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Tag 03CY

Chapter 18: Modules on Sites > Section 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 $\mathcal{O}$-modules. Let $\mathcal{O}^\#$ be the sheafification of $\mathcal{O}$ as a presheaf of rings, see Sites, Section 7.43. 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. $\square$

    The code snippet corresponding to this tag is a part of the file sites-modules.tex and is located in lines 814–846 (see updates for more information).

    \begin{lemma}
    \label{lemma-sheafification-presheaf-modules}
    Let $\mathcal{C}$ be a site.
    Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$
    Let $\mathcal{F}$ be a presheaf $\mathcal{O}$-modules.
    Let $\mathcal{O}^\#$ be the sheafification of $\mathcal{O}$ as a presheaf
    of rings, see Sites, Section \ref{sites-section-sheaves-algebraic-structures}.
    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.
    \end{lemma}
    
    \begin{proof}
    Omitted.
    \end{proof}

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