arrow[dashed, from=1-2, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCw0LCJBIl0sWzAsMiwiXFxidWxsZXQiXSxbMiwyLCJcXGJ1bGxldCJdLFsyLDQsIkIiXSxbMSwwLCJcXGJ1bGxldCJdLFsxLDBdLFsxLDNdLFsyLDBdLFsyLDNdLFs0LDEsIiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFs0LDIsIiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==}{Link
to Diagram}
\end{quote}

\begin{itemize}
\tightlist
\item
  If \(\mathsf{C}\in {\mathsf{Add}}\mathsf{Cat}\) then
  \(\mathsf{C} \left[ { \scriptstyle { {S}^{-1}} } \right] \in {\mathsf{Add}}\mathsf{Cat}\),
  where the calculus of fractions behaves as in ring localization:
  \({f_1\over s} + {f_2\over s} = {f_1+f_2\over s}\).
\item
  If \(I \leq \mathsf{C}\) is a full subcategorya nd \(S\) is
  \emph{compatible}, i.e.~\(S \cap\mathop{\mathrm{Mor}}(I)\) is
  localizing, then
  \(I \left[ { \scriptstyle { {S}^{-1}} } \right] \leq \mathsf{C} \left[ { \scriptstyle { {S}^{-1}} } \right]\)
  is a full subcategory.
\end{itemize}

\end{remark}

\begin{remark}

\(\mathsf{Ch}\mathsf{C}\) with \(S\) quasi-isomorphisms yields
\({\mathbb{D}}\mathsf{A}\coloneqq\mathsf{C} \left[ { \scriptstyle { {S}^{-1}} } \right]\).

\end{remark}

\begin{theorem}[?]

The collection \(S\) of quasi-isomorphisms is localizing.

\end{theorem}

\begin{corollary}[?]

\({\mathbb{D}}\mathsf{A}\) is additive and morphisms are roofs in
\(\mathsf{Ch}\mathsf{A}\).

\end{corollary}

\begin{theorem}[?]

\(I\) defined as \({\mathsf{ho}}\mathsf{Ch}\mathsf{C}^{\mathrm{inj}}\),
the homotopy category of complexes of injective objects, is compatible
with \(S\).

\end{theorem}

\begin{theorem}[?]

\(I \left[ { \scriptstyle { {S}^{-1}} } \right] \leq \mathsf{A} \left[ { \scriptstyle { {S}^{-1}} } \right] = {\mathbb{D}}\mathsf{A}\),
with an equivalence if \(\mathsf{A}\) has enough injectives.

\end{theorem}

\begin{warnings}

These last two theorems do \emph{not} hold just for
\(I = \mathsf{Ch}\mathsf{C}^{\mathrm{inj}}\).

\end{warnings}

\begin{remark}

An application: for
\(F\in {\mathsf{Ab}}\mathsf{Cat}(\mathsf{A}, \mathsf{B})\) additive
(with no left/right exactness conditions), there is a derived functor
\({\mathbb{D}}F\in \mathsf{{\mathbb{D}}^+ \mathsf{A}, {\mathbb{D}}^+ \mathsf{B}}\)
if \(\mathsf{A}\) has enough injectives. Note that
\({\mathbb{D}}\mathsf{A}\) is never abelian but admits a triangulated
structure.

\end{remark}

\begin{example}[?]

For \(X\in {\mathsf{sm}}\mathop{\mathrm{Proj}}{\mathsf{Var}}_{/ {k}}\),
the usual notation is
\({\mathbb{D}}(X) \coloneqq{\mathbb{D}}^b{\mathsf{Coh}}(X)\). Global
sections
\(\Gamma\in \mathsf{Cat}({\mathsf{Coh}}X\to {\mathsf{Ab}}{\mathsf{Grp}})\)
induce a derived functor
\({\mathbb{R}}\Gamma\in \mathsf{Cat}({\mathbb{D}}X \to {\mathbb{D}}^b{\mathsf{Ab}}{\mathsf{Grp}})\).
Note that \({\mathsf{Coh}}X \hookrightarrow{\mathbb{D}}(X)\) by
\({\mathcal{F}}\mapsto {\mathcal{F}}[0]\).

\end{example}

\begin{remark}

For \(X\mathop{\mathrm{Proj}}{\mathsf{Var}}_{/ {k}}\), recall
\({\mathsf{K}}_0 X \coloneqq{\mathsf{K}}_0 {\mathsf{Coh}}X\) where
\([b] = [a] + [c]\) for \(0\to a\to b\to c\), and
\({\mathsf{K}}^0 X \coloneqq{\mathsf{K}}^0 {\mathsf{Sh}}^{{\mathsf{loc}}{\mathrm{free}}}(X)\).
If \(X\) is smooth, these are isomorphic, but generally they are not if
\(X\) is singular. In general,
\({\mathbb{D}}X \coloneqq{\mathbb{D}}^+ {\mathsf{Coh}}X\) replaces
\({\mathsf{K}}_0(X)\), and
\({\mathbb{D}}^+ {\mathsf{Sh}}^{{\mathsf{loc}}{\mathrm{free}}}(X)\)
replaces \({\mathsf{K}}^0 X\).

\end{remark}

\begin{theorem}[?]

\({\mathbb{D}}\mathsf{A}\in {\mathsf{triang}}\mathsf{Cat}\).

\end{theorem}

\begin{remark}

Although these do not have SESs, there are distinguished triangles for
which any morphism \(X\to Y\) can be completed to
\(X\to Y\to Z\to { \Sigma^{\scriptstyle[1]} X }\). This can be
accomplished using mapping cylinders/cones:

\includegraphics{figures/2022-04-22_11-15-10.png}

\end{remark}

\begin{remark}

See tilting of complexes, exceptional sequences.

\end{remark}

\hypertarget{monday-april-25}{%
\section{Monday, April 25}\label{monday-april-25}}

\hypertarget{triangulated-categories}{%
\subsection{Triangulated categories}\label{triangulated-categories}}

\begin{definition}[Triangulated categories]

A \textbf{triangulated category} is an additive category
\(\mathsf{C}\in{\mathsf{Add}}\mathsf{Cat}\) with an additive
autoequivalence \(T: \mathsf{C}\to \mathsf{C}\) and a set of
distinguished triangles \(X\to Y\to Z\to TX\) satisfying

\begin{itemize}
\tightlist
\item
  \textbf{TR1}:

  \begin{itemize}
  \tightlist
  \item
    \(X \xrightarrow{\operatorname{id}} X \to 0 \to TX\) is
    distinguished,
  \item
    Any triangle isomorphic to a distinguished triangle is again
    distinguished,
  \item
    For every \(X \xrightarrow{u} Y\) there is a distinguished triangle
    \(X \xrightarrow{u} Y \to Z\to X[1]\). Idea: \(Z\approx Y/X\).
  \end{itemize}
\item
  \textbf{TR2}:

  \begin{itemize}
  \tightlist
  \item
    For every \(X\to Y\to Z\to X[1]\), there is a triangle
    \(Y\to Z\to X[1] \xrightarrow{Tu} Y[1]\).
  \end{itemize}
\item
  \textbf{TR3}:

  \begin{itemize}
  \tightlist
  \item
    Given 3 triangles
    \begin{align*}
    X&\to Y\to Z'\to
    Y&\to Z\to X'\to
    X&\to Z\to Y'\to
    \end{align*}
    there is a triangle \(Z'\to Y'\to X'\) making the relevant
    octahedral diagram commute.
  \end{itemize}
\end{itemize}

\begin{center}
\begin{tikzcd}
    & {Y'} \\
    {X'} && Z &&&& {} \\
    & Y &&&& {} \\
    {Z'} && X
    \arrow[from=4-3, to=3-2]
    \arrow[from=3-2, to=4-1]
    \arrow[from=3-2, to=2-3]
    \arrow[from=2-3, to=2-1]
    \arrow[from=4-3, to=2-3]
    \arrow[from=2-1, to=4-1]
    \arrow[from=2-1, to=3-2]
    \arrow[from=4-1, to=4-3]
    \arrow[dashed, from=2-3, to=1-2]
    \arrow[dashed, from=1-2, to=2-1]
    \arrow[dashed, from=4-1, to=1-2]
    \arrow[dashed, from=1-2, to=4-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsOCxbMiwzLCJYIl0sWzEsMiwiWSJdLFswLDMsIlonIl0sWzIsMSwiWiJdLFswLDEsIlgnIl0sWzUsMl0sWzYsMV0sWzEsMCwiWSciXSxbMCwxXSxbMSwyXSxbMSwzXSxbMyw0XSxbMCwzXSxbNCwyXSxbNCwxXSxbMiwwXSxbMyw3LCIiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbNyw0LCIiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMiw3LCIiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbNywwLCIiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=}{Link
to Diagram}
\end{quote}

This can equivalently be expressed as a braid lemma:

\begin{center}
\begin{tikzcd}
    X && Z && {X'} && {Z'[1]} \\
    \\
    & Y && {Y'} && {Y[1]} \\
    \\
    && {Z'} && {X[1]}
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-3, to=1-5]
    \arrow[from=1-5, to=1-7]
    \arrow[from=1-1, to=3-2]
    \arrow[from=3-2, to=1-3]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=1-3, to=3-4]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=3-4, to=1-5]
    \arrow[from=1-5, to=3-6]
    \arrow[from=3-6, to=1-7]
    \arrow[from=3-2, to=5-3]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=5-3, to=3-4]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=3-4, to=5-5]
    \arrow[from=5-3, to=5-5]
    \arrow[from=5-5, to=3-6]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

Equivalently, a 3x3 lemma holds:

\begin{center}
\begin{tikzcd}
    X & Y & Z & {X[1]} \\
    {X'} & {Y'} & {Z'} & {Y'[1]} \\
    {X''} & {Y''} & \textcolor{rgb,255:red,214;green,92;blue,92}{Z''} & {Z''[1]} \\
    {X[1]} & {Y[1]} & {Z[1]} & {X[2]}
    \arrow[from=1-1, to=1-2]
    \arrow[from=1-2, to=1-3]
    \arrow[from=1-3, to=1-4]
    \arrow[from=2-1, to=2-2]
    \arrow[from=2-2, to=2-3]
    \arrow[from=2-3, to=2-4]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=3-1, to=3-2]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=3-2, to=3-3]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=3-3, to=3-4]
    \arrow[from=4-1, to=4-2]
    \arrow[from=4-2, to=4-3]
    \arrow[from=4-3, to=4-4]
    \arrow[from=1-1, to=2-1]
    \arrow[from=2-1, to=3-1]
    \arrow[from=3-1, to=4-1]
    \arrow[from=1-2, to=2-2]
    \arrow[from=2-2, to=3-2]
    \arrow[from=3-2, to=4-2]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=1-3, to=2-3]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=2-3, to=3-3]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=3-3, to=4-3]
    \arrow[from=1-4, to=2-4]
    \arrow[from=2-4, to=3-4]
    \arrow[from=3-4, to=4-4]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

\end{definition}

\begin{theorem}[?]

For
\(\mathsf{A}\in{\mathsf{Ab}}\mathsf{Cat}, \mathbf{D} {\mathsf{A}} \in {\mathsf{triang}}\mathsf{Cat}\).

\end{theorem}

\begin{definition}[?]

For \(f\in \mathsf{Ch}\mathsf{A}(X, Y)\), there is a \emph{cone} complex
\({ \mathrm{Cone} }(f) = TX \oplus Y\) with differential
\(d_{{ \mathrm{Cone} }(f) } = { \begin{bmatrix}  {d_{X[1]}} & {0} \\  {f[1]} & {d_Y} \end{bmatrix} }\)
and a \emph{cylinder} complex \({ \mathrm{Cyl} }(f)\):

\includegraphics{figures/2022-04-25_11-03-11.png}

Note that
\(d_{{ \mathrm{Cone} }(f)} {\left[ {x_{i+1}, y_i} \right]} = {\left[ {-d_X x_{i+1}, f(x_{i+1}) + d_Y(y_i)} \right]}\),
and one can check \(d^2=0\).

\end{definition}

\begin{remark}

Any distinguished triangle \(X \xrightarrow{f} Y \to Z\to X[1]\) in
\(\mathbf{D} {\mathsf{A}}\) is isomorphic to a triangle of the form
\(X\to { \mathrm{Cyl} }(f) \to { \mathrm{Cone} }(f)\to X[1]\). For
\(\mathsf{Ch}\mathsf{A}\), define \(T^nA \coloneqq A[n]\), so
\((T^nA)_k = A[n]_k = A_{n+k}\), and
\({{\partial}}_{TA} \coloneqq(-1)^n {{\partial}}_A\).

\end{remark}

\hypertarget{wednesday-april-27}{%
\section{Wednesday, April 27}\label{wednesday-april-27}}

\hypertarget{cohomological-functors}{%
\subsection{Cohomological Functors}\label{cohomological-functors}}

\begin{remark}

Recall that for \(X \xrightarrow{f} Y\),
\(\operatorname{cone}(f) \approx X[1] \oplus Y\) and
\({ \mathrm{Cyl} }(f) \approx X \oplus X[1] \oplus Y\) with differential
\begin{align*}
d_{{ \mathrm{Cyl} }(f)} \coloneqq
\begin{bmatrix}
d_X & -1 & 
\\
 & d_X[1] & 
\\
 & f[1] & d_Y
\end{bmatrix}
\curvearrowright{\left[ {x_i, x_{i+1}, y_i} \right]} \in { \mathrm{Cyl} }(f)^i
.\end{align*}

\begin{quote}
Note: I use \(\approx\) above because these formulas hold levelwise, but
the SESs they fit into may not be split exact, so
\(\operatorname{cone}(f), { \mathrm{Cyl} }(f)\) may not such direct
sums.
\end{quote}

There are related exact triples, here the first and second rows:

\begin{center}
\begin{tikzcd}
    && Y && {{ \mathrm{Cone} }(f)} && {X[1]} \\
    \\
    X && {{ \mathrm{Cyl} }(f)} && {{ \mathrm{Cone} }(f)} \\
    \\
    X && Y
    \arrow[from=5-1, to=5-3]
    \arrow[from=3-1, to=3-3]
    \arrow[from=3-3, to=3-5]
    \arrow[from=1-3, to=1-5]
    \arrow[Rightarrow, no head, from=1-5, to=3-5]
    \arrow["\alpha", from=1-3, to=3-3]
    \arrow["\beta", from=3-3, to=5-3]
    \arrow[Rightarrow, no head, from=3-1, to=5-1]
    \arrow[from=1-5, to=1-7]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsOCxbMiwwLCJZIl0sWzIsNCwiWSJdLFs0LDAsIlxcQ29uZShmKSJdLFs2LDAsIlhbMV0iXSxbNCwyLCJcXENvbmUoZikiXSxbMiwyLCJcXEN5bChmKSJdLFswLDIsIlgiXSxbMCw0LCJYIl0sWzcsMV0sWzYsNV0sWzUsNF0sWzAsMl0sWzIsNCwiIiwxLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMCw1LCJcXGFscGhhIl0sWzUsMSwiXFxiZXRhIl0sWzYsNywiIiwxLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMiwzXV0=}{Link
to Diagram}
\end{quote}

Here \(\beta\alpha = \operatorname{id}_Y\) and
\(\alpha\beta \simeq\operatorname{id}_{{ \mathrm{Cyl} }(f)}\).

\end{remark}

\begin{definition}[Cohomological functors]

A functor \(H\in [\mathsf{C}, \mathsf{A}]\) with
\(\mathsf{C}\in {\mathsf{triang}}\mathsf{Cat}, \mathsf{A}\in {\mathsf{Ab}}\mathsf{Cat}\)
(where \(\mathsf{A}\) is not necessarily related to \(\mathsf{C}\)) is a
\textbf{cohomological functor} iff every distinguished triangle
\(A\to B\to C\in \mathsf{C}\) is sent to an exact sequence
\(HA\to HB\to HC\in \mathsf{A}\).

\end{definition}

\begin{corollary}[?]

If \(H\) is cohomological, there is an associated LES
\begin{align*}
\cdots \to HA \to HB\to HC \to H(A[1]) \to H(B[1]) \to \cdots
.\end{align*}

\end{corollary}

\begin{lemma}[?]

The functor \(H: \mathbf{D} {A} \to A\) where \(X\mapsto H^0(X)\) is
cohomological, noting that \(H^i(X)\) can be written as \(H^0(X[i])\).

\end{lemma}

\begin{definition}[Ext for triangulated categories]

\begin{align*}
\operatorname{Ext} ^i(X, Y) \coloneqq\mathop{\mathrm{Hom}}_{\mathsf{C}}(X, Y[1])
.\end{align*}

\end{definition}

\begin{lemma}[?]

\begin{align*}
\operatorname{Ext} _{\mathsf{A}}^i(X, Y) \cong \operatorname{Ext} _{\mathbf{D} {\mathsf{A}} }(\iota X, \iota Y)
\end{align*}
where \(\iota: \mathsf{A} \to \mathsf{Ch}\mathsf{A}\) is given by
\(\iota(A) = \cdots \to 0\to A\to 0 \to\cdots\) supported in degree
zero.

\end{lemma}

\begin{theorem}[?]

For all \(\mathsf{C}\in{\mathsf{triang}}\mathsf{Cat}\), for all
\(X,Y\in \mathsf{C}\) the (co)representable hom functors are
cohomological:
\begin{align*}
h_Y &\coloneqq\mathop{\mathrm{Hom}}_{\mathsf{C}}({-}, Y) \qquad\text{covariant} \\
({-})^X &\coloneqq\mathop{\mathrm{Hom}}_{\mathsf{C}}(X, {-}) \qquad\text{contravariant}
.\end{align*}

\end{theorem}

\begin{proof}[?]

