The Stacks project

Definition 59.93.1. Let $f : X \to S$ be a morphism of schemes. Let $K$ be an object of $D(X_{\acute{e}tale})$.

1. Let $\overline{x}$ be a geometric point of $X$ with image $\overline{s} = f(\overline{x})$. We say $f$ is locally acyclic at $\overline{x}$ relative to $K$ if for every geometric point $\overline{t}$ of $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$ the map (59.93.0.1) is an isomorphism¹.

2. We say $f$ is locally acyclic relative to $K$ if $f$ is locally acyclic at $\overline{x}$ relative to $K$ for every geometric point $\overline{x}$ of $X$.

3. We say $f$ is universally locally acyclic relative to $K$ if for any morphism $S' \to S$ of schemes the base change $f' : X' \to S'$ is locally acyclic relative to the pullback of $K$ to $X'$.

4. We say $f$ is locally acyclic if for all geometric points $\overline{x}$ of $X$ and any integer $n$ prime to the characteristic of $\kappa (\overline{x})$, the morphism $f$ is locally acyclic at $\overline{x}$ relative to the constant sheaf with value $\mathbf{Z}/n\mathbf{Z}$.

5. We say $f$ is universally locally acyclic if for any morphism $S' \to S$ of schemes the base change $f' : X' \to S'$ is locally acyclic.