\input{"preamble.tex"} \addbibresource{SheafCohomology.bib} \let\Begin\begin \let\End\end \newcommand\wrapenv[1]{#1} \makeatletter \def\ScaleWidthIfNeeded{% \ifdim\Gin@nat@width>\linewidth \linewidth \else \Gin@nat@width \fi } \def\ScaleHeightIfNeeded{% \ifdim\Gin@nat@height>0.9\textheight 0.9\textheight \else \Gin@nat@width \fi } \makeatother \setkeys{Gin}{width=\ScaleWidthIfNeeded,height=\ScaleHeightIfNeeded,keepaspectratio}% \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{\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=WzAsMTAsWzAsMCwiMCJdLFsyLDAsIlxca2VyIGYiXSxbNCwwLCJBIl0sWzYsMCwiQiJdLFs4LDAsIlxcY29rZXIgZiJdLFsxMCwwLCIwIl0sWzQsMiwiSSJdLFs0LDQsIjAiXSxbNiwyLCJJJyJdLFs2LDQsIjAiXSxbMCwxLCIiLDAseyJjb2xvdXIiOlsyNDAsNjAsNjBdfV0sWzEsMiwiIiwwLHsiY29sb3VyIjpbMjQwLDYwLDYwXX1dLFsyLDMsImYiLDAseyJjb2xvdXIiOlswLDYwLDYwXX0sWzAsNjAsNjAsMV1dLFszLDQsIiIsMCx7ImNvbG91ciI6WzMwLDYwLDYwXX1dLFs0LDUsIiIsMCx7ImNvbG91ciI6WzMwLDYwLDYwXX1dLFsyLDYsIiIsMCx7ImNvbG91ciI6WzI0MCw2MCw2MF19XSxbNiw3LCIiLDAseyJjb2xvdXIiOlsyNDAsNjAsNjBdfV0sWzksOCwiIiwwLHsiY29sb3VyIjpbMzAsNjAsNjBdfV0sWzgsMywiIiwwLHsiY29sb3VyIjpbMzAsNjAsNjBdfV0sWzYsOCwiXFxleGlzdHMgISIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==}{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=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}{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=WzAsOCxbNCwwLCJBX2kiXSxbNCw0LCJBX2oiXSxbMiwyLCJcXHByb2QgQV9pIl0sWzAsMiwiXFxjb2NvbGltIEFfaSJdLFs2LDIsIlxcY29wcm9kIEFfaSA9IFxcYmlnb3BsdXMgQV9pIl0sWzgsMiwiXFxjb2xpbSBBX2kiXSxbMCwzLCJcXHRzeyhhX2kpIFxcc3QgZl97aWp9KGFfaSkgPSBhX2ogfSJdLFs4LDMsIlxcYmlnb3BsdXMgQV9pLyBcXGdlbnN7YV9pIC0gZl97aWp9KGFfaSl9Il0sWzAsMSwiZl97aWp9Il0sWzAsNCwiIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoic3F1aWdnbHkifX19XSxbMSw0LCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJzcXVpZ2dseSJ9fX1dLFsyLDAsIiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6InNxdWlnZ2x5In19fV0sWzIsMSwiIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoic3F1aWdnbHkifX19XSxbMywwXSxbMywxXSxbMywyLCIiLDEseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFswLDVdLFsxLDVdLFs0LDUsIiIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dXQ==}{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=WzAsMTIsWzAsMCwiMCJdLFsyLDAsIkdCJyJdLFs0LDAsIkdCIl0sWzYsMCwiR0InJyJdLFs4LDAsIjAiXSxbNCwyLCJBIl0sWzAsNCwiMCJdLFsyLDQsIkInIl0sWzQsNCwiQiJdLFs2LDQsIkInJyJdLFs4LDQsIjAiXSxbNCw2LCJGQSJdLFs1LDJdLFs1LDMsIjAiLDJdLFswLDFdLFsxLDJdLFsyLDNdLFszLDRdLFs1LDEsIlxcdGhlcmVmb3JlIFxcZXhpc3RzIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzYsN10sWzcsOF0sWzgsOV0sWzksMTBdLFsxMSw4XSxbMTEsOSwiMCIsMl0sWzExLDcsIlxcZXhpc3RzIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzIxLDEzLCJGKFxcd2FpdCkiLDIseyJzaG9ydGVuIjp7InNvdXJjZSI6MjAsInRhcmdldCI6MjB9fV1d}{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=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}{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=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=WzAsMjEsWzIsNiwiSF8wKFxcZ2xvYnNlY3tYOyBcXHVse1N9KEYnKX0pIl0sWzQsNiwiSF8wKFxcZ2xvYnNlY3tYOyBcXHVse1N9KEYpfSkiXSxbNiw2LCJIXzAoXFxnbG9ic2Vje1g7IFxcdWx7U30oRicnKX0pIl0sWzIsOCwiMCJdLFs0LDgsIjAiXSxbNiw4LCIwIl0sWzIsNCwiSF8xKFxcZ2xvYnNlY3tYOyBcXHVse1N9KEYnKX0pIl0sWzQsNCwiSF8xKFxcZ2xvYnNlY3tYOyBcXHVse1N9KEYpfSkiXSxbNiw0LCJIXzEoXFxnbG9ic2Vje1g7IFxcdWx7U30oRicnKX0pIl0sWzIsMiwiSF8yKFxcZ2xvYnNlY3tYOyBcXHVse1N9KEYnKX0pIl0sWzQsMiwiSF8yKFxcZ2xvYnNlY3tYOyBcXHVse1N9KEYpfSkiXSxbNiwyLCJIXzIoXFxnbG9ic2Vje1g7IFxcdWx7U30oRicnKX0pIl0sWzIsMCwiXFx2ZG90cyJdLFs0LDAsIlxcdmRvdHMiXSxbNiwwLCJcXHZkb3RzIl0sWzAsNiwiMCJdLFs4LDYsIjAiXSxbMCw0LCIwIl0sWzgsNCwiMCIsWzAsNjAsNjAsMV1dLFswLDIsIjAiXSxbOCwyLCIwIixbMCw2MCw2MCwxXV0sWzMsMF0sWzQsMV0sWzUsMl0sWzAsNl0sWzEsN10sWzIsOF0sWzYsOV0sWzcsMTBdLFs4LDExXSxbOSwxMl0sWzEwLDEzXSxbMTEsMTRdLFsxNSwwXSxbMCwxXSxbMSwyXSxbNiw3XSxbNyw4XSxbMiwxNl0sWzksMTBdLFsxMCwxMV0sWzE3LDZdLFs4LDE4XSxbMTksOV0sWzExLDIwXSxbMiw2LCJcXGV4aXN0cyBcXGRlbHRhXzAiLDIseyJsYWJlbF9wb3NpdGlvbiI6MzAsImNvbG91ciI6WzI0MCw2MCw2MF0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX0sWzI0MCw2MCw2MCwxXV0sWzgsOSwiXFxleGlzdHMgXFxkZWx0YV8xIiwyLHsibGFiZWxfcG9zaXRpb24iOjMwLCJjb2xvdXIiOlsyNDAsNjAsNjBdLCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19LFsyNDAsNjAsNjAsMV1dLFsxMSwxMiwiXFxleGlzdHMgXFxkZWx0YV8yIiwyLHsibGFiZWxfcG9zaXRpb24iOjMwLCJjb2xvdXIiOlsyNDAsNjAsNjBdLCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19LFsyNDAsNjAsNjAsMV1dXQ==}{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}