The Stacks project

Lemma 15.104.14. Let $A \to B$ be a ring map. Then $A \to B$ is weakly étale in each of the following cases

  1. $B = S^{-1}A$ is a localization of $A$,

  2. $A \to B$ is étale,

  3. $B$ is a filtered colimit of weakly étale $A$-algebras.

Proof. An étale ring map is flat and the map $B \otimes _ A B \to B$ is also étale as a map between étale $A$-algebras (Algebra, Lemma 10.143.8). This proves (2).

Let $B_ i$ be a directed system of weakly étale $A$-algebras. Then $B = \mathop{\mathrm{colim}}\nolimits B_ i$ is flat over $A$ by Algebra, Lemma 10.39.3. Note that the transition maps $B_ i \to B_{i'}$ are flat by Lemma 15.104.11. Hence $B$ is flat over $B_ i$ for each $i$, and we see that $B$ is flat over $B_ i \otimes _ A B_ i$ by Algebra, Lemma 10.39.4. Thus $B$ is flat over $B \otimes _ A B = \mathop{\mathrm{colim}}\nolimits B_ i \otimes _ A B_ i$ by Algebra, Lemma 10.39.6.

Part (1) can be proved directly, but also follows by combining (2) and (3). $\square$

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