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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