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).

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03CY. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 18.11.1 as pdf