The Stacks project

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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

\[ \mathop{Mor}\nolimits _{\textit{PMod}(\mathcal{O})}(\mathcal{F}, i\mathcal{G}) = \mathop{Mor}\nolimits _{\textit{Mod}(\mathcal{O}^\# )}(\mathcal{F}^\# , \mathcal{G}) \]

An important case happens when $\mathcal{O}$ is already a sheaf of rings. In this case the formula reads

\[ \mathop{Mor}\nolimits _{\textit{PMod}(\mathcal{O})}(\mathcal{F}, i\mathcal{G}) = \mathop{Mor}\nolimits _{\textit{Mod}(\mathcal{O})}(\mathcal{F}^\# , \mathcal{G}) \]

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 10.106.14 and is proved in exactly the same way. $\square$


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