# II: Schemes ## II.1: Sheaves ### II.1.1. The constant sheaf #completed 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 $\mathcal{A}$ defined Solution (Part a) in the text is the sheaf associated to this presheaf. > Note: the constant sheaf $\mca$ defined in the text is: for $A$ an abelian group, give $A$ the discrete topolgy, and for any open $U$ let $\mca(U)$ be all continuous maps $U\to A$. > [!solution]- Solution > Let $\tilde\mca$ be the presheaf defined in the question and $\mca$ be the sheaf defined in the text. Define a map $F: \tilde \mca \to \mca^\pre$ by $F_U: \tilde \mca(U) \to \mca^\pre(U)$ where, noting $\tilde \mca(U) \da A$ for all $U$, we send $A\mapsto C^0(U, A)$. This is compatible with restriction and so defines a morphism of presheaves $F$. Checking stalks, $\tilde \mca_p = \mca^{\pre}_p = A$ for all $p$ and $F_p$ is an isomorphism $A\iso A$, so $F$ is an isomorphism of presheaves. By the universal property of sheafification, this lifts to an isomorphism of sheaves $(\tilde \mca)^+ \iso \mca$. ### II.1.2. Exact iff exact on stalks #completed a. For any morphism of sheaves $\varphi: \mcf \rightarrow \mathcal{G}$, show that for each point $P,(\operatorname{ker} \varphi)_P=$ $\operatorname{ker}\left(\varphi_P\right)$ and $(\operatorname{im} \varphi)_P=\operatorname{im}\left(\varphi_P\right)$. b. 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 $P$. c. Show that a sequence $$\ldots \mcf^{i-1} \mapsvia{\phi^{i-1}} \mcf^{i} \mapsvia{\phi^{i}} \mcf^{i+1} \rightarrow \ldots $$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. > [!solution]- Solution (Part a) > Taking stalks is always a filtered colimit, kernels are always finite limits, and **FCFinL**: filitered colimits commute with finite limits. > For the image, use $\im \phi \da \ker(\mcf \to \coker \phi)$ to write the image as a kernel and again apply **FCFinL**. > > Alternatively, a more direct proof: > $(\ker \phi)_p \subseteq \ker (\phi_p)$: take stalks to get a commutative diagram: > > \begin{tikzcd} > {(\ker \phi)(U)} & {\mcf(U)} & {\mcg(U)} \\ > {(\ker \phi)_p} & {\mcf_p} & {\mcg_p} > \arrow[hook, from=1-1, to=1-2] > \arrow[from=1-2, to=1-3] > \arrow[from=2-1, to=2-2] > \arrow[from=2-2, to=2-3] > \arrow["{\Res(U, p)}"', from=1-1, to=2-1] > \arrow[from=1-2, to=2-2] > \arrow[from=1-3, to=2-3] >\end{tikzcd} > > [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCIoXFxrZXIgXFxwaGkpKFUpIl0sWzEsMCwiXFxtY2YoVSkiXSxbMiwwLCJcXG1jZyhVKSJdLFsxLDEsIlxcbWNmX3AiXSxbMiwxLCJcXG1jZ19wIl0sWzAsMSwiKFxca2VyIFxccGhpKV9wIl0sWzAsMSwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMSwyXSxbNSwzXSxbMyw0XSxbMCw1LCJcXFJlcyhVLCBwKSIsMl0sWzEsM10sWzIsNF1d) > > Now $s\in (\ker \phi)_p \implies s$ is represented by some $(U, \tilde s\in (\ker \phi)(U))$ where $\Res(U, p)(\tilde s)= s$ and $\ro\phi U (\tilde s) = 0 \in \mcg(U)$. So $\Res(U, p)(\ro \phi U (\tilde s)) = \Res(U, p)(0) = 0$, and commutativity yields $\phi_p(s) = 0$. > > $\ker \phi_p \subseteq (\ker \phi)_p$: > A summary: > \begin{tikzcd} > s & {\mcf(U)} && {\mcg(U)} & {t\da \phi(U)(s)} \\ > {\ro{s}{V}} & {\mcf(V)} && {\mcg(V)} & {\ro{t}{V} = 0} \\ > & {\mcf_p} && {\mcg_p} \\ > {s_p} &&&& 0 > \arrow["{\phi(U)}", from=1-2, to=1-4] > \arrow["{\phi(V)}", from=2-2, to=2-4] > \arrow["{\phi_p}", from=3-2, to=3-4] > \arrow[maps