The proof uses the octahedral axiom TR3. To show that applying homs
yields a complex, show that the maps on homs square to zero using the
following:

\begin{center}
\begin{tikzcd}
    X && X && 0 && {X[1]} \\
    \\
    A && B && C && {A[1]} \\
    \\
    {[X, A]} && {[X, B]} && {[X, C]} \\
    && {}
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-3, to=1-5]
    \arrow[from=1-5, to=1-7]
    \arrow["u", from=3-1, to=3-3]
    \arrow["v", from=3-3, to=3-5]
    \arrow[from=3-5, to=3-7]
    \arrow["f"', from=1-1, to=3-1]
    \arrow["fu"', from=1-3, to=3-3]
    \arrow["{\exists 0}"', from=1-5, to=3-5]
    \arrow["{({-})^X(u)}", from=5-1, to=5-3]
    \arrow["{({-})^X(v)}", from=5-3, to=5-5]
    \arrow["{\therefore 0}"', curve={height=24pt}, dashed, from=5-1, to=5-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMTIsWzAsMCwiWCJdLFsyLDAsIlgiXSxbNCwwLCIwIl0sWzAsMiwiQSJdLFsyLDIsIkIiXSxbNCwyLCJDIl0sWzYsMiwiQVsxXSJdLFs2LDAsIlhbMV0iXSxbMCw0LCJbWCwgQV0iXSxbMiw0LCJbWCwgQl0iXSxbNCw0LCJbWCwgQ10iXSxbMiw1XSxbMCwxXSxbMSwyXSxbMiw3XSxbMyw0LCJ1Il0sWzQsNSwidiJdLFs1LDZdLFswLDMsImYiLDJdLFsxLDQsImZ1IiwyXSxbMiw1LCJcXGV4aXN0cyAwIiwyXSxbOCw5LCIoXFx3YWl0KV5YKHUpIl0sWzksMTAsIihcXHdhaXQpXlgodikiXSxbOCwxMCwiXFx0aGVyZWZvcmUgMCIsMix7ImN1cnZlIjo0LCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=}{Link
to Diagram}
\end{quote}

\end{proof}

\hypertarget{exceptional-collections}{%
\subsection{Exceptional Collections}\label{exceptional-collections}}

\begin{definition}[Exceptional collections]

For \(\mathsf{C}\in {\mathsf{triang}}\mathsf{Cat}\), an
\textbf{exceptional collection/sequence} is a chain of morphisms
\begin{align*}
{\mathcal{E}}_1\to {\mathcal{E}}_2 \to\cdots\to {\mathcal{E}}_n \in \mathsf{C}
\end{align*}

such that

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Self-Exts are supported only in degree zero,
  i.e.~\(\mathop{\mathrm{Hom}}({\mathcal{E}}_i, {\mathcal{E}}_i[k]) = 0\)
  for \(k\neq 0\).
\item
  There are no homs in the opposite direction,
  i.e.~\(\mathop{\mathrm{Hom}}({\mathcal{E}}_j, {\mathcal{E}}_i[m]) = 0\)
  for \(j > i\) and for any \(m\).
\end{enumerate}

\end{definition}

\begin{example}[?]

From a paper of Valery's: let \(X\) be a smooth projective surface with
\(H^1({\mathcal{O}}_X) = H^2({\mathcal{O}}_X) = 0\), which
cohomologically look like rational surfaces. Examples: \(X\) rational
with \({\left\lvert {n K_X} \right\rvert} = \emptyset\) (so ``negative''
canonical class), or \(X\) of general type with \(q=p_g=0\) and
\({\left\lvert {n K_X} \right\rvert}\) big for \(n \gg 0\). In these
cases, there are line bundles \({\mathcal{E}}\) with
\(\operatorname{Ext} ^i({\mathcal{E}}, {\mathcal{E}}) = H^1({\mathcal{O}}, {\mathcal{O}}) = H^i({\mathcal{O}}_X)\)
and one can use that
\(\operatorname{Ext} ({\mathcal{E}}_i, {\mathcal{E}}_j) = H^i({\mathcal{E}}_i \otimes{\mathcal{E}}_j^{-1})\).

\end{example}

\begin{theorem}[Beilinson, Bondal, Kapranov]

If \(\mathsf{C}' \leq \mathsf{C}\) is the full subcategory generated by
\({\mathcal{E}}_1,\cdots, {\mathcal{E}}_n\), then
\(\mathsf{C}' \simeq\mathbf{D}^b (Q)\) for \(Q\) a quiver. In
particular, if \(\left\{{{\mathcal{E}}_i}\right\}\) is a full
exceptional collection, \(\mathsf{C} = \mathsf{C}'\).

\end{theorem}

\hypertarget{friday-april-29}{%
\section{Friday, April 29}\label{friday-april-29}}

\hypertarget{applications-of-derived-categories}{%
\subsection{Applications of derived
categories}\label{applications-of-derived-categories}}

\begin{remark}

Some major work in this area:

\begin{itemize}
\tightlist
\item
  Beilinson-Gelfand-Gelfand (BGG)
\item
  Bendel-Kapranov
\item
  Mukai
\item
  Bendal-Orlov
\item
  Orlov
\item
  Kutznatsov
\item
  Kontsevich, Fukaya (homological mirror symmetry)
\item
  Beilinson-Bernstein-Gabber-Deligne on perverse sheaves
\item
  Bridgeland
\end{itemize}

\end{remark}

\begin{remark}

Results:

\begin{itemize}
\tightlist
\item
  BGG:
  \(\mathbf{D}^b ({\mathsf{Coh}}{\mathbb{P}}^n) \cong \mathbf{D}^b ({\mathsf{R}{\hbox{-}}\mathsf{Mod}})\)
  for \(R\) a certain ring.
\item
  BK, K: same for quadrics and grassmannians.
\end{itemize}

Recall that given \(\mathsf{T}\in{\mathsf{triang}}\mathsf{Cat}\) with an
exceptional collection \(\left\{{{\mathcal{E}}_i}\right\}\), they
generate a triangulated subcategory
\(\left\langle{{\mathcal{E}}_i}\right\rangle \leq \mathsf{T}\). It turns
out that
\(\left\langle{{\mathcal{E}}_i}\right\rangle\cong {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\)
for \(R = \bigoplus \mathop{\mathrm{End}}{\mathcal{E}}_i\). Beilinson
produces a collection
\(\left\{{{\mathcal{O}}, \Omega^1,\cdots, \Omega^{n-1}}\right\}\), but
an easier alternative is
\(\left\{{{\mathcal{O}}, {\mathcal{O}}(1), \cdots, {\mathcal{O}}(n-1)}\right\}\).
If the collection is full, then
\(\mathsf{T} \cong \left\langle{{\mathcal{E}}_i}\right\rangle\). As an
alternative to \(R\), one can take the corresponding quiver: make a
directed graph \({\mathcal{E}}_1\to{\mathcal{E}}_2\to\cdots\) where each
node has \(\mathop{\mathrm{End}}{\mathcal{E}}_i\) attached and each edge
\({\mathcal{E}}_i\to{\mathcal{E}}_j\) is assigned
\(\oplus_n \mathop{\mathrm{Hom}}({\mathcal{E}}_i, {\mathcal{E}}_j[n])\).
So the derived category corresponds to representations of this quiver.

Example: for \({\mathbb{P}}^1\), one obtains the following quiver:

\begin{center}
\begin{tikzcd}
    {\mathbb{C}}&& {\mathbb{C}}\\
    {\mathcal{O}}&& {{\mathcal{O}}(1)}
    \arrow["{{\mathbb{C}}\oplus {\mathbb{C}}}"', shift right=3, from=2-1, to=2-3]
    \arrow[shift left=3, from=2-1, to=2-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwxLCJcXE9PIl0sWzIsMSwiXFxPTygxKSJdLFswLDAsIlxcQ0MiXSxbMiwwLCJcXENDIl0sWzAsMSwiXFxDQ1xcb3BsdXMgXFxDQyIsMix7Im9mZnNldCI6M31dLFswLDEsIiIsMix7Im9mZnNldCI6LTN9XV0=}{Link
to Diagram}
\end{quote}

\end{remark}

\begin{proposition}[?]

If \(X\in{\mathsf{Alg}}{\mathsf{Var}}_{/ {k}}\) admits a full
exceptional collection, then the following also admit a full exceptional
collection:

\begin{itemize}
\tightlist
\item
  Any \({\mathbb{P}}^n{\hbox{-}}\)bundle \(Y = {\mathbb{P}}(V) \to X\),
  and
\item
  Any blowup \(Y = \operatorname{Bl}_Z X\) for \(Z\) a smooth
  subvariety.
\end{itemize}

\end{proposition}

\begin{corollary}[?]

Any rational smooth projective surface admits a full exceptional
collection, by running the MMP.

\end{corollary}

\begin{conjecture}

Given a smooth surface admitting a full exceptional collection, is it
rational? For a threefold, is it a blowup of something rational?

\end{conjecture}

\begin{definition}[Semiorthogonal decompositions]

Given \(\mathsf{T}\in {\mathsf{triang}}\mathsf{Cat}\) and
\(A\leq \mathsf{T}\) a full triangulated subcategory, one can define two
subcategories \({}^\perp A\) and \(A^\perp\):
\begin{align*}
A^\perp = \left\{{F {~\mathrel{\Big\vert}~}\mathop{\mathrm{Hom}}(F, A) = 1}\right\}
.\end{align*}

\end{definition}

\begin{remark}

For \(\mathsf{C} \in {\mathsf{triang}}\mathsf{Cat}\), one can take
\({\operatorname{HH}}\mathsf{C}\). For \(\mathsf{C} = \mathbf{D} (X)\),
the \({\operatorname{HH}}_0 D(X) \cong {\mathbb{Z}}^n \oplus A\) as a
group, for \(A\) some finite torsion group. If one has a full
exceptional collection, then
\(A = {\operatorname{HH}}_0( \left\langle{{\mathcal{E}}_1, \cdots, {\mathcal{E}}_n}\right\rangle^\perp )\).
As a corollary, the length \(m\) of an exceptional collection satisfies
\(m\leq \operatorname{rank}_{\mathbb{Z}}{\operatorname{HH}}_0 \mathbf{D} (X)\).

\end{remark}

\begin{conjecture}[Kaznutsov]

If \(\left\{{{\mathcal{E}}_1,\cdots, {\mathcal{E}}_n}\right\}\) is an
exceptional collection and
\(n = \operatorname{rank}_{\mathbb{Z}}{\operatorname{HH}}_0 \mathbf{D} (X)\),
then this is a \emph{full} exceptional collection.

\end{conjecture}

\begin{remark}

For surfaces of general type, a special Godeaux surface produces a
counterexample. There is a much easier counterexample coming from a
Burnist (?) surface -- generally fake \({\mathbb{P}}^2\), fake Fanos,
etc. See A-Orlov, Orlov-Gorheaise, Katgerov-?, ??

\end{remark}

\begin{remark}

Phantoms: categories with zero \({\operatorname{HH}}\), so no full
exceptional collections.

\end{remark}

\hypertarget{well-known-classical-results}{%
\subsection{Well-known classical
results}\label{well-known-classical-results}}

\begin{theorem}[Bondal-Orlov (very important!)]

If \(X\in {\mathsf{sm}}\mathop{\mathrm{proj}}{\mathsf{Var}}\) where
either \(K_X\) or \(-K_X\) is ample, then \(X\) can be recovered from
\(\mathbf{D} (X)\).

\end{theorem}

\begin{remark}

Having \(-K_X\) ample yields \textbf{Fano} varieties, and \(K_X\) ample
yields \textbf{general-type} surfaces.

\end{remark}

\begin{theorem}[Mukai]

If \(A \in {\mathsf{Ab}}{\mathsf{Var}}\), then
\(\mathbf{D} (A) \cong \mathbf{D} (A {}^{ \vee })\). Such pairs are
referred to as \textbf{Mukai partners}.

\end{theorem}

\begin{remark}

How to construct the equivalence
\(\mathbf{D} (A) \to \mathbf{D} (A {}^{ \vee })\): take the
Fourier-Mukai transform. Use the Poincare bundle
\(P_A \to A\times A {}^{ \vee }\), and construct the functor as a
push-pull over the span
\((A \leftarrow_{p_1} A\times A {}^{ \vee }\to_{p_2} A {}^{ \vee })\),
so
\begin{align*}
{\mathcal{F}}\mapsto (p_2)_* \qty{ (p_1)^* {\mathcal{F}}\otimes P_A) }
.\end{align*}

\end{remark}

\begin{remark}

The next in line: K3 surfaces. An easy example: take Kummer surfaces, so
\(A\to A/\pm 1\) and then blow up the 16 nodes.

\end{remark}

\hypertarget{monday-may-02}{%
\section{Monday, May 02}\label{monday-may-02}}

\hypertarget{calabi-yau-categories}{%
\subsection{Calabi-Yau Categories}\label{calabi-yau-categories}}

\begin{remark}

Recall that a collection \({\mathcal{E}}_i\) is \emph{exceptional} iff
\([{\mathcal{E}}_j, {\mathcal{E}}_i[n]] = 0\) if \(n\geq0\) and
\(j > i\). If there exists a full exceptional collection,
\(\mathbf{D} {X} \cong \mathbf{D} {(} {\mathsf{R}{\hbox{-}}\mathsf{Mod}})\)
for some \(R\). Recall that a variety is \emph{Fano} if \(-K_X\) is
ample.

\end{remark}

\begin{question}

Do full exceptional collections exist for Fano \(n{\hbox{-}}\)folds for
\(n=3\) or \(4\)?

\end{question}

\begin{answer}

Typically no.

\end{answer}

\begin{remark}

Let
\begin{align*}
X_3 \coloneqq V(f_3(x_0,\cdots, x_5)) \subseteq {\mathbb{P}}^5
,\end{align*}
then which \(X_3\) are rational? Note that
\(K_{X_3} = {\mathcal{O}}(-6+3) = {\mathcal{O}}(-3)\). Kuznatsov shows
that \(H^i({\mathcal{O}}_X) = {\mathbb{C}}[0]\) and
\(\operatorname{Ext} ^i({\mathcal{L}}, {\mathcal{L}}) = H^i({\mathcal{O}}_X)\).
One could look for exceptional collections of line bundles, so
\(\operatorname{Ext} ^i({\mathcal{L}}_j, {\mathcal{L}}_i) = H^m({\mathcal{L}}_i \otimes{\mathcal{L}}_j^{-1}) = 0\)
for all \(m\). On \({\mathbb{P}}^n\), take \({\mathcal{O}}(-k)\) for
\(1\leq k\leq n\) since \(K_{{\mathbb{P}}^n} = {\mathcal{O}}(-n-1)\).
For \(X_3\), there is enough vanishing that
\({\mathcal{O}}, {\mathcal{O}}(1), {\mathcal{O}}(2)\) are exceptional
(everything below the index 3 from above). Kuznatsov shows that the
``Kuznatsov component''
\(K = \left\langle{{\mathcal{O}}, {\mathcal{O}}(1), {\mathcal{O}}(2)}\right\rangle^\perp\)
is a \textbf{Calabi-Yau category} of dimension 2.

\end{remark}

\begin{remark}

If \(Y\) is a Calabi-Yau variety of dimension \(n\), so \(K_Y = 0\),
there is a Serre functor
\begin{align*}
S: \mathbf{D}^b (Y) &\to \mathbf{D}^b (Y) \\
F &\mapsto F\otimes\omega_Y[n]
.\end{align*}
Then (probably) \(S = T^n\), a shift by \(n\). A category is a
Calabi-Yau category of dimension \(n\) iff

\begin{itemize}
\tightlist
\item
  It has a Serre functor
\item
  \(S = T^n\)
\end{itemize}

One can also define fractional dimension using \(S^q\).

\end{remark}

\begin{conjecture}

\(X\) is rational iff \(K = \mathbf{D}^b (Y)\) for \(Y\in \mathsf{K3}\).

\end{conjecture}

\begin{remark}

A technique due to Clemens-Griffith for cubic threefolds. Let
\(X \subseteq {\mathbb{P}}^4\) be a smooth non-nodal curve. Consider the
intermediate Jacobian \(J_3(X)\), which is a PPAV for any smooth 3-fold.
Basic operations: blowing up a point \(p\) or a curve \(C\), since
blowing up a surface is the identity. Blowing up a point:
\(J_3(\operatorname{Bl}_p X) = J_3(X)\), so it doesn't change. For a
curve, \(J_3(\operatorname{Bl}_C X) = J(C) \oplus J_3(X)\). As a
corollary, if \(X\) is rational then \(J_3(X) = \bigoplus J(C_i)\) for
some curves \(C_i\). For non-rationality, show it's not the Jacobian of
a curve by considering the theta divisor.

\end{remark}

\begin{remark}

For 4-folds \(X\), one can now also blow up surfaces. The intermediate
cohomology carries a Hodge structure. Conjecture: \(X\) is rational iff
its Hodge structure looks like a K3.

\end{remark}

\begin{remark}

Older techniques for checking rationality: see log thresholds, generally
birational geometry e.g.~due to Manin. E.g. groups of birational
automorphism for quartic 4-folds are small. See another approach due to
Mumford using torsion in cohomology.

\end{remark}

\hypertarget{t-structures-and-hearts}{%
\subsection{T-Structures and Hearts}\label{t-structures-and-hearts}}

\begin{remark}

Note that it's possible for
\(\mathsf{A},\mathsf{B}\in {\mathsf{Ab}}\mathsf{Cat}\) to satisfy
\(\mathbf{D} {\mathsf{A}} { \, \xrightarrow{\sim}\, }\mathbf{D} {\mathsf{B}}\).

\end{remark}

\begin{example}[?]

Some examples:

\begin{itemize}
\tightlist
\item
  In the presence of a full exceptional collection
  \(\mathbf{D}^b ({\mathsf{Coh}}X) { \, \xrightarrow{\sim}\, }\mathbf{D}^b ({\mathsf{R}{\hbox{-}}\mathsf{Mod}})\).
\item
  Fourier-Mukai:
  \(\mathbf{D}^b ({\mathsf{Coh}}A) { \, \xrightarrow{\sim}\, }\mathbf{D}^b ({\mathsf{Coh}}A {}^{ \vee })\)
  for dual AVs.
\end{itemize}

\end{example}

\begin{example}[Perverse sheaves (BBD)]

Start with \(X\in {\mathsf{Var}}_{/ {{\mathbb{C}}}}\) Hausdorff
paracompact and constructible sheaves which come with stratifications
into closed subsets on which they restrict to locally constant sheaves.
Note that one can realize these sheaves as pullbacks from a poset
associated to the stratification. There are categories
\(\mathsf{Const}\) and \(\mathsf{Perv}\) with
\(\mathbf{D}^b (\mathsf{Const}) { \, \xrightarrow{\sim}\, }\mathbf{D}^b (\mathsf{Perv})\)
-- here perverse sheaves are complexes of constructible sheaves with
support conditions
\(h^j( { {{\mathcal{F}}}^{\scriptscriptstyle \bullet}} ) \leq -j\) and
\(h^j(D { {{\mathcal{F}}}^{\scriptscriptstyle \bullet}} ) \leq -j\) for
\(D\) the Verdier dual; this is a category closed under duality.

