II: Schemes

II.1: Sheaves

II.1.1. #to_work

Let \(A\) be an abelian group, and define the constant presheaf associated to \(A\) on the topological space \(X\) to be the presheaf \(U \mapsto A\) for all \(U \neq \varnothing\). with restriction maps the identity. Show that the constant sheaf \(\mathscr{A}\) defined in the text is the sheaf associated to this presheaf.

II.1.2. #to_work

• For any morphism of sheaves \(\varphi: {\mathcal{F}}\rightarrow \mathscr{G}\), show that for each point \(P,(\operatorname{ker} \varphi)_P=\) \(\operatorname{ker}\left(\varphi_P\right)\) and \((\operatorname{im} \varphi)_P=\operatorname{im}\left(\varphi_P\right)\).

• Show that \(\varphi\) is injective (respectively, surjective) if and only if the induced map on the stalks \(\varphi_P\) is injective (respectively, surjective) for all \(P\).

• Show that a sequence \begin{align*}\ldots {\mathcal{F}}^{i-1} \xrightarrow{\phi^{i-1}} {\mathcal{F}}^{i} \xrightarrow{\phi^{i}} {\mathcal{F}}^{i+1} \rightarrow \ldots \end{align*} of sheaves and morphisms is exact if and only if for each \(P \in X\) the corresponding sequence of stalks is exact as a sequence of abelian groups.

II.1.3. #to_work

• Let \(\varphi: {\mathcal{F}}\rightarrow \mathscr{G}\) 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 \(s \in \mathscr{G}\left(U \right)\), there is a covering \(\left\{U_i\right\}\) of \(U\), and there are elements \(t_i \in {\mathcal{F}}\left(U_i\right)\), such that \(\varphi\left(t_i\right)={ \left.{{s}} \right|_{{U_i}} }\) for all \(i\).

• Give an example of a surjective morphism of sheaves \(\varphi: {\mathcal{F}}\rightarrow \mathscr{G}\), and an open set \(U\) such that \(\varphi(U): {\mathcal{F}}(U) \rightarrow \mathscr{G}(U)\) is not surjective.

II.1.4. #to_work

• Let \(\varphi: {\mathcal{F}}\rightarrow \mathscr{G}\) be a morphism of presheaves such that \(\varphi(U): {\mathcal{F}}(U) \rightarrow \mathscr{G}(U)\) is injective for each \(U\). Show that the induced map \(\varphi^{+}: {\mathcal{F}}^{+} \rightarrow \mathscr{G}^{+}\)of associated sheaves is injective.

• Use part (a) to show that if \(\varphi: {\mathcal{F}}\rightarrow \mathscr{G}\) is a morphism of sheaves, then \(\operatorname{im}\varphi\) can be naturally identified with a subsheaf of \(\mathscr{G}\). as mentioned in the text.

II.1.5. #to_work

Show that a morphism of sheaves is an isomorphism if and only if it is both injective and surjective.

II.1.6. #to_work

• Let \({\mathcal{F}}^{\prime}\) be a subsheaf of a sheaf \(\tilde{{\mathcal{F}}}\). Show that the natural map of \(\tilde{{\mathcal{F}}}\) to the quotient sheaf \({\mathcal{F}}/{\mathcal{F}}'\) is surjective, and has kernel \({\mathcal{F}}^{\prime}\). Thus there is an exact sequence \begin{align*} 0 \rightarrow {\mathcal{F}}^{\prime} \rightarrow {\mathcal{F}}\rightarrow {\mathcal{F}}/ {\mathcal{F}}^{\prime} \rightarrow 0 .\end{align*}

• Conversely, if \begin{align*} 0 \rightarrow {\mathcal{F}}' \rightarrow {\mathcal{F}}\rightarrow {\mathcal{F}}^{\prime \prime} \rightarrow 0 \end{align*} is an exact sequence, show that \({\mathcal{F}}^{\prime}\) is isomorphic to a subsheaf of \({\mathcal{F}}\), and that \({\mathcal{F}}^{\prime \prime}\) is isomorphic to the quotient of \({\mathcal{F}}\) by this subsheaf.

