\input{"preamble.tex"} \addbibresource{Schemes.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{ Schemes } \\ {\normalsize Lectures by Phil Engel. University of Georgia, Fall 2021} \\ \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 Schemes taught by Phil Engel at the University of Georgia in Fall 2021, with material based on Hartshorne. 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:} 2021-09-15 } \end{flushleft} \newpage % Note: addsec only in KomaScript \addsec{Table of Contents} \tableofcontents \newpage \hypertarget{wednesday-august-18-sheaves}{% \section{Wednesday, August 18: Sheaves}\label{wednesday-august-18-sheaves}} \begin{remark} We'll be covering Hartshorne, chapter 2: \begin{itemize} \item Sections 1-5: Fundamental, sheaves, schemes, morphisms, constant sheaves. \item Sections 6-9: Divisors, linear systems of differentials, nonsingular varieties. \end{itemize} Note that most of the important material of this book is contained in the exercises! \end{remark} \begin{remark} Recall that a \textbf{topological space} $$X$$ is collection of \emph{open} sets $${\mathcal{U}}= \left\{{U_i \subseteq X}\right\}$$ which is closed under arbitrary unions and finite intersections, where $$X, \emptyset\in {\mathcal{U}}$$. \end{remark} \begin{definition}[Presheaf] A \textbf{presheaf of abelian groups} $${\mathcal{F}}$$ on $$X$$ a topological space is an assignment to every open $$U \subseteq X$$ an abelian group $${\mathcal{F}}(U)$$ and restriction morphisms $$\rho_{UV}: {\mathcal{F}}(U) \to {\mathcal{F}}(V)$$ for every inclusion $$V \subseteq U$$ satisfying \begin{itemize} \tightlist \item $${\mathcal{F}}(\emptyset) = 0$$ \item $$\rho_{UU}: {\mathcal{F}}(U) \to {\mathcal{F}}(U)$$ is $$\operatorname{id}_{{\mathcal{F}}(U)}$$. \item If $$W \subseteq V \subseteq U$$ are opens, then \begin{align*} \rho_{UW} = \rho_{VW} \circ \rho_{UV} .\end{align*} \end{itemize} We'll write $${\mathcal{F}}(U)$$ to be the \textbf{sections of $${\mathcal{F}}$$ over $$U$$}, also notated $${\mathsf{\Gamma}\qty{U; {\mathcal{F}}} }$$ and write the restrictions as $${ \left.{{s}} \right|_{{v}} } = \rho_{UV}(s)$$ for $$V \subseteq U$$. \end{definition} \begin{example}[Presheaf of continuous functions] Let $$X \coloneqq{\mathbb{R}}$$ with the standard topology and take $${\mathcal{F}}= C^0({-}; {\mathbb{R}})$$ (continuous real-valued functions) as the associated presheaf. So for $$U \subset {\mathbb{R}}$$ open, the sections are $${\mathcal{F}}(U) \coloneqq\left\{{f: U\to {\mathbb{R}}\text{ continuous}}\right\}$$. For restriction maps, given $$U \subseteq V$$ take the actual restriction of functions $$C^0(V; {\mathbb{R}}) \to C^0(U; {\mathbb{R}})$$. One needs to check the 3 conditions, but we can declare $$C^0(\emptyset; {\mathbb{R}}) = \left\{{0}\right\} = 0$$, and the others follow right away. \end{example} \begin{example}[Constant presheaves] The \textbf{constant presheaf} associated to $$A\in {\mathsf{Ab}}$$ on $$X\in {\mathsf{Top}}$$ is denote $$F = \underline{A}$$, where \begin{align*} \underline{A}(U) \coloneqq \begin{cases} A & U \neq \emptyset \\ 0 & U = \emptyset. \end{cases} \end{align*} and \begin{align*} \rho_{UV} \coloneqq \begin{cases} \operatorname{id}_A & V \neq \emptyset \\ 0 & V=\emptyset . \end{cases} .\end{align*} \end{example} \begin{warnings} The constant sheaf is not the sheaf of constant functions! Instead these are \emph{locally} constant functions. \end{warnings} \begin{remark} Let $${\mathsf{Open}}_{/ {X}}$$ denote the category of open sets of $$X$$, defined \begin{itemize} \tightlist \item $${\operatorname{Ob}}({\mathsf{Open}}_{/ {X}} ) \coloneqq\left\{{U_i}\right\}$$, so each object is an open set. \item $${\mathsf{Open}}_{/ {X}} (U, V)$$ is empty when $$V\not\subset U$$ and is the singleton inclusion $$\left\{{\iota: U\hookrightarrow V}\right\}$$ otherwise. \end{itemize} \end{remark} \begin{example}[Of $\Open\slice{X}$ ] Take $$X\coloneqq\left\{{p, q}\right\}$$ with the discrete topology to obtain a category with 4 objects: \begin{center} \begin{tikzcd} & {\left\{{p, q}\right\}} \\ {\left\{{p}\right\}} && {\left\{{q}\right\}} \\ & \emptyset \arrow[from=3-2, to=1-2] \arrow[from=3-2, to=2-1] \arrow[from=3-2, to=2-3] \arrow[from=2-1, to=1-2] \arrow[from=2-3, to=1-2] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNCxbMSwwLCJcXHRze3AsIHF9Il0sWzAsMSwiXFx0c3twfSJdLFsyLDEsIlxcdHN7cX0iXSxbMSwyLCJcXGVtcHR5c2V0Il0sWzMsMF0sWzMsMV0sWzMsMl0sWzEsMF0sWzIsMF1d}{Link to Diagram} \end{quote} Similarly, the indiscrete topology yields $$\emptyset \to \left\{{p, q}\right\}$$, a category with two objects. \end{example} \begin{remark} Then a presheaf is a contravariant functor $${\mathcal{F}}: {\mathsf{Open}}_{/ {X}} \to {\mathsf{Ab}}$$ which sends the cofinal/initial object $$\left\{{\emptyset}\right\} \in {\mathsf{Open}}_{/ {X}}$$ to the final/terminal object $$0 \in {\mathsf{Ab}}$$. More generally, we can replace $${\mathsf{Ab}}$$ with any category $$\mathsf{C}$$ admitting a final object: \begin{itemize} \tightlist \item $$\mathsf{C} \coloneqq\mathsf{CRing}$$ the category of commutative rings, which we'll use to define schemes. \item $$\mathsf{C} = {\mathsf{Grp}}$$, the full category of (potentially nonabelian) groups. \item $$\mathsf{C} \coloneqq{\mathsf{Top}}$$, arbitrary topological spaces. \end{itemize} \end{remark} \begin{example}[of presheaves] Let $$X\in {\mathsf{Var}}_{/ {k}}$$ a variety over $$k\in \mathsf{Field}$$ equipped with the Zariski topology, so the opens are complements of vanishing loci. Given $$U \subseteq X$$, define a presheaf of regular functions $${\mathcal{F}}\coloneqq{\mathcal{O}}$$ where \begin{itemize} \item $${\mathcal{O}}(U)$$ are the regular functions $$f:U\to k$$, i.e.~functions on $$U$$ which are locally expressible as a ratio $$f = g/h$$ with $$g, h\in k[x_1, \cdots, x_{n}]$$. \item Restrictions are restrictions of functions. \end{itemize} Taking $$X = {\mathbb{A}}^1_{/ {k}}$$, the Zariski topology is the cofinite topology, so every open $$U$$ is the complement of a finite set and $$U = \left\{{t_1, \cdots, t_m}\right\}^c$$. Then $${\mathcal{O}}(U) = \left\{{\phi: U\to k}\right\}$$ which is locally a fraction, and it turns out that these are all globally fractions and thus \begin{align*} {\mathcal{O}}(U) = \left\{{ {f \over g} {~\mathrel{\Big|}~}f,g\in k[t], g(t) \neq 0 \,\,\, \forall t\in U}\right\} = \left\{{{ f \over \prod (t-t_i)^{m_i}} {~\mathrel{\Big|}~}f\in k[t] }\right\} = k[t] \left[ { \scriptstyle { {S}^{-1}} } \right] ,\end{align*} where $$S = \left\langle{\prod t-t_i}\right\rangle$$ is the multiplicative set generated by the factors. This gives an abelian group since we can take least common denominators, and we have restrictions. \end{example} \begin{warnings} Note that there are two similar notations for localization which mean different things! For a multiplicative set $$S$$, the ring $$R \left[ { \scriptstyle { {S}^{-1}} } \right]$$ literally means localizing at that set. For $${\mathfrak{p}}\in \operatorname{Spec}R$$, the ring $$R \left[ { \scriptstyle { {{\mathfrak{p}}}^{-1}} } \right]$$ means localizing at the multiplicative set $$S \coloneqq{\mathfrak{p}}^c$$. \end{warnings} \hypertarget{friday-august-20}{% \section{Friday, August 20}\label{friday-august-20}} \begin{definition}[Sheaf] Recall the definition of a presheaf, and the main 3 properties: \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \item $$F( \emptyset) = 0$$, \item $$\rho_{UU} = \operatorname{id}_{{\mathcal{F}}(U)}$$ \item For all $$W \subseteq V \subseteq U$$, a cocycle condition: \begin{align*} \rho_{UW} = \rho_{VW} \circ \rho_{UV} .\end{align*} Write $$s_i \in {\mathcal{F}}(U_i)$$ to be a section. A presheaf is a \textbf{sheaf} if it additionally satisfies \item When restrictions are compatible on overlaps, so \begin{align*} { \left.{{s_i}} \right|_{{U_i \cap U_j}} } = { \left.{{s_j}} \right|_{{U_i \cap U_j}} } ,\end{align*} there exists a uniquely glued section $${\mathcal{F}}(\cup U_i)$$ such that $${ \left.{{s}} \right|_{{U_i}} } = s_i$$ for all $$i$$. \end{enumerate} \end{definition} \begin{example}[?] Take $$C^0({-}; {\mathbb{R}})$$ the sheaf of continuous real-valued functions on a topological space. For $$f_i: U_i \to {\mathbb{R}}$$ agreeing on overlaps, there is a continuous function $$f: \cup U_i\to {\mathbb{R}}$$ restricting to $$f_i$$ on each $$U_i$$ by just defining $$f(x) = f_i(x)$$ for $$x\in U_i$$, which is well-defined by agreement of the $$f_i$$ on overlaps. \end{example} \begin{example}[?] Let $$X$$ be a topological space and $$A\in \mathsf{CRing}$$, then take the constant sheaf $$\underline{A}$$ which maps to $$A$$ iff $$U\neq \emptyset$$ and 0 otherwise. This is not a sheaf, taking $$X = {\mathbb{R}}$$ and $$A = {\mathbb{Z}}/2$$. Let $$U_1 = (0, 1)$$ and $$U_2 = (2, 3)$$ and take $$s_1 = 0$$ on $$U_1$$ and $$s_2 = 1$$ on $$U_2$$. Using that $$U_1 \cap U_2 = \emptyset$$, so they trivially agree on overlaps, but there is no constant function on $$U_1 \cup U_2$$ restricting to 1 on $$U_2$$ and 0 on $$U_1$$ \end{example} \begin{definition}[Locally constant sheaves] The \textbf{(locally) constant sheaf} $$\underline{A}$$ on any $$X\in {\mathsf{Top}}$$ is defined as \begin{align*} \underline{A}(U) \coloneqq\left\{{ f: U\to A {~\mathrel{\Big|}~}f \text{ is locally constant} }\right\} .