\end{example}

\begin{remark}

On \(T{\hbox{-}}\)structures: write \(D = \mathbf{D} {\mathsf{A}}\),
then there are subcategories

\begin{itemize}
\tightlist
\item
  \(D^{\leq n} = D^{\leq 0}[n]\), complexes such that \(H^{> n} = 0\).
\item
  \(D^{\geq n} = D^{\geq 0}[n]\), complexes such that \(H^{< n} = 0\).
\end{itemize}

Then \(D^{\geq 0} \cap D^{\leq 0} = A\) is a category equivalent to
complexes supported in degree zero, since any such bounded complex is
quasi-isomorphic to such a complex. Some properties:

\begin{itemize}
\tightlist
\item
  \(A_0\in D^{<0}\) and \(A_1 \in D^{> 1}\) satisfy \([A_0, A_1] = 0\).
\item
  For all \(C\in \mathbf{D}^b (\mathsf{A})\), there exists
  \(A_0\to C\to A_1 \to A_0[1]\).
\end{itemize}

Note that there is a canonical truncation
\begin{align*}
\tau_{\leq 0}(\cdots \to C^{-1}\xrightarrow{d^0} C^0\to C^1\to \cdots) = (\cdots\to C^{-1}\to \ker d^0 \to 0)
.\end{align*}

\end{remark}

\hypertarget{bridgeland-stability}{%
\subsection{Bridgeland stability}\label{bridgeland-stability}}

\begin{remark}

Take \(X\) a smooth projective curve, let
\(D = \mathbf{D}^b ({\mathsf{Coh}}X) = \mathbf{D}^b { {\mathsf{Bun}}\qty{\operatorname{GL}_r} }\).
There is a notion of a semistable sheaf (all subsheaves have smaller
slopes
\(\mu({\mathcal{F}}) \coloneqq\deg {\mathcal{F}}/\operatorname{rank}{\mathcal{F}}\))
and an HN filtration where the quotients are semistable and the slopes
decrease. Bridgeland observed there is a central charge
\begin{align*}
Z: {\mathsf{Coh}}X &\to {\mathbb{C}}\\
{\mathcal{F}}&\mapsto -\deg {\mathcal{F}}+ i \operatorname{rank}{\mathcal{F}}
,\end{align*}
which can be used to recovered the heart
\(\mathsf{A} = {\mathsf{Coh}}X\). Idea: vary \(Z\) to get different
hearts, and \(\left\{{Z_i}\right\}\) form a complex analytic variety,
and one can form a new category of tilted complexes (complexes sitting
in two degrees).

\end{remark}

\hypertarget{useful-facts}{%
\section{Useful Facts}\label{useful-facts}}

\hypertarget{category-theory}{%
\subsection{Category Theory}\label{category-theory}}

\begin{remark}

\envlist

\begin{itemize}
\item
  Products: a collection of maps into factors \(Y\to X_i\) is the same
  as a map \(Y\to \prod X_i\). Products are easy to map \emph{into}.
  Products have projections \(\prod X_i \to X_i\).

  \begin{itemize}
  \tightlist
  \item
    Products are limits.
  \end{itemize}
\item
  Coproducts: a collection of out of factors \(X_i\to Y\) is the same as
  a map \(\coprod X_i \to Y\). Coproducts are easy to map \emph{out} of.
  Coproducts have injections \(X_i \to \coprod X_i\).

  \begin{itemize}
  \tightlist
  \item
    Coproducts are colimits.
  \end{itemize}
\item
  If \(\mathsf{C}\) has a zero object, there is a canonical map
  \(\coprod_{i\in I} X_i \to \prod_{j\in I} X_j\) given by assembling
  maps \(\delta_{ij}\).
\item
  \(\colim_{i\in I}({-})\) is generally not exact, but is exact if the
  colimit is filtered.

  \begin{itemize}
  \tightlist
  \item
    In any case, the functor of taking stalks
    \(({-})_x: {\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}}) \to {\mathsf{Ab}}{\mathsf{Grp}}\)
    is always exact.
  \end{itemize}
\item
  Left adjoints/colimits are characterized by morphisms on \(F(x)\), and
  right adjoints/limits by morphisms \emph{into} it.
\item
  Why RAPL and LAPC:
  \begin{align*}
  [\colim_i L(x_i), {-}] \cong \cocolim_i [L(x_i), {-}] \cong \cocolim_i [x_i, R({-})] \cong [\colim_i x_i, R({-})] = [L(\colim_i x_i), {-}]
  .\end{align*}
\item
  Right-derived functors are left Kan extensions.
\item
  Colimits are quotients of coproducts and \textbf{receive} maps from
  objects (i.e.~they are cocones). Taking colims is right exact. Limits
  send maps.
\end{itemize}

\end{remark}

\hypertarget{tor-and-ext}{%
\subsection{Tor and Ext}\label{tor-and-ext}}

\begin{remark}

\includegraphics{figures/2022-03-30_22-08-05.png}

Tor:

\begin{itemize}
\tightlist
\item
  \(\operatorname{Tor}\) commutes with arbitrary direct sums, colimits
  (direct limits), localization.
\item
  If \(M\) is flat over
  \(R, \operatorname{Tor}_{i}^{R}(M \otimes A, B) \cong M \otimes_{R} \operatorname{Tor}_{i}^{R}(A, B)\).
\item
  If \(S\) is a flat \(R\)-algebra,
  \(S \otimes_{R} \operatorname{Tor}_{i}^{R}(A, B) \cong \operatorname{Tor}_{i}^{S}\left(S \otimes_{R} A, S \otimes_{R} B\right)\).
\item
  \(\operatorname{Tor}_1^R(R/I, R/J) \cong {I \cap J \over IJ}\)
\item
  If \(I\) is an \(R{\hbox{-}}\)regular sequence
  \(I = \left\langle{x_1,\cdots, x_n}\right\rangle\), then
  \(\operatorname{Tor}_n^R(R/I, M) = (0 :_M I)\) is a colon ideal.
\item
  If \(A\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^\flat\) then
  \(\operatorname{Tor}_{\geq 1}^R(A, B) = 0\).
\item
  \(\operatorname{Tor}(A, B) \cong \operatorname{Tor}(B, A)\).
\end{itemize}

Ext:

\begin{itemize}
\tightlist
\item
  \(\tau_{\geq 1} { {\operatorname{Ext} }^{\scriptscriptstyle \bullet}} _R(A, B) = 0\)
  if either \(A\) is injective or \(B\) is projective.
\item
  The Koszul complex for \(k[x,y]\):
  \(K_x \otimes K_y = 0\to k[x,y]\to k[x,y]{ {}^{ \scriptscriptstyle\times^{2} } }\to k[x,y] \to k \to 0\)
  where \(K_x = 0\to k[x,y] \xrightarrow{\cdot x} k[x,y] \to 0\).
\item
  \({ {\operatorname{Ext} }^{\scriptscriptstyle \bullet}} _{k[x,y]}(k) = k \oplus \Sigma k{ {}^{ \scriptscriptstyle\times^{2} } } \oplus \Sigma^2 k\).
\end{itemize}

\end{remark}

\hypertarget{problem-set-1}{%
\section{Problem Set 1}\label{problem-set-1}}

\hypertarget{problem-1}{%
\subsection{Problem 1}\label{problem-1}}

\begin{problem}[1.1]

Recall that:

\begin{itemize}
\item
  A topology on a set \(X\) is \(T_{0}\) if any two points
  \(x, y \in X\) can be topologically distinguished (by open sets).
\item
  A topology is an Alexandrov if an intersection of any, possibly
  infinite, collection of open sets is open.
\item
  The \textbf{order topology} on a poset \((X, \leq)\) is defined in the
  following way: the open sets are the \emph{upper sets}, which satisfy
  the property
  \begin{align*}
  x \in U, x \leq y \Longrightarrow y \in U
  \end{align*}
  The closed sets are \textbf{lower sets}, which satisfy
  \begin{align*}
  x \in Z, x \geq y \Longrightarrow y \in U
  \end{align*}
\end{itemize}

Prove that a topology on \(X\) is an order topology
\(\Longleftrightarrow\) it is \(T_{0}\) and Alexandrov. As a corollary
conclude that any \(T_{0}\) topology on a finite set is an order
topology.

\end{problem}

\begin{proposition}

A topology \(\tau\) on \(X\) is an order topology
\(\Longleftrightarrow\) \(\tau\) is \(T_{0}\) and Alexandrov.

\end{proposition}

\begin{proof}

\(\impliedby\): Suppose \(X\) is a topological space and \(\tau\) is a
\(T_0\) Alexandrov topology on \(X\). For \(U \subseteq X\), write
\({ \operatorname{cl}} _X(U)\) for the closure in \(X\) of \(U\) with
respect to \(\tau\), define a poset \((P, \leq)\) where
\(P \coloneqq X\) with an ordering defined by
\begin{align*}
x\leq y \iff x\in { \operatorname{cl}} _X(y)
.\end{align*}
Regarding \(\tau\) now as a topology on \((P, \leq)\), the claim is that
this is an order topology on a poset. That this ordering defines a poset
is clear, since the ordering is:

\begin{itemize}
\tightlist
\item
  \textbf{Reflexive}: since \(x\) is contained in its closure,
  \(x\leq x\).
\item
  \textbf{Antisymmetric}: if \(x\leq y\) and \(y\leq x\), then \(x\) is
  a limit point of \(\left\{{y}\right\}\) and vice-versa. So every
  neighborhood of \(y\) contains \(x\) and similarly every neighborhood
  of \(x\) contains \(y\). Since \(X\) is \(T_0\) and topologically
  distinguishes points, this can only occur if \(x=y\).
\item
  \textbf{Transitive}: if \(x\leq y\) and \(y\leq z\), then
  \(x\in { \operatorname{cl}} _X(y)\) and
  \(y\in { \operatorname{cl}} _X(z)\). Since
  \({ \operatorname{cl}} _X(z)\) is a closed set containing \(y\) and
  \({ \operatorname{cl}} _X(y)\) is the \emph{smallest} closed set in
  \(X\) containing \(y\), we have
  \(x\in { \operatorname{cl}} _X(y) \subseteq { \operatorname{cl}} _X(z)\),
  so \(x\leq z\).
\end{itemize}

It thus suffices to show that if \(U \ni x\) is a neighborhood of \(x\)
and \(x\leq y\), then \(y\in U\) so that \(U\) is an upper set. By
definition of the closure of a set,
\begin{align*}
x\leq y \iff x\in { \operatorname{cl}} _X(y) \iff
\text{every neighborhood of $x$ intersects } \left\{{y}\right\}
,\end{align*}
so if \(U_\alpha \ni x\) is any neighborhood of \(x\), then
\(y\in U_\alpha\). Write \(\tilde U \coloneqq\cap_{\alpha} U_\alpha\)
for the neighborhood basis at \(x\), the intersection of all
neighborhoods of \(x\). Note that by construction, since
\(y\in U_\alpha\) for all \(\alpha\), \(y\in \tilde U\). Since \(\tau\)
is \(T_0\), \(\tilde U\) is an open set. Moreover, since \(U\) is a
neighborhood of \(x\), \(\tilde U \subseteq U\), so \(y\in U\).

\(\implies\): Suppose \((X, \leq)\) is a poset with an order topology
\(\tau\), so \(U\) is open iff whenever \(x\in U\) and \(x\leq y\) then
\(y\in U\). To see that \(\tau\) defines a \(T_0\) topology, let
\(x\neq y\) in \(X\). If \(x\) and \(y\) are not comparable, there is
nothing to show, so suppose either \(x< y\) or \(y< x\) -- without loss
of generality, relabeling if necessary, we can assume \(x< y\). Now
every neighborhood of \(x\) contains \(y\) by definition, but for
example
\begin{align*}
U_{\geq y} \coloneqq\left\{{z\in X{~\mathrel{\Big\vert}~}z\geq y}\right\}
\end{align*}
is neighborhood of \(y\) not containing \(x\), topologically
distinguishing \(x\) and \(y\).

To see that \(\tau\) is Alexandrov, it suffices to show that arbitrary
intersections of open sets are open. This follows from the fact that any
intersection of upper sets is again an upper set -- if
\(\left\{{ U_i}\right\}_{i\in I}\) is an arbitrary family of upper sets,
set \(U \coloneqq\cap_{i\in I} U_i\). Then if \(x\in U\) with
\(x\leq y\), \(x\in U_i\) for every \(i\) and so \(y\in U_i\) for every
\(i\), and thus \(y\in U\).

\end{proof}

\begin{corollary}[?]

If \(X\) is a finite set and \(\tau\) is a \(T_0\) topology on \(X\),
then \(\tau\) is an order topology.

\end{corollary}

\begin{proof}[of cor]

By the exercise, it suffices to show that any finite space is
Alexandrov. Let \((X, \tau)\) be a \(T_0\) space and let
\(\left\{{U_i}\right\}_{i\in I} \subseteq \tau\) be an arbitrary
collection of open sets -- we'll show
\(U\coloneqq\cap_{i\in I} U_i \in \tau\) is again open. This follows
immediately, since finite intersections of open sets are open in any
topology, and since \(X\) is finite and \(\tau \subseteq 2^X\) is
finite, \(I\) can only be a finite indexing set.

\end{proof}

\hypertarget{problem-2}{%
\subsection{Problem 2}\label{problem-2}}

\begin{problem}[1.2]

Recall that:

\begin{itemize}
\item
  A paracompact space is a topological space in which every open cover
  has an open refinement that is locally finite.
\item
  A partition of unity of a topological space \(X\) is a set
  \(f_{\alpha}\) of continous functions
  \(f_{\alpha}: X \rightarrow[0,1]\) such that for every point
  \(x \in X\) there exists an open neighborhood of \(x\) where all but
  finitely many \(f_{\alpha} \equiv 0\), and such that
  \(\sum f_{\alpha}=1\).
\end{itemize}

Prove that

\begin{itemize}
\item
  Any Hausdorff space is paracompact iff it admits a partition of unity
  subordinate to any open cover.
\item
  Any metric space is paracompact.
\end{itemize}

\begin{quote}
A sketch would suffice.
\end{quote}

\end{problem}

\begin{proposition}[?]

\(X\) is Hausdorff \(\iff X\) admits a partition of unity subordinate to
any open cover \({\mathcal{U}}\rightrightarrows X\).

\end{proposition}

\begin{proof}[?]

?

\end{proof}

\begin{proposition}[?]

Metric spaces are paracompact.

\end{proposition}

\begin{proof}[?]

Let \({\mathcal{U}}\rightrightarrows X\) be an open cover of a metric
space \(X\); we'll show \({\mathcal{U}}\) admits a locally finite
refinement. Without loss of generality, writing
\({\mathcal{U}}= \left\{{U_j}\right\}_{j\in J}\) for some index set
\(J\), we can assume the \(U_i\) are disjoint -- this follows by
invoking the axiom of choice to well-order the index set \(J\) and
setting
\begin{align*}
\tilde U_j \coloneqq U_j \setminus\displaystyle\bigcup_{k < j} U_k
.\end{align*}
Then
\(\tilde {\mathcal{U}}\coloneqq\left\{{\tilde U_j}\right\}_{j\in J}\)
refines \({\mathcal{U}}\) since \(\tilde U_j \subseteq U_j\), and still
covers \(X\). Moreover, we note that for every \(x\in X\), we can now
produce a \emph{minimal} index \(j(x)\) such that \(x\in U_{j(x)}\).

The idea is to now refine \({\mathcal{U}}\) to a cover \({\mathcal{V}}\)
by filling each disjoint annulus \(U_j\) with balls of small enough
radius. For ease of notation and to more clearly demonstrate the
following construction, suppose \(J \cong \left\{{0,1,\cdots}\right\}\)
is countable. For each \(n\in {\mathbb{Z}}_{\geq 0}\), let
\(\delta_n < {\varepsilon}_n\) be small to-be-determined real numbers
depending on \(n\), and define the following subsets of \(X\):

\begin{align*}
X_{0, n} \coloneqq\left\{{x\in U_0 {~\mathrel{\Big\vert}~}B_{{\varepsilon}(n) }(x) \subseteq U_0 }\right\} 
&&
V_{0, n} \coloneqq\displaystyle\bigcup_{x\in X_{0, n}} B_{\delta_n}(x) \subseteq X_{0, n} \subseteq U_0 \\
X_{1, n} \coloneqq\left\{{x\in U_1 {~\mathrel{\Big\vert}~}B_{{\varepsilon}(n) }(x) \subseteq U_1 }\right\} \setminus\displaystyle\bigcup_{\ell < n} V_{0, \ell} 
&&
V_{1, n} \coloneqq\displaystyle\bigcup_{x\in X_{1, n}} B_{\delta_n}(x) \subseteq X_{1, n} \subseteq U_1 \\
X_{2, n} \coloneqq\left\{{x\in U_2 {~\mathrel{\Big\vert}~}B_{{\varepsilon}(n) }(x) \subseteq U_2 }\right\} \setminus\displaystyle\bigcup_{\ell < n} V_{0, \ell} \setminus\displaystyle\bigcup_{\ell < n} V_{1, n} 
&&
V_{2, n} \coloneqq\displaystyle\bigcup_{x\in X_{2, n}} B_{\delta_n}(x) \subseteq X_{2, n} \subseteq U_2 \\
\vdots && \vdots \qquad \\
X_{j, n} \coloneqq\left\{{x\in U_j {~\mathrel{\Big\vert}~}B_{{\varepsilon}(n) }(x) \subseteq U_j }\right\} \setminus\displaystyle\bigcup_{k < j} \displaystyle\bigcup_{\ell < n} V_{k, \ell} 
&&
V_{j, n} \coloneqq\displaystyle\bigcup_{x\in X_{j, n}} B_{\delta_n}(x) \subseteq X_{j, n} \subseteq U_j 
.\end{align*}
Note that the last line prescribes a general formula which depends only
on the ordering and not on the countability of \(J\).

