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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