\end{align*} \end{definition} \begin{remark} As a general principle, this is a sheaf since this property can be verified locally. \end{remark} \begin{example}[?] Let $$C^0_{\mathrm{bd}}$$ be the presheaf of bounded continuous functions on $$S^1$$. This is not a sheaf, but one needs to go to infinitely many sets: take the image of $$[{1\over n}, {1\over n+1}]$$ with (say) $$f_n(x) = n$$ for each $$n$$. Then each $$f_n$$ is bounded (it's just constant), but the full collection is unbounded, so these can not glue to a bounded function. \end{example} \begin{definition}[Stalks] Let $${\mathcal{F}}\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)$$ and $$p\in X$$, then the \textbf{stalk} of $${\mathcal{F}}$$ at $$p$$ is defined as \begin{align*} {\mathcal{F}}_p(U) \coloneqq\lim_{U\ni p} \coloneqq\left\{{(s, U) {~\mathrel{\Big|}~}U\ni p \text{ open}, \, s\in {\mathcal{F}}(U)}\right\}/\sim ,\end{align*} where $$(s, U) \sim (t ,V)$$ iff there exists a $$W \ni p$$ with $$W \subset U \cap V$$ with $${ \left.{{s}} \right|_{{W}} } = { \left.{{t}} \right|_{{W}} }$$. An equivalence class $$[(s, U)] \in {\mathcal{F}}_p$$ is referred to as a \textbf{germ}. \end{definition} \begin{example}[?] Let $$C^\omega({-}; {\mathbb{R}})$$ be the sheaf of analytic functions, i.e.~those locally expressible as convergent power series. This is a sheaf because this condition can be checked locally. What is the stalk $$C_0^\omega$$ at zero? An example of a function in this germ is $$[(f(x) = {1\over 1-x}, (-1, 1))$$. A first guess is $${\mathbb{R}} { \left[ {t} \right] }$$, but the claim is that this won't work. Note that there is an injective map $$C_0^\omega \hookrightarrow{\mathbb{R}} { \left[ {t} \right] }$$ because $$f, g$$ have analytic power series expansions at zero, and if these expressions are equal then $${ \left.{{f}} \right|_{{I}} } = { \left.{{g}} \right|_{{I}} }$$ for some $$I$$ containing zero. This map won't be surjective because there are power series with a non-positive radius of convergence, for example taking $$f(t) \coloneqq\sum_{k=0}^\infty {kt}^k$$ which only converges at $$t=0$$. So the answer is that $$C_0^\omega \leq {\mathbb{R}} { \left[ {t} \right] }$$ is the subring of power series with positive radius of convergence. \end{example} \begin{definition}[Local ring of the structure sheaf, $\OO_p$] Let $$X \in {\mathsf{Alg}}{\mathsf{Var}}$$ and $${\mathcal{O}}$$ its sheaf of regular functions. For $$p\in X$$, the stalk $${\mathcal{O}}_p$$ is the \textbf{local ring} of $$X$$ at $$p$$. \end{definition} \begin{example}[?] For $$X \coloneqq{\mathbb{A}}^1_{/ {k}}$$ for $$k=\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu$$, the opens are cofinite sets and $${\mathcal{O}}(U) = \left\{{f/g {~\mathrel{\Big|}~}f, g\in k[t]}\right\}$$. Consider the stalk $${\mathcal{O}}_p$$. Applying the definition, we have \begin{align*} {\mathcal{O}}_p \coloneqq\left\{{(f/g, U) {~\mathrel{\Big|}~}p\in U,\, g\neq 0 \text {on } U}\right\} / \sim .\end{align*} Given any $$g\in k[t]$$ with $$g(p) \neq 0$$, there is a Zariski open set $$U = V(g)^c = D_g$$, the distinguished open associated to $$g$$, where $$g\neq 0$$ on $$U$$ by definition. Thus $$p\in U$$, and so any $$f/g\in \operatorname{ff}{k[t]}$$ with $$p\neq 0$$ defines an element $$(f/g, D_g) \in {\mathcal{O}}_p$$. Concretely: \begin{align*} { \left.{{f/g}} \right|_{{W}} } = { \left.{{f/g}} \right|_{{W'}} } \implies f/g = f'/g' \in \operatorname{ff}{k[t]} = k(t) ,\end{align*} and $$fg' = f'g$$ on the cofinite set $$W$$, making them equal as polynomials. We can thus write \begin{align*} {\mathcal{O}}_p = \left\{{f/g \in k(t) {~\mathrel{\Big|}~}g(p) \neq 0}\right\} = k[t] \left[ { \scriptstyle { {\left\langle{t-p}\right\rangle}^{-1}} } \right], \quad \left\langle{t-p}\right\rangle\in \operatorname{mSpec}k[t] ,\end{align*} recalling that $$k[t] \left[ { \scriptstyle { {{\mathfrak{p}}}^{-1}} } \right] = \left\{{f/g {~\mathrel{\Big|}~}g\not\in {\mathfrak{p}}}\right\}$$. Note that for $$X\in {\mathsf{Aff}}{\mathsf{Var}}$$, so $$X = V(f_i) = V(I)$$ for $$I$$ reduced, we have the coordinate ring $$k[X] = k[x_1, \cdots, x_{n}]/I = R$$, then $${\mathcal{O}}_p = R \left[ { \scriptstyle { {{\mathfrak{m}}_p}^{-1}} } \right]$$ where $${\mathfrak{m}}_p \coloneqq\left\{{f\in R {~\mathrel{\Big|}~}f(p) = 0}\right\}$$. \end{example} \begin{warnings} This doesn't quite hold for non-algebraically closed fields. Take $$f(x) x^p-x \in {\mathbb{F}}_p[x]$$, then $$f(x) \equiv 0$$ since every element in $${\mathbb{F}}_p$$ is a root. \end{warnings} \begin{remark} Next time: morphisms of sheaves/presheaves, and isomorphisms can be checked on stalks for sheaves. \end{remark} \hypertarget{monday-august-23}{% \section{Monday, August 23}\label{monday-august-23}} \begin{remark} Recall that the \textbf{stalk} of a presheaf $${\mathcal{F}}$$ at $$p$$ is defined as \begin{align*} {\mathcal{F}}_p \coloneqq\colim_{U\ni p} {\mathcal{F}}(U) = \left\{{ (s, U) {~\mathrel{\Big|}~}s\in {\mathcal{F}}(U) }\right\}_{/ {\sim}} .\end{align*} \end{remark} \begin{definition}[Morphisms of presheaves] Let $${\mathcal{F}}, {\mathcal{G}}\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)$$, then a \textbf{morphism} $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ is a collection $$\left\{{\phi(U): {\mathcal{F}}(U) \to {\mathcal{G}}(U)}\right\}$$ of morphisms of abelian groups for all $$U\in {\mathsf{Open}}(X)$$ such that for all $$V \subset U$$, the following diagram commutes: \begin{center} \begin{tikzcd} {{\mathcal{F}}(U)} && {{\mathcal{G}}(U)} \\ \\ {{\mathcal{F}}(V)} && {{\mathcal{G}}(V)} \arrow["{\phi(U)}", from=1-1, to=1-3] \arrow["{\phi(V)}", from=3-1, to=3-3] \arrow["{\operatorname{res}(UV)}"{description}, from=1-1, to=3-1] \arrow["{\operatorname{res}'(UV)}"{description}, from=1-3, to=3-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXG1jZihVKSJdLFswLDIsIlxcbWNmKFYpIl0sWzIsMCwiXFxtY2coVSkiXSxbMiwyLCJcXG1jZyhWKSJdLFswLDIsIlxccGhpKFUpIl0sWzEsMywiXFxwaGkoVikiXSxbMCwxLCJcXHJlcyhVVikiLDFdLFsyLDMsIlxccmVzJyhVVikiLDFdXQ==}{Link to Diagram} \end{quote} An \textbf{isomorphism} is a morphism with a two-sided inverse. \end{definition} \begin{remark} Note that if we regard a sheaf as a contravariant functor, a morphism is then just a natural transformation. \end{remark} \begin{remark} A morphism $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ defines a morphisms on stalks $$\phi_p: {\mathcal{F}}_p \to {\mathcal{G}}_p$$. \end{remark} \begin{example}[of a nontrivial morphism of sheaves] Let $$X \coloneqq{\mathbb{C}}^{\times}$$ with the classical topology, making it into a real manifold, and take $$C^0({-}; {\mathbb{C}}) \in {\mathsf{Sh}}(X, {\mathsf{Ab}})$$ be the sheaf of continuous functions and let $$C^0({-}; {\mathbb{C}})^{\times}$$ the sheaf of of nowhere zero continuous continuous functions. Note that this is a sheaf of abelian groups since the operations are defined pointwise. There is then a morphism \begin{align*} \exp({-}): C^0({-}; {\mathbb{C}}) &\to C^0({-}; {\mathbb{C}})^{\times}\\ f &\mapsto e^f && \text{ on open sets } U\subseteq X .\end{align*} Since exponentiating and restricting are operations done pointwise, the required square commutes, yielding a morphism of sheaves. \end{example} \begin{definition}[(co)kernel and image sheaves] Let $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ be morphisms of presheaves, then define the presheaves \begin{align*} \ker(\phi)(U) &\coloneqq\ker(\phi(U)) \\ \operatorname{coker}^{{\mathsf{pre}}}(\phi)(U) &\coloneqq{\mathcal{G}}(U) / \phi({\mathcal{F}}(U))\\ \operatorname{im}(\phi)(U) &\coloneqq\operatorname{im}(\phi(U)) \\ .\end{align*} \end{definition} \begin{warnings} If $${\mathcal{F}}, {\mathcal{G}}\in {\mathsf{Sh}}(X)$$, then for a morphism $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$, the image and cokernel presheaves need not be sheaves! \end{warnings} \begin{example}[?] Consider $$\ker \exp$$ where $$\exp: C^0({-}; {\mathbb{C}})\to C^0({-}; {\mathbb{C}})^{\times}\in {\mathsf{Sh}}({\mathbb{C}}^{\times})$$. One can check that $$\ker \exp = 2\pi i \underline{{\mathbb{Z}}}(U)$$, and so the kernel is actually a sheaf. We also have $$\operatorname{coker}^{{\mathsf{pre}}} \exp(U) \coloneqq C^0(U; {\mathbb{C}})/ \exp(C^0(U;{\mathbb{C}})^{\times})$$. On opens, $$\operatorname{coker}^{{\mathsf{pre}}} \exp(U) = \left\{{1}\right\} \iff$$ every nonvanishing continuous function $$g$$ on $$U$$ has a continuous logarithm, i.e.~$$g = e^f$$ for some $$f$$. Examples of opens with this property include any contractible (or even just simply connected) open set in $${\mathbb{C}}^{\times}$$. Consider $$U\coloneqq{\mathbb{C}}^{\times}$$ and $$z\in C^0({\mathbb{C}}^{\times}; {\mathbb{C}})^{\times}$$, which is a nonvanishing function. Then the equivalence class $$[z] \in \operatorname{coker}^{{\mathsf{pre}}} \exp({\mathbb{C}}^{\times})$$ is nontrivial, since $$z\neq e^f$$ for any $$f\in C^0({\mathbb{C}}^{\times}; {\mathbb{C}})$$, since any attempted definition of $$\log(z)$$ will have monodromy. on the other hand we can cover $${\mathbb{C}}^{\times}$$ by contractible opens $$\left\{{U_i}\right\}_{i\in I}$$ where $${ \left.{{[z]}} \right|_{{U_i}} } = 1 \in \operatorname{coker}^{{\mathsf{pre}}} \exp (U_i)$$ and similarly $${ \left.