In other words, for each fixed \(j_0\in J\), we consider all of those
\(x\in X\) such that \(j(x) = j_0\), so that for each such \(x\) we have
\(x\in U_{j_0}\) but \(x\not\in U_k\) for any \(k<j_0\). For a fixed
\(n\), we then consider those \(x\in U_{j_0}\) that are not too close to
the boundary, so that a ball of radius \({\varepsilon}_n\) fits entirely
in \(U_{j_0}\). We then shrink these balls to a smaller radius
\(\delta_n\) and take their union to form an open set \(V_{j_0, n}\) in
the new cover, and as \(n\to \infty\) these balls get smaller and fill
out all of \(U_{j_0}\). However, at each stage \(V_{j, n}\) we remove
redundancies by discarding sets \(V_{k, \ell}\) for \(k<j\) and
\(\ell < n\).

\begin{claim}

\({\mathcal{V}}\coloneqq\left\{{V_{j, n}}\right\}_{j\in J, n\in {\mathbb{Z}}_{\geq 0}}\)
is a locally finite refinement of \({\mathcal{U}}\).

\end{claim}

\begin{proof}[of claim]

There are several things that are clear from the construction:

\begin{itemize}
\tightlist
\item
  Each \(V_{j, n}\) is open, as they are arbitrary unions of open balls
  in a metric space.
\item
  \({\mathcal{V}}\) refines \({\mathcal{U}}\), since in fact
  \(V_{j, n} \subseteq U_j\) for every \(n\) and every \(j\).
\item
  \({\mathcal{V}}\) is a cover, since for any \(x\) one can pick
  \(j(x)\) minimally so that \(x\in U_{j(x)}\), and since \(x\) is an
  interior point and open balls form a basis for a metric space, for
  some \(n\) large enough we have
  \(x\in B_{\delta_n}(x) \subseteq V_{j(x), n}\).
\end{itemize}

So the content of this statement is that each \(x\in X\) is contained in
only finitely many opens from \({\mathcal{V}}\). Fix \(x\in X\) and pick
\(j(x)\) minimally as above, so \(x\in V_{j(x), n}\) for every \(n\) and
\(j(x)\) is the first such \(j\) where \(x\) is added. Then
\(x\in B_{\delta_n}(x')\) for some \(x'\) near \(x\), so choose \(n\)
and \(k\) so that
\(V_x \coloneqq B_{1\over 2^k}(x) \subset B_{\delta_n}(x) \subseteq V_{j, n}\).
The claim now is that \(V_x\) intersects only finitely many elements of
\({\mathcal{V}}\). A proof of this follows from using the triangle
inequality to show that \(B_{1/2^{n+k}}(x)\) does not intersect any
\(V_{\beta, \ell}\) for \(\ell \geq n+k\), and for \(\ell < n + k\) it
intersects these for at most one \(\beta\), leaving only finitely many
such \(\ell\).

\end{proof}

\end{proof}

\hypertarget{problem-3}{%
\subsection{Problem 3}\label{problem-3}}

\begin{problem}[1.3]

Let \(A=\mathbb{Z}\) be an abelian group. Let \(\mathcal{F}\) be a sheaf
on \(X\) such that every stalk \(\mathcal{F}_{x}=A .\) Does it follow
that \(\mathcal{F}\) is a constant sheaf?

\begin{itemize}
\item
  Show that the answer is no in general.
\item
  What if \(X\) is an irreducible algebraic variety with Zariski
  topology?
\item
  What if \(X=[0,1]\) with classical topology?
\end{itemize}

\end{problem}

\begin{proposition}[Part 1]

There is a sheaf \({\mathcal{F}}\) on a space \(X\) with stalks
satisfying \({\mathcal{F}}_x = {\mathbb{C}}\) for every \(x\in X\), but
\({\mathcal{F}}\) is not isomorphic to the constant sheaf
\(\underline{{\mathbb{C}}_X}\).

\end{proposition}

\begin{proof}[Part 1]

Let \(X = S^1\) in the Euclidean topology,
\(U = X\setminus\left\{{1}\right\}, Z = \left\{{1}\right\}\) and let the
two inclusions be
\begin{align*}
j: U &\to X, \\
i: Z &\to X
.\end{align*}
Now set
\begin{align*}
{\mathcal{F}}\coloneqq j_! \underline{{\mathbb{C}}_U} \oplus i_* \underline{{\mathbb{C}}_Z}
\end{align*}
We can then compute the stalks:
\begin{align*}
(j_! \underline{{\mathbb{C}}_U} )_x 
&= \colim_{W \ni x} (j_! \underline{{\mathbb{C}}_U} )(W) \\
&= \colim_{W \ni x} 
\begin{cases}
{\mathbb{C}}& W \subseteq U 
\\
0 & \text{else} .
\end{cases} \\
&= \colim
\begin{cases}
\cdots \to {\mathbb{C}}\to {\mathbb{C}}\to {\mathbb{C}}\to \cdots & x\in U 
\\
0 \to 0\to 0 \to 0 \to \cdots & x\not\in U
\end{cases}\\
&=
\begin{cases}
 {\mathbb{C}}& x\in U 
\\
0 & x\not\in U,
\end{cases}
\\
&=
\begin{cases}
{\mathbb{C}}& x \neq \left\{{1}\right\} 
\\
0 & x=\left\{{1}\right\}.
\end{cases}
\end{align*}
where we've used that since \(U\) is open, if \(x\in U\) then there is
an open neighborhood \(W\ni x\) completely contained in \(U\), making
the directed system eventually constant. Otherwise, if \(x\not\in U\),
then no neighborhood of \(x\) is completely contained in \(U\), and the
sections here are zero for every \(W\ni X\).

\begin{quote}
Note: this uses that the colimit of an eventually constant diagram is
isomorphic to whatever that constant object is, i.e.~it satisfies the
correct universal property.
\end{quote}

Similarly, noting that for \(W \subseteq X\),
\begin{align*}
i^{-1}(W) = 
\begin{cases}
\left\{{1}\right\} & \left\{{1}\right\} \in W  
\\
\emptyset & \left\{{1}\right\} \not\in W,
\end{cases}
\end{align*}
we have
\begin{align*}
i_* \underline{{\mathbb{C}}_Z} 
&= \colim_{W\ni x} \underline{{\mathbb{C}}_Z}(i^{-1}(W)) \\
&= \colim_{W\ni x} \underline{{\mathbb{C}}_Z}\qty{
\begin{cases}
\left\{{1}\right\} & \left\{{1}\right\} \in W
\\
\emptyset & \left\{{1}\right\}\not\in W
\end{cases}
} \\
&=
\colim
\begin{cases}
{\mathbb{C}}\to {\mathbb{C}}\to \cdots & x = \left\{{1}\right\} 
\\
\cdots \to 0\to 0\to \cdots & x\neq \left\{{1}\right\}
\end{cases}\\
&= 
\begin{cases}
{\mathbb{C}}& x = \left\{{1}\right\} 
\\
0 & x\neq \left\{{1}\right\},
\end{cases}
\end{align*}
where we've used that if \(U\) is open and \(x\in U\) with
\(x\neq \left\{{1}\right\}\), there eventually all small enough
neighborhoods \(W \ni x\) will not intersect
\(X\setminus U = \left\{{1}\right\}\). Thus
\begin{align*}
{\mathcal{F}}_x =
\begin{cases}
 0 \oplus {\mathbb{C}}& x = \left\{{1}\right\}
\\
{\mathbb{C}}\oplus 0 & x\neq \left\{{1}\right\},
\end{cases}
\end{align*}
and all of the stalks are one copy of \({\mathbb{C}}\), as in the
constant sheaf \(\underline{{\mathbb{C}}_X}\) on \(X\). However,
\({\mathcal{F}}\) and \(\underline{{\mathbb{C}}_X}\) do not have the
same sections: take \(W \subseteq S^1\) to be a connected open
neighborhood of \(\left\{{1}\right\}\), then
\begin{align*}
\left\{{1}\right\} \in W \implies i_* \underline{{\mathbb{C}}_Z}(W) = \underline{{\mathbb{C}}_Z}(i^{-1}(W) ) = \underline{{\mathbb{C}}_Z}(\left\{{1}\right\}) = {\mathbb{C}}
.\end{align*}
Note that
\(j^{-1}(W) = W \setminus\left\{{1}\right\} = W_1 {\textstyle\coprod}W_2\)
breaks into two connected components, so
\begin{align*}
j_! \underline{{\mathbb{C}}_U}(W) = \underline{{\mathbb{C}}_U}(W_1 {\textstyle\coprod}W_2) = {\mathbb{C}}{ {}^{ \scriptscriptstyle\oplus^{2} }  }
,\end{align*}
so
\begin{align*}
{\mathcal{F}}(W) = {\mathbb{C}}{ {}^{ \scriptscriptstyle\oplus^{2} }  } \oplus {\mathbb{C}}\neq {\mathbb{C}}= \underline{{\mathbb{C}}_X}(W) \implies {\mathcal{F}}\not\cong \underline{{\mathbb{C}}_X}
.\end{align*}

\end{proof}

\begin{proposition}[Parts 2 and 3]

If \(X\) is an irreducible algebraic variety or \(X = [0, 1]\) in the
Euclidean topology, the answer is still generally no.

\end{proposition}

\begin{proof}[Parts 2 and 3]

The previous example shows this, noting that
\(S^1 \cong \operatorname{Spec}{{\mathbb{R}}[x, y] \over \left\langle{x^2 + y^2 - 1}\right\rangle}\)
and \(f(x, y) = x^2+y^2-1\) does not factor in \({\mathbb{R}}[x, y]\),
making \(S^1\) irreducible in the Zariski topology.

For \(X = [0, 1]\), a modification of the previous example yields the
same conclusion: set \(Z = \left\{{1\over 2}\right\}\) and
\(U = X\setminus Z\); the same argument with the same sheaf goes
through.

\end{proof}

\hypertarget{problem-4}{%
\subsection{Problem 4}\label{problem-4}}

\begin{problem}[1.4]

Let \(X\) be a space with a poset topology (with increasing open sets).

\begin{itemize}
\item
  Prove that a sheaf \(F\) on \(X\) is the same as a collection
  \(F_{x}\) for \(x \in X\) and maps
  \(r_{x, y}: F_{x} \rightarrow F_{y}\) for all \(x \leq y\) satisfying
  \(r_{y, z} \circ r_{x, y}=r_{x, z}\).
\item
  Prove that for an open \(U \subset X\), the set of sections \(F(U)\)
  is \(\varprojlim_{U\ni x} F_{x}\).
\end{itemize}

\end{problem}

\begin{proposition}[?]

Let \((X, \leq)\) be a poset with the order topology, then a sheaf
\({\mathcal{F}}\in {\mathsf{Sh}}(X)\) on \(X\) is is equivalently a
functorial assignment on the corresponding poset category
\begin{align*}
{\mathcal{F}}: \mathsf{Poset}(X) &\to \mathsf{C} \\
x &\mapsto {\mathcal{F}}_x
,\end{align*}
where the objects of \(\mathsf{Poset}(X)\) are elements \(x\in X\) where
the hom spaces are defined as
\begin{align*}
\mathop{\mathrm{Hom}}_{\mathsf{Poset}(X)}(x, y)
\coloneqq
\begin{cases}
\left\{{{\operatorname{pt}}}\right\} &  x\leq y
\\
\emptyset & \text{else}.
\end{cases}
\end{align*}
Here
\(\mathsf{C} = {\mathsf{Ab}}{\mathsf{Grp}}, \mathsf{Ring}, {\mathsf{R}{\hbox{-}}\mathsf{Mod}}, {\mathsf{Alg}_{/k} }\),
etc.

\end{proposition}

\begin{proof}[?]

\(\implies\): Suppose that \({\mathcal{F}}\) is a sheaf on
\((X, \leq)\), and define
\begin{align*}
{\mathcal{G}}: \mathsf{Poset}(X) &\to \mathsf{C} \\
x &\mapsto {\mathcal{G}}_x \coloneqq{\mathcal{F}}(U_{\geq x}) \\
(\iota_{xy}: x \to y) &\mapsto (f_{xy}: {\mathcal{G}}_x \to {\mathcal{G}}_y) 
,\end{align*}
where we define \(f_{xy}\) using that
\begin{align*}
x\to y \in \mathsf{Poset}(X) \iff x\leq y \in X \implies U_{\geq y} \hookrightarrow U_{\geq x} \in {\mathsf{Open}}(X)
,\end{align*}
and since \({\mathcal{F}}\) is a contravariant functor, the latter
inclusion induces an morphism
\begin{align*}
f_{xy}: {\mathcal{F}}(U_{\geq x}) \to {\mathcal{F}}(U_{\geq y}) \qquad \in \mathsf{C}
.\end{align*}
Compatibility of the \(f_{xy}\) for \({\mathcal{G}}\) follow immediately
from the fact that \({\mathcal{F}}\) is a functor.

\(\impliedby\): Given a functorial assignment
\begin{align*}
{\mathcal{G}}: \mathsf{Poset}(X) &\to \mathsf{C} \\
x &\mapsto {\mathcal{G}}_x
,\end{align*}
we want to construct an associated sheaf
\begin{align*}
{\mathcal{F}}: {\mathsf{Open}}(X)^{\operatorname{op}}\to \mathsf{C}
.\end{align*}
By a result from class, it suffices to specify the sheaf on a basis
\({\mathcal{B}}\) for the order topology on \(X\), so let
\({\mathcal{B}}\coloneqq\left\{{U_{\geq x}}\right\}_{x\in X}\) be the
basis of up-sets. Define a presheaf by
\begin{align*}
{\mathcal{F}}^-(U_{\geq x}) \coloneqq{\mathcal{G}}_x
,\end{align*}
and take \({\mathcal{F}}\coloneqq({\mathcal{F}}^-)^+\).

\end{proof}

\begin{proposition}[?]

For \((X, \leq)\) a poset in the order topology, \(U \subseteq X\) open,
and \({\mathcal{F}}\) a sheaf on \(X\),
\begin{align*}
F(U) \cong \varprojlim_{x\in U} {\mathcal{F}}_x
.\end{align*}

\end{proposition}

\begin{proof}[?]

It suffices to show that if \({\mathcal{B}}\) is a basis for a topology,
\begin{align*}
{\mathcal{F}}(U) = \varprojlim_{V\in {\mathcal{B}}, V \subseteq U} {\mathcal{F}}(V)
,\end{align*}
which follows because this precisely describes a continuous section of
the \emph{espace étalé} over \(U \subseteq X\) as a compatible
collection of sections on \(U\) decomposed in a basis as
\(U = \cup B_i\). With this, we can then directly compute
\begin{align*}
{\mathcal{F}}(U) 
&= \varprojlim_{V\in {\mathcal{B}}, V \subseteq U} {\mathcal{F}}(V) \\
&= \varprojlim_{V_{\geq x} \subseteq U} {\mathcal{F}}(V_{\geq x}) \qquad \text{by the definition of }{\mathcal{B}}\\
&= \varprojlim_{V_{\geq x} \subseteq U} {\mathcal{F}}_x \qquad \text{since $V_{\geq x}$ is the smallest open containing $x$}\\
&= \varprojlim_{x\in U} {\mathcal{F}}_x \qquad \text{ since } V_{\geq x} \subseteq U \implies x\in U 
.\end{align*}

\end{proof}

\hypertarget{problem-set-2}{%
\section{Problem Set 2}\label{problem-set-2}}

\hypertarget{problem-1-1}{%
\subsection{Problem 1}\label{problem-1-1}}

\begin{proposition}[1.1]

The global sections functor is left-exact.

\end{proposition}

\begin{proof}

We'll use the fact that a sequence of sheaves is exact if and only if
the induced sequence on stalks is exact. Given this, let
\({\mathcal{F}}, {\mathcal{G}}, {\mathcal{H}}\) be sheaves of abelian
groups on \(X\), and consider the diagram induced by restriction
morphisms:

\begin{center}
\begin{tikzcd}
    0 && {\mathcal{F}}&& {\mathcal{G}}&& {\mathcal{H}}&& 0 \\
    \\
    && {{\mathcal{F}}(U)} && {{\mathcal{G}}(U)} && {{\mathcal{H}}(U)} \\
    \\
    0 && {{\mathcal{F}}_p} && {{\mathcal{G}}_p} && {{\mathcal{H}}_p} && 0
    \arrow[from=1-1, to=1-3]
    \arrow["f", from=1-3, to=1-5]
    \arrow["g", from=1-5, to=1-7]
    \arrow[from=1-7, to=1-9]
    \arrow[from=5-1, to=5-3]
    \arrow["{f_p}", from=5-3, to=5-5]
    \arrow["{g_p}", from=5-5, to=5-7]
    \arrow[from=5-7, to=5-9]
    \arrow[from=1-7, to=3-7]
    \arrow[from=3-7, to=5-7]
    \arrow[from=3-5, to=5-5]
    \arrow[from=1-3, to=3-3]
    \arrow[from=3-3, to=5-3]
    \arrow["{f_U}", from=3-3, to=3-5]
    \arrow[from=1-5, to=3-5]
    \arrow["{g_U}", from=3-5, to=3-7]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMTMsWzAsMCwiMCJdLFsyLDAsIlxcbWNmIl0sWzQsMCwiXFxtY2ciXSxbNiwwLCJcXG1jaCJdLFs4LDAsIjAiXSxbMiwyLCJcXG1jZihVKSJdLFsyLDQsIlxcbWNmX3AiXSxbNCwyLCJcXG1jZyhVKSJdLFs0LDQsIlxcbWNnX3AiXSxbNiwyLCJcXG1jaChVKSJdLFs2LDQsIlxcbWNoX3AiXSxbOCw0LCIwIl0sWzAsNCwiMCJdLFswLDFdLFsxLDIsImYiXSxbMiwzLCJnIl0sWzMsNF0sWzEyLDZdLFs2LDgsImZfcCJdLFs4LDEwLCJnX3AiXSxbMTAsMTFdLFszLDldLFs5LDEwXSxbNyw4XSxbMSw1XSxbNSw2XSxbNSw3LCJmX1UiXSxbMiw3XSxbNyw5LCJnX1UiXV0=}{Link
to Diagram}
\end{quote}

Note that we can take \(U = X\) in this diagram. If the top sequence of
sheaves is exact, there are isomorphisms of sheaves:

\begin{itemize}
\tightlist
\item
  \(\ker f = 0\) and
\item
  \(\operatorname{im}f = \ker g\).
\end{itemize}

\begin{claim}

\begin{align*}
\ker f_X = 0
,\end{align*}
making \(f_X\) injective and yielding exactness at the first position.