II.1.7. #to_work

Let \(\varphi: {\mathcal{F}}\rightarrow \mathscr{G}\) be a morphism of sheaves.

• Show that \(\operatorname{im}\varphi \cong {\mathcal{F}}/ \operatorname{ker} \varphi\).

• Show that \(\operatorname{coker} \varphi \cong \mathscr{G} / \operatorname{im} \varphi\).

II.1.8. #to_work

For any open subset \(U \subseteq X\), show that the functor \(\Gamma(U . \cdot)\) from sheaves on \(X\) to abelian groups is a left exact functor, i.e.. if \begin{align*} 0 \rightarrow {\mathcal{F}}^{\prime} \rightarrow {\mathcal{F}}\rightarrow {\mathcal{F}}^{\prime \prime} \end{align*} is an exact sequence of sheaves, then \begin{align*} 0 \rightarrow \Gamma\left(U, {\mathcal{F}}^{\prime}\right) \rightarrow \Gamma(U, {{\mathcal{F}}}) \rightarrow \Gamma\left(U, {\mathcal{F}}^{\prime \prime}\right) \end{align*} is an exact sequence of groups.

The functor \(\Gamma(U, {-})\) need not be exact: see \((\) Ex. 1.21) below.

II.1.9. Direct Sum. #to_work

Let \({\mathcal{F}}\) and \(\mathscr{G}\) be sheaves on \(X\). Show that the presheaf \(U \mapsto {\mathcal{F}}(U) \oplus\) \(\mathscr{G}(U)\) is a sheaf. It is called the direct sum of \({\mathcal{F}}\) and \(\mathscr{G}\), and is denoted by \({\mathcal{F}}\oplus \mathscr{G}\).

Show that it plays the role of direct sum and of direct product in the category of sheaves of abelian groups on \(X\).

II.1.10. Direct Limit. #to_work

Let \(\left\{{{\mathcal{F}}_i}\right\}\) be a direct system of sheaves and morphisms on \(X\). We define the direct limit of the system \(\left\{{{\mathcal{F}}_i}\right\}\), denoted \(\lim {\mathcal{F}}_i\), to be the sheaf associated to the presheaf \(U \mapsto \varinjlim {{\mathcal{F}}_i}(U)\).

Show that this is a direct limit in the category of sheaves on \(X\), i.e., that it has the following universal property: given a sheaf \(\mathscr{G}\), and a collection of morphisms \({\mathcal{F}}_i \rightarrow \mathscr{G}\). compatible with the maps of the direct system, then there exists a unique map \(\varinjlim {\mathcal{F}}_i \to \mathscr{G}\) such that for each \(i\), the original map \(\mathscr{F_i} \rightarrow \mathscr{G}\) is obtained by composing the maps \({\mathcal{F}}_i \rightarrow \varinjlim {\mathcal{F}}_i \rightarrow \mathscr{G}\).

II.1.11. #to_work

Let \(\left\{{\mathcal{F}}_i\right\}\) be a direct system of sheaves on a noetherian topological space \(X\). In this case show that the presheaf \(U \mapsto \varinjlim {\mathcal{F}}_i(U)\) is already a sheaf. In particular, \(\Gamma\left(X, \varinjlim {{\mathcal{F}}}_i\right) = \varinjlim \Gamma\left(X, {\mathcal{F}}_i\right)\)

II.1.12. Inverse Limit. #to_work

Let, \(\left\{{{\mathcal{F}}_i}\right\}\) be an inverse system of sheaves on \(X\). Show that the presheaf \(U \mapsto \varprojlim {\mathcal{F}}_i\left(U \right)\) is a sheaf. It is called the inverse limit of the system \(\left\{{\mathcal{F}}_i\right\}\), and is denoted by \(\varprojlim {{\mathcal{F}}}_i\). Show that it has the universal property of an inverse limit in the category of sheaves.

