The Stacks project

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

\[ H^ q(F_{\overline{x}, \overline{t}}, \underline{M}) = \mathop{\mathrm{colim}}\nolimits H^ q(F_{\overline{x}, \overline{t}}, \underline{M_ i}) \]

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

\[ H^ q(F_{\overline{x}, \overline{t}}, \underline{\mathbf{Z}/n_{i, j}\mathbf{Z}}) = \left\{ \begin{matrix} \mathbf{Z}/n_{i, j}\mathbf{Z} & \text{if} & q = 0 \\ 0 & \text{if} & q \not= 0 \end{matrix} \right. \]

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

\[ H^ q(F_{\overline{x}, \overline{t}}, \underline{M}) = \left\{ \begin{matrix} M & \text{if} & q = 0 \\ 0 & \text{if} & q \not= 0 \end{matrix} \right. \]

and we win. $\square$

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