\end{claim}

\begin{proof}[?]

Since the presheaf \(\ker f\) is in fact a sheaf, writing \(\mathbf 0\)
for the sheaf \(U\mapsto 0\), we have
\begin{align*}
\ker\qty{ {\mathcal{F}}(X) \xrightarrow{f_X} {\mathcal{G}}(X) }
= (\ker f)(X) = \mathbf{0}(X) = 0
\end{align*}

\end{proof}

\begin{claim}

\begin{align*}
\operatorname{im}f_X = \ker g_X
,\end{align*}
yielding exactness at the middle position.

\end{claim}

\begin{proof}[?]

\(\operatorname{im}f_X \subseteq \ker g_X\) follows from a diagram
chase:

\begin{center}
\begin{tikzcd}
    && \textcolor{rgb,255:red,92;green,92;blue,214}{a} && \textcolor{rgb,255:red,92;green,92;blue,214}{b } && \textcolor{rgb,255:red,92;green,92;blue,214}{\therefore g_X(b) = 0} \\
    && {{\mathcal{F}}(X)} && {{\mathcal{G}}(X)} && {{\mathcal{H}}(X)} \\
    \\
    0 && {{\mathcal{F}}_p} && {{\mathcal{G}}_p} && {{\mathcal{H}}_p} && 0 \\
    && \textcolor{rgb,255:red,92;green,92;blue,214}{a_p} &&&& \textcolor{rgb,255:red,92;green,92;blue,214}{\tilde a = (g_p\circ f_p)(a_p) = 0 \, \forall p}
    \arrow[from=4-1, to=4-3]
    \arrow["{f_p}", from=4-3, to=4-5]
    \arrow["{g_p}", from=4-5, to=4-7]
    \arrow[from=4-7, to=4-9]
    \arrow[from=2-7, to=4-7]
    \arrow[from=2-5, to=4-5]
    \arrow[from=2-3, to=4-3]
    \arrow["{f_X}", from=2-3, to=2-5]
    \arrow["{g_X}", from=2-5, to=2-7]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=18pt}, dashed, maps to, from=1-3, to=5-3]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=24pt}, dashed, maps to, from=5-3, to=5-7]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, dotted, maps to, from=1-3, to=1-5]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, dotted, maps to, from=1-5, to=1-7]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-24pt}, dotted, maps to, from=1-7, to=5-7]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

\begin{itemize}
\tightlist
\item
  Fix \(b\in \operatorname{im}f_X \subseteq {\mathcal{G}}(X)\), then by
  surjectivity choose a lift \(a\in {\mathcal{F}}(X)\).
\item
  Map \(a\) along
  \({\mathcal{F}}(X)\to {\mathcal{F}}_p \to {\mathcal{G}}_p \to {\mathcal{H}}_p\);
  by exactness the result is zero in \(C_p\).
\item
  By commutativity of the diagram, mapping \(a\) along
  \({\mathcal{F}}(X)\to {\mathcal{G}}(X)\to {\mathcal{H}}(X) \to {\mathcal{H}}_p\)
  also yields zero in \(C_p\).
\item
  Write \(\tilde b \coloneqq g_X(b)\); since the above argument holds
  for all \(p\in X\), \(g_X(b)\) is a section of \({\mathcal{H}}\) that
  is zero in every stalk. Thus by the sheaf property for
  \({\mathcal{H}}\), the section \(g_X(b)\) must be zero, and
  \(b\in \ker g_X\).
\end{itemize}

Similarly, \(\ker g_X \subseteq \operatorname{im}f_X\) follows from a
diagram chase:

\begin{itemize}
\tightlist
\item
  Fix \(b\in \ker g_X\), so the image of \(g_X\) in \({\mathcal{H}}(X)\)
  is zero. Then its image \(c_p\) along
  \({\mathcal{G}}(X) \to {\mathcal{H}}(X) \to {\mathcal{H}}_p\) is also
  zero.
\item
  By commutativity of the right square, \(g_p(b_p) = 0\) and so
  \(b_p\in \ker g_p = \operatorname{im}f_p\) by exactness of the bottom
  row.
\item
  Choose a lift \(a_p\in {\mathcal{F}}_p\) along \(f_p\), so
  \(f_p(a_p) = b_p\) Since \(a_p\) is a germ of \({\mathcal{F}}\), pick
  any global section \(a\in {\mathcal{F}}(X)\) restricting to \(a_p\)
  and making the square commute.
\item
  Since \(f_X(a)_p = f_p(a_p) = b_p\) for all \(p\), by uniqueness of
  gluing for \({\mathcal{G}}\) we have \(f_X(a) = b\) and
  \(b\in \operatorname{im}f_X\).
\end{itemize}

\end{proof}

\end{proof}

\begin{proposition}[1.2]

Taking global sections may fail to be right-exact.

\end{proposition}

\begin{proof}[?]

Consider the following poset and its corresponding category of open
sets:

\begin{center}
\begin{tikzcd}
    X &&&& {{\mathsf{Open}}(X)} \\
    a &&&& {\left\{{a,b,c}\right\}} \\
    \\
    b &&& {\left\{{a,b}\right\}} && {\left\{{b,c}\right\}} \\
    \\
    c &&& {\left\{{b}\right\}} && {\left\{{c}\right\}} \\
    \\
    &&&& \emptyset
    \arrow[from=8-5, to=6-6]
    \arrow[from=8-5, to=6-4]
    \arrow[from=6-4, to=4-6]
    \arrow[from=6-6, to=4-6]
    \arrow[from=6-4, to=4-4]
    \arrow[""{name=0, anchor=center, inner sep=0}, from=4-4, to=2-5]
    \arrow[from=4-6, to=2-5]
    \arrow["\leq", from=6-1, to=4-1]
    \arrow[""{name=1, anchor=center, inner sep=0}, "\leq", from=4-1, to=2-1]
    \arrow[shorten <=24pt, shorten >=24pt, Rightarrow, from=1, to=0]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMTEsWzQsMSwiXFx0c3thLGIsY30iXSxbMywzLCJcXHRze2EsYn0iXSxbNSwzLCJcXHRze2IsY30iXSxbMyw1LCJcXHRze2J9Il0sWzUsNSwiXFx0c3tjfSJdLFs0LDcsIlxcZW1wdHlzZXQiXSxbMCwxLCJhIl0sWzAsMywiYiJdLFswLDUsImMiXSxbMCwwLCJYIl0sWzQsMCwiXFxPcGVuKFgpIl0sWzUsNF0sWzUsM10sWzMsMl0sWzQsMl0sWzMsMV0sWzEsMF0sWzIsMF0sWzgsNywiXFxsZXEiXSxbNyw2LCJcXGxlcSJdLFsxOSwxNiwiIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==}{Link
to Diagram}
\end{quote}

Define the following two sheaves \({\mathcal{F}}, {\mathcal{G}}\) and a
morphism between them:

\includegraphics{figures/2022-02-18_10-18-05.png}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to diagram}
\end{quote}

Note that there are only three stalks to consider, none of which
coincide with global sections, so we can take the sheaf morphism to be
the identity on these to get a surjection on stalks. We then choose a
non-surjective map \({\mathcal{F}}(X) \to {\mathcal{G}}(X)\) given by
\((a, b) \mapsto (a, a+b)\), where e.g.~the image does not contain the
element \((1, 1)\).

One can check that the individual diagrams for \({\mathcal{F}}\) and
\({\mathcal{G}}\) commute, yielding a presheaf, and that existence and
uniqueness of gluing hold for both. Moreover, all of the squares formed
by the map \({\mathcal{F}}\to {\mathcal{G}}\) commute, so this does in
fact yield a morphism of sheaves.

\end{proof}

\hypertarget{problem-2-1}{%
\subsection{Problem 2}\label{problem-2-1}}

\begin{proposition}[?]

If a map \(f:X\to Y\) between posets is continuous, it is
order-preserving, i.e.~if \(x_1\leq x_2\) then \(f(x_1) \leq f(x_2)\).

\end{proposition}

\begin{proof}[?]

Continuity can be checked on a basis, so let
\(U_b = \left\{{y\in Y {~\mathrel{\Big\vert}~}y\geq b}\right\}\) be a
basic open upper set. Then \(f\) is continuous iff \(f^{-1}(U_a)\) is an
open set in \(X\). Being open means that for every
\(x_0 \in f^{-1}(U_a)\), \(x_1\geq x_0\implies x_1\in f^{-1}(U_a)\).
\begin{align*}
f \text{ is continuous } 
&\iff \forall U \text{ open in } Y, \, f^{-1}(U) \text{ is open in } X \\
&\iff \forall U_a \text{ a basic open in } Y, \, f^{-1}(U_a) \text{ is open in } X \\
&\iff \forall a\in Y,\, \forall x_0 \in f^{-1}(U_a),\, x_1\geq x_0\implies x_1\in f^{-1}(U_a) \\
&\iff \forall a\in Y,\, \forall x_0 \in f^{-1}(U_a),\, x_1\geq x_0\implies f(x_1) \in U_a \\
&\iff \forall a\in Y,\, \forall x_0 \in f^{-1}(U_a),\, x_1\geq x_0\implies f(x_1) \geq a\\
&\iff \forall a\in Y,\, \forall x_0\in X \text{ s.t. } f(x_0) \geq a,\, x_1\geq x_0\implies f(x_1) \geq a
.\end{align*}
Now taking \(x_0 = f^{-1}(a)\) for \(a\in \operatorname{im}f\) yields
\begin{align*}
\implies \forall a\in \operatorname{im}f,\quad x_1 \geq f^{-1}(a) \implies f(x_1) \geq a
.\end{align*}
Relabeling \(x_1 = f^{-1}(b)\),
\begin{align*}
&\implies \forall a\in \operatorname{im}f, \quad f^{-1}(b) \geq f^{-1}(a) \implies b \geq a \\
&\implies \forall \tilde a\in f^{-1}(Y), \quad \tilde b \geq \tilde a \implies f(\tilde b) \geq f(\tilde a)
.\end{align*}

\end{proof}

\begin{proposition}[?]

For
\({\mathcal{F}}\in {\mathsf{Sh}}_X, {\mathcal{G}}\in {\mathsf{Sh}}_Y, {\mathcal{H}}\in {\mathsf{Sh}}_U\)
with \(U \subseteq X\), \(X \xrightarrow{f} Y\), and
\(U \xhookrightarrow{j} X\),

\begin{itemize}
\tightlist
\item
  \(f_* {\mathcal{F}}\) is no additional data
\item
  \(f^{-1}{\mathcal{G}}= \left\{{{\mathcal{G}}_{f(x_0)}, \phi_{f(x_0), f(x_1) } {~\mathrel{\Big\vert}~}x_0, x_1\in X, x_0 \leq x_1}\right\}\).
\item
  \(j_! {\mathcal{H}}= ?\)
\end{itemize}

\end{proposition}

\begin{proof}

We'll use that
\({\mathcal{F}}\in {\mathsf{Sh}}(X, {\mathsf{Ab}}{\mathsf{Grp}})\) is
the same as the data of \(\left\{{{\mathcal{F}}_x, \phi_{xy}}\right\}\)
where \({\mathcal{F}}_x\) is a collection of groups and
\(\phi_{xy}: {\mathcal{F}}_x\to {\mathcal{F}}_y\) are group morphisms
for every \(x\leq y\). Thus the values of a sheaf on posets are entirely
determined by a functorial assignment of groups to the stalk at each
point, i.e.~an assignment of a group to each point. So it suffices to
determine what the stalks of these three sheaves are.

\begin{itemize}
\item
  For \(f_*{\mathcal{F}}\), noting that
  \begin{align*}
  (f_* {\mathcal{F}})(U_{\geq a}) = {\mathcal{F}}(f^{-1}(U_{\geq a})) = \cocolim_{x\in f^{-1}(U_{\geq a})} {\mathcal{F}}_x
  ,\end{align*}
  we see that this sheaf is completely determined by the data for
  \({\mathcal{F}}\).
\item
  For \(f^{-1}{\mathcal{G}}\), we can use the fact that for any sheaf,
  there is a formula on stalks:
  \begin{align*}
  (f^{-1}{\mathcal{G}})_p \cong {\mathcal{G}}_{f(p)}
  ,\end{align*}
  and so \(f^{-1}{\mathcal{G}}\) is the data
  \(\left\{{{\mathcal{G}}_x, \psi_{xy}}\right\}\) for every \(x\leq y\)
  with \(x,y\in \operatorname{im}f\).
\item
  For \(j_! {\mathcal{H}}, \cdots ?\)
\end{itemize}

\begin{quote}
Attempts to approach this: the general definition involves
sheafification, which seems hard to describe in general. On the other
hand, I haven't been able to work out what the sheaf space for a poset
should look like.
\end{quote}

\end{proof}

\hypertarget{problem-3-1}{%
\subsection{Problem 3}\label{problem-3-1}}

\begin{proposition}[?]

Let \({\mathcal{F}}\in {\mathsf{Sh}}(X)\) and let
\(\text{Ét}({\mathcal{F}}) \xrightarrow{\pi} X\) be its corresponding
sheaf space, so
\({\mathcal{F}}= \mathop{\mathrm{Sec}}_{\text{cts}}(\pi)\), and let
\({\mathcal{G}}= \mathop{\mathrm{Sec}}(\pi)\). Then
\begin{align*}
{\mathcal{G}}= \prod_{x\in X} x_* {\mathcal{F}}_x
\end{align*}
where \(x: \left\{{x}\right\} \hookrightarrow X\) is the inclusion of a
point and \({\mathcal{F}}\in {\mathsf{Sh}}(\left\{{x}\right\})\) is
regarded as a sheaf on a one-point space.

\end{proposition}

\begin{proof}

We'll use the fact that as a set,
\(\text{Ét}({\mathcal{F}}) = \coprod_{x\in X} {\mathcal{F}}_x\) is the
coproduct of all of the stalks of \({\mathcal{F}}\). We can compute the
sections of this sheaf as follows:
\begin{align*}
{\mathcal{G}}(U) 
&= \qty{ \prod_{x\in X} x_* {\mathcal{F}}_x}(U) \\
&= \prod_{x\in X} (x_* {\mathcal{F}}_x)(U) \\
&= \prod_{x\in X} {\mathcal{F}}_x(x^{-1}(U)) \\
&= \prod_{x\in X} {\mathcal{F}}_x\qty{
\begin{cases}
\left\{{x}\right\} & x\in U 
\\
\emptyset & x\not\in U.
\end{cases}
}  \\
&= \prod_{x\in X} 
\begin{cases}
{\mathcal{F}}_x & x\in U 
\\
0 & x\not\in U.
\end{cases} \\
&= \prod_{x\in U} {\mathcal{F}}_x 
.\end{align*}
We can now simply regard \({\mathcal{G}}(U)\) as the set of set-valued
functions
\(s: U\to \coprod_{x\in U} {\mathcal{F}}_x \subseteq \text{Ét}({\mathcal{F}})\)
by setting \(s(x) = \pi_x\qty{\prod_{x\in U} {\mathcal{F}}_x}\) to be
the \(x{\hbox{-}}\)coordinate in the direct product, where
\(\pi_x: \prod_{x\in U}{\mathcal{F}}_x \to {\mathcal{F}}_x\) is
projection onto the \(x{\hbox{-}}\)coordinate.

On the other hand, the data of a set-valued section
\(s \in \mathop{\mathrm{Sec}}( \text{Ét}({\mathcal{F}}) \xrightarrow{\pi} U)\)
is the following: for every \(x\in X\), a choice of an element
\begin{align*}
s(x) \in \pi^{-1}(x) = {\mathcal{F}}_x \subseteq \text{Ét}({\mathcal{F}})
,\end{align*}
with no other compatibility conditions, which is precisely the same as
the set-valued functions specified by \({\mathcal{G}}(U)\) above.

\end{proof}

\begin{proposition}[?]

The stalks \({\mathcal{G}}_p\) are given by
\begin{align*}
{\mathcal{G}}_p = \colim_{U\ni p} \prod_{x\in U} {\mathcal{F}}_x
,\end{align*}
the direct limit of the product of stalks of \({\mathcal{F}}\) along
neighborhoods of \(p\).

\end{proposition}

\begin{proof}

\begin{align*}
{\mathcal{G}}_p
&\coloneqq\colim_{U\ni p} {\mathcal{G}}(U) \\
&\coloneqq\colim_{U\ni p} \qty{ \prod_{x\in X} (\iota_x)_* {\mathcal{F}}_x }(U) \\
&= \colim_{U\ni p} \prod_{x\in X} \qty{ (\iota_x)_* {\mathcal{F}}_x} (U) \\
&\coloneqq\colim_{U\ni p} \prod_{x\in X} {\mathcal{F}}_x(\iota_x^{-1}(U)) \\
&= \colim_{U\ni p} \prod_{x\in X} {\mathcal{F}}_x\qty{
\begin{cases}
\left\{{x}\right\} & x\in U 
\\
\emptyset & \text{else}.
\end{cases}
} \\
&= \colim_{U\ni p} \prod_{x\in X} \qty{
\begin{cases}
{\mathcal{F}}_x & x\in U 
\\
0 & \text{else}.
\end{cases}
} \\
&= \colim_{U\ni p} \prod_{x\in U} {\mathcal{F}}_x
.\end{align*}

\end{proof}

\begin{proposition}[?]

There is an injective morphism of sheaves
\({\mathcal{F}}\hookrightarrow{\mathcal{G}}\).

\end{proposition}

\begin{proof}

For every open \(U \subseteq X\), define a map of sets on the function
spaces:
\begin{align*}
\Psi_U: \mathop{\mathrm{Sec}}_{\text{cts}}(\text{Ét}({\mathcal{F}}) \xrightarrow{\pi} U) &\to \mathop{\mathrm{Sec}}(\text{Ét}({\mathcal{F}}) \xrightarrow{\pi} U) \\
f &\mapsto f
,\end{align*}
which does nothing more than a forgetful map that regards a continuous
section as a set-valued section. This is evidently an injective map of
sets, since if \(f_1, f_2\) are continuous sections and \(f_1 = f_2\) as
set-valued functions, they continue to be equal when regarded as
continuous sections, so
\(\Psi_U(f_1) = \Psi_U(f_2) \implies f_1 = f_2\).