to, from=4-1, to=4-5] > \arrow[curve={height=-30pt}, dashed, from=4-1, to=1-1] > \arrow[curve={height=-30pt}, dashed, from=1-1, to=1-5] > \arrow[curve={height=-30pt}, dashed, from=1-5, to=4-5] > \arrow[curve={height=-12pt}, dashed, from=4-5, to=2-5] > \arrow[dashed, from=2-1, to=4-1] > \end{tikzcd} > > [Link to Diagram](% https://q.uiver.app/?q=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) > > Let $s_p\in \ker \phi_p$, then lift to $s_p = (U, s\in \mcf(U))$ with $\ro{s}{p} = s_p$. > Set $t \da \phi(U)(s)$, the commutativity yields $t_p\da \ro{t}{p} = 0\in \mcg_p$. > Lift $t_p$ to $(V, \ro t V \in \mcg(V))$ where $V \subseteq U$, $\ro t V = 0 \in \mcg(V)$, and $\Res(V, p)(\ro t V) = t_p = 0$. > Since $\phi$ is a morphism of sheaves, diagrams commute and $\phi(V)(\ro s V) = \ro t V = 0$, so $\ro{s}{V}\in \ker \phi(V)$. > Take stalks to get $\qty{\ro s V}_p \in (\ker \phi)_p$, and note $\qty{\ro s V}_p = s_p$. > > The diagram chase for images is similar. > [!solution]- Solution (Part b) > Injectivity: > $\implies$: if $\phi$ is injective then $\ker \phi = 0$ as sheaves, and by (a) $0 = (\ker \phi)_p = \ker (\phi_p)$ so $\phi_p$ is injective for all $p$. > $\impliedby$: write $0 = \ker (\phi_p) = (\ker \phi)_p$ for all $p$. For each $p$, pass to a germ $(U_p, s(p) \in K(U_p))$ where $K\da \ker \phi$ as a sheaf and $s(p) = \ro{s(p)}{p} = 0$. Cover $X$ by $\ts{U_p \st p\in X}$ and glue the sections $\ts{s(p) \st p\in X}$ to a global section $s$; then since $K$ is a sheaf, $s = 0$ and $K = 0$. > [!solution]- Solution (Part c) > Exactness means $\ker \phi^i = \im \phi^{i-1}$ as sheaves. > $\implies$: if $\ker \phi^i = \im \phi^{i-1}$ as sheaves then they have the same stalks. But then $\ker \phi^i_p = (\ker \phi^i)_p = (\im \phi^{i-1})_p = \im \phi^{i-1}_p$, which is exactness on stalks. > $\impliedby$: if exact on stalks, just do the same computation in a different order: $(\ker \phi^i)_p = \ker \phi^i_p = \im \phi^{i-1}_p = (\im \phi^{i-1})_p$. > ? > Alternatively: > $$ > \ker \phi^i = \im \phi^{i-1} \iff (\ker \phi^i)_p = (\im \phi^{i-1})_p \,\forall p \iff \ker (\phi^i_p) = \im (\phi^{i-1}_p)\, \forall p > ,$$ > using the previous parts. ### II.1.3: Handling surjectivity. #completed a. Let $\varphi: \mcf \rightarrow \mathcal{G}$ 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 $s \in \mathcal{G}\left(U \right)$, there is a covering $\left\{U_i\right\}$ of $U$, and there are elements $t_i \in \mcf\left(U_i\right)$, such that $\varphi\left(t_i\right)=\ro{s}{U_i}$ for all $i$. b. Give an example of a surjective morphism of sheaves $\varphi: \mcf \rightarrow \mathcal{G}$, and an open set $U$ such that $\varphi(U): \mcf(U) \rightarrow \mathcal{G}(U)$ is not surjective. > [!solution]- Solution (Part a) > $\implies$: > If $\phi$ is surjective then $\phi_p$ is surjective for all $p$, so for every $p\in X$ pick $t_p\in \mcg_p$ and $s_p\in \mcf_p$ with $\phi_p(s_p) = t_p$. As in II.1.2 above, for each $p$ this relationship can be lifted to a small enough open set $U$: there is some $s\in U, t\in U$ with $\ro s p = s_p, \ro t p = t_p$, and $\phi(U)(s) = t$. Do this for every $p$ to get an open cover by the opens $U$. > > $\impliedby$: > If the condition holds, this yields surjectivity on all stalks, hence surjectivity of $\phi$. > [!solution]- Solution (Part b) > Use the standard holomorphic example $\ul{\ZZ} \injects (\OO_\CC, +) \surjectsvia{\exp} ( \OO_{\CC}\units, \cdot)$. > Then $U\da \cstar$ is counterexample: let $f(z) = z \in \OO_\CC^\units(U)$; there is no $g\in \OO_\CC(U)$ such that $\exp(g) = z$ on all of $U$, since this is equivalent to $g = \log(z)$ which is multivalued on $U$ and thus not holomorphic there. > However, for any point $p\in \cstar$, pick a disc $V\ni p$ small enough to avoid the origin -- then $\log$ is locally defined, $\exp$ is surjective on sections over $V$ and all smaller sets, and thus surjective on the stalk at $p$ (and thus all stalks $p\neq 0$). ### II.1.4. #completed a. Let $\varphi: \mcf \rightarrow \mathcal{G}$ be a morphism of presheaves such that $\varphi(U): \mcf(U) \rightarrow \mathcal{G}(U)$ is injective for each $U$. Show that the induced map $\varphi^{+}: \mcf^{+} \rightarrow \mathcal{G}^{+}$of associated sheaves is injective. b. Use part (a) to show that if $\varphi: \mcf \rightarrow \mathcal{G}$ is a morphism of sheaves, then $\im \varphi$ can be naturally identified with a subsheaf of $\mathcal{G}$. as mentioned in the text. > [!solution]- Solution > **Part a:** > Injective on all sections implies injective on all stalks iff injective as a morphism of sheaves, using II.2.2b above. > **Part b:** > Define a morphism of presheaves $\psi: \im \phi \injects \mcg$ by $\psi(U): \im \phi(U) \injects \mcg(U)$, which is an injection of subgroups for every $U$. By (a), the induced map on sheaves is injective, and restrictions on $\im \phi$ are induced by restrictions on $\mcg$, so $\im \phi \leq \mcg$ is a subsheaf. ### II.1.5. #completed Show that a morphism of sheaves is an isomorphism if and only if it is both injective and surjective. > [!solution]- Solution > Use that $\phi: \mcf \to \mcg$ is an isomorphism iff $0 \to \mcf \mapsvia{\phi} \mcg \to 0$ is exact iff exact on stalks. ### II.1.6. #to-work a. Let $\mcf^{\prime}$ be a subsheaf of a sheaf $\tilde{\mcf}$. Show that the natural map of $\tilde{\mcf}$ to the quotient sheaf $\mcf/\mcf'$ is surjective, and has kernel $\mcf^{\prime}$. Thus there is an exact sequence $$ 0 \rightarrow \mcf^{\prime} \rightarrow \mcf \rightarrow \mcf / \mcf^{\prime} \rightarrow 0 .$$ b. Conversely, if \[ 0 \rightarrow \mcf' \rightarrow \mcf \rightarrow \mcf^{\prime \prime} \rightarrow 0 \] is an exact sequence, show that $\mcf^{\prime}$ is isomorphic to a subsheaf of $\mcf$, and that $\mcf^{\prime \prime}$ is isomorphic to the quotient of $\mcf$ by this subsheaf. ### II.1.7. #to-work Let $\varphi: \mcf \rightarrow \mathcal{G}$ be a morphism of sheaves. a. Show that $\im \varphi \cong \mcf / \operatorname{ker} \varphi$. b. Show that $\operatorname{coker} \varphi \cong \mathcal{G} / \operatorname{im} \varphi$. ### 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 \[ 0 \rightarrow \mcf^{\prime} \rightarrow \mcf \rightarrow \mcf^{\prime \prime} \] is an exact sequence of sheaves, then \[ 0 \rightarrow \Gamma\left(U, \mcf^{\prime}\right) \rightarrow \Gamma(U, {\mcf}) \rightarrow \Gamma\left(U, \mcf^{\prime \prime}\right) \] is an exact sequence of groups. The functor $\Gamma(U, \wait)$ need not be exact: see $($ Ex. 1.21) below. ### II.1.9. Direct Sum. #to-work Let $\mcf$ and $\mathcal{G}$ be sheaves on $X$. Show that the presheaf $U \mapsto \mcf(U) \oplus$ $\mathcal{G}(U)$ is a sheaf. It is called the **direct sum** of $\mcf$ and $\mathcal{G}$, and is denoted by $\mcf \oplus \mathcal{G}$. 