Elkik’s Theorem 7

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In the title I am referring to a result in the paper “Solutions d’équations à coefficients dans un anneau hensélien”.

I’ve completely forgotten how to make mathematical symbols appear in blog posts. So this blog post will use ugly workarounds to get the mathematics across. Sorry!

Let A be a Noetherian ring. Let I be an ideal of A. Let A —> B be a ring map such that B is I-adically complete and such that B/IB is of finite type over A/I. Then we say A —> B is rig-smooth if the “completed” naive cotangent complex NL_{B/A}^\wedge 0AJL has the following property: over the complement of V(IB) in Spec(B) it only has cohomology in degree 0 and this cohomology is a finite locally free module there. (I intend to add this definition to the Stacks project soon.)

Elkik’s Theorem 7: If I is a principal ideal and A —> B is rig-smooth, then B is the I-adic completion of a finite type A-algebra.

Question: Does this theorem also hold if I is not principal?

This question was asked by Temkin in this paper in Remark 3.1.3 part (iii).

I do not know any application of this if it were true. Moreover, I am having a very hard time even coming up with a possible scenario in which you could apply this and not one of the many closely related things that I do know how to prove. But this is of course just one of those things one goes through as a mathematician: once a question has been asked it is hard to stop thinking about it.

Anyway, I would be grateful for any ideas of comments you might have about the question. Please either email me or leave a comment on this post. Thanks!

Let me list the things I think I can prove (caveat emptor):

  1. It is true if I is principal but more generally when the complement of V(IB) in Spec(B) is affine
  2. It is true if B is rig-etale, see 0AKA
  3. It is true if A is an excellent ring
  4. There exists an ideal J = (b_1, …, b_m) in B such that V(J) = V(IB) and such that the I-adic completion of the affine blow-up algebras B[J/b_i] are completions of finite type A-algebras

The final statement says that locally on the rigid space associated to the formal scheme Spf(B) we have the algebraizability (spelling?) that we want. The second statement is what we use in the Stacks project to reprove a strong version of Artin’s algebraization of dilatations. I am hoping to upgrade the relevant chapter of the Stacks project and add a discussion of Artin’s theorem on contractions.

Thanks for reading!