The Stacks project

Remark 88.7.3. Let $I$ be an ideal of a Noetherian ring $A$. Let $B$ be an object of (88.2.0.2) which is rig-smooth over $(A, I)$. It is shown in [Theorem 1.2, gabber-zavyalov] that $B$ is isomorphic to the $I$-adic completion of a finite type $A$-algebra. This result supercedes the following list of partial results:

1. If $A$ is a G-ring, then the result follows from Proposition 88.6.3.

2. If $B$ is rig-étale over $(A, I)$, then the result follows from Lemma 88.10.2.

3. If $I$ is principal, then the result follows from [III Theorem 7, Elkik].