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 $\ts{\mcf_i}$ be a direct system of sheaves and morphisms on $X$. We define the direct limit of the system $\ts{\mcf_i}$, denoted $\lim \mcf_i$, to be the sheaf associated to the presheaf $U \mapsto \varinjlim {\mcf_i}(U)$. 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 $\mathcal{G}$, and a collection of morphisms $\mcf_i \rightarrow \mathcal{G}$. compatible with the maps of the direct system, then there exists a unique map $\varinjlim \mcf_i \to \mathcal{G}$ such that for each $i$, the original map $\mathcal{F_i} \rightarrow \mathcal{G}$ is obtained by composing the maps $\mcf_i \rightarrow \varinjlim \mcf_i \rightarrow \mathcal{G}$. ### II.1.11. #to-work Let $\left\{\mcf_i\right\}$ be a direct system of sheaves on a noetherian topological space $X$. In this case show that the presheaf $U \mapsto \varinjlim \mcf_i(U)$ is already a sheaf. In particular, $\Gamma\left(X, \varinjlim {\mcf}_i\right) = \varinjlim \Gamma\left(X, \mcf_i\right)$ ### II.1.12. Inverse Limit. #to-work Let, $\ts{\mcf_i}$ be an inverse system of sheaves on $X$. Show that the presheaf $U \mapsto \varprojlim \mcf_i\left(U \right)$ is a sheaf. It is called the **inverse limit** of the system $\left\{\mcf_i\right\}$, and is denoted by $\varprojlim {\mcf}_i$. 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 $\mcf$ on $X$, we define a topological space $\operatorname{\operatorname{Spe}}(\mcf)$, called the **espace etale** of $\mcf$, as follows[^II.1.13.this_exc]. As a set, $\operatorname{Spe} = \Union_{p\in X} \mcf_P$. We define a projection map $\pi: \operatorname{Spe} ({\mcf}) \rightarrow X$ by sending $s \in \mcf_p$ to $P$. For each open set $U \subseteq X$ and each section $s \in \mcf\left(U \right)$, we obtain a map $\bar{s}: U \rightarrow \operatorname{Spe}(\left.\mcf\right)$ by sending $P \mapsto s_P$, its germ at $P$. This map has the property that $\pi \circ \bar{s}= \id$, in other words, it is a "section" of $\pi$ over $U$. We now make $\operatorname{Spe}(\mcf)$ into a topological space by giving it the strongest topology such that all the maps $\bar{s}: U \rightarrow \operatorname{Spe}(\mcf)$ for all ${U}$. and all $s \in \mcf\left(U\right)$, are continuous. Now show that the sheaf $\mcf^{+}$associated to $\mcf$ can be described as follows: for any open set $U \subseteq X, \mcf^{+}\left(U \right)$ is the set of continuous sections of $\operatorname{Spe}(\mcf)$ over $U$. In particular, the original presheaf $\mcf$ was a sheaf if and only if for each $U, \mcf(U)$ is equal to the set of all continuous sections of $\operatorname{Spe}(\mcf)$ over $U$. [^II.1.13.this_exc]: This exercise is included only to establish the connection between our definition of a sheaf and another definition often found in the literature. See for example Godement (1. Ch. II, 1.2). ### II.1.14. Support. #to-work Let $\mcf$ be a sheaf on $X$, and let $s \in \mcf(U)$ be a section over an open set $U$. The **support of $s$**, denoted $\supp s$, is defined to be $\left\{P \in U \mid s_P \neq 0\right\}$, where $s_P$ denotes the germ of $s$ in the stalk $\mcf_P$. Show that Supp $s$ is a closed subset of $U$. We define the support of $\mcf$, $\supp \mcf$, to be $\left\{P \in X \mid \cdot \mcf_P \neq 0\right\}$. It need not be a closed subset. ### II.1.15. Sheaf $\sheafhom$. #to-work Let $\mcf, \mathcal{G}$ be sheaves of abelian groups on $X$. For any open set $U \subseteq X$, show that the set $\Hom(\ro{\mcf}{U}, \ro{\mathcal G}{U})$ of morphisms of the restricted sheaves has a natural structure of abelian group. Show that the presheaf $U \mapsto \operatorname{Hom}\left(\left.