18.11 Sheafification of presheaves of modules
Lemma 18.11.1. Let \mathcal{C} be a site. Let \mathcal{O} be a presheaf of rings on \mathcal{C}. Let \mathcal{F} be a presheaf of \mathcal{O}-modules. Let \mathcal{O}^\# be the sheafification of \mathcal{O} as a presheaf of rings, see Sites, Section 7.44. Let \mathcal{F}^\# be the sheafification of \mathcal{F} as a presheaf of abelian groups. There exists a unique map of sheaves of sets
\mathcal{O}^\# \times \mathcal{F}^\# \longrightarrow \mathcal{F}^\#
which makes the diagram
\xymatrix{ \mathcal{O} \times \mathcal{F} \ar[r] \ar[d] & \mathcal{F} \ar[d] \\ \mathcal{O}^\# \times \mathcal{F}^\# \ar[r] & \mathcal{F}^\# }
commute and which makes \mathcal{F}^\# into a sheaf of \mathcal{O}^\# -modules. In addition, if \mathcal{G} is a sheaf of \mathcal{O}^\# -modules, then any morphism of presheaves of \mathcal{O}-modules \mathcal{F} \to \mathcal{G} (into the restriction of \mathcal{G} to a \mathcal{O}-module) factors uniquely as \mathcal{F} \to \mathcal{F}^\# \to \mathcal{G} where \mathcal{F}^\# \to \mathcal{G} is a morphism of \mathcal{O}^\# -modules.
Proof.
Omitted.
\square
This actually means that the functor i : \textit{Mod}(\mathcal{O}^\# ) \to \textit{PMod}(\mathcal{O}) (combining restriction and including sheaves into presheaves) and the sheafification functor of the lemma {}^\# : \textit{PMod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}^\# ) are adjoint. In a formula
\mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O})}(\mathcal{F}, i\mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O}^\# )}(\mathcal{F}^\# , \mathcal{G})
An important case happens when \mathcal{O} is already a sheaf of rings. In this case the formula reads
\mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O})}(\mathcal{F}, i\mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O})}(\mathcal{F}^\# , \mathcal{G})
because \mathcal{O} = \mathcal{O}^\# in this case.
Lemma 18.11.2. Let \mathcal{C} be a site. Let \mathcal{O} be a presheaf of rings on \mathcal{C} The sheafification functor
\textit{PMod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}^\# ), \quad \mathcal{F} \longmapsto \mathcal{F}^\#
is exact.
Proof.
This is true because it holds for sheafification \textit{PAb}(\mathcal{C}) \to \textit{Ab}(\mathcal{C}). See the discussion in Section 18.3.
\square
Let \mathcal{C} be a site. Let \mathcal{O}_1 \to \mathcal{O}_2 be a morphism of sheaves of rings on \mathcal{C}. In Section 18.9 we defined a restriction functor and a change of rings functor on presheaves of modules associated to this situation.
If \mathcal{F} is a sheaf of \mathcal{O}_2-modules then the restriction \mathcal{F}_{\mathcal{O}_1} of \mathcal{F} is clearly a sheaf of \mathcal{O}_1-modules. We obtain the restriction functor
\textit{Mod}(\mathcal{O}_2) \longrightarrow \textit{Mod}(\mathcal{O}_1)
On the other hand, given a sheaf of \mathcal{O}_1-modules \mathcal{G} the presheaf of \mathcal{O}_2-modules \mathcal{O}_2 \otimes _{p, \mathcal{O}_1} \mathcal{G} is in general not a sheaf. Hence we define the tensor product sheaf \mathcal{O}_2 \otimes _{\mathcal{O}_1} \mathcal{G} by the formula
\mathcal{O}_2 \otimes _{\mathcal{O}_1} \mathcal{G} = (\mathcal{O}_2 \otimes _{p, \mathcal{O}_1} \mathcal{G})^\#
as the sheafification of our construction for presheaves. We obtain the change of rings functor
\textit{Mod}(\mathcal{O}_1) \longrightarrow \textit{Mod}(\mathcal{O}_2)
Lemma 18.11.3. 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 _{\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 Lemma 18.9.2 and the fact that \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_2}( \mathcal{O}_2 \otimes _{\mathcal{O}_1} \mathcal{G}, \mathcal{F} ) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_2}( \mathcal{O}_2 \otimes _{p, \mathcal{O}_1} \mathcal{G}, \mathcal{F} ) because \mathcal{F} is a sheaf.
\square
Lemma 18.11.4. Let \mathcal{C} be a site. Let \mathcal{O} \to \mathcal{O}' be an epimorphism of sheaves of rings. Let \mathcal{G}_1, \mathcal{G}_2 be \mathcal{O}'-modules. Then
\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}'}(\mathcal{G}_1, \mathcal{G}_2) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{G}_1, \mathcal{G}_2).
In other words, the restriction functor \textit{Mod}(\mathcal{O}') \to \textit{Mod}(\mathcal{O}) is fully faithful.
Proof.
This is the sheaf version of Algebra, Lemma 10.107.14 and is proved in exactly the same way.
\square
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