{{1}} \right|_{{\operatorname{id}}} } = 1 \in \operatorname{coker}^{{\mathsf{pre}}} \exp(U_i)$$, showing that the cokernel fails the unique gluing axiom and is not a sheaf. \end{example} \begin{definition}[Sheafification] Given any $${\mathcal{F}}\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)$$ there exists an $${\mathcal{F}}^+ \in {\mathsf{Sh}}(X)$$ and a morphism of presheaves $$\theta: {\mathcal{F}}\to {\mathcal{F}}^+$$ such that for any $${\mathcal{G}}\in {\mathsf{Sh}}(X)$$ with a morphism $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ there exists a unique $$\psi: {\mathcal{F}}^+ \to {\mathcal{G}}$$ making the following diagram commute: \begin{center} \begin{tikzcd} {\mathcal{F}}&& {\mathcal{G}}\\ \\ && {{\mathcal{F}}^+} \arrow["\theta"', from=1-1, to=3-3] \arrow["\phi", from=1-1, to=1-3] \arrow["{\exists! \psi}"', from=3-3, to=1-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXG1jZiJdLFsyLDIsIlxcbWNmXisiXSxbMiwwLCJcXG1jZyJdLFswLDEsIlxcdGhldGEiLDJdLFswLDIsIlxccGhpIl0sWzEsMiwiXFxleGlzdHMhIFxccHNpIiwyXV0=}{Link to Diagram} \end{quote} The sheaf $${\mathcal{F}}^+ \in {\mathsf{Sh}}(X)$$ is called the \textbf{sheafification} of $${\mathcal{F}}$$. This is an example of an adjunction of functors: \begin{align*} \mathop{\mathrm{Hom}}_{ \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)}({\mathcal{F}}, {\mathcal{G}}^{\mathsf{pre}}) \cong \mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(X)}({\mathcal{F}}^+, {\mathcal{G}}) ,\end{align*} where we use the forgetful functor $${\mathcal{G}}\to {\mathcal{G}}^{\mathsf{pre}}$$. This yields an adjoint pair \begin{align*} \adjunction{a}{b}{c}{d} .\end{align*} \end{definition} \begin{proof}[of existence of sheafification] We construct it directly as $${\mathcal{F}}^+ \coloneqq\left\{{s:U \to {\textstyle\coprod}_{p\in U} {\mathcal{F}}_p }\right\}$$ such that \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item $$s(p) \in {\mathcal{F}}_p$$, \item The germs are compatible locally, so for all $$p\in U$$ there is a $$V\supseteq p$$ such that for some $$t\in {\mathcal{F}}(V)$$, $$s(p) = t_p$$ for all $$p$$ in $$V$$. \end{enumerate} \begin{slogan} Collections of germs that are locally compatible. \end{slogan} So about any point, there should be an actual function specializing to all germs in an open set. \end{proof} \begin{remark} The point will be that $$\operatorname{coker}\exp$$ will be zero as a sheaf, since it'll be zero on a sufficiently small set. \end{remark} \hypertarget{wednesday-august-25}{% \section{Wednesday, August 25}\label{wednesday-august-25}} \begin{remark} Recall the definition of sheafification: let $${\mathcal{F}}\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X; {\mathsf{Ab}}{\mathsf{Grp}})$$. Construct a sheaf $${\mathcal{F}}^+\in {\mathsf{Sh}}(X, {\mathsf{Ab}}{\mathsf{Grp}})$$ and a morphism $$\theta: {\mathcal{F}}\to {\mathcal{F}}^+$$ of presheaves satisfying the appropriate universal property: \begin{center} \begin{tikzcd} {{\mathcal{F}}^+} \\ \\ {\mathcal{F}}&& {\mathcal{G}}\\ \\ {} \arrow["\psi", from=3-1, to=3-3] \arrow["\theta", from=3-1, to=1-1] \arrow["{\exists \tilde \psi}", dashed, from=1-1, to=3-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNCxbMCwyLCJcXG1jZiJdLFswLDRdLFswLDAsIlxcbWNmXisiXSxbMiwyLCJcXG1jZyJdLFswLDMsIlxccHNpIl0sWzAsMiwiXFx0aGV0YSJdLFsyLDMsIlxcZXhpc3RzIFxcdGlsZGUgXFxwc2kiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=}{Link to Diagram} \end{quote} So any presheaf morphism to a sheaf factors through the sheafification uniquely (via $$\theta$$). Note that this is a instance of a general free/forgetful adjunction. We can construct it as \begin{align*} {\mathcal{F}}^+(U) \coloneqq\left\{{s:U\to {\textstyle\coprod}_{p\in U} {\mathcal{F}}_p,\quad s(p) \in {\mathcal{F}}_p, \cdots}\right\} .\end{align*} where the addition condition is that for all $$q\in U$$ there exists a $$V\nu q$$ and $$t\in {\mathcal{F}}(V)$$ such that $$t_p = s(p)$$ for all $$p\in V$$. Note that $$\theta$$ is defined by $$\theta(U)(s) = \left\{{s:p\to s_p}\right\}$$, the function assigning points to germs with respect to the section $$s$$. Idea: this is like replacing an analytic function on an interval with the function sending a point $$p$$ to its power series expansion at $$p$$. \end{remark} \begin{example}[?] Recall $$\exp: C^0 \to (C^0)^{\times}$$ on $${\mathbb{C}}^{\times}$$, then $$\operatorname{coker}^{\mathsf{pre}}(\exp)(U) = \left\{{1}\right\}$$ on contractible $$U$$, using that one can choose a logarithm on such a set. However $$\operatorname{coker}^{\mathsf{pre}}(\exp)({\mathbb{C}}^{\times}) \neq \left\{{1}\right\}$$ since $$[z]\in (C^0)^{\times}({\mathbb{C}}^{\times})/\exp(C^0({\mathbb{C}}^{\times}))$$. \end{example} \begin{remark} Letting $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ be a morphisms of sheaves, then we defined $$\operatorname{coker}(\phi) \coloneqq(\operatorname{coker}^{\mathsf{pre}}(\phi))^+$$ and $$\operatorname{im}(\phi) \coloneqq(\operatorname{im}^{\mathsf{pre}}(\phi))^+$$. Then \begin{align*} \operatorname{coker}^{\mathsf{pre}}(\exp) &\to \operatorname{coker}(\exp) \\ s\in {\mathcal{F}}(U) &\mapsto s(p) = s_p .\end{align*} The claim is that $$[z]_p = 1$$ for all $$p\in {\mathbb{C}}^{\times}$$, since we can replace $$[([z], {\mathbb{C}}^{\times})]$$ with $$([z]_U, U)$$ for $$U$$ contractible. \end{remark} \begin{example}[?] A useful example to think about: $$X = \left\{{p, q}\right\}$$ with \begin{itemize} \tightlist \item $${\mathcal{F}}(p) = A$$ \item $${\mathcal{F}}(q) = B$$ \item $${\mathcal{F}}(X) = 0$$ \end{itemize} Then local sections don't glue to a global section, so this isn't a sheaf, but it is a presheaf. The sheafification satisfies $${\mathcal{F}}^+(X) = A\times B$$. \end{example} \hypertarget{subsheaves}{% \subsection{Subsheaves}\label{subsheaves}} \begin{definition}[Subsheaves, injectivity, surjectivity] $${\mathcal{F}}'$$ is a \textbf{subsheaf} of $${\mathcal{F}}$$ if \begin{itemize} \tightlist \item $${\mathcal{F}}'(U) \leq {\mathcal{F}}(U)$$ for all $$U$$, \item $$\mathop{\mathrm{Res}}'(U, V) = { \left.{{ \mathop{\mathrm{Res}}(U, V) }} \right|_{{{\mathcal{F}}'(U)}} }$$. \end{itemize} $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ is \textbf{injective} iff $$\ker \phi = 0$$, \textbf{surjective} if $$\operatorname{im}(\phi) = {\mathcal{G}}$$ or $$\operatorname{coker}\phi = 0$$. \end{definition} \begin{exercise}[?] Check that $$\ker \phi$$ already satisfies the sheaf property. \end{exercise} \begin{definition}[Exact sequences of sheaves] Let $$\cdots \to {\mathcal{F}}^{i-1} \xrightarrow{\phi^{i-1}} {\mathcal{F}}^i \xrightarrow{\phi^i} {\mathcal{F}}^{i+1}\to \cdots$$ be a sequence of morphisms in $${\mathsf{Sh}}(X)$$, this is \textbf{exact} iff $$\ker \phi^i = \operatorname{im}\phi^{i-1}$$. \end{definition} \begin{lemma}[?] $$\ker \phi$$ is a sheaf. \end{lemma} \begin{proof}[?] By definition, $$\ker(\phi)(U) \coloneqq\ker \qty{ \phi(U): {\mathcal{F}}(U) \to {\mathcal{G}}(U) }$$, satisfying part (a) in the definition of presheaves. We can define restrictions $${ \left.{{\mathop{\mathrm{Res}}(U, V)}} \right|_{{\ker(\phi)(U)}} } \subseteq \ker(\phi)(V)$$. Use the commutative diagram for the morphism $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$. Now checking gluing: Let $$s_i \in \ker(\phi)(U_i)$$ such that $$\mathop{\mathrm{Res}}(s_i, U_i \cap U_j) = \mathop{\mathrm{Res}}(s_j, U_i \cap U_j)$$ for all $$i, j$$. This holds by viewing $$s_i \in {\mathcal{F}}(U_i)$$, so $$\exists ! s\in {\mathcal{F}}(\displaystyle\bigcup_i U_i)$$ such that $$\mathop{\mathrm{Res}}(s, U_i) = s_i$$. We want to show $$s\in \ker(\phi)\qty{\displaystyle\bigcup U_i}$$, so consider \begin{align*} t\coloneqq\phi\qty{ \displaystyle\bigcup_i U_i}(s) \in {\mathcal{G}}\qty{\displaystyle\bigcup U_i} ,\end{align*} which is zero. Now \begin{align*} \mathop{\mathrm{Res}}(t, U_i) = \phi(U_i)(\mathop{\mathrm{Res}}(s, U_i)) = \phi(U_i)(s_i) = 0 \end{align*} by assumption, using the commutative diagram. By unique gluing for $${\mathcal{G}}$$, we have $$t=0$$, since $$0$$ is also a section restricting to $$0$$ everywhere. \end{proof} \begin{definition}[Quotients] For $${\mathcal{F}}' \leq {\mathcal{F}}$$ a subsheaf, define the \textbf{quotient} $${\mathcal{F}}/{\mathcal{F}}' \coloneqq(({\mathcal{F}}/{\mathcal{F}}')^{\mathsf{pre}})^+$$ where \begin{align*} ({\mathcal{F}}/{\mathcal{F}}')^{\mathsf{pre}}(U) \coloneqq{\mathcal{F}}(U)/ {\mathcal{F}}'(U) .\end{align*} \end{definition} \hypertarget{friday-august-27}{% \section{Friday, August 27}\label{friday-august-27}} \begin{theorem}[Sheaf isomorphism iff isomorphism on stalks] Let $$\phi:{\mathcal{F}}\to{\mathcal{G}}$$ be a morphism in $${\mathsf{Sh}}(X)$$, then $$\phi$$ is an isomorphism iff $$\phi_p: {\mathcal{F}}_p \to{\mathcal{G}}_p$$ is an isomorphism for all $$p\in X$$. \end{theorem} \begin{proof}[$\implies$] Suppose $$\phi$$ is an isomorphism, so there exists a $$\psi: {\mathcal{G}}\to {\mathcal{F}}$$ which is a two-sided inverse for $$\phi$$. Then $$\psi_p$$ is a two-sided inverse to $$\phi_p$$, making it an isomorphism. This follows directly from the formula: \begin{align*} \phi_p: {\mathcal{F}}_p &\to {\mathcal{G}}_p \\ (s, U) & \mapsto (\phi(U)(s), U) .\end{align*} \end{proof} \begin{proof}[$\impliedby$] It suffices to show $$\phi(U): {\mathcal{F}}(U) \to {\mathcal{G}}(U)$$ is an isomorphism for all $$U$$. This is because we could define $$\psi(U):{\mathcal{G}}(U) \to {\mathcal{F}}(U)$$ and set $$\phi^{-1}(U) \coloneqq\psi(U)$$, then reversing the arrows in the diagram for a sheaf morphism again yields a commutative diagram. \begin{claim} $$\phi(U)$$ is injective. \end{claim} For $$s\in {\mathcal{F}}(U)$$, we want to show $$\phi(U)(s) = 0$$ implies $$s=0$$. Consider the germs $$(s, U) \in {\mathcal{F}}_p$$ for $$p\in U$$, we have $$\phi_p(s, U) = (0, U) = 0\in {\mathcal{F}}_p$$. So $$S_p = 0$$ for all $$p\in U$$. Since we have a germ, there exists $$V_p \ni p$$ open such that $${ \left.{{s}} \right|_{{V_p}} } = 0$$. Noting that $$\left\{{V_p {~\mathrel{\Big|}~}p\in U}\right\}\rightrightarrows U$$, by unique gluing we get an $$s$$ where $${ \left.