\newpage # II: Schemes > Note: there are many, many important notions tucked away in the exercises in this section. ## II.1: Sheaves $\star$ :::{.remark} \envlist - **Presheaves** $F$ of abelian groups: contravariant functors $F\in \Fun(\Open(X), \Ab\Grp)$. - Assigns every open $U \subseteq X$ some $F(U) \in \Ab\Grp$ - For $\iota_{VU}: V \subseteq U$, restriction morphisms $\phi_{UV}: F(U) \to F(V)$. - $F(\emptyset) = 0$, so $F(\initial) = \terminal$. - $\phi_{UU} = \id_{F(U)}$ - $W \subseteq V \subseteq U \implies \phi_{UW} = \phi_{VW} \circ \phi_{UV}$. - **Sections**: elements $s\in F(U)$ are sections of $F$ over $U$. Also notation $\Gamma(U; F)$ and $H^0(U; F)$, and the restrictions are written $\ro s V \da \phi_{UV}(s)$ for $s\in F(U)$. - **Sheaves**: presheaves $F$ which are completely determined by local data. Additional requirements on open covers $\mcv\covers U$: - If $s\in F(U)$ with $\ro{s}{V_i} = 0$ for all $i$ then $s\equiv 0 \in F(U)$. - Given $s_i\in F(V_i)$ where $\ro{s_i}{V_{ij}} = \ro{s_j}{V_{ij}} \in F(V_{ij})$ then $\exists s\in F(U)$ such that $\ro{s}{V_i} = s_i$ for each $i$, which is unique by the previous condition. - **Constant sheaf**: for $A\in \Ab\Grp$, define the constant sheaf \[ \ul{A}(U) \da \Top(U, A^\disc) .\] - **Stalks**: $F_p \da \colim_{U\ni p} F(U)$ along the system of restriction maps. - These are represented by pairs $(U, s)$ with $U\ni p$ an open neighborhood and $s\in F(U)$, modulo $(U, s)\sim (V, t)$ when $\exists W \subseteq U \intersect V$ with $\ro s w = \ro t w$. - **Germs**: a germ of a section of $F$ at $p$ is an elements of the stalk $F_p$. - **Morphisms of presheaves**: natural transformations $\eta\in \Mor_{\Fun}(F, G)$, i.e. for every $U, V$, components $\eta_U, \eta_V$ fitting into a diagram \begin{tikzcd} {\Open(X)} &&& \Ab\Grp \\ U && {F(U)} && {G(U)} \\ \\ V && {F(V)} && {G(V)} \arrow["{\eta_V}", from=4-3, to=4-5] \arrow["{\eta_U}", from=2-3, to=2-5] \arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{Res}_F(U, V)}", from=2-3, to=4-3] \arrow["{\mathrm{Res}_G(U, V)}", from=2-5, to=4-5] \arrow[""{name=1, anchor=center, inner sep=0}, hook, from=4-1, to=2-1] \arrow["{F, G}", shorten <=15pt, shorten >=15pt, Rightarrow, from=1, to=0] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwxLCJVIl0sWzAsMywiViJdLFsyLDEsIkYoVSkiXSxbNCwxLCJHKFUpIl0sWzIsMywiRihWKSJdLFs0LDMsIkcoVikiXSxbMCwwLCJcXE9wZW4oWCkiXSxbMywwLCJcXEFiXFxHcnAiXSxbNCw1LCJcXGV0YV9WIl0sWzIsMywiXFxldGFfVSJdLFsyLDQsIlxcbWF0aHJte1Jlc31fRihVLCBWKSJdLFszLDUsIlxcbWF0aHJte1Jlc31fRyhVLCBWKSJdLFsxLDAsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEyLDEwLCJGLCBHIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==) - A morphism of sheaves is exactly a morphism of the underlying presheaves. - Morphisms of sheaves $\eta: F\to G$ induce morphisms of rings on the stalks $\eta_p: F_p \to G_p$. - Morphisms of sheaves are isomorphisms iff isomorphisms on all stalks, see exercise below. - **Kernels, cokernels, images**: for $\phi: F\to G$, sheafify the assignments to kernels/cokernels/images on open sets. - **Sheafification**: for any $F\in \Presh(X)$, there is a unique $F^+\in \Sh(X)$ and a morphism $\theta: F\to F^+$ of presheaves such that any sheaf presheaf morphism $F\to G$ factors as $F\to F^+ \to G$. - The construction: $F^+(U) = \Top(U, \disjoint_{p\in U} F_p)$ are all functions $s$ into the union of stalks, subject to $s(p) \in F_p$ for all $p\in U$ and for each $p\in U$, there is a neighborhood $V\contains U \ni p$ and $t\in F(V)$ such that for all $q\in V$, the germ $t_q$ is equal to $s(q)$. - Note that the stalks are the same: $(F^+)_p = F_p$, and if $F$ is already a sheaf then $\theta$ is an isomorphism. - **Subsheaves**: $F'\leq F$ iff $F'(U) \leq F(U)$ is a subgroup for every $U$ and the restrictions on $F'$ are induced by restrictions from $F$. - If $F'\leq F$ then $F'_p \leq F_p$. - **Injectivity**: $\phi: F\to G$ is injective iff the sheaf kernel $\ker \phi = 0$ as a subsheaf of $F$. - $\phi$ is injective iff injective on all sections. - $\im \phi\leq G$ is a subsheaf. - **Surjectivity**: $\phi: F\to G$ is surjective iff $\im \phi = G$ as a subsheaf. - **Exactness**: a sequence of sheaves $(F_i, \phi_i:F_i\to F_{i+1})$ is exact iff $\ker \phi_i = \im \phi^{i-1}$ as subsheaves of $F_i$. - $\phi:F\to G$ is injective iff $0\to F \mapsvia{\phi} G$ is exact. - $\phi: F\to G$ is surjective iff $F \mapsvia{\phi} G \to 0$ is exact. - Sequences of sheaves are exact iff exact on stalks. - **Quotient sheaves**: $F/F'$ is the sheafification of $U\mapsto F(U) / F'(U)$. - **Cokernels**: for $\phi: F\to G$, $\coker \phi$ is sheafification of $U\mapsto \coker( F(U) \mapsvia{\phi(U)} G(U))$. - **Direct images**: for $f \in \Top(X, Y)$, the sheaf defined on sections by $(f_* F)(V) \da F(f\inv(V))$ for any $V \subseteq Y$. Yields a functor $f_*: \Sh(X) \to \Sh(Y)$. - **Inverse images**: denoted $f\inv G$, the sheafification of $U \mapsto \colim_{V\contains f(U)} G(V)$, i.e. take the limit from above of all open sets $V$ of $Y$ containing the image $f(U)$. Yields a functor $f\inv: \Sh(Y) \to \Sh(X)$. - **Restriction of a sheaf**: for $F\in \Sh(X)$ and $Z \subseteq X$ with $\iota:Z \injects X$ the inclusion, define $i\inv F\in \Sh(Z)$ to be the restriction. Also denoted $\ro{F}{Z}$. This has the same stalks: $(\ro{F}{Z})_p = F_p$. - For any $U \subseteq X$, the global sections functor $\Gamma(U; \wait): \Sh(X)\to \Ab\Grp$ is left-exact (proved in exercises). - **Limits of sheaves**: for $\ts{F_i}$ a direct system of sheaves, $\colim_{i} F_i$ has underlying presheaf $U\mapsto \colim_i F_i(U)$. If $X$ is Noetherian, then this is already a sheaf, and commutes with sections: $\Gamma(X; \colim_i F_i) = \colim_i \Gamma(X; F_i)$. - Inverse limits exist and are defined similarly. - **The espace étalé**: define $\Et(F) = \disjoint_{p\in X} F_p$ and a projection $\pi: \Et(F) \to X$ by sending $s\in F_p$ to $p$. For each $U \subseteq X$ and $s\in F(U)$, there is a local section $\bar{s}: U\to \Et(F)$ where $p\mapsto s_p$, its germ at $p$; this satisfies $\pi \circ \bar s = \id_U$. Give $\Et(F)$ the strongest topology such that the $\bar{s}$ are all continuous. Then $F^+(U) \da \Top(U, \Et(F))$ is the set of continuous sections of $\Et(F)$ over $U$. - **Support**: for $s\in F(U)$, $\supp(s) \da \ts{p\in U \st s_p \neq 0}$ where $s_p$ is the germ of $s$ in $F_p$. This is closed. - This extends to $\supp(F) \da \ts{p\in X \st F_p \neq 0}$, which need not be closed. - **Sheaf hom**: $U\mapsto \Hom(\ro{F}{U}, \ro{G}{U})$ forms a sheaf of local morphisms and is denoted $\sheafhom(F, G)$. - **Flasque sheaves**: a sheaf is flasque iff $V\injects U \implies F(U) \surjects F(V)$. - **Skyscraper sheaves**: for $A\in \Ab\Grp$ and $p\in X$, define \[ i_p(A)(U) = \begin{cases} A & p\in U \\ 0 & \text{otherwise}. \end{cases} .