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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