{{s}} \right|_{{V_p}} } = 0$$ for all $$V_p$$, so $$s\equiv 0$$ on $$U$$. \begin{claim} $$\phi(U)$$ is surjective. \end{claim} Let $$t\in {\mathcal{G}}(U)$$, and consider germs $$t_p\in {\mathcal{G}}_p$$. There exists a unique $$s_p\in {\mathcal{F}}_p$$ with $$\phi_p(s_p) = t_p$$, since $$\phi_p$$ is an isomorphism of stalks by assumption. Use that $$s_p$$ is a germ to get an equivalence class $$(s_p, V)$$ where $$V \subseteq U$$. We have $$\phi(V)(s(p), V) \sim (t, U)$$, noting that $$s$$ depends on $$p$$. Having equivalent germs means there exists a $$W(p) \subseteq V$$ with $$p\in W$$ with $$\phi(W(p)) \qty{{ \left.{{s(p)}} \right|_{{W}} }} = { \left.{{t}} \right|_{{W(p)}} }$$. We want to glue these $$\left\{{ { \left.{{s(p)}} \right|_{{W(p)}} } {~\mathrel{\Big|}~}p\in U }\right\}$$ together. It suffices to show they agree on intersections. Taking $$p, q\in U$$, both $${ \left.{{s(p)}} \right|_{{W(p) \cap W(q)}} }$$ and $${ \left.{{s(q)}} \right|_{{W(p) \cap W(q)}} }$$ map to $${ \left.{{t}} \right|_{{W(p) \cap W(q)}} }$$ under $$\phi(W(p) \cap W(q) )$$. Injectivity will force these to be equal, so $$\exists ! s \in {\mathcal{F}}(U)$$ with $${ \left.{{s}} \right|_{{W(p)}} } = s(p)$$. We want to now show that $$\phi(U)(s) = t$$. Using commutativity of the square, we have $$\phi(U)(s) { \left.{{}} \right|_{{W(p)}} } = \phi(W(p)) \qty{{ \left.{{s}} \right|_{{W(p)}} } }$$. This equals $$\phi(W(p))(s(p)) = { \left.{{t}} \right|_{{W(p)}} }$$. Therefore $$\phi(U)(s)$$ and $$t$$ restrict to sections $$\left\{{w(p) {~\mathrel{\Big|}~}p\in U}\right\}$$. Using unique gluing for $${\mathcal{G}}$$ we get $$\phi(U)(s) = t$$. \end{proof} \begin{remark} Note: we only needed to check overlaps because of exactness of the following sequence: \begin{align*} 0 \to{\mathcal{F}}(U) \to \prod_{i\in I} {\mathcal{F}}(U_i) \to \prod_{i1\). For example, any irreducible subvariety of $${\mathbb{A}}^2$$ yields a generic point. \todo[inline]{Krull's dimension theorem?} \end{example} \begin{exercise}[?] Try to draw $$\operatorname{Spec}{\mathbb{Z}}$$ and $$\operatorname{Spec}{\mathbb{Z}}[t]$$. \end{exercise} \begin{remark} We'll now try a naive definition of schemes, which we'll find won't quite work. \end{remark} \begin{definition}[A wrong definition of a scheme!] A \textbf{scheme} is a ringed space $$(X, {\mathcal{O}}_X)$$ which is locally an affine scheme, i.e.~there exists an open cover $${\mathcal{U}}\rightrightarrows X$$ such that there is a collection of rings $$A_i$$ with \begin{align*} (U_i, { \left.{{{\mathcal{O}}_{X}}} \right|_{{U_i}} } ) { { \, \xrightarrow{\sim}\, }}(\operatorname{Spec}A_i, {\mathcal{O}}_{\operatorname{Spec}A_i}) .\end{align*} \end{definition} \begin{remark} This produces the right objects, but not the correct morphisms. This is a subtle issue! With this definition, a morphism of schemes would be a morphism of ringed spaces $$(f, f^\#)$$ with $$f\in {\mathsf{Top}}(X, Y)$$ and $$f^\# \in {\mathsf{Sh}}_{/ {Y}} ({\mathcal{O}}_Y, f_* {\mathcal{O}}_X)$$, where $$f^\#$$ is supposed to capture pullback of functions''. The issue: $$f^\#$$ may not notice that $$p \xrightarrow{f} f(p)$$! In particular, it may not preserve the fact that $$f(p) = 0$$. \begin{figure} \centering \resizebox{\columnwidth}{!}{% \begin{tikzpicture} \fontsize{42pt}{1em} \node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/Schemes/sections/figures}{2021-09-15_11-49.pdf_tex} }; \end{tikzpicture} } \end{figure} Hartshorne exercises for how this issue can actually arise. \end{remark} \begin{remark} Let $$(f, f^\#)$$ be a map of ringed spaces, then there is an induced map \begin{align*} f_p^\#: {\mathcal{O}}_{Y, f(p)} &\to {\mathcal{O}}_{X, p} \\ (U, s) &\mapsto (f^{-1}(U), f^\#(U)(s)) .\end{align*} \end{remark} \begin{definition}[Locally ringed space] A \textbf{locally ringed space} $$(X, {\mathcal{O}}_X)$$ is a ringed space for which the stalks $${\mathcal{O}}_{X, p} \in \mathsf{Loc}\mathsf{Ring}$$ are local rings, i.e.~there exists a unique maximal ideal. \end{definition} \begin{example}[of a locally ringed space] For $$(X, {\mathcal{O}}_X) \coloneqq(\operatorname{Spec}A, {\mathcal{O}}_{\operatorname{Spec}A})$$, we saw that $${\mathcal{O}}_{X, p} = A \left[ { \scriptstyle { {p}^{-1}} } \right]$$, which is local. \end{example} \begin{definition}[Morphisms of locally ringed spaces] A \textbf{morphism of locally ringed spaces} is a morphism of ringed spaces \begin{align*} (f, F^\#): (X, {\mathcal{O}}_X) \to (Y, {\mathcal{O}}_Y) \end{align*} such that $$f^\#_p: {\mathcal{O}}_{Y, f(p)} \to {\mathcal{O}}_{X, p}$$ is a homomorphism of local rings, i.e.~$$f^\#({\mathfrak{m}}_{f(p)}) \subseteq {\mathfrak{m}}_p$$. \begin{quote} Here we should also require that $$f^\# \neq 0$$. \end{quote} \end{definition} \begin{remark} Morally: this extra condition enforces that pulling back functions vanishing at $$f(p)$$ yields functions which vanish at $$p$$. \end{remark} \begin{remark} Alternatively one could require that $$(f^\#)^{-1}({\mathfrak{m}}_p) = {\mathfrak{m}}_{f(p)}$$, and (claim) this is equivalent to the above definition. Use that $$(f^\#)^{-1}({\mathfrak{m}}_p)$$ is a prime ideal containing $${\mathfrak{m}}_p$$. \end{remark} \begin{example}[of a locally ringed space] Take $$(X, {\mathcal{O}}_X) \coloneqq({\mathbb{R}}, C^0({-}; {\mathbb{R}}))$$. Why this is in $$\mathsf{Loc}\mathsf{RingSp}$$: write a stalk as \begin{align*} C_p^0 = \left\{{(f, I) {~\mathrel{\Big|}~}I\ni p \text{ an interval}, f\in {\mathsf{Top}}(I, {\mathbb{R}})}\right\}_{/ {\sim}} .\end{align*} Why is this local? Take $${\mathfrak{m}}_p \coloneqq\left\{{(f, I) {~\mathrel{\Big|}~}f(p) = 0}\right\}$$, which is maximal since $$C_p^0/{\mathfrak{m}}\cong {\mathbb{R}}$$ is a field. Then $${\mathfrak{m}}_p^c = \left\{{(f, I) {~\mathrel{\Big|}~}f(p) \neq 0}\right\}$$, and any continuous function that isn't zero at $$p$$ is nonzero in some neighborhood $$J \supseteq I$$, so $${ \left.{{f}} \right|_{{J}} }\neq 0$$ anywhere. Then $$(f, I) \sim ({ \left.{{f}} \right|_{{J}} }, J)$$, which is invertible in the ring, so any element in the complement is a unit. \end{example} \begin{example}[?] Consider \begin{align*} ({\mathbb{R}}, C^0({-}; {\mathbb{R}})) \xrightarrow{(f, f^\#)} ({\mathbb{R}}, C^\infty({-}; {\mathbb{R}})) .\end{align*} Take $$f = \operatorname{id}$$ and the inclusion \begin{align*} f^\# : C^\infty({-}; {\mathbb{R}})\hookrightarrow\operatorname{id}_* C^0({-}; {\mathbb{R}}) = C^0({-}; {\mathbb{R}}) .\end{align*} Then \begin{align*} f_p^\#: C_p^\infty({-}; {\mathbb{R}}) \to C_p^0({-}; {\mathbb{R}}) .\end{align*} satisfies $$f_p^\#({\mathfrak{m}}^\infty_{\operatorname{id}(p)}) = {\mathfrak{m}}^0_p$$. We also have $$(f_p^\#)^{-1}({\mathfrak{m}}_p^0) = {\mathfrak{m}}_p^\infty$$, since the germs are just equal. \end{example} \begin{definition}[Scheme] A \textbf{scheme} $$(X, {\mathcal{O}}_X)$$ is a locally ringed space which is locally isomorphic to $$(\operatorname{Spec}A, {\mathcal{O}}_{\operatorname{Spec}A})$$ in $$\mathsf{Loc}\mathsf{RingSp}$$. A \textbf{morphism of schemes} is a morphism in $$\mathsf{Loc}\mathsf{RingSp}$$. \end{definition} \begin{remark} Note that we can restrict to opens, since this doesn't change the stalks. \end{remark} \begin{remark} As a proof of concept that this is a good notion, we'll see that there's a fully faithful contravariant functor $$\operatorname{Spec}: \mathsf{CRing}\to {\mathsf{Sch}}$$, so \begin{align*} \operatorname{Spec}(\mathop{\mathrm{Mor}}_\mathsf{Ring}(B, A)) = \mathop{\mathrm{Mor}}_{\mathsf{Sch}}( \operatorname{Spec}A, \operatorname{Spec}B) .\end{align*} \end{remark} \hypertarget{appendix}{% \section{Appendix}\label{appendix}} \begin{remark} A bunch of stuff I always forget! \end{remark} \begin{definition}[Classical AG] \envlist \begin{itemize} \item A \textbf{section} is just an element $$s\in {\mathcal{F}}(U)$$. \item A \textbf{stalk} of a (pre)sheaf $${\mathcal{F}}$$ at a point $$p$$ is defined as \begin{align*} {\mathcal{F}}_p \coloneqq\colim_{p\ni U_i} ({\mathcal{F}}(U_i), \operatorname{res}_{ij}) .\end{align*} \item A \textbf{germ} $$\tilde f_p$$ at a point $$p$$ is an element in a stalk $${\mathcal{F}}_p$$. It can concretely be described as \begin{align*} \tilde f_p = [(U\ni p, s\in {\mathcal{F}}(U))]/\sim && (U, s)\sim (V, t) \iff \exists W \subseteq U \cap V,\, { \left.{{s}} \right|_{{W}} } = { \left.{{t}} \right|_{{W}} } .\end{align*} \end{itemize} \end{definition} \begin{definition}[Colimit of a diagram] Given a diagram $$J$$ in a category $$\mathsf{C}$$, regard it as a functor $$F: \mathsf{J}\to \mathsf{C}$$ where $$\mathsf{J}$$ is the diagram category of $$J$$. Then the \textbf{colimit} of $$J$$ is defined as the initial object in the category of co-cones over $$F$$. \begin{itemize} \item A \textbf{co-cone} of $$F$$ is an $$N\in {\operatorname{Ob}}(\mathsf{C})$$ and a family of morphisms $$\left\{{ \psi_X: F(X)\to N{~\mathrel{\Big|}~}X\in {\operatorname{Ob}}(\mathsf{J})}\right\}$$. \item The \textbf{category of co-cones} over $$F$$ is the comma category $$F \downarrow \Delta$$, where $$\Delta: \mathsf{C} \to {\mathsf{Fun}}(\mathsf{J}, \mathsf{C})$$ is the diagonal functor sending $$N\in {\operatorname{Ob}}(\mathsf{C})$$ to the constant functor to $$N$$: \begin{align*} \Delta(N):\mathsf{J} &\to \mathsf{C} \\ X &\mapsto N .