Proposition 59.93.2. Let $f : X \to S$ be a smooth morphism of schemes. Then $f$ is universally locally acyclic.
Proof. Since the base change of a smooth morphism is smooth, it suffices to show that smooth morphisms are locally acyclic. Let $\overline{x}$ be a geometric point of $X$ with image $\overline{s} = f(\overline{x})$. Let $\overline{t}$ be a geometric point of $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, f(\overline{x})})$. Since we are trying to prove a property of the ring map $\mathcal{O}^{sh}_{S, \overline{s}} \to \mathcal{O}^{sh}_{X, \overline{x}}$ (see discussion following Definition 59.93.1) we may and do replace $f : X \to S$ by the base change $X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$. Thus we may and do assume that $S$ is the spectrum of a strictly henselian local ring and that $\overline{s}$ lies over the closed point of $S$.
We will apply Lemma 59.86.5 to the diagram
\[ \xymatrix{ X \ar[d]_ f & X_{\overline{t}} \ar[l]^ h \ar[d]^ e \\ S & \overline{t} \ar[l]_ g } \]
and the sheaf $\mathcal{F} = \underline{M}$ where $M = \mathbf{Z}/n\mathbf{Z}$ for some integer $n$ prime to the characteristic of the residue field of $\overline{x}$. We know that the map $f^{-1}R^ qg_*\mathcal{F} \to R^ qh_*e^{-1}\mathcal{F}$ is an isomorphism by smooth base change, see Theorem 59.89.2 (the assumption on torsion holds by our choice of $n$). Thus Lemma 59.86.5 gives us the middle equality in
\[ H^ q(F_{\overline{x}, \overline{t}}, \underline{M}) = H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _ S \overline{t}, \underline{M}) = H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \times _ S \overline{t}, \underline{M}) = H^ q(\overline{t}, \underline{M}) \]
For the outer two equalities we use that $S = \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$. Since $\overline{t}$ is the spectrum of a separably closed field we conclude that
\[ H^ q(F_{\overline{x}, \overline{t}}, \underline{M}) = \left\{ \begin{matrix} M
& \text{if}
& q = 0
\\ 0
& \text{if}
& q \not= 0
\end{matrix} \right. \]
which is what we had to show (see discussion following Definition 59.93.1). $\square$
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