The Stacks project

Lemma 15.111.5. Let $A \subset B$ be an extension of discrete valuation rings. The following are equivalent

  1. $A \to B$ is formally smooth in the $\mathfrak m_ B$-adic topology, and

  2. $A \to B$ is weakly unramified and $\kappa _ B/\kappa _ A$ is a separable field extension.

Proof. This follows from Proposition 15.40.5 and Algebra, Proposition 10.158.9. $\square$

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