The Stacks project

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6.6 Presheaves of modules

Suppose that $\mathcal{O}$ is a presheaf of rings on $X$. We would like to define the notion of a presheaf of $\mathcal{O}$-modules over $X$. In analogy with Definition 6.4.4 we are tempted to define this as a sheaf of sets $\mathcal{F}$ such that for every open $U \subset X$ the set $\mathcal{F}(U)$ is endowed with the structure of an $\mathcal{O}(U)$-module compatible with restriction mappings (of $\mathcal{F}$ and $\mathcal{O}$). However, it is customary (and equivalent) to define it as in the following definition.

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

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

    commutes.

  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 $\textit{PMod}(\mathcal{O})$.

Suppose that $\mathcal{O}_1 \to \mathcal{O}_2$ is a morphism of presheaves of rings on $X$. In this case, if $\mathcal{F}$ is a presheaf of $\mathcal{O}_2$-modules then we can think of $\mathcal{F}$ as a presheaf of $\mathcal{O}_1$-modules by using the composition

\[ \mathcal{O}_1 \times \mathcal{F} \to \mathcal{O}_2 \times \mathcal{F} \to \mathcal{F}. \]

We sometimes denote this by $\mathcal{F}_{\mathcal{O}_1}$ to indicate the restriction of rings. We call this the restriction of $\mathcal{F}$. We obtain the restriction functor

\[ \textit{PMod}(\mathcal{O}_2) \longrightarrow \textit{PMod}(\mathcal{O}_1) \]

On the other hand, given a presheaf of $\mathcal{O}_1$-modules $\mathcal{G}$ we can construct a presheaf of $\mathcal{O}_2$-modules $\mathcal{O}_2 \otimes _{p, \mathcal{O}_1} \mathcal{G}$ by the rule

\[ \left(\mathcal{O}_2 \otimes _{p, \mathcal{O}_1} \mathcal{G}\right)(U) = \mathcal{O}_2(U) \otimes _{\mathcal{O}_1(U)} \mathcal{G}(U) \]

The index $p$ stands for “presheaf” and not “point”. This presheaf is called the tensor product presheaf. We obtain the change of rings functor

\[ \textit{PMod}(\mathcal{O}_1) \longrightarrow \textit{PMod}(\mathcal{O}_2) \]

Lemma 6.6.2. With $X$, $\mathcal{O}_1$, $\mathcal{O}_2$, $\mathcal{F}$ and $\mathcal{G}$ as above there exists a canonical bijection

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_1}(\mathcal{G}, \mathcal{F}_{\mathcal{O}_1}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_2}( \mathcal{O}_2 \otimes _{p, \mathcal{O}_1} \mathcal{G}, \mathcal{F} ) \]

In other words, the restriction and change of rings functors are adjoint to each other.

Proof. This follows from the fact that for a ring map $A \to B$ the restriction functor and the change of ring functor are adjoint to each other. $\square$


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