Sheaf of modules - Knowlpedia

Definition

Let $(X,\mathcal O_X)$ be a ringed space. A sheaf of $\mathcal O_X$ -modules, or an $\mathcal O_X$ -module, is a sheaf $\mathcal F$ such that $\mathcal F(U)$ is an $\mathcal O_X(U)$ -module for every open $U\subseteq X$ , and restriction commutes with scalar multiplication:

(a s)|_V=(a|_V)(s|_V) \qquad(V\subseteq U).

Equivalently, $\mathcal F$ is a module object over the ring object $\mathcal O_X$ in sheaves of abelian groups.

Morphisms and stalks

A morphism $\varphi:\mathcal F\to\mathcal G$ of $\mathcal O_X$ -modules is a morphism of sheaves whose map on every open set is $\mathcal O_X(U)$ -linear. It induces an $\mathcal O_{X,x}$ -linear map

\varphi_x:\mathcal F_x\longrightarrow\mathcal G_x

on every stalk. Kernels are computed sectionwise, while a cokernel is the sheafification of the sectionwise presheaf cokernel. These constructions make the category of $\mathcal O_X$ -modules an abelian category.

Examples

The structure sheaf $\mathcal O_X$ is a module over itself, as is every finite direct sum $\mathcal O_X^{\oplus r}$ . Ideals of $\mathcal O_X$ form subsheaves of modules. On a smooth manifold, smooth sections of a vector bundle form a sheaf of modules over the sheaf of smooth functions.

A sheaf of modules is not merely one module with a topology attached. Its sections and scalar rings vary over all open sets and retain gluing data that the global module $\mathcal F(X)$ can lose.

References

The Stacks Project Authors, The Stacks Project. Tag 01AG. Relevant: sheaves of modules on ringed spaces.
Robin Hartshorne, Algebraic Geometry, Springer, 1977. DOI record. Relevant: Chapter II, §1, sheaves of modules.