18.9 Presheaves of modules
Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$. At this point we have not yet defined a presheaf of $\mathcal{O}$-modules. Thus we do so right now.
Definition 18.9.1. Let $\mathcal{C}$ be a category, and let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$.
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 object $U$ of $\mathcal{C}$ 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 $\mathcal{F}(U)$.
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.
The set of $\mathcal{O}$-module morphisms as above is denoted $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G})$.
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 the category $\mathcal{C}$. 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
\[ U \longmapsto \left(\mathcal{O}_2 \otimes _{p, \mathcal{O}_1} \mathcal{G}\right)(U) = \mathcal{O}_2(U) \otimes _{\mathcal{O}_1(U)} \mathcal{G}(U) \]
where $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, with obvious restriction mappings. 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 18.9.2. With $\mathcal{C}$, $\mathcal{O}_1 \to \mathcal{O}_2$, $\mathcal{F}$ 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 defined above 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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