The Stacks project

Lemma 59.78.1. Let $X$ be a quasi-compact and quasi-separated scheme.

  1. If $\mathcal{F}$ is a torsion abelian sheaf on $X_{\acute{e}tale}$, then $H^ n_{\acute{e}tale}(X, \mathcal{F})$ is a torsion abelian group for all $n$.

  2. If $K$ in $D^+(X_{\acute{e}tale})$ has torsion cohomology sheaves, then $H^ n_{\acute{e}tale}(X, K)$ is a torsion abelian group for all $n$.

Proof. To prove (1) we write $\mathcal{F} = \bigcup \mathcal{F}[n]$ where $\mathcal{F}[d]$ is the $d$-torsion subsheaf. By Lemma 59.51.4 we have $H^ n_{\acute{e}tale}(X, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ n_{\acute{e}tale}(X, \mathcal{F}[d])$. This proves (1) as $H^ n_{\acute{e}tale}(X, \mathcal{F}[d])$ is annihilated by $d$.

To prove (2) we can use the spectral sequence $E_2^{p, q} = H^ p_{\acute{e}tale}(X, H^ q(K))$ converging to $H^ n_{\acute{e}tale}(X, K)$ (Derived Categories, Lemma 13.21.3) and the result for sheaves. $\square$

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