These \(\Psi_U\) assemble to a morphism of sheaves
\(\Psi: {\mathcal{F}}\to {\mathcal{G}}\), and since
\((\ker \Psi)^- = \mathbf{0}\) vanishes as a presheaf and the kernel
presheaf is a sheaf, we have \(\ker \Psi = \mathbf{0}\).

\end{proof}

\hypertarget{problem-set-3}{%
\section{Problem Set 3}\label{problem-set-3}}

\hypertarget{problem-1-2}{%
\subsection{Problem 1}\label{problem-1-2}}

\begin{problem}[Problem 1]

Let \(I\) be an index category, \(\mathcal{A}\) an abelian category, and
\(\mathcal{A}^{I}\) be the category of functors
\(F: I \rightarrow \mathcal{A}\). Prove that the functor
\begin{align*}
\cocolim_{i \in I}: \mathcal{A}^{I} \rightarrow \mathcal{A}, \quad F \mapsto \cocolim_{i \in I} F_{i}
\end{align*}
is left exact. (By duality, the functor \(\colim_{i \in I}\) is right
exact.)

What is this functor in the case when \(I\) is a poset and \(F_{i}\) is
a collection of stalks on the space \(X=I\) with poset topology?

\end{problem}

\begin{solution}[Part 1]

It suffices to show that \(\cocolim_{i\in I}\) is a right adjoint
functor, and right adjoints are left exact by general homological
algebra.

\begin{claim}

There is an adjunction
\begin{align*}
\adjunction{\Delta}{\cocolim_{i\in I}}{\mathsf{A}}{\mathsf{A}^{\mathsf{I}}}
,\end{align*}
where \(\Delta\) is the diagonal functor:
\begin{align*}
\Delta: \mathsf{A} &\to \mathsf{A}^{\mathsf{I}} \\
X &\mapsto \Delta_X \\
(X \xrightarrow{f} Y) &\mapsto (\Delta_X \xrightarrow{\eta_f } \Delta_Y)
\end{align*}
where

\begin{itemize}
\tightlist
\item
  The constant functor \(\Delta_X: \mathsf{I} \to \mathsf{C}\) is
  defined on objects \(i\in \mathsf{I}\) as \(\Delta_X(i) \coloneqq X\)
  and on morphisms \(i\xrightarrow{\iota_{ij}} j\) as
  \(\Delta_f(\iota_{ij}) = X \xrightarrow{\operatorname{id}_X} X\).
\item
  \(\eta_f\) is a natural transformation of functors with components
  given by \(f\):
\end{itemize}

\begin{center}
\begin{tikzcd}
    {\mathsf{I}} && {\mathsf{C}} \\
    i && {\Delta_X(i) } && {\Delta_Y(i)} && X && Y \\
    \\
    j && {\Delta_X(j)} && {\Delta_Y(j)} && X && Y
    \arrow[""{name=0, anchor=center, inner sep=0}, "{\iota_{ij}}"', from=2-1, to=4-1]
    \arrow["{\eta_f(i)}", from=2-3, to=2-5]
    \arrow[""{name=1, anchor=center, inner sep=0}, "{\Delta_X(\iota_{ij})}", from=2-3, to=4-3]
    \arrow["{\eta_f(j)}"', from=4-3, to=4-5]
    \arrow[""{name=2, anchor=center, inner sep=0}, "{\Delta_Y(\iota_{ij})}"', from=2-5, to=4-5]
    \arrow[""{name=3, anchor=center, inner sep=0}, "{\operatorname{id}_X}", from=2-7, to=4-7]
    \arrow["{\operatorname{id}_Y}"', from=2-9, to=4-9]
    \arrow["f", from=2-7, to=2-9]
    \arrow["f", from=4-7, to=4-9]
    \arrow[shorten <=13pt, shorten >=13pt, Rightarrow, no head, from=2, to=3]
    \arrow["{\Delta: \mathsf{I}\to \mathsf{C}}", shorten <=14pt, shorten >=14pt, Rightarrow, from=0, to=1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

\end{claim}

Why this claim is true: this follows immediately from the fact that
there is a natural isomorphism
\begin{align*}
\mathop{\mathrm{Hom}}_{\mathsf{A}}(X, \lim F)  { \, \xrightarrow{\sim}\, }\mathop{\mathrm{Hom}}_{\mathsf{A}^{\mathsf{I}}}(\Delta_X, F)
,\end{align*}
i.e.~maps from an object \(X\) into the limit of \(F\) are equivalent to
natural transformations between the constant functor \(\Delta_X\) and
\(F\). This follows from the fact that a morphism \(X\to \lim F\) in
\(\mathsf{A}\) is the data of a family of compatible maps
\(\left\{{f_i}\right\}_{i\in \mathsf{I}}\) over the essential image of
\(F\):

\begin{center}
\begin{tikzcd}
    {\mathsf{I}} &&& {\mathsf{A}} \\
    i &&& {F(i)} \\
    &&&&&& {\lim_i F} &&& X \\
    j &&& {F(j)}
    \arrow[""{name=0, anchor=center, inner sep=0}, from=4-1, to=2-1]
    \arrow[""{name=1, anchor=center, inner sep=0}, from=4-4, to=2-4]
    \arrow[from=3-7, to=2-4]
    \arrow[from=3-7, to=4-4]
    \arrow["{f_i}"{description}, curve={height=12pt}, from=3-10, to=2-4]
    \arrow["{f_j}"{description}, curve={height=-18pt}, from=3-10, to=4-4]
    \arrow["{\exists !}", dashed, from=3-10, to=3-7]
    \arrow["F", shorten <=10pt, shorten >=10pt, Rightarrow, from=0, to=1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsOCxbMCwxLCJpIl0sWzAsMywiaiJdLFswLDAsIlxcY2F0e0l9Il0sWzMsMSwiRihpKSJdLFszLDMsIkYoaikiXSxbMywwLCJcXGNhdHtBfSJdLFs2LDIsIlxcbGltX2kgRiJdLFs5LDIsIlgiXSxbMSwwXSxbNCwzXSxbNiwzXSxbNiw0XSxbNywzLCJmX2kiLDEseyJjdXJ2ZSI6Mn1dLFs3LDQsImZfaiIsMSx7ImN1cnZlIjotM31dLFs3LDYsIlxcZXhpc3RzICEiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbOCw5LCJGIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjEwLCJ0YXJnZXQiOjEwfX1dXQ==}{Link
to Diagram}
\end{quote}

On the other hand, a natural transformation \(\Delta_X \to F\) is
precisely the same data:

\begin{center}
\begin{tikzcd}
    {\mathsf{I}} &&& {\mathsf{A}} \\
    i &&& {F(i)} &&&&&& {\Delta_X(i) = X} \\
    \\
    j &&& {F(j)} &&&&&& {\Delta_X(j) = X}
    \arrow["g", from=4-1, to=2-1]
    \arrow["{F(g)}", from=4-4, to=2-4]
    \arrow["{f_i}"{description}, curve={height=12pt}, from=2-10, to=2-4]
    \arrow["{F(g) = \operatorname{id}_X}", Rightarrow, no head, from=2-10, to=4-10]
    \arrow["{f_j}"{description}, curve={height=-18pt}, from=4-10, to=4-4]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsOCxbMCwxLCJpIl0sWzAsMywiaiJdLFswLDAsIlxcY2F0e0l9Il0sWzMsMSwiRihpKSJdLFszLDMsIkYoaikiXSxbMywwLCJcXGNhdHtBfSJdLFs5LDEsIlxcRGVsdGFfWChpKSA9IFgiXSxbOSwzLCJcXERlbHRhX1goaikgPSBYIl0sWzEsMCwiZyJdLFs0LDMsIkYoZykiXSxbNiwzLCJmX2kiLDEseyJjdXJ2ZSI6Mn1dLFs2LDcsIkYoZykgPSBcXGlkX1giLDAseyJsZXZlbCI6Miwic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dLFs3LDQsImZfaiIsMSx7ImN1cnZlIjotM31dXQ==}{Link
to Diagram}
\end{quote}

\end{solution}

\begin{solution}[Part 2]

If \(\mathsf{I} = {\mathsf{Open}}(X)\) where \(X\) is given the order
topology and \(F: {\mathsf{Open}}{X} \to \mathsf{A}\) is a functor
specified by stalks, \(\lim\) sends \(F\) to the universal object
\(\lim F\) living over the essential image of \(F\) in \(\mathsf{A}\):

\begin{center}
\begin{tikzcd}[column sep=tiny]
  & X &&&& {\mathsf{I} = {\mathsf{Open}}(X)} &&& {} && {\mathsf{A}} \\
  & 3 &&&& {X = \left\{{1,2,3}\right\}} &&&&& {F(X)} &&& \textcolor{rgb,255:red,214;green,92;blue,92}{\lim F} \\
  &&&&& {} \\
  1 && 2 && {\left\{{1, 3}\right\}} && {\left\{{2, 3}\right\}} &&& {F_1} && {F_2} \\
  \\
  &&&&& {\left\{{3}\right\}} &&&&& {F_3} \\
  &&&&& \emptyset &&&&& 0
  \arrow[from=4-1, to=2-2]
  \arrow[from=4-3, to=2-2]
  \arrow[from=6-6, to=4-5]
  \arrow[from=6-6, to=4-7]
  \arrow[from=4-5, to=2-6]
  \arrow[from=4-7, to=2-6]
  \arrow[from=4-10, to=6-11]
  \arrow[from=4-12, to=6-11]
  \arrow[from=7-6, to=6-6]
  \arrow[from=2-11, to=4-10]
  \arrow[from=2-11, to=4-12]
  \arrow["{F\in \mathsf{A}^{\mathsf{I}}}"', Rightarrow, from=1-6, to=1-11]
  \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=2-14, to=7-11]
  \arrow[from=6-11, to=7-11]
  \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=2-14, to=4-12]
  \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=2-14, to=4-10]
  \arrow[color={rgb,255:red,214;green,92;blue,92}, dashed, from=2-14, to=2-11]
  \arrow[squiggly, from=1-2, to=1-6]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

The object corresponding to global sections \(F(X) \in \mathsf{A}\)
seems to also satisfies this universal property, so a conjecture would
be that this construction recovers
\(\lim F \cong F(X) \coloneqq{{\Gamma}\qty{X; F} }\).

\end{solution}

\hypertarget{problem-2-2}{%
\subsection{Problem 2}\label{problem-2-2}}

\begin{problem}[Problem 2]

In the category of abelian groups compute
\(\operatorname{Tor}_{i}^{\mathbb{Z}}\left(\mathbb{Z}_{n}, M\right)\),
the left derived functors of \(N \mapsto N \otimes_{\mathbb{Z}}M\).

\end{problem}

\begin{solution}

\envlist

\begin{claim}

\begin{align*}
\operatorname{Tor}^{{\mathbb{Z}}}_1({\mathbb{Z}}/n{\mathbb{Z}}, M) \cong \ker(M \xrightarrow{\times n} M)
\cong \left\{{m\in M {~\mathrel{\Big\vert}~}nm = 0_M}\right\}
,\end{align*}
which is the kernel of multiplication by \(n\), and
\(\operatorname{Tor}^{i>1}_{{\mathbb{Z}}}({\mathbb{Z}}/n{\mathbb{Z}}, M) = 0\).

\end{claim}

Why this is true: in \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), free
implies flat, and \(\operatorname{Tor}\) is balanced and can thus be
resolved in either variable, so this can be computed by tensoring a free
resolution of \({\mathbb{Z}}/n{\mathbb{Z}}\) and using the long exact
sequence in \(\operatorname{Tor}\):

\begin{center}
\begin{tikzcd}[column sep=small]
    &&& 0 & {\mathbb{Z}}&& {\mathbb{Z}}&& {{\mathbb{Z}}/n{\mathbb{Z}}} & 0 \\
    \\
    {} &&&& {{\mathbb{Z}}\otimes_{\mathbb{Z}}M \cong M} && {{\mathbb{Z}}\otimes_{\mathbb{Z}}M\cong M} && {{\mathbb{Z}}/n{\mathbb{Z}}\otimes_{\mathbb{Z}}M} & 0 \\
    \\
    &&&& \textcolor{rgb,255:red,214;green,92;blue,92}{\operatorname{Tor}_1^{{\mathbb{Z}}}({\mathbb{Z}}, {\mathbb{Z}}) = 0} && \textcolor{rgb,255:red,214;green,92;blue,92}{\operatorname{Tor}_1^{{\mathbb{Z}}}({\mathbb{Z}}, {\mathbb{Z}}) = 0} && {\operatorname{Tor}_1^{{\mathbb{Z}}/n{\mathbb{Z}}}({\mathbb{Z}}, M)} \\
    \\
    &&&& \textcolor{rgb,255:red,214;green,92;blue,92}{\operatorname{Tor}_2^{{\mathbb{Z}}}({\mathbb{Z}}, {\mathbb{Z}}) = 0} && \textcolor{rgb,255:red,214;green,92;blue,92}{\operatorname{Tor}_2^{{\mathbb{Z}}}({\mathbb{Z}}, {\mathbb{Z}}) = 0} && {\operatorname{Tor}_2^{{\mathbb{Z}}/n{\mathbb{Z}}}({\mathbb{Z}}, M)}
    \arrow[from=1-4, to=1-5]
    \arrow[""{name=0, anchor=center, inner sep=0}, "{\times n}", hook, from=1-5, to=1-7]
    \arrow[two heads, from=1-7, to=1-9]
    \arrow[from=1-9, to=1-10]
    \arrow[from=3-9, to=3-10]
    \arrow[two heads, from=3-7, to=3-9]
    \arrow[""{name=1, anchor=center, inner sep=0}, "{(\times n)\otimes\operatorname{id}_M}", from=3-5, to=3-7]
    \arrow[from=5-9, to=3-5]
    \arrow[from=5-5, to=5-7]
    \arrow[from=5-7, to=5-9]
    \arrow[from=7-9, to=5-5]
    \arrow[from=7-7, to=7-9]
    \arrow[from=7-5, to=7-7]
    \arrow["{({-})\otimes_{\mathbb{Z}}M}", shorten <=9pt, shorten >=13pt, Rightarrow, from=0, to=1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

In the resulting long exact sequence, since \({\mathbb{Z}}\) is free,
thus flat, thus tor-acyclic, the first two columns vanish in degrees
\(d\geq 1\). As a result, in degrees \(d\geq 2\), the terms
\(\operatorname{Tor}_d^{\mathbb{Z}}({\mathbb{Z}}/n{\mathbb{Z}}, M)\) are
surrounded by zeros and thus zero, meaning that only
\(\operatorname{Tor}_1\) survives. By exactness,
\(\operatorname{Tor}_1({\mathbb{Z}}/n{\mathbb{Z}}, M)\) is isomorphic to
the kernel of the next map in the sequence, which is precisely
\(\ker(M \xrightarrow{\times n} M)\) after applying the canonical
isomorphism
\begin{align*}
{\mathbb{Z}}\otimes_{\mathbb{Z}}M &\to M \\
n \otimes m &\mapsto nm
.\end{align*}

\end{solution}

\hypertarget{problem-3-2}{%
\subsection{Problem 3}\label{problem-3-2}}

\begin{problem}[Problem 3]

Let \(k\) be a field and \(R=k[x, y]\). In the category of \(R\)-modules
compute

\begin{itemize}
\tightlist
\item
  \(\operatorname{Ext}_R^{n}(R, m)\)
\item
  \(\operatorname{Ext}_R^{n}(m, R)\), and
\item
  \(\operatorname{Tor}^R_{n}(m, m)\),
\end{itemize}

where \(m=(x, y)\) is the maximal ideal at the origin.