II.1.13. Espace Etale of a Presheaf. #to_work

Given a presheaf \({\mathcal{F}}\) on \(X\), we define a topological space \(\operatorname{\operatorname{Spe}}({\mathcal{F}})\), called the espace etale of \({\mathcal{F}}\), as follows ¹ . As a set, \(\operatorname{Spe} = \bigcup_{p\in X} {\mathcal{F}}_P\). We define a projection map \(\pi: \operatorname{Spe} ({{\mathcal{F}}}) \rightarrow X\) by sending \(s \in {\mathcal{F}}_p\) to \(P\). For each open set \(U \subseteq X\) and each section \(s \in {\mathcal{F}}\left(U \right)\), we obtain a map \(\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu: U \rightarrow \operatorname{Spe}(\left.{\mathcal{F}}\right)\) by sending \(P \mapsto s_P\), its germ at \(P\).

This map has the property that \(\pi \circ \mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu= \operatorname{id}\), in other words, it is a “section” of \(\pi\) over \(U\). We now make \(\operatorname{Spe}({\mathcal{F}})\) into a topological space by giving it the strongest topology such that all the maps \(\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu: U \rightarrow \operatorname{Spe}({\mathcal{F}})\) for all \({U}\). and all \(s \in {\mathcal{F}}\left(U\right)\), are continuous.

Now show that the sheaf \({\mathcal{F}}^{+}\)associated to \({\mathcal{F}}\) can be described as follows: for any open set \(U \subseteq X, {\mathcal{F}}^{+}\left(U \right)\) is the set of continuous sections of \(\operatorname{Spe}({\mathcal{F}})\) over \(U\).

In particular, the original presheaf \({\mathcal{F}}\) was a sheaf if and only if for each \(U, {\mathcal{F}}(U)\) is equal to the set of all continuous sections of \(\operatorname{Spe}({\mathcal{F}})\) over \(U\).

II.1.14. Support. #to_work

Let \({\mathcal{F}}\) be a sheaf on \(X\), and let \(s \in {\mathcal{F}}(U)\) be a section over an open set \(U\). The support of \(s\), denoted \(\mathop{\mathrm{supp}}s\), is defined to be \(\left\{P \in U \mathrel{\Big|}s_P \neq 0\right\}\), where \(s_P\) denotes the germ of \(s\) in the stalk \({\mathcal{F}}_P\). Show that Supp \(s\) is a closed subset of \(U\).

We define the support of \({\mathcal{F}}\), \(\mathop{\mathrm{supp}}{\mathcal{F}}\), to be \(\left\{P \in X \mathrel{\Big|}\cdot {\mathcal{F}}_P \neq 0\right\}\). It need not be a closed subset.

II.1.15. Sheaf \(\mathop{\mathcal{H}\! \mathit{om}}\). #to_work

Let \({\mathcal{F}}, \mathscr{G}\) be sheaves of abelian groups on \(X\). For any open set \(U \subseteq X\), show that the set \(\mathop{\mathrm{Hom}}({ \left.{{{\mathcal{F}}}} \right|_{{U}} }, { \left.{{\mathscr G}} \right|_{{U}} })\) of morphisms of the restricted sheaves has a natural structure of abelian group. Show that the presheaf \(U \mapsto \operatorname{Hom}\left(\left.{\mathcal{F}}\right|_U,\left.\mathscr{G}\right|_U\right)\) is a sheaf. It is called the sheaf of local morphisms of \({\mathcal{F}}\) into \(\mathscr{G}\), “sheaf hom” for short, and is denoted \(\mathop{\mathcal{H}\! \mathit{om}}({\mathcal{F}}, \mathscr{G})\).

II.1.16. Flasque Sheaves. #to_work

A sheaf \({\mathcal{F}}\) on a topological space \(X\) is flasque if for every inclusion \(V \subseteq U\) of open sets, the restriction map \({\mathcal{F}}(U) \rightarrow {\mathcal{F}}(V)\) is surjective.

