The Stacks project

Lemma 15.104.12. Let $A \to B$ be a ring map such that $B \otimes _ A B \to B$ is flat. Then $\Omega _{B/A} = 0$, i.e., $B$ is formally unramified over $A$.

Proof. Let $I \subset B \otimes _ A B$ be the kernel of the flat surjective map $B \otimes _ A B \to B$. Then $I$ is a pure ideal (Algebra, Definition 10.108.1), so $I^2 = I$ (Algebra, Lemma 10.108.2). Since $\Omega _{B/A} = I/I^2$ (Algebra, Lemma 10.131.13) we obtain the vanishing. This means $B$ is formally unramified over $A$ by Algebra, Lemma 10.148.2. $\square$

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