## 6.10 Sheaves of modules

Definition 6.10.1. Let $X$ be a topological space. Let $\mathcal{O}$ be a sheaf of rings on $X$.

A *sheaf of $\mathcal{O}$-modules* is a presheaf of $\mathcal{O}$-modules $\mathcal{F}$, see Definition 6.6.1, such that the underlying presheaf of abelian groups $\mathcal{F}$ is a sheaf.

A *morphism of sheaves of $\mathcal{O}$-modules* is a morphism of presheaves of $\mathcal{O}$-modules.

Given sheaves of $\mathcal{O}$-modules $\mathcal{F}$ and $\mathcal{G}$ we denote $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G})$ the set of morphism of sheaves of $\mathcal{O}$-modules.

The category of sheaves of $\mathcal{O}$-modules is denoted $\textit{Mod}(\mathcal{O})$.

This definition kind of makes sense even if $\mathcal{O}$ is just a presheaf of rings, although we do not know any examples where this is useful, and we will avoid using the terminology “sheaves of $\mathcal{O}$-modules” in case $\mathcal{O}$ is not a sheaf of rings.

## Comments (0)