Lemma 59.84.4. Let $\Lambda $ be a Noetherian ring, let $M$ be a finite $\Lambda $-module which is annihilated by an integer $n > 0$, let $k$ be an algebraically closed field, let $X$ be a separated, finite type scheme of dimension $\leq 1$ over $k$, and let $j : U \to X$ be an open immersion. Then

$H^ q_{\acute{e}tale}(X, j_!\underline{M})$ is a finite $\Lambda $-module if $n$ is prime to $\text{char}(k)$,

$H^ q_{\acute{e}tale}(X, j_!\underline{M})$ is a finite $\Lambda $-module if $X$ is proper.

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