The Stacks project

Lemma 38.37.7. Let $\mathcal{F}$ be a sheaf on one of the sites $(\mathit{Sch}/S)_ h$ constructed in Definition 38.34.12. Then for any almost blow up square (38.37.0.1) in the category $(\mathit{Sch}/S)_ h$ the diagram

\[ \xymatrix{ \mathcal{F}(E) & \mathcal{F}(X') \ar[l] \\ \mathcal{F}(Z) \ar[u] & \mathcal{F}(X) \ar[u] \ar[l] } \]

is cartesian in the category of sets.

Proof. Since $Z \amalg X' \to X$ is a surjective proper morphism of finite presentation we see that $\{ Z \amalg X' \to X\} $ is an h covering (Lemma 38.34.6). We have

\[ (Z \amalg X') \times _ X (Z \amalg X') = Z \amalg E \amalg E \amalg X' \times _ X X' \]

Since $\mathcal{F}$ is a Zariski sheaf we see that $\mathcal{F}$ sends disjoint unions to products. Thus the sheaf condition for the covering $\{ Z \amalg X' \to X\} $ says that $\mathcal{F}(X) \to \mathcal{F}(Z) \times \mathcal{F}(X')$ is injective with image the set of pairs $(t, s')$ such that (a) $t|_ E = s'|_ E$ and (b) $s'$ is in the equalizer of the two maps $\mathcal{F}(X') \to \mathcal{F}(X' \times _ X X')$. Next, observe that the obvious morphism

\[ E \times _ Z E \amalg X' \longrightarrow X' \times _ X X' \]

is a surjective proper morphism of finite presentation as $b$ induces an isomorphism $X' \setminus E \to X \setminus Z$. We conclude that $\mathcal{F}(X' \times _ X X') \to \mathcal{F}(E \times _ Z E) \times \mathcal{F}(X')$ is injective. It follows that (a) $\Rightarrow $ (b) which means that the lemma is true. $\square$


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