The Stacks project

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. $\square$

Comments (2)

Comment #3603 by David Holmes on

Tiny typos: in the first line, missing period between caligraphic C and 'Let', and missing 'of' between 'presheaf' and caligraphic O.

While I'm bothering you about this lemma, I will also suggest adding the word 'unique' in the sentence 'There exists a map of sheaves of sets' just before the commutative diagram. The uniqueness seems clear (since it can be checked locally), and it makes the later part of the lemma make more sense; the last line would not make sense if F^{hash} admitted multiple O^{hash}-module structures. But probably I'm just nitpicking (or missing something).

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