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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