6.20 Sheafification of presheaves of modules
Lemma 6.20.1. Let X be a topological space. Let \mathcal{O} be a presheaf of rings on X. Let \mathcal{F} be a presheaf \mathcal{O}-modules. Let \mathcal{O}^\# be the sheafification of \mathcal{O}. Let \mathcal{F}^\# be the sheafification of \mathcal{F} as a presheaf of abelian groups. There exists a 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})
Let X be a topological space. Let \mathcal{O}_1 \to \mathcal{O}_2 be a morphism of sheaves of rings on X. In Section 6.6 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 6.20.2. 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 _{\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 6.6.2 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 6.20.3. Let X be a topological space. Let \mathcal{O} \to \mathcal{O}' be a morphism of sheaves of rings on X. Let \mathcal{F} be a sheaf \mathcal{O}-modules. Let x \in X. We have
\mathcal{F}_ x \otimes _{\mathcal{O}_ x} \mathcal{O}'_ x = (\mathcal{F} \otimes _\mathcal {O} \mathcal{O}')_ x
as \mathcal{O}'_ x-modules.
Proof.
Follows directly from Lemma 6.14.2 and the fact that taking stalks commutes with sheafification.
\square
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