The Stacks project

Lemma 18.37.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $p$ be a point of $\mathcal{C}$. Let $U$ be an object of $\mathcal{C}$. For $\mathcal{G}$ in $\textit{Mod}(\mathcal{O}_ U)$ we have

\[ (j_{U!}\mathcal{G})_ p = \bigoplus \nolimits _ q \mathcal{G}_ q \]

where the coproduct is over the points $q$ of $\mathcal{C}/U$ lying over $p$, see Sites, Lemma 7.35.2.

Proof. We use the description of $j_{U!}\mathcal{G}$ as the sheaf associated to the presheaf $V \mapsto \bigoplus \nolimits _{\varphi \in \mathop{Mor}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V/_\varphi U)$ of Lemma 18.19.2. The stalk of $j_{U!}\mathcal{G}$ at $p$ is equal to the stalk of this presheaf, see Lemma 18.35.3. Let $u : \mathcal{C} \to \textit{Sets}$ be the functor corresponding to $p$ (see Sites, Section 7.32). Hence we see that

\[ (j_{U!}\mathcal{G})_ p = \mathop{\mathrm{colim}}\nolimits _{(V, y)} \bigoplus \nolimits _{\varphi : V \to U} \mathcal{G}(V/_\varphi U) \]

where the colimit is taken in the category of abelian groups. To a quadruple $(V, y, \varphi , s)$ occurring in this colimit, we can assign $x = u(\varphi )(y) \in u(U)$. Hence we obtain

\[ (j_{U!}\mathcal{G})_ p = \bigoplus \nolimits _{x \in u(U)} \mathop{\mathrm{colim}}\nolimits _{(\varphi : V \to U, y), \ u(\varphi )(y) = x} \mathcal{G}(V/_\varphi U). \]

This is equal to the expression of the lemma by the description of the points $q$ lying over $x$ in Sites, Lemma 7.35.2. $\square$


Comments (0)


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Your email address will not be published.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 0710. Beware of the difference between the letter 'O' and the digit '0'.