The Stacks project

Lemma 68.9.3. Let $S$ be a scheme. Let $f : X \to Y$ be an ├ętale morphism of algebraic spaces over $S$. Let $Z \subset Y$ be a closed subspace such that $f^{-1}(Z) \to Z$ is an isomorphism of algebraic spaces. Let $\mathcal{F}$ be an abelian sheaf on $X$. Then

\[ \mathcal{H}^ q_ Z(\mathcal{F}) = \mathcal{H}^ q_{f^{-1}(Z)}(f^{-1}\mathcal{F}) \]

as abelian sheaves on $Z = f^{-1}(Z)$ and we have $H^ q_ Z(Y, \mathcal{F}) = H^ q_{f^{-1}(Z)}(X, f^{-1}\mathcal{F})$.

Proof. Because $f$ is ├ętale an injective resolution of $\mathcal{F}$ pulls back to an injective resolution of $f^{-1}\mathcal{F}$. Hence it suffices to check the equality for $\mathcal{H}_ Z(-)$ which follows from the definitions. The proof for cohomology with supports is the same. Some details omitted. $\square$


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