• Show that a constant sheaf on an irreducible topological space is flasque. See (I, §1) for irreducible topological spaces.

• If \begin{align*} 0 \rightarrow {\mathcal{F}}^{\prime} \rightarrow {\mathcal{F}}\rightarrow {\mathcal{F}}^{\prime \prime} \rightarrow 0 \end{align*} is an exact sequence of sheaves, and if \({\mathcal{F}}^{\prime}\) is flasque, then for any open set \(L\), the sequence \begin{align*} 0 \rightarrow {\mathcal{F}}^{\prime}(U) \rightarrow {\mathcal{F}}(U) \rightarrow {\mathcal{F}}^{\prime \prime}\left(U\right) \rightarrow 0 \end{align*} of abelian groups is also exact.

• If \begin{align*} 0 \rightarrow {\mathcal{F}}^{\prime} \rightarrow {\mathcal{F}}\rightarrow {\mathcal{F}}^{\prime \prime} \rightarrow 0 \end{align*} is an exact sequence of sheaves, and if \({\mathcal{F}}^{\prime}\) and \({\mathcal{F}}\) are flasque, then \({\mathcal{F}}^{\prime \prime}\) is flasque.

• If \(f: X \rightarrow Y\) is a continuous map, and if \({\mathcal{F}}\) is a flasque sheaf on \(X\), then \(f_* {\mathcal{F}}\) is a flasque sheaf on \(Y\).

• Let \({\mathcal{F}}\) be any sheaf on \(X\). We define a new sheaf \(\mathscr{G}\), called the sheaf of discontinuous sections of \({\mathcal{F}}\) as follows. For each open set \(U \subseteq X, \mathscr{G}(U)\) is the set of maps \(s: U \rightarrow \bigcup_{P \in U} {\mathcal{F}}_P\) such that for each \(P \in U, s(P) \in {\mathcal{F}}_P\). Show that \(\mathscr{G}\) is a flasque sheaf, and that there is a natural injective morphism of \({\mathcal{F}}\) to \(\mathscr{G}\).

II.1.17. Skyscraper Sheaves. #to_work

Let \(X\) be a topological space, let \(P\) be a point, and let \(A\) be an abelian group. Define a sheaf \(i_P(A)\) on \(X\) as follows: \(i_P(A)\left(U\right)=A\) if \(P \in U^{\prime}, 0\) otherwise. Verify that the stalk of \(i_P(A)\) is \(A\) at every point \(Q \in\{P\}^{-}\), and 0 elsewhere, where \(\{P\}^{-}\)denotes the closure of the set consisting of the point \(P\). Hence the name “skyscraper sheaf.”

Show that this sheaf could also be described as \(i_*(A)\), where \(A\) denotes the constant sheaf \(A\) on the closed subspace \(\{P\}^{-}\), and \(i:\{P\}^{-} \rightarrow X\) is the inclusion.

II.1.18. Adjoint Property of \(f^{-1}\). #to_work

Let \(f: X \rightarrow Y\) be a continuous map of topological spaces. Show that for any sheaf \({\mathcal{F}}\) on \(X\) there is a natural map \(f^{-1} f_* {\mathcal{F}}\rightarrow {\mathcal{F}}\), and for any sheaf \(\mathscr{G}\) on \(Y\) there is a natural map \(\mathscr{G} \rightarrow f_* f^{-1} \mathscr{G}\).

Use these maps to show that there is a natural bijection of sets, for any sheaves \({\mathcal{F}}\) on \(X\) and \(\mathscr{G}\) on \(Y\), \begin{align*} \operatorname{Hom}_X\left(f^{-1} \mathscr{G}, {\mathcal{F}}\right)=\operatorname{Hom}_Y\left(\mathscr{G}, f_* {\mathcal{F}}\right) . \end{align*} Hence we say that \(f^{-1}\) is a left adjoint of \(f_*\), and that \(f_*\) is a right adjoint of \(f^{-1}\).

