The Stacks project

Lemma 18.38.1. 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 $V \mapsto \bigoplus \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V/_\varphi U)$ of Lemma 18.19.2. The stalk of $j_{U!}\mathcal{G}$ at $p$ is equal to the stalk of this presheaf, see Lemma 18.36.3. 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. $\square$

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