\end{problem}

\begin{solution}[Problem 3]

Note that \(R\) is a free \(R{\hbox{-}}\)module, and so
\(\operatorname{Ext} _R^n(R, M) = 0\) for any \(R{\hbox{-}}\)module
\(M\). This is because \(\operatorname{Ext}\) can be computed using a
free resolution of either variable. For
\(\operatorname{Ext} _R^n(R, m)\), compute this as
\({\mathbb{R}}\mathop{\mathrm{Hom}}_R({-}, m)\) evaluated at \(R\). Take
the free resolution
\begin{align*}
\cdots \to 0 \to R \xrightarrow{\operatorname{id}_R} R \to 0
,\end{align*}
delete the augmentation and apply the contravariant
\(\mathop{\mathrm{Hom}}_R({-}, m)\) to obtain
\begin{align*}
0 \to \mathop{\mathrm{Hom}}_R(R, m) \cong m \to 0 \to \cdots
,\end{align*}
and take homology to obtain
\begin{align*}
\operatorname{Ext} _R^0(R, m) \cong m, \qquad \operatorname{Ext} _R^{>0}(R, m) = 0
.\end{align*}

Compute \(\operatorname{Ext} _R(m, R)\) as
\({\mathbb{R}}\mathop{\mathrm{Hom}}(m, {-})\) applied to \(R\) proceeds
similarly: using the same resolution, applying covariant
\(\mathop{\mathrm{Hom}}_R(m, {-})\) yields
\begin{align*}
0 \to \mathop{\mathrm{Hom}}_R(m, R) \to 0 \to \cdots
,\end{align*}
and taking homology yields
\begin{align*}
\operatorname{Ext} _R^0(m, R) \cong \mathop{\mathrm{Hom}}_R(m, R) \qquad \operatorname{Ext} _R^{>0}(m, R) = 0
.\end{align*}

For the \(\operatorname{Tor}\) calculation, we can use the Koszul
resolution of \(m\):
\begin{align*}
0 \to k[x, y] \xrightarrow{\cdot {\left[ {x, y} \right]} } 
k[x, y] \oplus k[x,y] \xrightarrow{ \cdot { \, {}^{t}{ \left(  {\left[ {-y, x} \right]} \right) } } } \left\langle{x, y}\right\rangle \to 0
,\end{align*}
so the differentials are \(t\mapsto {\left[ {tx, ty} \right]}\) and
\({\left[ {u, v} \right]} \mapsto -uy + vx\) respectively. More
succinctly, this resolution is
\begin{align*}
0 \to R \xrightarrow{d_1} R{ {}^{ \scriptscriptstyle\oplus^{2} }  } \xrightarrow{d_2}  m \to 0
,\end{align*}
so we can delete \(m\) and apply \(({-})\otimes_R m\) to obtain
\begin{align*}
0 \to R\otimes_R m \xrightarrow{d_1 \otimes\operatorname{id}_m}  R{ {}^{ \scriptscriptstyle\oplus^{2} }  }\otimes_R m \to 0\\
\end{align*}
which simplifies to
\begin{align*}
{ {C}_{\scriptscriptstyle \bullet}} \coloneqq 0 \to m \xrightarrow{\tilde d_1 \coloneqq{\left[ {x, y} \right]}}  m \oplus m \to 0 \\
\end{align*}
and thus we can compute \(\operatorname{Tor}\) as the homology of this
complex. We have
\begin{align*}
\operatorname{Tor}_0^R(m,m)&=
H^0({ {C}_{\scriptscriptstyle \bullet}} ) \\
&= \operatorname{coker}\tilde d_1 \\
&= {m \oplus m \over xm \oplus ym} \\
&\cong {m\over xm} \oplus {m\over ym} \\
&= {\left\langle{x, y}\right\rangle \over \left\langle{x^2, y}\right\rangle } \oplus {\left\langle{x, y}\right\rangle \over \left\langle{x, y^2}\right\rangle} \\
&= \left\{{f(x, y) \coloneqq c_1 x \in k[x,y] {~\mathrel{\Big\vert}~}c_1\in k }\right\} \oplus \left\{{g(x, y) \coloneqq c_1 y \in k[x,y] {~\mathrel{\Big\vert}~}c_1\in k }\right\} \\
&\cong k \oplus k
\\ \\
\operatorname{Tor}_1^R(m,m)&=
H^1({ {C}_{\scriptscriptstyle \bullet}} ) \\
&= \ker \tilde d_1 \\
&= \left\{{t\in \left\langle{x,y}\right\rangle {~\mathrel{\Big\vert}~}{\left[ {tx, ty} \right]} = {\left[ {0, 0} \right]} }\right\} \\
&= 0 \\ \\ \\
\operatorname{Tor}{\geq 2}^R(m,m)&=
H^{\geq 2} ({ {C}_{\scriptscriptstyle \bullet}} ) \\
&= 0
.\end{align*}

\end{solution}

\hypertarget{problem-4-1}{%
\subsection{Problem 4}\label{problem-4-1}}

\begin{problem}[Problem 4]

Let
\(0 \rightarrow F^{\prime} \rightarrow F \rightarrow F^{\prime \prime} \rightarrow 0\)
be a short exact triple of sheaves and assume that \(F^{\prime}\) is
flasque. Prove that the sequence
\begin{align*}
0 \rightarrow \Gamma\left(F^{\prime}\right) \rightarrow \Gamma(F) \rightarrow \Gamma\left(F^{\prime \prime}\right) \rightarrow 0
\end{align*}
of the spaces of global sections is exact.

\end{problem}

\begin{solution}[Using cohomology]

\envlist

\begin{claim}

Flasque sheaves are \(F{\hbox{-}}\)acyclic for the functor global
sections functor \(F({-}) \coloneqq{{\Gamma}\qty{X; {-}} }\).

\end{claim}

\begin{proof}[of claim]

Proved in class.

\end{proof}

Applying the functor \({{\Gamma}\qty{X; {-}} }\) to the given short
exact sequence of sheaves produces a long exact sequence of abelian
groups in its right-derived functors. Using the claim above, we have
\({\mathbb{R}}^i {{\Gamma}\qty{X; {\mathcal{F}}'} } = 0\) for
\(i\geq 1\), and thus we have the following:

\begin{center}
\begin{tikzcd}[column sep=small]
    0 && {{\mathcal{F}}'} && {\mathcal{F}}&& {{\mathcal{F}}''} && 0 \\
    \\
    0 && {{{\Gamma}\qty{X; {\mathcal{F}}'} }} && {{{\Gamma}\qty{X; {\mathcal{F}}} }} && {{{\Gamma}\qty{X; {\mathcal{F}}''} }} \\
    \\
    && {{\mathbb{R}}^1 {{\Gamma}\qty{X; {\mathcal{F}}'} } = 0} && {{\mathbb{R}}^1 {{\Gamma}\qty{X; {\mathcal{F}}} }} && {{\mathbb{R}}^1 {{\Gamma}\qty{X; {\mathcal{F}}''} }} \\
    \\
    && {{\mathbb{R}}^2 {{\Gamma}\qty{X; {\mathcal{F}}'} } = 0} && {{\mathbb{R}}^2 {{\Gamma}\qty{X; {\mathcal{F}}} }} && \cdots
    \arrow[from=1-1, to=1-3]
    \arrow[""{name=0, anchor=center, inner sep=0}, from=1-3, to=1-5]
    \arrow[from=1-5, to=1-7]
    \arrow[from=1-7, to=1-9]
    \arrow[from=3-1, to=3-3]
    \arrow[""{name=1, anchor=center, inner sep=0}, from=3-3, to=3-5]
    \arrow[from=3-5, to=3-7]
    \arrow[from=3-7, to=5-3]
    \arrow[from=5-3, to=5-5]
    \arrow[from=5-7, to=7-3]
    \arrow[from=7-3, to=7-5]
    \arrow["\sim", from=7-5, to=7-7]
    \arrow["\sim", from=5-5, to=5-7]
    \arrow["{{{\Gamma}\qty{X; {-}} }}", shorten <=9pt, shorten >=9pt, Rightarrow, from=0, to=1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

In particular, since
\({\mathbb{R}}^1 {{\Gamma}\qty{X; {\mathcal{F}}'} } = 0\), the first row
forms the desired short exact sequence. As a corollary, we also obtain
\({\mathbb{R}}^i {{\Gamma}\qty{X; {\mathcal{F}}} } \cong {\mathbb{R}}^i {{\Gamma}\qty{X; {\mathcal{F}}''} }\)
for all \(i\geq 1\).

\end{solution}

\begin{solution}[Direct]

First, we'll modify the notation slightly and give names to the maps
involved. We'll use the following convention for restrictions of sheaf
morphisms to opens and stalks:

\begin{center}
\begin{tikzcd}
    0 && A && B && C && 0 \\
    & \textcolor{rgb,255:red,92;green,92;blue,214}{0} && \textcolor{rgb,255:red,92;green,92;blue,214}{A(X)} && \textcolor{rgb,255:red,92;green,92;blue,214}{B(X)} && \textcolor{rgb,255:red,92;green,92;blue,214}{C(X)} \\
    {} & \textcolor{rgb,255:red,92;green,92;blue,214}{0} & {} & \textcolor{rgb,255:red,92;green,92;blue,214}{A(U)} & {} & \textcolor{rgb,255:red,92;green,92;blue,214}{B(U)} && \textcolor{rgb,255:red,92;green,92;blue,214}{C(U)} \\
    \\
    0 && {A_x} && {B_x} && {C_x} && 0
    \arrow[from=1-1, to=1-3]
    \arrow["f", from=1-3, to=1-5]
    \arrow["g", from=1-5, to=1-7]
    \arrow[from=1-7, to=1-9]
    \arrow[from=5-1, to=5-3]
    \arrow["{f_x}", from=5-3, to=5-5]
    \arrow["{g_x}", from=5-5, to=5-7]
    \arrow[from=5-7, to=5-9]
    \arrow[curve={height=-12pt}, from=1-3, to=5-3]
    \arrow[curve={height=-12pt}, from=1-5, to=5-5]
    \arrow[curve={height=-12pt}, from=1-7, to=5-7]
    \arrow[from=1-3, to=2-4]
    \arrow[from=1-5, to=2-6]
    \arrow[from=1-7, to=2-8]
    \arrow["F"{pos=0.3}, color={rgb,255:red,92;green,92;blue,214}, from=2-4, to=2-6]
    \arrow["G"{pos=0.3}, color={rgb,255:red,92;green,92;blue,214}, from=2-6, to=2-8]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, from=2-2, to=2-4]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, from=3-2, to=3-4]
    \arrow["{{ \left.{{F}} \right|_{{U}} }}"{pos=0.3}, color={rgb,255:red,92;green,92;blue,214}, from=3-4, to=3-6]
    \arrow["{{ \left.{{G}} \right|_{{U}} }}"{pos=0.3}, color={rgb,255:red,92;green,92;blue,214}, from=3-6, to=3-8]
    \arrow[from=2-4, to=3-4]
    \arrow[from=2-6, to=3-6]
    \arrow[from=2-8, to=3-8]
    \arrow[from=3-4, to=5-3]
    \arrow[from=3-6, to=5-5]
    \arrow[from=3-8, to=5-7]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

Given \(c\in C(X)\), our goal is to produce a \(b\in B(X)\) such that
\(g(b) = c\), and the strategy will be to use surjectivity at stalks to
produce a maximal section of \(B\) mapping to \(c\), and argue that it
must be a section over all of \(X\). This will proceed by showing that
if a lift is not maximal, sections over open sets that are missed can be
extended using that \(A\) is flasque, contradicting maximality.

Write \({ \left.{{c}} \right|_{{x}} }\) for the image of \(c\) in the
stalk \(C_x\); by surjectivity of \(g_x: B_x \twoheadrightarrow C_x\) we
can find a germ \(b_x\) with \(g_x(b_x) = c_x\). The germ lifts to some
set \(U\ni x\) and some \(b\in B(U)\) with
\(b\mapsto { \left.{{c}} \right|_{{U}} }\) under
\({ \left.{{F}} \right|_{{U}} }: B(U) \to C(U)\). So define a poset of
all such lifts:
\begin{align*}
P \coloneqq\left\{{ (U, b \in B(U)) {~\mathrel{\Big\vert}~}{ \left.{{F}} \right|_{{U}} }(b) = { \left.{{c}} \right|_{{U}} } }\right\} \\
\qquad \text{ where } (U_1, b_1) \leq (U_2, b_2) \iff U_1 \subseteq U_2 \text{ and} { \left.{{b_2}} \right|_{{U_1}} } = b_1
.\end{align*}
As noted above, \(P\) is nonempty, and every chain
\(\left\{{(U_i, b_i)}\right\}_{i\in I}\) has an upper bound given by
\((\tilde U, \tilde b)\) where \(\tilde U \coloneqq\cup_{i\in I} U_i\)
and \(\tilde b\) is the unique glued section of \(B\) restricting to all
of the \(b_i\), which exists by the sheaf property for \(B\). Thus
Zorn's lemma applies, and (reusing notation) we can assume \((U, b)\) is
maximal with respect to this property.

The claim is that \(U\) must be all of \(X\). Toward a contradiction,
suppose not -- then pick any \(x\in X\setminus U\), and again using
surjectivity on stalks at \(x\), produce an open set \(V\ni x\) and a
section \(b'\in B(V)\) with
\({ \left.{{G}} \right|_{{V}} }(b') = { \left.{{c}} \right|_{{V}} }\).
Now on the overlap \(W\coloneqq U \cap V\), both \(b\) and \(b'\) map to
\({ \left.{{c}} \right|_{{W}} }\), and so
\begin{align*}
{ \left.{{G}} \right|_{{W}} }({ \left.{{b}} \right|_{{W}} } - { \left.{{b'}} \right|_{{W}} }) = { \left.{{c}} \right|_{{W}} } { \left.{{c}} \right|_{{W}} } = 0 \implies b-b'\in \ker { \left.{{G}} \right|_{{W}} } = \operatorname{im}{ \left.{{F}} \right|_{{W}} }
,\end{align*}
where we've used exactness in the middle spot in the exact sequence
\(A(W) \to B(W)\to C(W)\). So there is some \(\alpha \in A(W)\) with
\({ \left.{{F}} \right|_{{W}} }(\alpha) = { \left.{{b}} \right|_{{W}} } - { \left.{{b'}} \right|_{{W}} }\),
and since \(A\) is flasque this can be extended to a global section
\(\tilde\alpha\in A(X)\). Write
\(\tilde \beta \coloneqq F(\tilde \alpha) \in B(X)\) with
\({ \left.{{\tilde \beta}} \right|_{{W}} } = { \left.{{b}} \right|_{{W}} }- { \left.{{b'}} \right|_{{W}} }\)
in \(B(W)\). We can now glue \(\tilde \beta\) to a section over
\(U \cup V\) which extends the original section \(b\): setting
\(\widehat{b} \coloneqq\tilde \beta + b'\) yields
\begin{align*}
{ \left.{{\widehat{b}}} \right|_{{W}} } = \qty{{ \left.{{b}} \right|_{{W}} } - { \left.{{b'}} \right|_{{W}} }} + b' = { \left.{{b}} \right|_{{W}} }
,\end{align*}
so this section over \(U \cup V\) agrees with \(b\) on the overlap
\(W = U \cap V\), and thus by existence and uniqueness of gluing (using
the sheaf property of \(B\)) \(\widehat{b} \in B(U \cup V)\) is a
section extending \(b\) over a set that strictly contains \(U\). This
contradicts the maximality of the pair \((U, b)\).

\end{solution}

\hypertarget{problem-5}{%
\subsection{Problem 5}\label{problem-5}}

\begin{problem}[Problem 5]

For a sheaf \(F\) on \(X\), let
\begin{align*}
S(F)=\prod_{x \in X}\left(i_{x}\right)_{*} F_{x}, \quad i_{x}: x \rightarrow X
\end{align*}
be the sheaf of all, possibly discontinuous section of the étale space
of \(F\). The canonical flasque resolution of \(F\) is
\begin{align*}
\underline{S}(F) \coloneqq 0 \rightarrow F \to S\left(F_{0}\right) \rightarrow S\left(F_{1}\right) \rightarrow S\left(F_{2}\right) \rightarrow \ldots
\end{align*}
where \(F_{0}=F\) and \(F_{i}\) are defined inductively as
\(F_{i+1}=S\left(F_{i}\right) / F_{i}\). Some books define cohomology
groups \(\mathbf{H}^{n}(X, F)\) as the cohomology groups of the complex
\begin{align*}
0 \rightarrow \Gamma\left(S\left(F_{0}\right)\right) \rightarrow \Gamma\left(S\left(F_{1}\right)\right) \rightarrow \Gamma\left(S\left(F_{2}\right)\right) \rightarrow \ldots
\end{align*}
Prove that they coincide with the cohomology defined by other means by
showing that this gives an exact \(\delta\)-functor and that
\(\mathbf{H}^{n}\) are effaceable for \(n>0\) through the following
steps:

\begin{enumerate}
\def\labelenumi{(\arabic{enumi})}
\item
  A homomorphism \(F \rightarrow G\) induces a canonical homomorphism of
  resolutions \(\underline{S}(F) \rightarrow \underline{S}(G) .\)
\item
  A short exact triple
  \(0 \rightarrow F^{\prime} \rightarrow F \rightarrow F^{\prime \prime} \rightarrow 0\)
  induces a short exact triple of complexes
  \(0 \rightarrow \underline{S}\left(F^{\prime}\right) \rightarrow \underline{S}(F) \rightarrow \underline{S}\left(F^{\prime \prime}\right) \rightarrow 0\).
\item
  Applying \(\Gamma\) to it gives a short exact triple of complexes,
  i.e.~\(0 \rightarrow S\left(F_{n}^{\prime}\right) \rightarrow\)
  \(S\left(F_{n}\right) \rightarrow S\left(F_{n}^{\prime \prime}\right) \rightarrow 0\)
  is exact. (You can assume the previous problem.)
\item
  \(\left(\mathbf{H}^{n}\right)\) is an exact \(\delta\)-functor.
\item
  For \(n>0, \mathbf{H}^{n}(F) \rightarrow \mathbf{H}^{n}(S(F))\) is the
  zero map.
\end{enumerate}

Conclude by Grothendieck's universality theorem.

\end{problem}

\begin{solution}[Part 1]

This follows readily from the fact that a morphism \(f: F\to G\) of
sheaves on \(X\) induces group morphisms \(f_x: F_x \to G_x\) on stalks
for every \(x\in X\). Letting \(y\in X\) be arbitrary, there is a
morphism
\begin{align*}
\phi_{y}: \prod_{x\in X} F_x \xrightarrow{\pi_y} F_y \xrightarrow{f_y} G_y
\end{align*}
where \(\pi_y\) is the canonical projection out of the product. By the
universal property of the product, the \(\phi_y\) assemble to a morphism
\begin{align*}
S(f): \prod_{x\in X} F_x\to \prod_{y\in X} G_y
.\end{align*}

So there is a morphism \(S(F_0) \to S(G_0)\) at the first stage of the
complex. This induces a morphism on the quotient sheaves
\(S(F_0)/F_0 \to S(G_0)/G_0\), and thus by the same argument as above, a
morphism on the second stage \(S(S(F_0)/F_0) \to S(S(G_0)/G_0)\), i.e.~a
morphism \(S(F_1) \to S(G_1)\). Continuing inductively yields levelwise
morphisms \(S(F_i) \to S(G_i)\). The claim is that these assemble to a
chain map

\begin{center}
\begin{tikzcd}
    &&&& \textcolor{rgb,255:red,214;green,92;blue,92}{S(F)/F} \\
    0 & F && {S(F_0) = S(F)} && {S(F_1) = S(S(F)/F)} & \cdots \\
    \\
    0 & G && {S(G_0) = S(G)} && {S(G_1) = S(S(G)/G)} & \cdots \\
    &&&& \textcolor{rgb,255:red,214;green,92;blue,92}{S(G)/G}
    \arrow[from=2-2, to=2-4]
    \arrow[from=2-4, to=2-6]
    \arrow[from=4-2, to=4-4]
    \arrow[from=4-4, to=4-6]
    \arrow["{S(f)}", color={rgb,255:red,214;green,92;blue,92}, from=2-4, to=4-4]
    \arrow["{S_1(F)}", from=2-6, to=4-6]
    \arrow[from=2-6, to=2-7]
    \arrow[from=4-6, to=4-7]
    \arrow[draw={rgb,255:red,214;green,92;blue,92}, dashed, two heads, from=4-4, to=5-5]
    \arrow["{S({-})}"', dashed, hook, from=5-5, to=4-6]
    \arrow["{S({-})}", dashed, hook, from=1-5, to=2-6]
    \arrow[draw={rgb,255:red,214;green,92;blue,92}, dashed, two heads, from=2-4, to=1-5]
    \arrow[from=2-1, to=2-2]
    \arrow[from=4-1, to=4-2]
    \arrow["f", from=2-2, to=4-2]
    \arrow[draw={rgb,255:red,214;green,92;blue,92}, curve={height=-18pt}, dashed, from=1-5, to=5-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

To see this is true, it is enough to show that the first square
commutes, i.e.~that applying \(S({-})\) to a morphism of sheaves
produces a commuting square. This is because every other square has a
factorization as indicated, where the square in red naturally commutes
since it involves canonically induced maps on quotients/cokernels, and
the other half of the square arises by applying the \(S\) construction
to some morphism of sheaves.