\] Also denoted $\iota_*(A)$ where $\iota: \cl_X(\ts p) \injects X$ is the inclusion. - The stalks are \[ (i_p(A))_q = \begin{cases} A & q\in \cl_X(\ts{p}) \\ 0 & \text{otherwise}. \end{cases} .\] - **Extension by zero**: if $\iota: Z\injects X$ is the inclusion of a closed set and $U\da X\sm Z$ with $j: U\to X$, then for $F\in \Sh(Z)$, the sheaf $\iota_* F\in \Sh(X)$ is the extension of $F$ by zero outside of $Z$. The stalks are \[ (\iota_* F)_p = \begin{cases} F_p & p\in Z \\ 0 & \text{otherwise}. \end{cases} .\] - For the open $U$, extension by zero is $j_! F$ which has presheaf $V \mapsto F(V)$ if $V \subseteq U$ and 0 otherwise. The stalks are \[ (j_! F)_p = \begin{cases} F_p & p\in U \\ 0 & \text{otherwise}. \end{cases} .\] - **Sheaf of ideals**: for $Y \subseteq X$ closed and $U \subseteq X$ open, $\mci_Y(U)$ has presheaf $U \mapsto$ the ideal in $\OO_X(U)$ of regular functions vanishing on all of $Y \intersect U$. This is a subsheaf of $\OO_X$. - **Gluing sheaves**: given $\mcu \covers X$ and sheaves $F_i\in \Sh(U_i)$, one can glue to a unique $F\in \Sh(X)$ if one is given morphisms $\phi_{ij}\ro{F_i}{U_{ij}} \iso \ro{F_j}{U_{ij}}$ where $\phi_{ii} = \id$ and $\phi_{ik} = \phi_{jk} \circ \phi_{ij}$ on $U_{ijk}$. ::: :::{.warnings} Some common mistakes: - Kernel presheaves are already sheaves, but not cokernels or images. See exercise below. - $\phi: F\to G$ is injective iff injective on sections, but this is not true for surjectivity. - The sheaves $f\inv G$ and $f^* G$ are different! See III.5 for the latter. - Global sections need not be right-exact. ::: :::{.exercise title="Regular functions on varieties form a sheaf"} For $X\in \Var\slice k$, define the ring $\OO_X(U)$ of literal regular functions $f_i: U\to k$ where restriction morphisms are induced by literal restrictions of functions. Show that $\OO_X$ is a sheaf of rings on $X$. > Hint: Locally regular implies regular, and regular + locally zero implies zero. ::: :::{.exercise title="?"} Show that for every connected open subset $U \subseteq X$, the constant sheaf satisfies $\ul{A}(U) = A$, and if $U$ is open with open connected component so the $\ul{A}(U) = A\prodpower{\size \pi_0 U}$. ::: :::{.exercise title="?"} Show that if $X\in\Var\slice k$ and $\OO_X$ is its sheaf of regular functions, then the stalk $\OO_{X, p}$ is the *local ring of $p$* on $X$ as defined in Ch. I. ::: :::{.exercise title="Prop 1.1"} Let $\phi: F\to G$ be a morphism in $\Sh(X)$ and show that $\phi$ is an isomorphism iff $\phi_p$ is an isomorphism on stalks for all $p\in X$. Show that this is false for presheaves. ::: :::{.exercise title="?"} Show that for $\phi\in \Mor_{\Sh(X)}(F, G)$, $\ker \phi$ is a sheaf, but $\coker \phi, \im \phi$ are not in general. ::: :::{.exercise title="?"} Show that if $\phi: F\to G$ is surjective then the maps on sections $\phi(U): F(U) \to G(U)$ need not all be surjective. :::