II.1.19. Extending a Sheaf by Zero. #to_work

Let \(X\) be a topological space, let \(Z\) be a closed subset, let \(i: Z \rightarrow X\) be the inclusion, let \(U=X-Z\) be the complementary open subset, and let \(j: U \rightarrow X\) be its inclusion.

• Let \({\mathcal{F}}\) be a sheaf on \(Z\). Show that the stalk \(\left(i_* {\mathcal{F}}\right)_P\) of the direct image sheaf on \(X\) is \({\mathcal{F}}_P\) if \(P \in Z, 0\) if \(P \notin Z\). Hence we call \(i_* \cdot {\mathcal{F}}\) the sheaf obtained by extending \({\mathcal{F}}\) by zero outside \(Z\). By abuse of notation we will sometimes write \({\mathcal{F}}\) instead of \(i_* {\mathcal{F}}\), and say “consider \({\mathcal{F}}\) as a sheaf on \(X\),” when we mean "consider \(i_* {\mathcal{F}}\).

• Now let \({\mathcal{F}}\) be a sheaf on \(U\). Let \(j_!({{\mathcal{F}}})\) be the sheaf on \(X\) associated to the presheaf \(V \mapsto {\mathcal{F}}(V)\) if \(V \subseteq U, V \mapsto 0\) otherwise. Show that the stalk \((j_!({\mathcal{F}}))_P\) is equal to \({\mathcal{F}}_P\) if \(P \in U, 0\) if \(P \notin U\), and show that \(j_! {\mathcal{F}}\) is the only sheaf on \(X\) which has this property, and whose restriction to \(U\) is \({\mathcal{F}}\). We call \(j_! {\mathcal{F}}\) the sheaf obtained by extending \({\mathcal{F}}\) by zero outside \(U\).

• Now let \({\mathcal{F}}\) be a sheaf on \(X\). Show that there is an exact sequence of sheaves on \(X\), \begin{align*} 0 \to j_!({ \left.{{{\mathcal{F}}}} \right|_{{U}} }) \to {\mathcal{F}}\to i_*({ \left.{{{\mathcal{F}}}} \right|_{{Z}} })\to 0 \end{align*}

II.1.20. Subsheaf with Supports. #to_work

Let \(Z\) be a closed subset of \(X\), and let \({\mathcal{F}}\) be a sheaf on \(X\). We define \(\Gamma_Z(X, {\mathcal{F}})\) to be the subgroup of \(\Gamma(X, {\mathcal{F}})\) consisting of all sections whose support (Ex. 1.14) is contained in \(Z\).

• Show that the presheaf \(V \mapsto \Gamma_{\mathrm{Z} \cap V}\left(V,\left.{\mathcal{F}}\right|_V\right)\) is a sheaf. It is called the subsheaf of \({\mathcal{F}}\) with supports in \(Z\), and is denoted by \(\mathscr{H}_Z^0({\mathcal{F}})\).

• Let \(U=X-Z\), and let \(j: U \rightarrow X\) be the inclusion. Show there is an exact sequence of sheaves on \(X\) \begin{align*} 0 \rightarrow \mathscr{H}_Z^0(\tilde{{\mathcal{F}}}) \rightarrow {\mathcal{F}}\rightarrow i_*({ \left.{{{\mathcal{F}}}} \right|_{{U}} }) \end{align*} Furthermore, if \({\mathcal{F}}\) is flasque, the map \({\mathcal{F}}\rightarrow i_*({ \left.{{{\mathcal{F}}}} \right|_{{U}} })\) is surjective.

II.1.21. Some Examples of Sheaves on Varieties. #to_work

Let \(X\) be a variety over an algebraically closed field \(k\), as in Ch. I. Let \({\mathcal{O}}_X\) be the sheaf of regular functions on \(X\) (See (1.0 .1)).

