Exercise 2, § 3

Let X be a topological space, let $\mathscr{F}$ be a sheaf of abelian groups on X.

a) Let $s\in \mathscr{F}(X)$. Prove that there exists a smallest closed subset $F\subset$ X such that $s|(X-F) = 0$. It is denoted supp($s$) and is called the support of $s$. b) Let $\mathscr{G}$ be a sheaf of abelian groups on X and let $u:\mathscr{F}\rightarrow \mathscr{G}$ be a morphism of sheaves of abelian groups. Prove that, for every section $s\in \mathscr{F}(X)$, one has supp($u_X(s)$)$\subset$ supp($s$).

c) Conversely, let $v:\mathscr{F}(X)\rightarrow \mathscr{G}(X)$ be a morphism of abelian groups such that supp($v(s)$)$\subset$ supp($s$) for every $s\in \mathscr{F}(X)$. If the sheaf $\mathscr{F}$ is fine, prove that there exists a unique morphism of sheaves $u:\mathscr{F}\rightarrow \mathscr{G}$ such that $v=u_X$.