Lemma 61.17.3 (0F69)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 61: Pro-étale Cohomology
Section 61.17: Comparing big and small topoi
Lemma 61.17.3 (cite)

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Lemma 61.17.3. Let $f : T \to S$ be a morphism of schemes. For $K$ in $D(S_{pro\text{-}\acute{e}tale})$ we have

$H^ n_{pro\text{-}\acute{e}tale}(S, \pi _ S^{-1}K) = H^ n(S_{pro\text{-}\acute{e}tale}, K)$

and

$H^ n_{pro\text{-}\acute{e}tale}(T, \pi _ S^{-1}K) = H^ n(T_{pro\text{-}\acute{e}tale}, f_{small}^{-1}K).$

For $M$ in $D((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})$ we have

$H^ n_{pro\text{-}\acute{e}tale}(T, M) = H^ n(T_{pro\text{-}\acute{e}tale}, i_ f^{-1}M).$

Proof. To prove the last equality represent $M$ by a K-injective complex of abelian sheaves and apply Lemma 61.17.1 and work out the definitions. The second equality follows from this as $i_ f^{-1} \circ \pi _ S^{-1} = f_{small}^{-1}$ . The first equality is a special case of the second one. $\square$

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