The Stacks project

The implication (4) $\Rightarrow $ (1) is the main result of [Andre-smooth].

Theorem 15.49.3 (André). Let $A \to B$ be a local homomorphism of Noetherian local rings. Let $k$ be the residue field of $A$ and $\overline{B} = B \otimes _ A k$ the special fibre. Assume $A \to A^\wedge $ is regular1. The following are equivalent

  1. $A \to B$ is regular,

  2. $A \to B$ is flat and $\overline{B}$ is geometrically regular over $k$,

  3. $A \to B$ is flat and $k \to \overline{B}$ is formally smooth in the $\mathfrak m_{\overline{B}}$-adic topology, and

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

Proof. We have seen the equivalence of (2), (3), and (4) in Proposition 15.40.5. It is clear that (1) implies (2). Thus we assume (4) holds and we prove (1).

By Lemma 15.37.4 we see that $A^\wedge \to B^\wedge $ is formally smooth. By Proposition 15.49.2 we see that $A^\wedge \to B^\wedge $ is regular. By assumption $A \to A^\wedge $ is regular, hence $A \to B^\wedge $ is regular by Lemma 15.41.4. Since $B \to B^\wedge $ is faithfully flat, we conclude that $A \to B$ is regular by Lemma 15.41.7. $\square$

[1] For example, if $A$ is a G-ring or (quasi-)excellent, see Sections 15.50 and 15.52.

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