Lemma 15.104.7 (092H)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.104: Weakly étale ring maps
Lemma 15.104.7 (cite)

Lemma 15.104.7. Let $A \to B$ and $A \to A'$ be ring maps. Let $B' = B \otimes _ A A'$ be the base change of $B$ .

If $B \otimes _ A B \to B$ is flat, then $B' \otimes _{A'} B' \to B'$ is flat.
If $A \to B$ is weakly étale, then $A' \to B'$ is weakly étale.

Proof. Assume $B \otimes _ A B \to B$ is flat. The ring map $B' \otimes _{A'} B' \to B'$ is the base change of $B \otimes _ A B \to B$ by $A \to A'$ . Hence it is flat by Algebra, Lemma 10.39.7. This proves (1). Part (2) follows from (1) and the fact (just used) that the base change of a flat ring map is flat. $\square$

The Stacks project

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