The Stacks project

Definition 6.6.1. Let $X$ be a topological space, and let $\mathcal{O}$ be a presheaf of rings on $X$.

  1. A presheaf of $\mathcal{O}$-modules is given by an abelian presheaf $\mathcal{F}$ together with a map of presheaves of sets

    \[ \mathcal{O} \times \mathcal{F} \longrightarrow \mathcal{F} \]

    such that for every open $U \subset X$ the map $\mathcal{O}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)$ defines the structure of an $\mathcal{O}(U)$-module structure on the abelian group $\mathcal{F}(U)$.

  2. A morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves of $\mathcal{O}$-modules is a morphism of abelian presheaves $\varphi : \mathcal{F} \to \mathcal{G}$ such that the diagram

    \[ \xymatrix{ \mathcal{O} \times \mathcal{F} \ar[r] \ar[d]_{\text{id} \times \varphi } & \mathcal{F} \ar[d]^{\varphi } \\ \mathcal{O} \times \mathcal{G} \ar[r] & \mathcal{G} } \]


  3. The set of $\mathcal{O}$-module morphisms as above is denoted $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G})$.

  4. The category of presheaves of $\mathcal{O}$-modules is denoted $\textit{PMod}(\mathcal{O})$.

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