\end{align*} \item The \textbf{comma category} generalizes slice categories: given categories and functors \begin{align*} \mathsf{A} \mapsto{S} \mathsf{C} \mapsfrom{T} \mathsf{B} ,\end{align*} the comma category $$S\downarrow T$$ is given by triples $$(A, B, h: S(A)\to T(B))$$ making the obvious diagrams commute: \end{itemize} \begin{center} \begin{tikzcd} {A_0} && {A_1} &&& {S(A_0)} && {S(A_1)} \\ && {} \\ {B_0} && {B_1} &&& {T(B_0)} && {T(B_1)} \arrow["{S(f)}", from=1-6, to=1-8] \arrow["{h_1}", from=1-8, to=3-8] \arrow["{h_1}"', from=1-6, to=3-6] \arrow["{T(g)}"', from=3-6, to=3-8] \arrow["f", from=1-1, to=1-3] \arrow["g", from=3-1, to=3-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsOSxbNSwwLCJTKEFfMCkiXSxbNSwyLCJUKEJfMCkiXSxbNywwLCJTKEFfMSkiXSxbNywyLCJUKEJfMSkiXSxbMCwwLCJBXzAiXSxbMiwwLCJBXzEiXSxbMCwyLCJCXzAiXSxbMiwyLCJCXzEiXSxbMiwxXSxbMCwyLCJTKGYpIl0sWzIsMywiaF8xIl0sWzAsMSwiaF8xIiwyXSxbMSwzLCJUKGcpIiwyXSxbNCw1LCJmIl0sWzYsNywiZyJdXQ==}{Link to Diagram} \end{quote} Taking $$\mathsf{C} = A$$, $$S = \operatorname{id}_{\mathsf{A}}$$, and $$\mathsf{B} \coloneqq{\operatorname{pt}}$$ to be a 1-object category with only the identity morphism forces $$X\coloneqq T({\operatorname{pt}}) \in {\operatorname{Ob}}(\mathsf{A})$$ to be a single object and $$(\mathsf{A} \downarrow X)$$ is the usual slice category over $$X$$. \end{definition} \hypertarget{problem-sets}{% \section{Problem Sets}\label{problem-sets}} \hypertarget{problem-set-1}{% \subsection{Problem Set 1}\label{problem-set-1}} \begin{remark} All problems are sourced from Hartshorne. \end{remark} \hypertarget{chapter-2-section-1}{% \subsection{Chapter 2, Section 1}\label{chapter-2-section-1}} \begin{remark} List of useful facts used: \begin{itemize} \tightlist \item Morphisms of sheaves commute with restrictions: if $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ then for any $$s\in {\mathcal{F}}(U)$$ and $$V \subseteq U$$, $$\mathop{\mathrm{Res}}(U, V)(\phi(s)) = \phi(\mathop{\mathrm{Res}}(U, V)(s))$$. \item $$\phi$$ is an isomorphism iff $$\phi_p$$ are all isomorphisms. \item Elements of stalks $${\mathcal{F}}_p:$$ equivalence classes $$[U, s\in {\mathcal{F}}(U)]$$. \item The induced map on stalks: $$\phi_p([U, s]) \coloneqq[U, \phi(U)(s)]$$. \item A surjection of sheaves need not induce a surjection on sections. \item The colimit diagram: \end{itemize} \begin{center} \begin{tikzcd} & \bullet \\ \vdots && \vdots \\ {U_1} && {F(U_1)} \\ &&&& {\forall O} && {\colim_{i} F(U_i)} \\ {U_2} && {F(U_2)} \\ \vdots && \vdots \\ & \bullet \arrow["f", from=5-1, to=3-1] \arrow["{F(f)}"', from=3-3, to=5-3] \arrow["{\psi_2}"', from=5-3, to=4-7] \arrow["{\psi_1}", from=3-3, to=4-7] \arrow["{\psi'_1}"', from=3-3, to=4-5] \arrow["{\psi'_2}", from=5-3, to=4-5] \arrow["{\exists !}", dashed, from=4-5, to=4-7] \arrow[dotted, no head, from=1-2, to=7-2] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsMTIsWzAsMiwiVV8xIl0sWzAsNCwiVV8yIl0sWzIsMiwiRihVXzEpIl0sWzIsNCwiRihVXzIpIl0sWzAsMSwiXFx2ZG90cyJdLFswLDUsIlxcdmRvdHMiXSxbMiw1LCJcXHZkb3RzIl0sWzIsMSwiXFx2ZG90cyJdLFs2LDMsIlxcY29saW1fe2l9IEYoVV9pKSJdLFs0LDMsIlxcZm9yYWxsIE8iXSxbMSwwLCJcXGJ1bGxldCJdLFsxLDYsIlxcYnVsbGV0Il0sWzEsMCwiZiJdLFsyLDMsIkYoZikiLDJdLFszLDgsIlxccHNpXzIiLDJdLFsyLDgsIlxccHNpXzEiXSxbMiw5LCJcXHBzaSdfMSIsMl0sWzMsOSwiXFxwc2knXzIiXSxbOSw4LCJcXGV4aXN0cyAhIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzEwLDExLCIiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkb3R0ZWQifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dXQ==}{Link to Diagram} \end{quote} \begin{itemize} \item Colimits are initial co-cones, where $$I$$ is initial if $$I\to X$$ for any $$X$$. AKA direct limits. \item Filtered colimits commute with finite limits. \begin{itemize} \tightlist \item In particular, monomorphisms are pullbacks, so finite limits, and stalks are filtered colimits. So injections of sheaves induce injections on stalks. \end{itemize} \end{itemize} \end{remark} \begin{remark} Recommended problems: \begin{itemize} \tightlist \item 1.1 \item 1.2 \item 1.3 \item 1.4 \item 1.5 \end{itemize} \end{remark} \begin{problem}[1.1] Let $$A$$ be an abelian group, and define the \emph{constant presheaf} associated to $$A$$ on the topological space $$X$$ to be the presheaf $$U \mapsto A$$ for all $$U \neq \emptyset$$, with restriction maps the identity. Show that the constant sheaf $${\mathcal{A}}$$ defined in Hartshorne is the sheafification of this presheaf. \end{problem} \begin{solution} Let $$X\in {\mathsf{Top}}$$ be a space. Recapping the definitions, define the constant presheaf as \begin{align*} \underline{A}^{\mathsf{pre}}(U) \coloneqq \begin{cases} A & U\neq \emptyset \\ 0 & \text{else}. \end{cases} \quad \operatorname{res}^1(U, V) \coloneqq \begin{cases} \operatorname{id}_A & U\neq \emptyset \\ 0 & \text{else}. \end{cases} .\end{align*} Then define the constant \emph{sheaf} as \begin{align*} \underline{A}(U) \coloneqq\mathop{\mathrm{Hom}}_{{\mathsf{Top}}}(U, A)\quad \operatorname{res}^2(U, V)(f) \coloneqq{ \left.{{f}} \right|_{{V}} } .\end{align*} We're then tasked with finding a morphism of sheaves \begin{align*} \Psi: (\underline{A}^{\mathsf{pre}})^+ \xrightarrow{\sim} \mathop{\mathrm{Hom}}_{{\mathsf{Top}}}({-}, A) ,\end{align*} which we'll also want to have an inverse morphism and this define an isomorphism in $${\mathsf{Sh}}(X)$$. We'll use the implicitly stated fact in Hartshorne that $$\mathop{\mathrm{Hom}}_{{\mathsf{Top}}}(U, A) = A^{\oplus n}$$ where $$n \coloneqq\# \pi_0(X)$$ is the number of connected components of $$U$$. Suppose first that $$n=1$$, so $$X$$ is connected, and define the following morphism of groups: \begin{align*} \Psi_U: (\underline{A}^{\mathsf{pre}})(U) = A &\longrightarrow\mathop{\mathrm{Hom}}_{\mathsf{Top}}(U, A)\\ a_0 &\mapsto \left\{ { \begin{aligned} \varphi_{a_0}: U \to A \\ x \mapsto a_0, \end{aligned} } \right. \end{align*} which maps a group element $$a_0$$ to the constant function on $$U$$ sending every point to $$a_0 \in A$$. The claim is that the following diagram commutes in the category $$\underset{ \mathsf{pre} } {\mathsf{Sh} }(X)$$ (in both directions) for all $$U$$ and $$V$$: \begin{center} \begin{tikzcd} && {f(U)} && f \\ && {a_0} && { \begin{aligned} \varphi_{a_0}: U &\to A \\ x &\mapsto a_0 \end{aligned} } \\ U && {(\underline{A}^{\mathsf{pre}})(U) = A} && {\mathop{\mathrm{Hom}}_{\mathsf{Top}}(U, A)} \\ \\ V && {(\underline{A}^{\mathsf{pre}})(V) = A} && {\mathop{\mathrm{Hom}}_{\mathsf{Top}}(V, A)} \\ && {a_0} && { \begin{aligned} \varphi_{a_0}: V &\to A \\ x &\mapsto a_0 \end{aligned} } \\ && {f(V)} && f \arrow[hook, from=5-1, to=3-1] \arrow["{\Psi_U}", from=3-3, to=3-5] \arrow["{\Psi_V}", from=5-3, to=5-5] \arrow["{\operatorname{res}^1(U, V)}"', from=3-3, to=5-3] \arrow["{\operatorname{res}^2(U, V)}", from=3-5, to=5-5] \arrow[maps to, from=2-3, to=2-5] \arrow[maps to, from=1-5, to=1-3] \arrow[maps to, from=7-5, to=7-3] \arrow[maps to, from=6-3, to=6-5] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=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}{Link to Diagram} \end{quote} Here we've specified simultaneously what $$\Psi$$ and $$\Psi^{-1}$$ prescribe on opens $$U, V$$, and abuse notation slightly by writing $$\mathop{\mathrm{Hom}}_{{\mathsf{Top}}}({-}, A)$$ for the sheaf it represents and its underlying presheaf. \begin{itemize} \item That this commutes follows readily, since running the diagram counterclockwise yields $$\operatorname{res}^1(U, V) = \operatorname{id}_A$$, so the composition \begin{align*} (A \xrightarrow{\operatorname{res}^1(U, V)} A \xrightarrow{\Psi_V} \mathop{\mathrm{Hom}}(V, A)) = (A \xrightarrow{\Psi_V} \mathop{\mathrm{Hom}}(V, A)) \end{align*} sends an element $$a_0\in A$$ to the constant function $$\varphi_{a_0, V}: V\to A$$. Running the diagram clockwise yields \begin{align*} (A \xrightarrow{\Psi_U} \mathop{\mathrm{Hom}}(U, A) \xrightarrow{\operatorname{res}^2(U, V)} \mathop{\mathrm{Hom}}(V, A)) ,\end{align*} which sends $$a_0$$ to the constant function $$\varphi_{a_0, U}: U\to A$$ sending everything to $$a_0$$, which then gets sent to $${ \left.{{\varphi_{a_0, U}}} \right|_{{V}} }: V\to A$$ sending everything to $$a$$. Since $${ \left.{{\varphi_{a_0}}} \right|_{{V}} }(x) = \varphi_{a_0, V}(x) = a$$ for every $$x\in U$$, these functions are equal. \item That the reverse maps $$\Psi_U^{-1}$$ are well-defined follows from the fact that $$U$$ is connected: the continuous image of a connected set is connected. Since $$A$$ is given the discrete topology, any subset with 2 or more elements in disconnected, so each function $$f\in \mathop{\mathrm{Hom}}(U, A)$$ is necessarily a constant function and $$f(U) = \left\{{a}\right\}$$ is a singleton. \item $$\Psi_U, \Psi_U^{-1}$$ clearly compose to the identity in either order, so $$\Psi_U$$ defines an isomorphism of abelian groups. \end{itemize} As a consequence, we get a well-defined morphism of presheaves $$\underline{A}^{\mathsf{pre}}({-}) \to { \left.{{ \mathop{\mathrm{Hom}}({-}, A)}} \right|_{{ \underset{ \mathsf{pre} } {\mathsf{Sh} }}} }$$, and by the sheafification adjunction we can lift this to a morphism of sheaves: \begin{align*} \adjunction{{\mathcal{F}}\mapsto {\mathcal{F}}^+ }{{\mathcal{G}}\mapsto { \left.{{{\mathcal{F}}}} \right|_{{ \underset{ \mathsf{pre} } {\mathsf{Sh} }}} } }{ \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)}{{\mathsf{Sh}}(X)} ,\end{align*} which reads \begin{align*} \mathop{\mathrm{Hom}}_{ \underset{ \mathsf{pre} } {\mathsf{Sh} }}({\mathcal{F}}, { \left.{{{\mathcal{G}}}} \right|_{{ \underset{ \mathsf{pre} } {\mathsf{Sh} }}} }) &\xrightarrow{\sim} \mathop{\mathrm{Hom}}_{{\mathsf{Sh}}}({\mathcal{F}}^+, {\mathcal{G}}) \\ \Psi &\mapsto \tilde \Psi ,\end{align*} and since $$\Psi$$ was an isomorphism, so is $$\tilde \Psi$$. \begin{quote} It remains to handle the $$n\geq 2$$, case when (say) $$U = U_1 {\textstyle\coprod}U_2$$ has more than 1 connected component. Actually, is it even true that adjunctions preserve isomorphisms\ldots? Todo: help?? \end{quote} \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} Alternatively, consider the map $$\Psi$$ defined on presheaves -- by the universal property, we get some sheaf morphism $$\tilde\Psi$$, which we can show is an isomorphism by showing its induced map on stalks is an isomorphism. This amounts to showing the following map is a group isomorphism: \begin{align*} \Psi_p: (\underline{A}^{\mathsf{pre}}({-}))_p \xrightarrow{\sim} \mathop{\mathrm{Hom}}_{\mathsf{Top}}({-}, A)_p .