The Stacks project

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.

  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. $\square$

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

  1. 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 $n$.

  2. 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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DDB. Beware of the difference between the letter 'O' and the digit '0'.