Note: there are many, many important notions tucked away in the exercises in this section.
II.1: Sheaves
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Presheaves \(F\) of abelian groups: contravariant functors \(F\in {\mathsf{Fun}}({\mathsf{Open}}(X), {\mathsf{Ab}}{\mathsf{Grp}})\).
- Assigns every open \(U \subseteq X\) some \(F(U) \in {\mathsf{Ab}}{\mathsf{Grp}}\)
- For \(\iota_{VU}: V \subseteq U\), restriction morphisms \(\phi_{UV}: F(U) \to F(V)\).
- \(F(\emptyset) = 0\), so \(F({ \mathscr \emptyset^{\scriptscriptstyle \downarrow} }) = { \mathscr{1}_{\scriptscriptstyle \uparrow} }\).
- \(\phi{UU} = \operatorname{id}_{F(U)}\)
- \(W \subseteq V \subseteq U \implies \phi_{UW} = \phi_{VW} \circ \phi_{UV}\).
- 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 \({ \left.{{s}} \right|_{{V}} } \coloneqq\phi_{UV}(s)\) for \(s\in F(U)\).
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Sheaves: presheaves \(F\) which are completely determined by local data. Additional requirements on open covers \({\mathcal{V}}\rightrightarrows U\):
- If \(s\in F(U)\) with \({ \left.{{s}} \right|_{{V_i}} } = 0\) for all \(i\) then \(s\equiv 0 \in F(U)\).
- Given \(s_i\in F(V_i)\) where \({ \left.{{s_i}} \right|_{{V_{ij}}} } = \so{s_j}{V_{ij}} \in F(V_{ij})\) then \(\exists s\in F(U)\) such that \({ \left.{{s}} \right|_{{V_i}} } = s_i\) for each \(i\), which is unique by the previous condition.
- Constant sheaf: for \(A\in \AbGrp\), define the constant sheaf \begin{align*} \underline{A}(U) \coloneqq{\mathsf{Top}}(U, A^{\operatorname{disc}}) .\end{align*}
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Stalks: \(F_p \coloneqq\colim_{U\ni p} F(U)\) 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 \(\exists W \subseteq U \cap V\) with \({ \left.{{s}} \right|_{{w}} } = { \left.{{t}} \right|_{{w}} }\).
- Germs: a germ of a section of \(F\) at \(p\) is an elements of the stalk \(F_p\).
- Morphisms of presheaves: natural transformations \(\eta\in \mathop{\mathrm{Mor}}_{{\mathsf{Fun}}}(F, G)\), i.e. for every \(U, V\), components \(\eta_U, \eta_V\) fitting into a diagram
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A morphism of sheaves is exactly a morphism of the underlying presheaves.
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Morphisms of sheaves \(\eta: F\to G\) induce morphisms of rings on the stalks \(\eta_p: F_p \to G_p\).
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Morphisms of sheaves are isomorphisms iff isomorphisms on all stalks, see exercise below.
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Kernels, cokernels, images: for \(\phi: F\to G\), sheafify the assignments to kernels/cokernels/images on open sets.
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Sheafification: for any \(F\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)\), there is a unique \(F^+\in {\mathsf{Sh}}(X)\) 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: \(F^+(U) = {\mathsf{Top}}(U, {\textstyle\coprod}_{p\in U} F_p)\) 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 \(V\supseteq U \ni p\) 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.
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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 \(F'_p \leq F_p\).
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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.
- \(\operatorname{im}\phi\leq G\) is a subsheaf.
- Surjectivity: \(\phi: F\to G\) is surjective iff \(\operatorname{im}\phi = G\) as a subsheaf.
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Exactness: a sequence of sheaves \((F_i, \phi_i:F_i\to F_{i+1})\) is exact iff \(\ker \phi_i = \operatorname{im}\phi^{i-1}\) as subsheaves of \(F_i\).
- \(\phi:F\to G\) is injective iff \(0\to F \xrightarrow{\phi} G\) is exact.
- \(\phi: F\to G\) is surjective iff \(F \xrightarrow{\phi} G \to 0\) is exact.
- Sequences of sheaves are exact iff exact on stalks.
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Quotient sheaves: \(F/F'\) is the sheafification of \(U\mapsto F(U) / F'(U)\).
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Cokernels: for \(\phi: F\to G\), \(\operatorname{coker}\phi\) is sheafification of \(U\mapsto \operatorname{coker}( F(U) \xrightarrow{\phi(U)} G(U))\).
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Direct images: for \(f \in {\mathsf{Top}}(X, Y)\), the sheaf defined on sections by \((f_* F)(V) \coloneqq F(f^{-1}(V))\) for any \(V \subseteq Y\). Yields a functor \(f_*: {\mathsf{Sh}}(X) \to {\mathsf{Sh}}(Y)\).
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Inverse images: denoted \(f^{-1}G\), the sheafification of \(U \mapsto \colim_{V\supseteq f(U)} G(V)\), i.e. take the limit from above of all open sets \(V\) of \(Y\) containing the image \(f(U)\). Yields a functor \(f^{-1}: {\mathsf{Sh}}(Y) \to {\mathsf{Sh}}(X)\).
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Restriction of a sheaf: for \(F\in {\mathsf{Sh}}(X)\) and \(Z \subseteq X\) with \(\iota:Z \hookrightarrow X\) the inclusion, define \(i^{-1}F\in {\mathsf{Sh}}(Z)\) to be the restriction. Also denoted \({ \left.{{F}} \right|_{{Z}} }\). This has the same stalks: \(({ \left.{{F}} \right|_{{Z}} })_p = F_p\).
