# Elkik’s Theorem 7

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.