Let \(\varphi: \mcf \to \mcg\) be a morphism of presheaves such that \(\varphi(U): \mcf(U) \to \mcg(U)\) is injective for each \(U\). Show that the induced map \(\varphi^{+}: \mcf^{+} \to \mcg^{+}\) of associated sheaves is injective.
Use part (a) to show that if \(\varphi: \mcf \to \mcg\) is a morphism of sheaves, then \(\im \varphi\) can be naturally identified with a subsheaf of \(\mcg\).
Solution
Part a. A stalk is a filtered colimit of the groups of sections, and a filtered colimit of injections of abelian groups is injective. So if \(\phi(U)\) is injective for every \(U\), then \(\phi_P\) is injective for every \(P\). Sheafification does not change stalks, so \((\phi^+)_P = \phi_P\) is injective for every \(P\), and by the stalkwise criterion for injectivity of a morphism of sheaves, \(\phi^+\) is injective.
Part b. The presheaf image \(U \mapsto \im\qty{\phi(U)}\) is a subgroup of \(\mcg(U)\) for every \(U\), and the inclusions \[ \psi(U): \im\qty{\phi(U)} \injects \mcg(U) \] are compatible with restriction, so they assemble into a morphism of presheaves \(\psi\) into the sheaf \(\mcg\). Each \(\psi(U)\) is injective, so by part (a) the induced map on associated sheaves is injective. Since \(\mcg\) is already a sheaf, the target of the sheafified map is \(\mcg\) itself, and the source is \(\im \phi\) by definition. The restriction maps of \(\im \phi\) are those induced from \(\mcg\), so \(\im \phi\) is identified with a subsheaf of \(\mcg\).