Lemma 63.15.1 (0GLH)—The Stacks project

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Lemma 63.15.1. Let $k$ be an algebraically closed field. Let $X$ be a finite type separated scheme over $k$ . Let $\Lambda$ be a Noetherian ring. Let $K$ be an object of $D^+_{tors, c}(X_{\acute{e}tale}, \Lambda )$ or of $D_ c(X_{\acute{e}tale}, \Lambda )$ in case $\Lambda$ is torsion. Then $H^ i_ c(X, K)$ is a finite $\Lambda$ -module for all $i \in \mathbf{Z}$ .

Proof. Immediate consequence of Theorem 63.14.5 and the definition of compactly supported cohomology in Section 63.12. $\square$

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