\mcf\right|_U,\left.\mathcal{G}\right|_U\right)$ is a sheaf. It is called the **sheaf of local morphisms of $\mcf$ into $\mathcal{G}$, "sheaf hom" for short**, and is denoted $\sheafhom(\mcf, \mathcal{G})$. ### II.1.16. Flasque Sheaves. #to-work A sheaf $\mcf$ on a topological space $X$ is **flasque** if for every inclusion $V \subseteq U$ of open sets, the restriction map $\mcf(U) \rightarrow \mcf(V)$ is surjective. a. Show that a constant sheaf on an irreducible topological space is flasque. See (I, ยง1) for irreducible topological spaces. b. If \[ 0 \rightarrow \mcf^{\prime} \rightarrow \mcf \rightarrow \mcf^{\prime \prime} \rightarrow 0 \] is an exact sequence of sheaves, and if $\mcf^{\prime}$ is flasque, then for any open set $L$, the sequence \[ 0 \rightarrow \mcf^{\prime}(U) \rightarrow \mcf(U) \rightarrow \mcf^{\prime \prime}\left(U\right) \rightarrow 0 \] of abelian groups is also exact. c. If \[ 0 \rightarrow \mcf^{\prime} \rightarrow \mcf \rightarrow \mcf^{\prime \prime} \rightarrow 0 \] is an exact sequence of sheaves, and if $\mcf^{\prime}$ and $\mcf$ are flasque, then $\mcf^{\prime \prime}$ is flasque. d. If $f: X \rightarrow Y$ is a continuous map, and if $\mcf$ is a flasque sheaf on $X$, then $f_* \mcf$ is a flasque sheaf on $Y$. e. Let $\mcf$ be any sheaf on $X$. We define a new sheaf $\mathcal{G}$, called the sheaf of **discontinuous sections** of $\mcf$ as follows. For each open set $U \subseteq X, \mathcal{G}(U)$ is the set of maps $s: U \rightarrow \bigcup_{P \in U} \mcf_P$ such that for each $P \in U, s(P) \in \mcf_P$. Show that $\mathcal{G}$ is a flasque sheaf, and that there is a natural injective morphism of $\mcf$ to $\mathcal{G}$. ### 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 $\mcf$ on $X$ there is a natural map $f^{-1} f_* \mcf \rightarrow \mcf$, and for any sheaf $\mathcal{G}$ on $Y$ there is a natural map $\mathcal{G} \rightarrow f_* f^{-1} \mathcal{G}$. Use these maps to show that there is a natural bijection of sets, for any sheaves $\mcf$ on $X$ and $\mathcal{G}$ on $Y$, $$ \operatorname{Hom}_X\left(f^{-1} \mathcal{G}, \mcf\right)=\operatorname{Hom}_Y\left(\mathcal{G}, f_* \mcf\right) . $$ 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. a. Let $\mcf$ be a sheaf on $Z$. Show that the stalk $\left(i_* \mcf\right)_P$ of the direct image sheaf on $X$ is $\mcf_P$ if $P \in Z, 0$ if $P \notin Z$. Hence we call $i_* \cdot \mcf$ the sheaf obtained by **extending $\mcf$ by zero outside $Z$.** By abuse of notation we will sometimes write $\mcf$ instead of $i_* \mcf$, and say "consider $\mcf$ as a sheaf on $X$," when we mean "consider $i_* \mcf$. b. Now let $\mcf$ be a sheaf on $U$. Let $j_!({\mcf})$ be the sheaf on $X$ associated to the presheaf $V \mapsto \mcf(V)$ if $V \subseteq U, V \mapsto 0$ otherwise. Show that the stalk $(j_!(\mcf))_P$ is equal to $\mcf_P$ if $P \in U, 0$ if $P \notin U$, and show that $j_! \mcf$ is the only sheaf on $X$ which has this property, and whose restriction to $U$ is $\mcf$. We call $j_! \mcf$ the sheaf obtained by **extending $\mcf$ by zero outside $U$.** c. Now let $\mcf$ be a sheaf on $X$. Show that there is an exact sequence of sheaves on $X$, $$ 0 \to j_!