## 59.78 Torsion sheaves

A brief section on torsion abelian sheaves and their étale cohomology. Let $\mathcal{C}$ be a site. We have shown in Cohomology on Sites, Lemma 21.19.8 that any object in $D(\mathcal{C})$ whose cohomology sheaves are torsion sheaves, can be represented by a complex all of whose terms are torsion.

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

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 $n$.

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.
$\square$

Lemma 59.78.2. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes.

If $\mathcal{F}$ is a torsion abelian sheaf on $X_{\acute{e}tale}$, then $R^ nf_*\mathcal{F}$ is a torsion abelian sheaf on $Y_{\acute{e}tale}$ for all $n$.

If $K$ in $D^+(X_{\acute{e}tale})$ has torsion cohomology sheaves, then $Rf_*K$ is an object of $D^+(Y_{\acute{e}tale})$ whose cohomology sheaves are torsion abelian sheaves.

**Proof.**
Proof of (1). Recall that $R^ nf_*\mathcal{F}$ is the sheaf associated to the presheaf $V \mapsto H^ n_{\acute{e}tale}(X \times _ Y V, \mathcal{F})$ on $Y_{\acute{e}tale}$. See Cohomology on Sites, Lemma 21.7.4. If we choose $V$ affine, then $X \times _ Y V$ is quasi-compact and quasi-separated because $f$ is, hence we can apply Lemma 59.78.1 to see that $H^ n_{\acute{e}tale}(X \times _ Y V, \mathcal{F})$ is torsion.

Proof of (2). Recall that $R^ nf_*K$ is the sheaf associated to the presheaf $V \mapsto H^ n_{\acute{e}tale}(X \times _ Y V, K)$ on $Y_{\acute{e}tale}$. See Cohomology on Sites, Lemma 21.20.6. If we choose $V$ affine, then $X \times _ Y V$ is quasi-compact and quasi-separated because $f$ is, hence we can apply Lemma 59.78.1 to see that $H^ n_{\acute{e}tale}(X \times _ Y V, K)$ is torsion.
$\square$

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