However, this square can be readily seen to commute using the following:
first regard the sections of \({\mathcal{F}}\) as continuous sections of
its espace étale \(\text{Ét}_F \xrightarrow{\pi} X\) and regarding
sections of \(S(F)\) as arbitrary (potentially discontinuous) sections
of \(\pi\). Then \({\mathcal{F}}\leq S(F)\) is clearly a subsheaf and
\(F\to S(F)\) is an inclusion of spaces of sections.

\end{solution}

\begin{solution}[Part 2]

By part 1, it is clear there are morphisms
\(\underline{S}(F') \to \underline{S}(F) \to \underline{S}(F'')\) of
complexes of sheaves, yielding a double complex:

\begin{center}
\begin{tikzcd}
    && \vdots && \vdots && \vdots \\
    \\
    && {S(F'_1)} && {S(F_1)} && {S(F''_1)} \\
    \\
    && {S(F'_0)} && {S(F_0)} && {S(F''_0)} \\
    \\
    0 && {F'} && F && {F''} && 0 \\
    \\
    && 0 && 0 && 0
    \arrow[from=9-3, to=7-3]
    \arrow[from=9-5, to=7-5]
    \arrow[from=9-7, to=7-7]
    \arrow[from=7-3, to=5-3]
    \arrow[from=7-5, to=5-5]
    \arrow[from=7-7, to=5-7]
    \arrow[from=5-3, to=3-3]
    \arrow[from=5-5, to=3-5]
    \arrow[from=5-7, to=3-7]
    \arrow[from=3-3, to=1-3]
    \arrow[from=3-5, to=1-5]
    \arrow[from=3-7, to=1-7]
    \arrow[from=7-1, to=7-3]
    \arrow[from=7-3, to=7-5]
    \arrow[from=7-5, to=7-7]
    \arrow[from=5-3, to=5-5]
    \arrow[from=5-5, to=5-7]
    \arrow[from=7-7, to=7-9]
    \arrow[from=3-3, to=3-5]
    \arrow[from=3-5, to=3-7]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMTcsWzIsNiwiRiciXSxbNCw2LCJGIl0sWzYsNiwiRicnIl0sWzIsOCwiMCJdLFs0LDgsIjAiXSxbNiw4LCIwIl0sWzIsNCwiUyhGJ18wKSJdLFs0LDQsIlMoRl8wKSJdLFs2LDQsIlMoRicnXzApIl0sWzIsMiwiUyhGJ18xKSJdLFs0LDIsIlMoRl8xKSJdLFs2LDIsIlMoRicnXzEpIl0sWzIsMCwiXFx2ZG90cyJdLFs0LDAsIlxcdmRvdHMiXSxbNiwwLCJcXHZkb3RzIl0sWzAsNiwiMCJdLFs4LDYsIjAiXSxbMywwXSxbNCwxXSxbNSwyXSxbMCw2XSxbMSw3XSxbMiw4XSxbNiw5XSxbNywxMF0sWzgsMTFdLFs5LDEyXSxbMTAsMTNdLFsxMSwxNF0sWzE1LDBdLFswLDFdLFsxLDJdLFs2LDddLFs3LDhdLFsyLDE2XSxbOSwxMF0sWzEwLDExXV0=}{Link
to Diagram}
\end{quote}

It suffices to show injectivity, exactness, and surjectivity
respectively along each horizontal row. Exactness is a local condition,
so it suffices to show exactness on stalks.

\begin{claim}

For any open \(U\), the following sequence at the first stage of the
complex is exact:
\begin{align*}
0\to S(F')(U) \to S(F)(U) \to S(F'')(U)\to 0
.\end{align*}

\end{claim}

\begin{proof}[of claim]

This follows because \(S(F')(U) = \prod_{x\in U} F'_x\) and similarly
for \(F, F''\), and so if \(f: F' \to F\) is injective on sheaves, then
\(f_x: F_x' \to F_x\) is injective on stalks.

\end{proof}

Now apply the functor \(\colim_{U\ni p}({-})\) to this exact sequence
and use that taking stalks is exact (despite not generally being a
\emph{filtered} colimit) to conclude
\begin{align*}
0\to S(F')_x \to S(F)_x \to S(F'')_x \to 0
.\end{align*}
is exact for all \(x\in X\), thus making the following sequence exact:
\begin{align*}
0\to S(F_0') \to S(F_0) \to S(F_0'')\to 0
\end{align*}

Our double complex is now the following:

\begin{center}
\begin{tikzcd}
    && \vdots && \vdots && \vdots \\
    \\
    \textcolor{rgb,255:red,214;green,92;blue,92}{?} && {S(F'_1)} && {S(F_1)} && {S(F''_1)} && \textcolor{rgb,255:red,214;green,92;blue,92}{?} \\
    \\
    0 && {S(F'_0)} && {S(F_0)} && {S(F''_0)} && 0 \\
    \\
    0 && {F'} && F && {F''} && 0 \\
    \\
    && 0 && 0 && 0
    \arrow[from=9-3, to=7-3]
    \arrow[from=9-5, to=7-5]
    \arrow[from=9-7, to=7-7]
    \arrow[from=7-3, to=5-3]
    \arrow[from=7-5, to=5-5]
    \arrow[from=7-7, to=5-7]
    \arrow[from=5-3, to=3-3]
    \arrow[from=5-5, to=3-5]
    \arrow[from=5-7, to=3-7]
    \arrow[from=3-3, to=1-3]
    \arrow[from=3-5, to=1-5]
    \arrow[from=3-7, to=1-7]
    \arrow[from=7-1, to=7-3]
    \arrow[from=7-3, to=7-5]
    \arrow[from=7-5, to=7-7]
    \arrow[from=5-3, to=5-5]
    \arrow[from=5-5, to=5-7]
    \arrow[from=7-7, to=7-9]
    \arrow[from=3-3, to=3-5]
    \arrow[from=3-5, to=3-7]
    \arrow[from=5-1, to=5-3]
    \arrow[from=5-7, to=5-9]
    \arrow[draw={rgb,255:red,214;green,92;blue,92}, from=3-1, to=3-3]
    \arrow[draw={rgb,255:red,214;green,92;blue,92}, from=3-7, to=3-9]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

To see that
\begin{align*}
0\to S(F_k') \to S(F_k) \to S(F_k'')\to 0
\end{align*}
is exact for all \(k\), we can truncate this complex:

\begin{center}
\begin{tikzcd}
    && 0 && 0 && 0 \\
    \\
    \textcolor{rgb,255:red,214;green,92;blue,92}{?} && {S(F'_0)/F'_0} && {S(F_0)/F_0} && {S(F''_0)/F''_0} && \textcolor{rgb,255:red,214;green,92;blue,92}{?} \\
    \\
    0 && {S(F'_0)} && {S(F_0)} && {S(F''_0)} && 0 \\
    \\
    0 && {F'} && F && {F''} && 0 \\
    \\
    && 0 && 0 && 0
    \arrow[from=9-3, to=7-3]
    \arrow[from=9-5, to=7-5]
    \arrow[from=9-7, to=7-7]
    \arrow[from=7-3, to=5-3]
    \arrow[from=7-5, to=5-5]
    \arrow[from=7-7, to=5-7]
    \arrow[from=5-3, to=3-3]
    \arrow[from=5-5, to=3-5]
    \arrow[from=5-7, to=3-7]
    \arrow[from=3-3, to=1-3]
    \arrow[from=3-5, to=1-5]
    \arrow[from=3-7, to=1-7]
    \arrow[from=7-1, to=7-3]
    \arrow[from=7-3, to=7-5]
    \arrow[from=7-5, to=7-7]
    \arrow[from=5-3, to=5-5]
    \arrow[from=5-5, to=5-7]
    \arrow[from=7-7, to=7-9]
    \arrow[from=3-3, to=3-5]
    \arrow[from=3-5, to=3-7]
    \arrow[from=5-1, to=5-3]
    \arrow[from=5-7, to=5-9]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, from=3-1, to=3-3]
    \arrow[color={rgb,255:red,214;green,92;blue,92}, from=3-7, to=3-9]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

The row highlighted in red is exact by the Nine Lemma, regarding each
row as a chain complex, and since applying \(S({-})\) is exact, by
applying this to the top row we obtain

\begin{center}
\begin{tikzcd}
    && \vdots && \vdots && \vdots \\
    \\
    0 && {S(F_1')} && {S(F_1)} && {S(F''_1)} && 0 \\
    \\
    0 && {S(F'_0)} && {S(F_0)} && {S(F''_0)} && 0 \\
    \\
    0 && {F'} && F && {F''} && 0 \\
    \\
    && 0 && 0 && 0
    \arrow[from=9-3, to=7-3]
    \arrow[from=9-5, to=7-5]
    \arrow[from=9-7, to=7-7]
    \arrow[from=7-3, to=5-3]
    \arrow[from=7-5, to=5-5]
    \arrow[from=7-7, to=5-7]
    \arrow[from=5-3, to=3-3]
    \arrow[from=5-5, to=3-5]
    \arrow[from=5-7, to=3-7]
    \arrow[from=3-3, to=1-3]
    \arrow[from=3-5, to=1-5]
    \arrow[from=3-7, to=1-7]
    \arrow[from=7-1, to=7-3]
    \arrow[from=7-3, to=7-5]
    \arrow[from=7-5, to=7-7]
    \arrow[from=5-3, to=5-5]
    \arrow[from=5-5, to=5-7]
    \arrow[from=7-7, to=7-9]
    \arrow[from=3-3, to=3-5]
    \arrow[from=3-5, to=3-7]
    \arrow[from=5-1, to=5-3]
    \arrow[from=5-7, to=5-9]
    \arrow[from=3-1, to=3-3]
    \arrow[from=3-7, to=3-9]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

The remaining rows are exact by repeating this argument inductively, and
regarding the columns as complexes, we obtain the desired exact
sequences of complexes by deleting the first row.

\end{solution}

\begin{solution}[Part 3]

Note: there may be a typo in the statement of this problem, so what I
will show is that the following sequence of complexes is exact:
\begin{align*}
0 
\to {{\Gamma}\qty{X; \underline{S}(F')} }
\to {{\Gamma}\qty{X; \underline{S}(F)} }
\to {{\Gamma}\qty{X; \underline{S}(F'')} }
\to 0
.\end{align*}

Take the double complex from part (2) and apply the functor
\({{\Gamma}\qty{X; {-}} }\) to obtain the following double complex:

\begin{center}
\begin{tikzcd}[column sep=small]
    && \vdots && \vdots && \vdots \\
    \\
    0 && {{{\Gamma}\qty{X; S(F_1')} }} && {{{\Gamma}\qty{X; S(F_0)} }} && {{{\Gamma}\qty{X; S(F''_1)} }} && \textcolor{rgb,255:red,214;green,92;blue,92}{0} \\
    \\
    0 && {{{\Gamma}\qty{X; S(F'_0)} }} && {{{\Gamma}\qty{X; S(F_0)} }} && {{{\Gamma}\qty{X; S(F''_0) } }} && \textcolor{rgb,255:red,214;green,92;blue,92}{0} \\
    \\
    0 && {{{\Gamma}\qty{X; F'} }} && {{{\Gamma}\qty{X; F} }} && {{{\Gamma}\qty{X; F''} }} && {{\mathbb{R}}^1{{\Gamma}\qty{X; F'} }} \\
    \\
    && 0 && 0 && 0
    \arrow[from=9-3, to=7-3]
    \arrow[from=9-5, to=7-5]
    \arrow[from=9-7, to=7-7]
    \arrow[from=7-3, to=5-3]
    \arrow[from=7-5, to=5-5]
    \arrow[from=7-7, to=5-7]
    \arrow[from=5-3, to=3-3]
    \arrow[from=5-5, to=3-5]
    \arrow[from=5-7, to=3-7]
    \arrow[from=3-3, to=1-3]
    \arrow[from=3-5, to=1-5]
    \arrow[from=3-7, to=1-7]
    \arrow[from=7-1, to=7-3]
    \arrow[from=7-3, to=7-5]
    \arrow[from=7-5, to=7-7]
    \arrow[from=5-3, to=5-5]
    \arrow[from=5-5, to=5-7]
    \arrow[from=7-7, to=7-9]
    \arrow[from=3-3, to=3-5]
    \arrow[from=3-5, to=3-7]
    \arrow[from=5-1, to=5-3]
    \arrow[draw={rgb,255:red,214;green,92;blue,92}, from=5-7, to=5-9]
    \arrow[from=3-1, to=3-3]
    \arrow[draw={rgb,255:red,214;green,92;blue,92}, from=3-7, to=3-9]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

Here the bottom row continues in the long exact sequence for the
right-derived functors of \({{\Gamma}\qty{X; {-}} }\), i.e.~sheaf
cohomology. Since the desired sequence of complexes involved truncating
this double complex by deleting the first row, consider everything from
row two upward. That these levelwise maps assemble to a map of complexes
is just a consequence of functoriality of \({{\Gamma}\qty{X;{-}} }\),
and left exactness preserves the zeros in the left-most column, so it
suffices to show that the right-most column (highlighted in red) is zero
as claimed.

However, this follows from the previous problem if the sheaves
\(S(F_n')\) are all flasque. This is immediate since they are sheaves of
discontinuous sections, and such a section on \(U\) can always be
extended to a global section by simply assigning any other values on
\(X\setminus U\) -- any choice works, since no compatibility
(e.g.~continuity) is required.

\end{solution}

\begin{solution}[Part 4]

It is a general theorem in homological algebra that a short exact
sequence of chain complexes induces a long exact sequence in cohomology.
In this case, if we take the vertical homology of the above double
complex, by the snake lemma there are connecting morphisms:

\begin{center}
\begin{tikzcd}[column sep=small]
    && \vdots && \vdots && \vdots \\
    \\
    0 && {H_2({{\Gamma}\qty{X; \underline{S}(F')} })} && {H_2({{\Gamma}\qty{X; \underline{S}(F)} })} && {H_2({{\Gamma}\qty{X; \underline{S}(F'')} })} && 0\\
    \\
    0 && {H_1({{\Gamma}\qty{X; \underline{S}(F')} })} && {H_1({{\Gamma}\qty{X; \underline{S}(F)} })} && {H_1({{\Gamma}\qty{X; \underline{S}(F'')} })} && 0 \\
    \\
    0 && {H_0({{\Gamma}\qty{X; \underline{S}(F')} })} && {H_0({{\Gamma}\qty{X; \underline{S}(F)} })} && {H_0({{\Gamma}\qty{X; \underline{S}(F'')} })} && 0 \\
    \\
    && 0 && 0 && 0
    \arrow[from=9-3, to=7-3]
    \arrow[from=9-5, to=7-5]
    \arrow[from=9-7, to=7-7]
    \arrow[from=7-3, to=5-3]
    \arrow[from=7-5, to=5-5]
    \arrow[from=7-7, to=5-7]
    \arrow[from=5-3, to=3-3]
    \arrow[from=5-5, to=3-5]
    \arrow[from=5-7, to=3-7]
    \arrow[from=3-3, to=1-3]
    \arrow[from=3-5, to=1-5]
    \arrow[from=3-7, to=1-7]
    \arrow[from=7-1, to=7-3]
    \arrow[from=7-3, to=7-5]
    \arrow[from=7-5, to=7-7]
    \arrow[from=5-3, to=5-5]
    \arrow[from=5-5, to=5-7]
    \arrow[from=7-7, to=7-9]
    \arrow[from=3-3, to=3-5]
    \arrow[from=3-5, to=3-7]
    \arrow[from=5-1, to=5-3]
    \arrow[from=5-7, to=5-9]
    \arrow[from=3-1, to=3-3]
    \arrow[from=3-7, to=3-9]
    \arrow["{\exists \delta_0}"'{pos=0.3}, color={rgb,255:red,92;green,92;blue,214}, dashed, from=7-7, to=5-3]
    \arrow["{\exists \delta_1}"'{pos=0.3}, color={rgb,255:red,92;green,92;blue,214}, dashed, from=5-7, to=3-3]
    \arrow["{\exists \delta_2}"'{pos=0.3}, color={rgb,255:red,92;green,92;blue,214}, dashed, from=3-7, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

\end{solution}

\begin{solution}[Part 5]

This holds because flasque sheaves are \(F{\hbox{-}}\)acyclic for
\(F({-}) = {{\Gamma}\qty{X; {-}} }\), so we can conclude that
\(\mathbf{H}^n(S(F)) = 0\) for \(n > 0\) since the sheaves \(S(F)\) are
always flasque for any sheaf \(F\).

\begin{quote}
Note: I realized at the last minute that this argument may not actually
work, since this \(\mathbf{H}^n\) a priori has nothing to do with
\({\mathbb{R}}\Gamma(X;{-})\) computed via injective resolutions.
\end{quote}

\end{solution}

\addsec{ToDos}
\listoftodos[List of Todos]
\cleardoublepage

% Hook into amsthm environments to list them.
\addsec{Definitions}
\renewcommand{\listtheoremname}{}
\listoftheorems[ignoreall,show={definition}, numwidth=3.5em]
\cleardoublepage

\addsec{Theorems}
\renewcommand{\listtheoremname}{}
\listoftheorems[ignoreall,show={theorem,proposition}, numwidth=3.5em]
\cleardoublepage

\addsec{Exercises}
\renewcommand{\listtheoremname}{}
\listoftheorems[ignoreall,show={exercise}, numwidth=3.5em]
\cleardoublepage

\addsec{Figures}
\listoffigures
\cleardoublepage



\newpage
\printbibliography[title=Bibliography]


\end{document}
