## 109.32 Sheaves

A morphism $f : X \to Y$ of a category $\mathcal{C}$ is an monomorphism if for every pair of morphisms $a, b : W \to X$ we have $f \circ a = f \circ b \Rightarrow a = b$. A monomorphism in the category of sets is an injective map of sets.

Exercise 109.32.1. Carefully prove that a map of sheaves of sets is an monomorphism (in the category of sheaves of sets) if and only if the induced maps on all the stalks are injective.

A morphism $f : X \to Y$ of a category $\mathcal{C}$ is an isomorphism if there exists a morphism $g : Y \to X$ such that $f \circ g = \text{id}_ Y$ and $g \circ f = \text{id}_ X$. An isomorphism in the category of sets is a bijective map of sets.

Exercise 109.32.2. Carefully prove that a map of sheaves of sets is an isomorphism (in the category of sheaves of sets) if and only if the induced maps on all the stalks are bijective.

A morphism $f : X \to Y$ of a category $\mathcal{C}$ is an epimorphism if for every pair of morphisms $a, b : Y \to Z$ we have $a \circ f = b \circ f \Rightarrow a = b$. An epimorphism in the category of sets is a surjective map of sets.

Exercise 109.32.3. Carefully prove that a map of sheaves of sets is an epimorphism (in the category of sheaves of sets) if and only if the induced maps on all the stalks are surjective.

Exercise 109.32.4. Let $f : X \to Y$ be a map of topological spaces. Prove pushforward $f_\ast$ and pullback $f^{-1}$ for sheaves of sets form an adjoint pair of functors.

Exercise 109.32.5. Let $j : U \to X$ be an open immersion. Show that

1. Pullback $j^{-1} : \mathop{\mathit{Sh}}\nolimits (X) \to \mathop{\mathit{Sh}}\nolimits (U)$ has a left adjoint $j_{!} : \mathop{\mathit{Sh}}\nolimits (U) \to \mathop{\mathit{Sh}}\nolimits (X)$ called extension by the empty set.

2. Characterize the stalks of $j_{!}({\mathcal G})$ for $\mathcal{G} \in \mathop{\mathit{Sh}}\nolimits (U)$.

3. Pullback $j^{-1} : \textit{Ab}(X) \to \textit{Ab}(U)$ has a left adjoint $j_{!} : \textit{Ab}(U) \to \textit{Ab}(X)$ called extension by zero.

4. Characterize the stalks of $j_{!}({\mathcal G})$ for $\mathcal{G} \in \textit{Ab}(U)$.

Observe that extension by zero differs from extension by the empty set!

Exercise 109.32.6. Let $X = \mathbf{R}$ with the usual topology. Let $\mathcal{O}_ X = \underline{\mathbf{Z}/2\mathbf{Z}}_ X$. Let $i : Z = \{ 0\} \to X$ be the inclusion and let $\mathcal{O}_ Z = \underline{\mathbf{Z}/2\mathbf{Z}}_ Z$. Prove the following (the first three follow from the definitions but if you are not clear on the definitions you should elucidate them):

1. $i_*\mathcal{O}_ Z$ is a skyscraper sheaf.

2. There is a canonical surjective map from $\underline{\mathbf{Z}/2\mathbf{Z}}_ X \to i_*\underline{\mathbf{Z}/2\mathbf{Z}}_ Z$. Denote the kernel $\mathcal{I} \subset \mathcal{O}_ X$.

3. $\mathcal{I}$ is an ideal sheaf of $\mathcal{O}_ X$.

4. The sheaf $\mathcal{I}$ on $X$ cannot be locally generated by sections (as in Modules, Definition 17.8.1.)

Exercise 109.32.7. Let $X$ be a topological space. Let ${\mathcal F}$ be an abelian sheaf on $X$. Show that ${\mathcal F}$ is the quotient of a (possibly very large) direct sum of sheaves all of whose terms are of the form

$j_{!}(\underline{{\mathbf Z}}_ U)$

where $U \subset X$ is open and $\underline{{\mathbf Z}}_ U$ denotes the constant sheaf with value ${\mathbf Z}$ on $U$.

Remark 109.32.8. Let $X$ be a topological space. In the category of abelian sheaves the direct sum of a family of sheaves $\{ {\mathcal F}_ i\} _{i\in I}$ is the sheaf associated to the presheaf $U \mapsto \oplus {\mathcal F}_ i(U)$. Consequently the stalk of the direct sum at a point $x$ is the direct sum of the stalks of the ${\mathcal F}_ i$ at $x$.

Exercise 109.32.9. Let $X$ be a topological space. Suppose we are given a collection of abelian groups $A_ x$ indexed by $x \in X$. Show that the rule $U \mapsto \prod _{x \in U} A_ x$ with obvious restriction mappings defines a sheaf $\mathcal{G}$ of abelian groups. Show, by an example, that usually it is not the case that $\mathcal{G}_ x = A_ x$ for $x \in X$.

Exercise 109.32.10. Let $X$, $A_ x$, $\mathcal{G}$ be as in Exercise 109.32.9. Let $\mathcal{B}$ be a basis for the topology of $X$, see Topology, Definition 5.5.1. For $U \in \mathcal{B}$ let $A_ U$ be a subgroup $A_ U \subset \mathcal{G}(U) = \prod _{x \in U} A_ x$. Assume that for $U \subset V$ with $U, V \in \mathcal{B}$ the restriction maps $A_ V$ into $A_ U$. For $U \subset X$ open set

$\mathcal{F}(U) = \left\{ (s_ x)_{x \in U} \middle | \begin{matrix} \text{ for every }x\text{ in }U\text{ there exists } V \in \mathcal{B} \\ x \in V \subset U\text{ such that } (s_ y)_{y \in V} \in A_ V \end{matrix} \right\}$

Show that $\mathcal{F}$ defines a sheaf of abelian groups on $X$. Show, by an example, that it is usually not the case that $\mathcal{F}(U) = A_ U$ for $U \in \mathcal{B}$.

Exercise 109.32.11. Give an example of a topological space $X$ and a functor

$F : \mathop{\mathit{Sh}}\nolimits (X) \longrightarrow \textit{Sets}$

which is exact (commutes with finite products and equalizers and commutes with finite coproducts and coequalizers, see Categories, Section 4.23), but there is no point $x \in X$ such that $F$ is isomorphic to the stalk functor $\mathcal{F} \mapsto \mathcal{F}_ x$.

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