• Let \(Y\) be a closed subset of \(X\). For each open set \(U \subseteq X\), let \(\mathscr{I}_Y(U)\) be the ideal in the ring \({\mathcal{O}}_X(U)\) consisting of those regular functions which vanish at all points of \(Y \cap U\). Show that the presheaf \(U \mapsto \mathscr{I}_Y(U)\) is a sheaf. It is called the sheaf of ideals \({\mathcal{I}}_Y\) of \(Y\), and it is a subsheaf of the sheaf of rings \({\mathcal{O}}_X\).

• If \(Y\) is a subvariety, then the quotient sheaf \(C_1 \mathcal{T}_{,}\)is isomorphic to \(i_*\left(C_1\right)\), where \(i: Y \rightarrow X\) is the inclusion, and \({\mathcal{O}}_Y\) is the sheaf of regular functions on \(Y\).

• Now let \(X=\mathbf{P}^1\), and let \(Y\) be the union of two distinct points \(P, Q \in X\). Then there is an exact sequence of sheaves on \(X\) where \({\mathcal{F}}= i_* {\mathcal{O}}_P \oplus i_* {\mathcal{O}}_Q\): \begin{align*} 0 \to {\mathcal{I}}_Y \to {\mathcal{O}}_X \to {\mathcal{F}}\to 0 \end{align*} Show however that the induced map on global sections \(\Gamma(X; {\mathcal{O}}_X) \to \Gamma(X; {\mathcal{F}})\) is not surjective. This shows that the global section functor \(\Gamma(X, \cdot)\) is not exact (cf. (Ex. 1.8) which shows that it is left exact).

• Again let \(X=\mathbf{P}^1\), and let \({\mathcal{O}}\) be the sheaf of regular functions. Let \(\mathscr{K}\) be the constant sheaf on \(X\) associated to the function field \(K\) of \(X\). Show that there is a natural injection \({\mathcal{O}}\rightarrow \mathscr{K}\). Show that the quotient sheaf \(\mathscr K / {\mathcal{O}}\) is isomorphic to the direct sum of sheaves \(\sum_{P \in X} i_P\left(I_P\right)\), where \(I_P\) is the group \(K / {\mathcal{O}}_P\), and \(i_P\left(I_P\right)\) denotes the skyscraper sheaf (Ex. 1.17) given by \(I_P\) at the point \(P\).

• Finally show that in the case of (d) the sequence \begin{align*} 0 \rightarrow \Gamma(X, \mathcal{O}) \rightarrow \Gamma(X, \mathscr{K}) \rightarrow \Gamma(X, \mathscr K/{\mathcal{O}}) \rightarrow 0 \end{align*} is exact. ²

II.1.22. Glueing Sheaves. #to_work

Let \(X\) be a topological space, let \(\mathfrak{U} = \left\{{U_i}\right\}\) be an open cover of \(X\), and suppose we are given for each \(i\) a sheaf \({{\mathcal{F}}}_i\) on \(U_i\), and for each \(i, j\) an isomorphism \begin{align*} \phi_{ij}: { \left.{{{\mathcal{F}}_i}} \right|_{{U_i \cap U_j}} } \to { \left.{{{\mathcal{F}}_j}} \right|_{{U_i \cap U_j}} } \end{align*} such that

• for each \(i, \varphi_{ii}=\mathrm{id}\), and
• for each \(i, j, k\), \(\phi_{ik} = \phi_{jk} \circ \phi_{ij}\) on \(U_i \cap U_j \cap U_k\).

Then there exists a unique sheaf \({\mathcal{F}}\) on \(X\), together with isomorphisms \(\psi_i:\left.{\mathcal{F}}\right|_{U_i} { \, \xrightarrow{\sim}\, }{\mathcal{F}}_i\), such that for each \(i, j, \psi_j= \varphi_{i j} \circ \psi_i\) on \(U_i \cap U_j\). We say loosely that \({\mathcal{F}}\) is obtained by glueing the sheaves \({\mathcal{F}}_i\) via the isomorphisms \(\varphi_i\).