The Stacks project

Lemma 29.36.17. Let

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ S' \ar[r]^ g & S } \]

be a cartesian diagram of schemes. Let $W \subset X$, resp. $W' \subset X'$ be the open subscheme of points where $f$, resp. $f'$ is étale. Then $W' = (g')^{-1}(W)$ if

  1. $f$ is flat and locally of finite presentation, or

  2. $f$ is locally of finite presentation and $g$ is flat.

Proof. Assume first that $f$ locally of finite type. Consider the set

\[ T = \{ x \in X \mid f\text{ is unramified at }x\} \]

and the corresponding set $T' \subset X'$ for $f'$. Then $T' = (g')^{-1}(T)$ by Lemma 29.35.15.

Thus case (1) follows because in case (1) $T$ is the (open) set of points where $f$ is étale by Lemma 29.36.16.

In case (2) let $x' \in W'$. Then $g'$ is flat at $x'$ (Lemma 29.25.7) and $g \circ f'$ is flat at $x'$ (Lemma 29.25.5). It follows that $f$ is flat at $x = g'(x')$ by Lemma 29.25.13. On the other hand, since $x' \in T'$ (Lemma 29.34.5) we see that $x \in T$. Hence $f$ is étale at $x$ by Lemma 29.36.15. $\square$

Comments (2)

Comment #3211 by Dario Weißmann on

Typo: should be

There are also:

  • 3 comment(s) on Section 29.36: Étale morphisms

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