The Stacks project

Lemma 57.87.2. Let $(A, I)$ be a henselian pair. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism of schemes. Let $i : Z \to X$ be the closed immersion of $X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/I)$ into $X$. For any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $\Gamma (X, \mathcal{F}) = \Gamma (Z, i_{small}^{-1}\mathcal{F})$.

Proof. This follows from Lemma 57.79.2 and 57.87.1 and the fact that any scheme finite over $X$ is proper over $\mathop{\mathrm{Spec}}(A)$. $\square$


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