\end{align*} First we identify the LHS: \begin{align*} (\underline{A}^{\mathsf{pre}}({-}))_p \coloneqq\colim_{U\ni p} \underline{A}^{\mathsf{pre}}(U) = \colim_{U\ni p} A = A .\end{align*} (todo: show $$A$$ satisfies the universal property for a colimit) Identifying the RHS, we have equivalence classes $$[U\ni p, s: U\to A]$$ \begin{itemize} \item Injectivity: that $$\Psi_p$$ is injective follows from the fact that $$\ker \psi_p \coloneqq\left\{{a\in A {~\mathrel{\Big|}~}\Psi_p(a) = e}\right\}$$, where $$e$$ is the identity in the right-hand side stalk, which is represented by the class $$[U, f_e:U\to A]$$ where $$f_e(x) \coloneqq e_A$$, the identity of $$A$$, for every $$x\in U$$. \item Surjectivity: that $$\Psi_p$$ is surjective follows from the fact that every fixed $$f: U\to A$$ for $$A$$ discrete is constant on connected components. Use that $$p$$ is contained in a connected component $$U_1 \ni p$$, then $$[U, f] \sim [U_1, { \left.{{f}} \right|_{{U_1}} }] \coloneqq[U_1, g]$$ to get that $$g$$ is now a constant function of $$U_1$$. So $$g(x) = a$$ for some $$a\in A$$, so $$g = \Psi_p(a)$$ is in the image. \end{itemize} \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} Alternatively: \begin{itemize} \tightlist \item Show that $$\underline{A}$$ satisfies the universal property of $$(\underline{A}^{\mathsf{pre}})^+$$: we need to produce a morphism $$\theta: (\underline{A}^{\mathsf{pre}}) \to \underline{A}$$ such that for any $${\mathcal{G}}\in {\mathsf{Sh}}(X)$$ and morphism of presheaves $$\varphi: \underline{A}^{\mathsf{pre}}\to { \left.{{{\mathcal{G}}}} \right|_{{{\mathsf{pre}}}} }$$ we can produce a unique morphism $$\tilde \varphi$$ of sheaves making the following diagram commute: \end{itemize} \begin{center} \begin{tikzcd} {\underline{A}^{{\mathsf{pre}}}} && {{ \left.{{{\mathcal{G}}}} \right|_{{{\mathsf{pre}}}} }} && {\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)} \\ \\ {\underline{A}} && {\mathcal{G}}&& {\in{\mathsf{Sh}}(X)} \arrow["\varphi", from=1-1, to=1-3] \arrow["\theta"', from=1-1, to=3-1] \arrow["{{ \left.{{{-}}} \right|_{{{\mathsf{pre}}}} }}"', from=3-3, to=1-3] \arrow["{\exists! \tilde \varphi}"', dashed, from=3-1, to=3-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNixbMCwyLCJcXHVse0F9Il0sWzAsMCwiXFx1bHtBfV57XFxwcmV9Il0sWzIsMCwiXFxyb3tcXG1jZ317XFxwcmV9Il0sWzIsMiwiXFxtY2ciXSxbNCwwLCJcXGluIFxcUHJlc2goWCkiXSxbNCwyLCJcXGluXFxTaChYKSJdLFsxLDIsIlxcdGhldGEiXSxbMSwwLCJcXHRoZXRhIiwyXSxbMywyLCJcXHJve1xcd2FpdH17XFxwcmV9IiwyXSxbMCwzLCJcXGV4aXN0cyEgXFx0aWxkZSBcXHRoZXRhIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d}{Link to Diagram} \end{quote} \begin{itemize} \item To define $$\tilde \varphi$$, it suffices to define morphisms of the form \begin{align*} \tilde\varphi(U): \underline{A}(U) &\to{\mathcal{G}}(U) \\ f & \mapsto \tilde\varphi(U)(f) \end{align*} \item Take a map $$f\in \underline{A}(U) \coloneqq\mathop{\mathrm{Hom}}_{\mathsf{Top}}(U, A)$$. Write $$U \coloneqq{\textstyle\coprod}U_i$$ as a union of connected components. Use that $$f$$ is constant on connected components since $$A$$ is discrete, so $$f(U_i) = a_i$$ for some elements $$a_i \in A \in {\mathsf{Ab}}{\mathsf{Grp}}$$. \item Plug the $$U_i$$ into $$\underline{A}^{\mathsf{pre}}$$ to get morphisms \begin{align*} \varphi(U_i): \underline{A}^{\mathsf{pre}}(U_i)= A \to { \left.{{{\mathcal{G}}}} \right|_{{{\mathsf{pre}}}} }(U_i) && \in {\mathsf{Ab}}{\mathsf{Grp}} \end{align*} \item Write $$b_i \coloneqq\varphi(U_i)(a_i) \in { \left.{{{\mathcal{G}}}} \right|_{{{\mathsf{pre}}}} }(U_i) = {\mathcal{G}}(U_i)$$. \end{itemize} \begin{center} \begin{tikzcd} {a_i} &&& {b_i \coloneqq\varphi(U_i)(a_i)} \\ & {\underline{A}^{{\mathsf{pre}}}(U_i) = A} && {{ \left.{{{\mathcal{G}}}} \right|_{{{\mathsf{pre}}}} }(U_i)} && {\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)} \\ \\ & {\mathop{\mathrm{Hom}}(U_i, A)} && {{\mathcal{G}}(U_i)} && {\in{\mathsf{Sh}}(X)} \\ & f && {b_i} && {} \arrow["{\varphi(U_i)}", from=2-2, to=2-4] \arrow["\theta"', from=2-2, to=4-2] \arrow["{{ \left.{{{-}}} \right|_{{{\mathsf{pre}}}} }}"', from=4-4, to=2-4] \arrow["{\exists! \tilde \varphi}"', dashed, from=4-2, to=4-4] \arrow[curve={height=-24pt}, dashed, maps to, from=5-2, to=1-1] \arrow[dashed, maps to, from=1-1, to=1-4] \arrow[maps to, from=5-2, to=5-4] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsMTEsWzEsMywiXFxIb20oVV9pLCBBKSJdLFsxLDEsIlxcdWx7QX1ee1xccHJlfShVX2kpID0gQSJdLFszLDEsIlxccm97XFxtY2d9e1xccHJlfShVX2kpIl0sWzMsMywiXFxtY2coVV9pKSJdLFs1LDEsIlxcaW4gXFxQcmVzaChYKSJdLFs1LDMsIlxcaW5cXFNoKFgpIl0sWzUsNF0sWzEsNCwiZiJdLFswLDAsImFfaSJdLFszLDAsImJfaSBcXGRhIFxcdmFycGhpKFVfaSkoYV9pKSJdLFszLDQsImJfaSJdLFsxLDIsIlxcdmFycGhpKFVfaSkiXSxbMSwwLCJcXHRoZXRhIiwyXSxbMywyLCJcXHJve1xcd2FpdH17XFxwcmV9IiwyXSxbMCwzLCJcXGV4aXN0cyEgXFx0aWxkZSBcXHZhcnBoaSIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFs3LDgsIiIsMCx7ImN1cnZlIjotNCwic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9LCJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbOCw5LCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn0sImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFs3LDEwLCIiLDIseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV1d}{Link to Diagram} \end{quote} \begin{itemize} \item Since $${\mathcal{G}}$$ is in fact a sheaf, by unique gluing there exists a unique element $$b \in {\mathcal{G}}(U)$$ such that $${ \left.{{b}} \right|_{{U_i}} } = b_i$$. So define $$\tilde\varphi(U)(f) \coloneqq b$$. \item Now define the map $$\theta: \underline{A}^{\mathsf{pre}}(U_i) \to \mathop{\mathrm{Hom}}(U_i, A)$$ sending $$a_i$$ to the constant function $$f_{i}(x)\coloneqq a_i$$. Since $$\underline{A}$$ is a sheaf, there is a well defined $$F\in \mathop{\mathrm{Hom}}(U, A)$$ such that $${ \left.{{F}} \right|_{{U_i}} } = f_i$$. So for $$a\in \underline{A}^{\mathsf{pre}}(U)$$ set $$\theta(a) = F \in \underline{A}(U)$$. \item This makes the relevant diagram commute: if $$a\in A = \underline{A}^{\mathsf{pre}}(U)$$, then $$b\coloneqq\phi(U)(a) \in {\mathcal{G}}(U)$$. On the other hand, $$\theta(a)$$ is the constant function $$f_a: x\mapsto a$$ (on every connected component of $$U$$), and setting $$F \coloneqq\tilde\phi(f_a)\in {\mathcal{G}}(U)$$, we have $$F \coloneqq b$$. \end{itemize} \end{solution} \begin{problem}[1.2] \begin{enumerate} \def\labelenumi{(\alph{enumi})} \item For any morphism of sheaves $$\varphi: {\mathcal{F}}\rightarrow {\mathcal{G}}$$, show that for each point $$p$$ that $$\ker (\varphi)_{p}=$$ $$\operatorname{ker}\left(\varphi_{p}\right)$$ and $$\operatorname{im}(\varphi)_{p} = \operatorname{im}\left(\varphi_{p}\right)$$. \item Show that $$\varphi$$ is injective (resp. surjective) if and only if the induced map on the stalks $$\varphi_{p}$$ is injective (resp. surjective) for all $$p$$. \item Show that a sequence of sheaves and morphisms \begin{align*} \cdots {\mathcal{F}}^{i-1} \xrightarrow{\varphi^{i-1}} {\mathcal{F}}^i \xrightarrow{\varphi^{i}} {\mathcal{F}}^{i+1} \to \cdots \end{align*} is exact if and only if for each $$P \in X$$ the corresponding sequence of stalks is exact as a sequence of abelian groups. \end{enumerate} \end{problem} \begin{proof}[of 1, kernels] \envlist \begin{itemize} \tightlist \item Write $$K\in {\mathsf{Sh}}(X)$$ for the kernel sheaf $$U \mapsto \ker \qty{ {\mathcal{F}}(U) \xrightarrow{\phi(U)} {\mathcal{G}}(U) }$$, \item We then want to show $$K_p = \ker\qty{{\mathcal{F}}_p \xrightarrow{\phi_p} {\mathcal{G}}_p}$$, an equality of sets in $${\mathsf{Ab}}{\mathsf{Grp}}$$. So we just do it! \begin{itemize} \tightlist \item Addendum: this works because both are subsets of the same abelian group, $${\mathcal{F}}_p$$. \end{itemize} \item We can write \begin{align*} \phi_p: {\mathcal{F}}_p &\to {\mathcal{G}}_p \\ [U, s] &\mapsto [U, \phi(U)(s)] ,\end{align*} and note that the zero element in a stalk is the equivalence class $$[U, 0]$$ where $$0\in {\mathsf{Ab}}{\mathsf{Grp}}$$ is the zero object. Thus \begin{align*} \ker \phi_p &\coloneqq\left\{{ x\in {\mathcal{F}}_p {~\mathrel{\Big|}~}\phi_p(x) = 0 \in {\mathcal{G}}_p }\right\} \\ & = \left\{{ [U, s] \in {\mathcal{F}}_p {~\mathrel{\Big|}~}[U, \phi(U)(s)] = [U, 0] }\right\} \\ & = \left\{{ [U, s] \in {\mathcal{F}}_p {~\mathrel{\Big|}~}\phi(U)(s) = 0 }\right\} \\ & = \left\{{ [U, s] \in {\mathcal{F}}_p {~\mathrel{\Big|}~}s \in \ker \phi(U) }\right\} \\ &= \left\{{ [U, s] {~\mathrel{\Big|}~}s\in \ker{\qty{{\mathcal{F}}(U) \xrightarrow{\phi(U)} {\mathcal{G}}(U) } } }\right\} \\ &\coloneqq\left\{{[U, s] {~\mathrel{\Big|}~}s\in K(U)}\right\} \\ &\coloneqq K_p .\end{align*} \end{itemize} \end{proof} \begin{proof}[of 1, images] \envlist \begin{itemize} \tightlist \item Write $${\mathcal{I}}$$ for the sheaf $$\operatorname{im}\phi$$ which sends $$U\mapsto \operatorname{im}\qty{{\mathcal{F}}(U) \xrightarrow{\phi(U)} {\mathcal{G}}(U)}$$. \item We want to show $${\mathcal{I}}_p = \operatorname{im}\qty{{\mathcal{F}}_p \xrightarrow{\phi_p} {\mathcal{G}}_p}$$, where both are subsets of $${\mathcal{G}}_p$$. \item So we show set equality: \begin{align*} \operatorname{im}(\phi_p) &= \left\{{ y\in {\mathcal{G}}_p {~\mathrel{\Big|}~}\exists x\in {\mathcal{F}}_p,\, \phi_p(x) = y }\right\} \\ &= \left\{{ [U, t] \in {\mathcal{G}}_p {~\mathrel{\Big|}~}\exists [U, s] \in {\mathcal{F}}_p,\, \phi_p([U, s]) = [U, t] }\right\} \\ &= \left\{{ [U, t] \in {\mathcal{G}}_p {~\mathrel{\Big|}~}\exists s\in {\mathcal{F}}(U),\, \phi(U)(s) = t }\right\} \\ &= \left\{{ [U, t] \in {\mathcal{G}}_p {~\mathrel{\Big|}~}t\in \operatorname{im}\qty{{\mathcal{F}}(U) \xrightarrow{\phi(U)} {\mathcal{G}}(U) }}\right\} \\ &\coloneqq\left\{{ [U, t] \in {\mathcal{G}}_p {~\mathrel{\Big|}~}t \in {\mathcal{I}}(U) }\right\} \\ &\coloneqq{\mathcal{I}}_p .