Lemma 59.84.4 (0GJF)—The Stacks project

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.

Proof. Since $\dim (X) \leq 1$ there is an open $V \subset X$ which is disjoint from $U$ such that $X' = U \cup V$ is dense open in $X$ (details omitted). If $j' : X' \to X$ denotes the inclusion morphism, then we see that $j_!\underline{M}$ is a direct summand of $j'_!\underline{M}$ . Hence it suffices to prove the lemma in case $U$ is open and dense in $X$ . This case follows from Lemmas 59.84.3 and 59.84.1. $\square$

The Stacks project

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