(\ro{\mcf} U) \to \mcf \to i_*(\ro {\mcf} Z)\to 0 $$ ### II.1.20. Subsheaf with Supports. #to-work Let $Z$ be a closed subset of $X$, and let $\mcf$ be a sheaf on $X$. We define $\Gamma_Z(X, \mcf)$ to be the subgroup of $\Gamma(X, \mcf)$ consisting of all sections whose support (Ex. 1.14) is contained in $Z$. a. Show that the presheaf $V \mapsto \Gamma_{\mathrm{Z} \cap V}\left(V,\left.\mcf\right|_V\right)$ is a sheaf. It is called the **subsheaf of $\mcf$ with supports in $Z$**, and is denoted by $\mathcal{H}_Z^0(\mcf)$. b. Let $U=X-Z$, and let $j: U \rightarrow X$ be the inclusion. Show there is an exact sequence of sheaves on $X$ $$ 0 \rightarrow \mathcal{H}_Z^0(\tilde{\mcf}) \rightarrow \mcf \rightarrow i_*(\ro{\mcf} U) $$ Furthermore, if $\mcf$ is flasque, the map $\mcf \rightarrow i_*(\ro{\mcf} U)$ 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 $\OO_X$ be the sheaf of regular functions on $X$ (See (1.0 .1)). a. Let $Y$ be a closed subset of $X$. For each open set $U \subseteq X$, let $\mathcal{I}_Y(U)$ be the ideal in the ring $\OO_X(U)$ consisting of those regular functions which vanish at all points of $Y \cap U$. Show that the presheaf $U \mapsto \mathcal{I}_Y(U)$ is a sheaf. It is called the sheaf of ideals $\mci_Y$ of $Y$, and it is a subsheaf of the sheaf of rings $\OO_X$. b. 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 $\OO_Y$ is the sheaf of regular functions on $Y$. c. 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 $\mcf = i_* \OO_P \oplus i_* \OO_Q$: $$ 0 \to \mci_Y \to \OO_X \to \mcf\to 0 $$ Show however that the induced map on global sections $\Gamma(X; \OO_X) \to \Gamma(X; \mcf)$ 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). d. Again let $X=\mathbf{P}^1$, and let $\OO$ be the sheaf of regular functions. Let $\mathcal{K}$ be the constant sheaf on $X$ associated to the function field $K$ of $X$. Show that there is a natural injection $\OO \rightarrow \mathcal{K}$. Show that the quotient sheaf $\mathcal K / \OO$ is isomorphic to the direct sum of sheaves $\sum_{P \in X} i_P\left(I_P\right)$, where $I_P$ is the group $K / \OO_P$, and $i_P\left(I_P\right)$ denotes the skyscraper sheaf (Ex. 1.17) given by $I_P$ at the point $P$. e. Finally show that in the case of (d) the sequence $$ 0 \rightarrow \Gamma(X, \mathcal{O}) \rightarrow \Gamma(X, \mathcal{K}) \rightarrow \Gamma(X, \mathcal K/\OO) \rightarrow 0 $$ is exact.[^cousin_problem] [^cousin_problem]: This is an analogue of what is called the *first Cousin problem* in several complex variables. See Gunning and Rossi (1, p. 248) ### II.1.22. Glueing Sheaves. #to-work Let $X$ be a topological space, let $\mathfrak{U} = \ts{U_i}$ be an open cover of $X$, and suppose we are given for each $i$ a sheaf ${\mcf}_i$ on $U_i$, and for each $i, j$ an isomorphism $$ \phi_{ij}: \ro{\mcf_i}{U_i \intersect U_j} \to \ro{\mcf_j}{U_i \intersect U_j} $$ such that 1. for each $i, \varphi_{ii}=\mathrm{id}$, and 2. for each $i, j, k$, $\phi_{ik} = \phi_{jk} \circ \phi_{ij}$ on $U_i \intersect U_j \intersect U_k$. Then there exists a unique sheaf $\mcf$ on $X$, together with isomorphisms $\psi_i:\left.\mcf\right|_{U_i} \iso \mcf_i$, such that for each $i, j, \psi_j= \varphi_{i j} \circ \psi_i$ on $U_i \cap U_j$. We say loosely that $\mcf$ is obtained by **glueing** the sheaves $\mcf_i$ via the isomorphisms $\varphi_i$.