The Stacks project

Lemma 59.84.6. Let $\Lambda $ be a Noetherian ring, let $k$ be an algebraically closed field, let $X$ be a separated, finite type scheme of dimension $\leq 1$ over $k$, let $j : U \to X$ be an open immersion with $U$ connected, let $\ell $ be a prime number, let $n > 0$, and let $\mathcal{G}$ be a finite type, locally constant sheaf of $\Lambda $-modules on $U_{\acute{e}tale}$ annihilated by $\ell ^ n$. Then

  1. $H^ q_{\acute{e}tale}(X, j_!\mathcal{G})$ is a finite $\Lambda $-module if $\ell $ is prime to $\text{char}(k)$,

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

Proof. Let $f : V \to U$ be a finite étale morphism of degree prime to $\ell $ as in Lemma 59.66.4. The discussion in Section 59.66 gives maps

\[ \mathcal{G} \to f_*f^{-1}\mathcal{G} \to \mathcal{G} \]

whose composition is an isomorphism. Hence it suffices to prove the finiteness of $H^ q_{\acute{e}tale}(X, j_!f_*f^{-1}\mathcal{G})$. By Zariski's Main theorem (More on Morphisms, Lemma 37.42.3) we can choose a diagram

\[ \xymatrix{ V \ar[r]_{j'} \ar[d]_ f & Y \ar[d]^{\overline{f}} \\ U \ar[r]^ j & X } \]

with $\overline{f} : Y \to X$ finite and $j'$ an open immersion with dense image. Since $f$ is finite and $V$ dense in $Y$ we have $V = U \times _ X Y$. By Lemma 59.70.9 we have

\[ j_!f_*f^{-1}\mathcal{G} = \overline{f}_*j'_!f^{-1}\mathcal{G} \]

By Lemma 59.84.2 it suffices to consider $j'_!f^{-1}\mathcal{G}$. The existence of the filtration given by Lemma 59.66.4, the fact that $j'_!$ is exact, and Lemma 59.84.5 reduces us to the case $\mathcal{F} = j'_!\underline{M}$ for a finite $\Lambda $-module $M$ which is Lemma 59.84.4. $\square$


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