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For any \(U \subseteq X\), the global sections functor \(\Gamma(U; {-}): {\mathsf{Sh}}(X)\to {\mathsf{Ab}}{\mathsf{Grp}}\) is left-exact (proved in exercises).
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Limits of sheaves: for \(\left\{{F_i}\right\}\) a direct system of sheaves, \(\colim_{i} F_i\) has underlying presser \(U\mapsto \colim_i F_i(U)\). 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.
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The espace étalé: define \(\text{Ét}(F) = \disjoin_{p\in X} F_p\) and a projection \(\ppi: \text{Ét}(F) \to X\) by sending \(s\in F_p\) to \(p\). For each \(U \subseteq X\) and \(s\in F(U)\), there is a local section \(\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu: U\to \text{Ét}(F)\) where \(p\mapsto s_p\), its germ at \(p\); this satisfies \(\pi \circ \mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu = \operatorname{id}_U\). Give \(\text{Ét}(F)\) the strongest topology such that the \(\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu\) are all continuous. Then \(F^+(U) \coloneqq{\mathsf{Top}}(U, \text{Ét}(F))\) is the set of continuous sections of \(\text{Ét}(F)\) over \(U\).
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Support: for \(s\in F(U)\), \(\mathop{\mathrm{supp}}(s) \coloneqq\left\{{p\in U {~\mathrel{\Big\vert}~}s_p \neq 0}\right\}\) where \(s_p\) is the germ of \(s\) in \(F_p\). This is closed.
- This extends to \(\mathop{\mathrm{supp}}(F) \coloneqq\left\{{p\in X {~\mathrel{\Big\vert}~}F_p \neq 0}\right\}\), which need not be closed.
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Sheaf hom: \(U\mapsto \mathop{\mathrm{Hom}}({ \left.{{F}} \right|_{{U}} }, { \left.{{G}} \right|_{{U}} })\) forms a sheaf of local morphisms and is denoted \(\mathop{\mathcal{H}\! \mathit{om}}(F, G)\).
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Flasque sheaves: a sheaf is flasque iff \(V\hookrightarrow U \implies F(U) \twoheadrightarrow F(V)\).
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Skyscraper sheaves: for \(A\in {\mathsf{Ab}}{\mathsf{Grp}}\) and \(p\in X\), define \(i_p(A)\) by \(U\mapsto A\) if \(p\in U\) and 0 otherwise. Also denoted \(\iota_*(A)\) where \(\iota: { \operatorname{cl}} _X(\left\{{p}\right\}) \hookrightarrow X\) is the inclusion.
- The stalks are \((i_p(A))_q = A\) if \(q\in { \operatorname{cl}} _X(\left\{{p}\right\})\) and 0 otherwise.
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Extension by zero: if \(\iota: Z\hookrightarrow X\) is the inclusion of a closed set and \(U\coloneqq X\setminus Z\) with \(j: U\to X\), then for \(F\in {\mathsf{Sh}}(Z)\), the sheaf \(\iota_* F\in {\mathsf{Sh}}(X)\) is the extension of \(F\) by zero outside of \(Z\). The stalks \((\iota_* F)_p\) are \(F_p\) is \(p\in Z\) and 0 otherwise.
- 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 \((j_! F)_p\) are \(F_p\) if \(p\in U\) and 0 otherwise.
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Sheaf of ideals: for \(Y \subseteq X\) closed and \(U \subseteq X\) open, \({\mathcal{I}}_Y(U)\) has presheaf \(U \mapsto\) the ideal in \({\mathcal{O}}_X(U)\) of regular functions vanishing on all of \(Y \cap U\). This is a subsheaf of \({\mathcal{O}}_X\).
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Gluing sheaves: given \({\mathcal{U}}\rightrightarrows X\) and sheaves \(F_i\in {\mathsf{Sh}}(U_i)\), one can glue to a unique \(F\in {\mathsf{Sh}}(X)\) if one is given morphisms \(\phi_{ij}{ \left.{{F_i}} \right|_{{U_{ij}}} } { \, \xrightarrow{\sim}\, }{ \left.{{F_j}} \right|_{{U_{ij}}} }\) where \(\phi_{ii} = \operatorname{id}\) and \(\phi_{ik} = \phi_{jk} \circ \phi_{ij}\) on \(U_{ijk}\).
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 \(f^{-1}G\) and \(f^* G\) are different! See III.5 for the latter.
- Global sections need not be right-exact.
For $X\in {\mathsf{Var}}_{/ {k}} $, define the ring \({\mathcal{O}}_X(U)\) of literal regular functions \(f_i: U\to k\) where restriction morphisms are induced by literal restrictions of functions. Show that \({\mathcal{O}}_X\) is a sheaf of rings on \(X\).
Hint: Locally regular implies regular, and regular + locally zero implies zero.
Show that for every connected open subset \(U \subseteq X\), the constant sheaf satisfies \(\underline{A}(U) = A\), and if \(U\) is open with open connected component so the \(\underline{A}(U) = A{ {}^{ \scriptscriptstyle\times^{{\sharp}\pi_0 U} } }\).
Show that if $X\in{\mathsf{Var}}_{/ {k}} $ and \({\mathcal{O}}_X\) is its sheaf of regular functions, then the stalk \({\mathcal{O}}_{X, p}\) is the local ring of \(p\) on \(X\) as defined in Ch. I.
Let \(\phi: F\to G\) be a morphism in \({\mathsf{Sh}}(X)\) 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.
Show that for \(\phi\in \mathop{\mathrm{Mor}}_{{\mathsf{Sh}}(X)}(F, G)\), \(\ker \phi\) is a sheaf, but \(\operatorname{coker}\phi, \operatorname{im}\phi\) are not in general.
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.