The Stacks project

Lemma 7.27.1. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. If the topology on $\mathcal{C}$ is subcanonical, see Definition 7.12.2, and if $\mathcal{G}$ is a sheaf on $\mathcal{C}/U$, then

\[ j_{U!}(\mathcal{G})(V) = \coprod \nolimits _{\varphi \in \mathop{Mor}\nolimits _\mathcal {C}(V, U)} \mathcal{G}(V \xrightarrow {\varphi } U), \]

in other words sheafification is not necessary in Lemma 7.25.2.

Proof. Let $\mathcal{V} = \{ V_ i \to V\} _{i \in I}$ be a covering of $V$ in the site $\mathcal{C}$. We are going to check the sheaf condition for the presheaf $\mathcal{H}$ of Lemma 7.25.2 directly. Let $(s_ i, \varphi _ i)_{i \in I} \in \prod _ i \mathcal{H}(V_ i)$, This means $\varphi _ i : V_ i \to U$ is a morphism in $\mathcal{C}$, and $s_ i \in \mathcal{G}(V_ i \xrightarrow {\varphi _ i} U)$. The restriction of the pair $(s_ i, \varphi _ i)$ to $V_ i \times _ V V_ j$ is the pair $(s_ i|_{V_ i \times _ V V_ j/U}, \text{pr}_1 \circ \varphi _ i)$, and likewise the restriction of the pair $(s_ j, \varphi _ j)$ to $V_ i \times _ V V_ j$ is the pair $(s_ j|_{V_ i \times _ V V_ j/U}, \text{pr}_2 \circ \varphi _ j)$. Hence, if the family $(s_ i, \varphi _ i)$ lies in $\check{H}^0(\mathcal{V}, \mathcal{H})$, then we see that $\text{pr}_1 \circ \varphi _ i = \text{pr}_2 \circ \varphi _ j$. The condition that the topology on $\mathcal{C}$ is weaker than the canonical topology then implies that there exists a unique morphism $\varphi : V \to U$ such that $\varphi _ i$ is the composition of $V_ i \to V$ with $\varphi $. At this point the sheaf condition for $\mathcal{G}$ guarantees that the sections $s_ i$ glue to a unique section $s \in \mathcal{G}(V \xrightarrow {\varphi } U)$. Hence $(s, \varphi ) \in \mathcal{H}(V)$ as desired. $\square$


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