\end{align*} \end{itemize} \end{proof} \begin{proof}[of 2, injectivity] $$\implies$$: \begin{itemize} \item Use that injectivity of a morphism $$\phi$$ of sheaves is \emph{defined} to hold exactly when $$\ker \phi = 0$$ is the constant zero sheaf. \item Now use (1): \begin{align*} 0 = \ker(\phi) \implies 0 = \ker(\phi)_p = \ker(\phi_p) && \forall p .\end{align*} \item If $$\ker \phi = 0$$, so $$\phi$$ is injective, then $$\ker \phi_p = 0$$ for all $$p$$, so $$\ker \phi_p$$ is injective. \end{itemize} $$\impliedby$$: \begin{itemize} \tightlist \item Conversely, suppose $$\ker \phi_p = 0$$ for all $$p$$; we want to show $$\ker \phi(U) = 0$$ for all $$U$$. \item So fix $$U\ni p$$, we want to show \begin{align*} s\in K(U) \coloneqq\ker\qty{{\mathcal{F}}(U) \xrightarrow{\phi(U)} {\mathcal{G}}(U)} \implies s = 0 \in {\mathcal{F}}(U) .\end{align*} \item We have $$\phi(U)(s) = 0$$, so \begin{align*} \phi_p([U, s]) \coloneqq[U, \phi(U)(s)] = [U, 0] \in {\mathcal{G}}_p \implies [U, s] \in \ker (\phi_p) .\end{align*} \item By injectivity of $$\phi_p$$, we have $$[U, s] = 0 \in {\mathcal{F}}_p$$. \item So there is some open $$U_p$$ with $$U \supseteq U_p \ni p$$ and $$\mathop{\mathrm{Res}}(U, U_p)(s) = 0$$ in $${\mathcal{F}}(U_p)$$. \item Then $$\left\{{U_p }\right\}_{p\in U} \rightrightarrows U$$, and since $${\mathcal{F}}$$ is a sheaf, by existence of gluing these glue to an $$F \in {\mathcal{F}}(U)$$ with $$\mathop{\mathrm{Res}}(U, U_p)(F) = 0$$ for each $$p$$. By uniqueness of gluing, $$0 = F = s$$. \end{itemize} \end{proof} \begin{proof}[of 2, surjectivity] $$\implies$$: \begin{itemize} \tightlist \item Suppose $$\phi$$ is surjective, then by definition $$\operatorname{im}\phi = {\mathcal{G}}$$ is an equality of sheaves. \item So $$(\operatorname{im}\phi)(U) = {\mathcal{G}}(U)$$ for all $$U$$. \item Let $$[U, t]\in {\mathcal{G}}_p$$, so $$t\in {\mathcal{G}}(U)$$. \item Then $$t\in (\operatorname{im}\phi)(U)$$, so there exists an $$s\in {\mathcal{F}}(U)$$ such that $$\phi(U)(s) = t$$. \item Then $$[U, s] \mapsto [U, \phi(U)(s)] = [U, t]$$, under $$\phi_p$$, making $$\phi_p$$ surjective. \end{itemize} $$\impliedby$$: \begin{itemize} \item Suppose $$\phi_p$$ is surjective for all $$p$$, fix $$U$$, and let $$t \in {\mathcal{G}}(U)$$. We want to produce an $$s\in {\mathcal{F}}(U)$$ such that $$\phi(U)(s) = t$$. \item For $$p\in U$$, the image of $$t$$ in the stalk of $${\mathcal{G}}$$ is of the form $$[U_p, t_p] \in {\mathcal{G}}_p$$ where $$t_p \in {\mathcal{G}}(U_p)$$. \item Since $$\phi_p: {\mathcal{F}}_p \twoheadrightarrow{\mathcal{G}}_p$$, we can find some pair $$[U_p, s_p]$$ mapping to $$[U_p, t_p]$$ under $$\phi_p$$, so $$\phi(U_p)(s_p) = t_p$$. \begin{itemize} \tightlist \item Note that $$\mathop{\mathrm{Res}}(U, U_p)(t) = t_p$$. \item Note: may need to pull back to some $$\tilde U_p$$, then take a common refinement in both germs? \end{itemize} \item Now $$\left\{{U_p}\right\}_{p\in U}\rightrightarrows U$$, so using existence of gluing for $${\mathcal{F}}$$ we have some $$s\in {\mathcal{F}}(U)$$ with $$\mathop{\mathrm{Res}}(U, U_p)(s) = s_p$$ for all $$p$$. \item Claim: $$\phi(U)(s) = t$$. \begin{align*} \mathop{\mathrm{Res}}(U, U_p)( \phi(s) ) &= \phi(\mathop{\mathrm{Res}}(U, U_p)(s)) \\ &= \phi(s_p) \\ &= t_p \\ &= \mathop{\mathrm{Res}}(U, U_p)(t) && \forall p\in U ,\end{align*} so $$\phi(s) = t$$ by uniqueness of gluing of $${\mathcal{G}}$$. \end{itemize} \end{proof} \begin{proof}[of 3, exactness] $$\implies$$: Assuming exactness of sheaves, \begin{align*} \ker({\mathcal{F}}^{i+1}) = \operatorname{im}({\mathcal{F}}^{i}) \iff \ker({\mathcal{F}}^{i+1})_p = \operatorname{im}({\mathcal{F}}^{i})_p && \forall p .\end{align*} $$\impliedby$$: Assuming exactness on stalks, write \begin{align*} \ker({\mathcal{F}}^{i+1})_p &= \ker({\mathcal{F}}^{i+1}_p) && \text{by 1 } \\ &= \operatorname{im}({\mathcal{F}}^{i}_p) && \text{exactness, by assumption} \\ &= \operatorname{im}({\mathcal{F}}^{i})_p && \text{by 1} .\end{align*} \end{proof} \begin{problem}[1.3] \begin{enumerate} \def\labelenumi{(\alph{enumi})} \item Let $$\varphi: {\mathcal{F}}\to{\mathcal{G}}$$ be a morphism of sheaves on $$X$$. Show that $$\varphi$$ is surjective if and only if the following condition holds: For every open set $$U \subseteq X$$, and for every $$s\in {\mathcal{G}}(U)$$, there is a cover $$\left\{{U_i}\right\}$$ of $$U$$ and elements $$t_i \in {\mathcal{F}}(U_i)$$ such that $$\varphi(t_i) = { \left.{{s}} \right|_{{U_i}} }$$ for all $$i$$. \item Give an example of a surjective morphism of sheaves $$\varphi: {\mathcal{F}}\rightarrow {\mathcal{G}}$$, and an open set $$U$$ such that $$\varphi(U): {\mathcal{F}}(U) \rightarrow {\mathcal{G}}(U)$$ is not surjective. \end{enumerate} \end{problem} \begin{proof}[of 1] $$\implies$$: \begin{itemize} \tightlist \item If $$\phi: {\mathcal{F}}\twoheadrightarrow{\mathcal{G}}$$, then $$\phi_p: {\mathcal{F}}_p \twoheadrightarrow{\mathcal{G}}_p$$ for all $$p$$, since $$\operatorname{im}(\phi_p) = (\operatorname{im}\phi)_p = {\mathcal{G}}_p$$, using problem 1.2. \item Fix $$U \subseteq X$$ and $$s\in {\mathcal{G}}(U)$$, we want \begin{itemize} \tightlist \item To produce a cover $$\left\{{U_i}\right\}\rightrightarrows U$$, \item To find $$t_i\in {\mathcal{F}}(U_i)$$, and \item To show that $$\phi(t_i) = \mathop{\mathrm{Res}}(U, U_i)(s)$$ for all $$i$$. \end{itemize} \item Fix $$p$$, and take the image of $$s$$ in the stalk of $${\mathcal{G}}$$ to get $$[U_p, s_p] \in {\mathcal{G}}_p$$ with $$s_p \in {\mathcal{G}}(U_p)$$ and $$\mathop{\mathrm{Res}}(U, U_p)(s) = s_p$$. Note that $$\left\{{U_p}\right\}_{p\in U}\rightrightarrows U$$. \item By surjectivity on stalks, these pull back to $$[U_p, t_p]\in {\mathcal{F}}_p$$ with $$t_p \in {\mathcal{F}}(U_p)$$ and $$\phi_p([U_p, t_p]) \coloneqq[U_p, \phi(U_p)(t_p)] = [U_p, s_p]$$. \item Then $$s_p \in \operatorname{im}({\mathcal{F}}(U_p) \xrightarrow{\phi(U_p)} {\mathcal{G}}(U_p ))$$ and $$\phi(t_p) = s_p = \mathop{\mathrm{Res}}(U, U_p)(s)$$. \end{itemize} $$\impliedby$$: \begin{itemize} \tightlist \item If $$\left\{{U_i}\right\}\rightrightarrows U$$ with $$\phi(t_i) = \mathop{\mathrm{Res}}(U, U_i)(s)$$ for all $$i$$, then the $$t_i$$ glue to a unique section $$t\in {\mathcal{F}}(U)$$ since $${\mathcal{F}}$$ is a sheaf. \item Moreover $$\mathop{\mathrm{Res}}(U, U_i)( \phi(t) ) = \phi(\mathop{\mathrm{Res}}(U, U_i)(t)) = \phi(t_i) = \mathop{\mathrm{Res}}(U, U_i)(s)$$ for all $$i$$, and by unique gluing for $${\mathcal{G}}$$ we have $$\phi(t) = s$$. \item So $$\phi(U): {\mathcal{F}}(U) \to {\mathcal{G}}(U)$$ is surjective for all $$U$$, making $$\operatorname{im}(\phi(U)) = {\mathcal{G}}(U)$$ \item So $$\operatorname{im}\phi = {\mathcal{G}}$$ as sheaves since they make the same assignment to every open set $$U$$, making $$\phi: {\mathcal{F}}\to{\mathcal{G}}$$ surjective by definition. \end{itemize} \end{proof} \begin{proof}[of 2] \envlist \begin{itemize} \item Take $$X \coloneqq\left\{{a,b,c}\right\}$$ a 3-point space with the topology $$\tau_X \coloneqq\left\{{\emptyset, \left\{{a}\right\}, \left\{{b}\right\}, \left\{{a,b}\right\}, X}\right\}$$. \item Take $${\mathcal{F}}\coloneqq\underline{A}$$ for some nontrivial $$A\in {\mathsf{Ab}}{\mathsf{Grp}}$$. We have the stalks \begin{itemize} \tightlist \item $${\mathcal{F}}_a = A$$ \item $${\mathcal{F}}_b = A$$ \item $${\mathcal{F}}_c = A$$ \end{itemize} \item Take $${\mathcal{G}}\coloneqq\underline{A}(a) \times \underline{A}(b)$$, the skyscraper sheaves at $$a$$ and $$b$$ respectively, where \begin{align*} \underline{A}(x)(U) \coloneqq \begin{cases} A & x\in U \\ 0 & \text{ else} . \end{cases} \end{align*} Note that the stalks are given by $$\underline{A}(x)_x = A$$ and $$\underline{A}(x)_y = 0$$ for $$y\neq x$$, so \begin{itemize} \tightlist \item $${\mathcal{G}}_a = A\times 0$$ \item $${\mathcal{G}}_b = 0 \times A$$ \item $${\mathcal{G}}_c = 0 \times 0$$. \end{itemize} \item Now define $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ by specifying $$\phi(U):{\mathcal{F}}(U) \to {\mathcal{G}}(U)$$ for all $$U$$ in the following way: \end{itemize} \begin{center} \begin{tikzcd} & {{\mathsf{Open}}(X)} &&&& {\underline{A}} &&&& {\underline{A}(a) \times \underline{A}(b)} \\ & {X = \left\{{a,b,c}\right\}} &&&& A &&&& {A^{\times 2}} \\ & {\left\{{a,b}\right\}} &&&& A &&&& {A^{\times 2}} \\ {\left\{{a}\right\}} && {\left\{{b}\right\}} && A && A && {A\times 0} && {0\times A} \\ & \emptyset &&&& 0 &&&& 0 \arrow[from=5-2, to=4-1] \arrow[from=5-2, to=4-3] \arrow[from=4-1, to=3-2] \arrow[from=4-3, to=3-2] \arrow[from=3-2, to=2-2] \arrow[from=2-6, to=3-6] \arrow[from=3-6, to=4-5] \arrow[from=3-6, to=4-7] \arrow[from=4-7, to=5-6] \arrow[from=4-5, to=5-6] \arrow[from=2-10, to=3-10] \arrow[from=3-10, to=4-9] \arrow[from=3-10, to=4-11] \arrow[from=4-11, to=5-10] \arrow[from=4-9, to=5-10] \arrow[color={rgb,255:red,92;green,214;blue,214}, curve={height=-18pt}, dashed, from=2-6, to=2-10] \arrow[color={rgb,255:red,92;green,214;blue,214}, curve={height=-18pt}, dashed, from=3-6, to=3-10] \arrow[color={rgb,255:red,92;green,214;blue,214}, curve={height=-18pt}, dashed, from=4-5, to=4-9] \arrow[color={rgb,255:red,92;green,214;blue,214}, curve={height=-18pt}, dashed, from=4-7, to=4-11] \arrow[color={rgb,255:red,92;green,214;blue,214}, curve={height=-18pt}, dashed, from=5-6, to=5-10] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=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}{Link to Diagram} \end{quote} \begin{itemize} \item Note that the induced maps on stalks are surjective, since $$\phi_p: A \to A, 0$$ is either the identity or the zero map. But e.g.