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 $(A, I)$. As far as we know, it is an open question as to whether $B$ is isomorphic to the $I$-adic completion of a finite type $A$-algebra. Here are some things we do know:

  1. If $A$ is a G-ring, then the answer is yes by Proposition 88.6.3.

  2. If $B$ is rig-étale over $(A, I)$, then the answer is yes by Lemma 88.10.2.

  3. If $I$ is principal, then the answer is yes by [III Theorem 7, Elkik].

  4. In general there exists an ideal $J = (b_1, \ldots , b_ s) \subset B$ such that $V(J) \subset V(IB)$ and such that the $I$-adic completion of each of the affine blowup algebras $B[\frac{J}{b_ i}]$ are isomorphic to the $I$-adic completion of a finite type $A$-algebra.

To see the last statement, choose $b_1, \ldots , b_ s$ as in Lemma 88.4.2 part (4) and use the properties mentioned there to see that Lemma 88.7.2 applies to each completion $(B[\frac{J}{b_ i}])^\wedge $. Part (4) tells us that “rig-locally a rig-smooth formal algebraic space is the completion of a finite type scheme over $A$” and it tells us that “there is an admissible formal blowing up of $\text{Spf}(B)$ which is affine locally algebraizable”.


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