The Stacks project

Lemma 86.28.5. With assumptions and notation as in Theorem 86.27.4 let $f : X' \to X$ correspond to $g : W \to X_{/T}$. Then $f$ is étale if and only if $g$ is étale.

Proof. If $f$ is étale, then $g$ is étale by Lemma 86.23.2. Conversely, assume $g$ is étale. Since $f$ is an isomorphism over $U$ we see that $f$ is étale over $U$. Thus it suffices to prove that $f$ is étale at any point of $X'$ lying over $T$. Denote $Z \subset X$ the reduced closed subspace whose underlying topological space is $|Z| = T \subset |X|$, see Properties of Spaces, Definition 64.12.5. Letting $Z_ n \subset X$ be the $n$th infinitesimal neighbourhood we have $X_{/T} = \mathop{\mathrm{colim}}\nolimits Z_ n$. Since $X'_{/T} = W \to X_{/T}$ we conclude that $f^{-1}(Z_ n) = X' \times _ X Z_ n \to Z_ n$ is étale by the assumed étaleness of $g$. By More on Morphisms of Spaces, Lemma 74.20.3 we conclude that $f$ is étale at points lying over $T$. $\square$

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