~for $$\left\{{a, b}\right\}$$ we have $$A\mapsto A^{\times 2}$$, which can not be surjective. \begin{quote} Question: what is this map? Apparently its image is the diagonal\ldots? \end{quote} \end{itemize} \end{proof} \begin{problem}[1.4] \begin{enumerate} \def\labelenumi{(\alph{enumi})} \item Let $$\varphi: {\mathcal{F}}\to {\mathcal{G}}$$ be a morphism of presheaves such that $$\varphi(U): {\mathcal{F}}(U) \to {\mathcal{G}}(U)$$ is injective for each $$U$$. Show that the induced map $$\varphi^+: {\mathcal{F}}^+ \to {\mathcal{G}}^+$$of associated sheaves is injective. \item Use part (a) to show that if $$\varphi: {\mathcal{F}}\to{\mathcal{G}}$$ is a morphism of sheaves, then $$\operatorname{im}\varphi$$ can be naturally identified with a subsheaf of $${\mathcal{G}}$$, as mentioned in the text. \end{enumerate} \end{problem} \begin{proof}[of a] \envlist \begin{itemize} \tightlist \item $$\phi: {\mathcal{F}}\to {\mathcal{G}}$$ is injective iff $$\phi_p:{\mathcal{F}}_p \to {\mathcal{G}}_p$$ is injective for all $$p$$. \item Sheafification induces a map $$\phi^+: {\mathcal{F}}^+_p \to {\mathcal{G}}^+_p$$ \item The sheafification has the same stalks, so $${\mathcal{F}}^+_p = {\mathcal{F}}_p$$ and $${\mathcal{G}}^+_p = {\mathcal{G}}_p$$. \item So in fact $$\phi^+_p = \phi_p$$. Since $$\phi^+_p$$ is thus injective on all stalks, $$\phi^+$$ is injective on sheaves. \end{itemize} \end{proof} \begin{proof}[of b] \envlist \begin{itemize} \tightlist \item Noting that on opens $$(\operatorname{im}\phi)(U) \subseteq {\mathcal{G}}(U)$$ is an inclusion of abelian groups, so define a morphism of sheaves by $$\iota(U): (\operatorname{im}\phi)(U) \to {\mathcal{G}}(U)$$ using this inclusion. \begin{itemize} \tightlist \item By definition, it suffices to show $$\ker \iota = 0$$ as a sheaf. \item By 1.2.2, it suffices to show $$(\ker \iota)_p = 0$$ on all stalks. \item By 1.2.1, $$(\ker \iota)_p = \ker(\iota_p)$$, so it suffices to show $$\iota_p$$ is injective for all $$p$$. \end{itemize} \item Now use that \begin{align*} \ker(\iota_p) = \colim_{U\ni p} (\ker \phi)(\iota(U)) = \colim_{U\ni p} 0 = 0 ,\end{align*} since all of the $$\iota(U)$$ are injective, so 0 satisfies the universal property for this colimit. So we're done. \end{itemize} \end{proof} \begin{problem}[1.5] Show that a morphism of sheaves is an isomorphism if and only if it is both injective and surjective. \end{problem} \begin{proof}[?] \envlist Problem: surjections of sheaves don't induce surjections ons sections! \begin{itemize} \item $$\phi:{\mathcal{F}}\to{\mathcal{G}}$$ being injective means that $$(\ker \phi) = 0$$ as sheaves, and surjective means $$(\operatorname{im}\phi) = {\mathcal{G}}$$. \item Thus $$\phi(U): {\mathcal{F}}(U) \to {\mathcal{G}}(U)$$ is injective, since $$(\ker \phi)(U) = 0(U) = 0$$, and surjective since $$\operatorname{im}(\phi(U)) = (\operatorname{im}\phi)(U) = {\mathcal{G}}(U)$$. This $$\phi(U)$$ is an isomorphism in abelian groups, and has an left and right inverse $$\phi^{-1}(U): {\mathcal{G}}(U) \to {\mathcal{F}}(U)$$. \item So we have a diagram: \end{itemize} \begin{center} \begin{tikzcd} {{\mathcal{F}}(U)} && {{\mathcal{G}}(U)} && {{\mathcal{F}}(U)} \\ \\ {{\mathcal{F}}(V)} && {{\mathcal{G}}(V)} && {{\mathcal{F}}(V)} \arrow["{\phi(U)}", from=1-1, to=1-3] \arrow["{\phi(V)}", from=3-1, to=3-3] \arrow["{\mathop{\mathrm{Res}}_{{\mathcal{F}}}(U, V)}"', from=1-1, to=3-1] \arrow["{\mathop{\mathrm{Res}}_{{\mathcal{G}}}(U, V)}"{description}, from=1-3, to=3-3] \arrow["{\phi(U)^{-1}}", from=1-3, to=1-5] \arrow["{\phi(V)^{-1}}"', from=3-3, to=3-5] \arrow["{\mathop{\mathrm{Res}}_{{\mathcal{F}}}(U, V)}", from=1-5, to=3-5] \arrow["{\operatorname{id}_{{\mathcal{F}}(U)}}"{description}, curve={height=-30pt}, from=1-1, to=1-5] \arrow["{\operatorname{id}_{{\mathcal{F}}(V)}}"{description}, curve={height=30pt}, from=3-1, to=3-5] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNixbMCwwLCJcXG1jZihVKSJdLFswLDIsIlxcbWNmKFYpIl0sWzIsMCwiXFxtY2coVSkiXSxbMiwyLCJcXG1jZyhWKSJdLFs0LDAsIlxcbWNmKFUpIl0sWzQsMiwiXFxtY2YoVikiXSxbMCwyLCJcXHBoaShVKSJdLFsxLDMsIlxccGhpKFYpIl0sWzAsMSwiXFxSZXNfe1xcbWNmfShVLCBWKSIsMl0sWzIsMywiXFxSZXNfe1xcbWNnfShVLCBWKSIsMV0sWzIsNCwiXFxwaGkoVSlcXGludiJdLFszLDUsIlxccGhpKFYpXFxpbnYiLDJdLFs0LDUsIlxcUmVzX3tcXG1jZn0oVSwgVikiXSxbMCw0LCJcXGlkX3tcXG1jZihVKX0iLDEseyJjdXJ2ZSI6LTV9XSxbMSw1LCJcXGlkX3tcXG1jZihWKX0iLDEseyJjdXJ2ZSI6NX1dXQ==}{Link to Diagram} \end{quote} \begin{itemize} \tightlist \item Both squares form a morphism of sheaves, so the right square assembles to $$\phi^{-1}: {\mathcal{G}}\to{\mathcal{F}}$$ \item Moreover $$(\phi^{-1}\circ \phi)({\mathcal{F}})(U) = \operatorname{id}_{{\mathcal{F}}(U)}$$ and similarly in the other order, so $$\phi^{-1}\circ \phi= \operatorname{id}_{{\mathcal{F}}}$$ Similarly $$(\phi\circ \phi^{-1})({\mathcal{G}})(U) = \operatorname{id}_{{\mathcal{G}}(U)}$$ and $$(\phi^{-1}\circ \phi) = \operatorname{id}_{{\mathcal{G}}}$$. \item Then by definition an isomorphism of sheaves is a morphism with a two-sided inverse, so we're done. \end{itemize} \end{proof} \hypertarget{problem-set-2}{% \subsection{Problem Set 2}\label{problem-set-2}} \hypertarget{ii.1}{% \subsection{II.1}\label{ii.1}} \begin{exercise}[II.1.8] For any open $$U \subseteq X$$ show that the functor \begin{align*} {\mathsf{\Gamma}\qty{U, {-}} }: {\mathsf{Sh}}(X) \to {\mathsf{Ab}}{\mathsf{Grp}} \end{align*} is left-exact, but need not be exact. \end{exercise} \begin{exercise}[II.1.14] Let $${\mathcal{F}}\in {\mathsf{Sh}}(X)$$ and $$s\in {\mathcal{F}}(U)$$ be a section, and define \begin{align*} \mathop{\mathrm{supp}}s &\coloneqq\left\{{p\in U {~\mathrel{\Big|}~}s_p \neq 0}\right\} \subseteq U \\ \mathop{\mathrm{supp}}{\mathcal{F}}&\coloneqq\left\{{p\in X{~\mathrel{\Big|}~}{\mathcal{F}}_p\neq 0}\right\} \subseteq U ,\end{align*} where $$s_p$$ denotes the germ of $$s$$ in the stalk $${\mathcal{F}}_p$$. Show that $$\mathop{\mathrm{supp}}s$$ is closed in $$U$$ but $$\mathop{\mathrm{supp}}{\mathcal{F}}$$ need not be. \end{exercise} \begin{exercise}[II.1.17] Let $$X\in {\mathsf{Top}}, A\in {\mathsf{Ab}}{\mathsf{Grp}}, p\in X$$ and define the skyscraper sheaf as \begin{align*} \iota_p(A)(U) \coloneqq \begin{cases} A & p\in U \\ 0 & \text{else}. \end{cases} .\end{align*} Show that the stalk $$\iota_p(A)_q = A$$ when $$q\in { \operatorname{cl}} _X(\left\{{p}\right\})$$ and 0 otherwise, and that there is an equality of sheaves $$\iota_p(A) = \iota_*(\underline{A})$$ where $$\iota: { \operatorname{cl}} _X(\left\{{p}\right\}) \hookrightarrow X$$ is the inclusion. \end{exercise} \hypertarget{ii.2}{% \subsection{II.2}\label{ii.2}} \begin{exercise}[II.2.1] Let $$A\in \mathsf{Ring}$$ and $$X\coloneqq\operatorname{Spec}(A)$$, and for $$f\in A$$ let $$D(f) \coloneqq V(\left\langle{f}\right\rangle)^c$$. Show that there is an isomorphism of ringed spaces \begin{align*} (D(f), { \left.{{{\mathcal{O}}_X}} \right|_{{D(f)}} }) \xrightarrow{\sim} \operatorname{Spec}(A_f) .\end{align*} \end{exercise} \begin{exercise}[II.2.3] Note that $$(X, {\mathcal{O}}_X)\in {\mathsf{Sch}}$$ is \textbf{reduced} iff $${\mathcal{O}}_X(U)$$ has no nilpotents, and for $$A\in \mathsf{Ring}$$ define $$A^{{ \text{red} }}\coloneqq A/\sqrt{0}$$ to be $$A$$ modulo its ideal of nilpotents. \begin{enumerate} \def\labelenumi{\alph{enumi}.} \item Show that $$X$$ is reduced iff for every $$p\in X$$, the local ring $${\mathcal{O}}_{X, p}$$ has no nilpotents. \item Let $${\mathcal{O}}_X^{{ \text{red} }}$$ be the sheafification of $$U \mapsto {\mathcal{O}}_X(U)^{ \text{red} }$$. Show that $$X_{ \text{red} }\coloneqq(X, {\mathcal{O}}_X^{ \text{red} })$$ is a scheme, and there is a morphism of schemes $$X_{ \text{red} }\xrightarrow{{ \text{red} }} X$$ which induces a homeomorphism $${\left\lvert {X_{ \text{red} }} \right\rvert}\to {\left\lvert {X} \right\rvert}$$ on underlying topological spaces. \item Let $$X \xrightarrow{f} Y\in {\mathsf{Sch}}$$ with $$X$$ reduced. Show that there is a unique morphism $$X \xrightarrow{g} Y_{ \text{red} }$$ such that $$f$$ is the composition \begin{align*} (X \xrightarrow{f} Y) = (X \xrightarrow{g} Y_{ \text{red} }\xrightarrow{{ \text{red} }} Y ) .\end{align*} \end{enumerate} \end{exercise} \begin{exercise}[II.2.5] Describe $$\operatorname{Spec}{\mathbb{Z}}$$ and show it is terminal in $${\mathsf{Sch}}$$, i.e.~each $$X\in {\mathsf{Sch}}$$ admits a unique morphism $$X\to \operatorname{Spec}{\mathbb{Z}}$$. \end{exercise} \begin{exercise}[II.2.7] Let $$X\in {\mathsf{Sch}}$$ and for $$x\in X$$ let $${\mathcal{O}}_x$$ be the local ring at $$x$$ and $${\mathfrak{m}}_x$$ its maximal ideal. Let $$\kappa(x) \coloneqq{\mathcal{O}}_x/{\mathfrak{m}}_x$$ be the residue field at $$x$$. Then for $$k$$ any field, show that giving a morphism $$\operatorname{Spec}(k) \to X \in {\mathsf{Sch}}$$ is equivalent to giving a point $$x\in X$$ and an inclusion $$\kappa(x) \hookrightarrow k$$. \begin{align*} x^2 - y^q = 1 && x^p - y^2 = 1 .\end{align*} \begin{align*} x^2 - y^q = 1 && x^p - y^2 = 1 .\end{align*} \end{exercise} \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 \printbibliography[title=Bibliography] \end{document}