Lemma 59.93.3. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a locally constant abelian sheaf on $X_{\acute{e}tale}$ such that for every geometric point $\overline{x}$ of $X$ the abelian group $\mathcal{F}_{\overline{x}}$ is a torsion group all of whose elements have order prime to the characteristic of the residue field of $\overline{x}$. If $f$ is locally acyclic, then $f$ is locally acyclic relative to $\mathcal{F}$.

**Proof.**
Namely, let $\overline{x}$ be a geometric point of $X$. Since $\mathcal{F}$ is locally constant we see that the restriction of $\mathcal{F}$ to $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}})$ is isomorphic to the constant sheaf $\underline{M}$ with $M = \mathcal{F}_{\overline{x}}$. By assumption we can write $M = \mathop{\mathrm{colim}}\nolimits M_ i$ as a filtered colimit of finite abelian groups $M_ i$ of order prime to the characteristic of the residue field of $\overline{x}$. Consider a geometric point $\overline{t}$ of $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, f(\overline{x})})$. Since $F_{\overline{x}, \overline{t}}$ is affine, we have

by Lemma 59.51.4. For each $i$ we can write $M_ i = \bigoplus \mathbf{Z}/n_{i, j}\mathbf{Z}$ as a finite direct sum for some integers $n_{i, j}$ prime to the characteristic of the residue field of $\overline{x}$. Since $f$ is locally acyclic we see that

See discussion following Definition 59.93.1. Taking the direct sums and the colimit we conclude that

and we win. $\square$

## 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Your email address will not be published.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.

## Comments (0)