Let \(\varphi: \mcf \to \mcg\) 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 every \(s \in \mcg(U)\), there is a covering \(\ts{U_i}\) of \(U\) and elements \(t_i \in \mcf(U_i)\) such that \(\varphi(t_i) = \ro{s}{U_i}\) for all \(i\).
Give an example of a surjective morphism of sheaves \(\varphi: \mcf \to \mcg\) and an open set \(U\) such that \(\varphi(U): \mcf(U) \to \mcg(U)\) is not surjective.
Solution
Part a. (\(\Rightarrow\)) Suppose \(\phi\) is surjective, so \(\phi_P\) is surjective for every \(P\). Fix an open \(U\) and a section \(s \in \mcg(U)\). For each \(P \in U\) the germ \(s_P \in \mcg_P\) lifts to some \(t_P \in \mcf_P\) with \(\phi_P(t_P) = s_P\). As in the stalk arguments of the previous exercise, this identity of germs is already an identity of sections over a small enough neighbourhood: there is an open \(U_P \ni P\) with \(U_P \subseteq U\) and a section \(t \in \mcf(U_P)\) with \(\phi(U_P)(t) = \ro{s}{U_P}\). Doing this for every \(P \in U\) produces the required cover \(\ts{U_P}\) together with the lifts.
(\(\Leftarrow\)) Conversely, suppose the condition holds and fix \(P\) and a germ \(s_P \in \mcg_P\), represented by \((U, s)\). The condition gives a cover \(\ts{U_i}\) of \(U\) and lifts \(t_i \in \mcf(U_i)\) with \(\phi(t_i) = \ro{s}{U_i}\). Choose an index \(i\) with \(P \in U_i\); then the germ of \(t_i\) at \(P\) maps to \(s_P\). So \(\phi_P\) is surjective for all \(P\), and hence \(\phi\) is surjective.
Part b. Take the exponential sequence of sheaves on \(\CC\): \[ 0 \to \ul{\ZZ} \injects (\OO_\CC, +) \surjectsvia{\exp} (\unitsof{\OO_\CC}, \cdot) \to 0. \] Let \(U \da \cstar\) and let \(f(z) = z\), a nowhere-vanishing holomorphic function, so \(f \in \unitsof{\OO_\CC}(U)\). There is no \(g \in \OO_\CC(U)\) with \(\exp(g) = z\) on all of \(U\): such a \(g\) would be a single-valued holomorphic branch of \(\log z\) on the punctured plane, and \(\log\) is multivalued there. So \(\exp(U)\) is not surjective on sections.
Nevertheless \(\exp\) is surjective as a morphism of sheaves. For any \(P \in \cstar\), choose a disc \(V \ni P\) small enough to avoid the origin. On \(V\) a holomorphic branch of \(\log\) exists, so \(\exp\) is surjective on sections over \(V\) and over every smaller open set, hence surjective on the stalk at \(P\).