The Stacks project

Lemma 7.35.3. Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$ given by $u : \mathcal{C} \to \textit{Sets}$. Let $U$ be an object of $\mathcal{C}$. For any sheaf $\mathcal{G}$ on $\mathcal{C}/U$ we have

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

where the coproduct is over the points $q$ of $\mathcal{C}/U$ associated to elements $x \in u(U)$ as in Lemma 7.35.1.

Proof. We use the description of $j_{U!}\mathcal{G}$ as the sheaf associated to the presheaf $V \mapsto \coprod \nolimits _{\varphi \in \mathop{Mor}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V/_\varphi U)$ of Lemma 7.25.2. Also, the stalk of $j_{U!}\mathcal{G}$ at $p$ is equal to the stalk of this presheaf, see Lemma 7.32.5. Hence we see that

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

To each element $(V, y, \varphi , s)$ of this colimit, we can assign $x = u(\varphi )(y) \in u(U)$. Hence we obtain

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

This is equal to the expression of the lemma by our construction of the points $q$. $\square$


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