This is a follow up of Example wanted. There I ask for two examples.
Firstly, I ask for a Noetherian local domain A such that its completion A* has an isolated singularity and such that Spec(A) does not have a resolution of singularities.
I now think such an example cannot exist. Namely, conjecturally resolution for Spec(A*) would proceed by blowing up nonsingular centers each lying about the closed point, which would transfer over to Spec(A) thereby giving a resolution for Spec(A).
Secondly, I ask for a Noetherian local ring A and a proper morphism Y —> Spec(A*) of algebraic spaces which is an iso above the puctured spectrum U* which is NOT the base change of a similar morphism X —> Spec(A).
As Jason pointed out in the comments on the aforementioned blog post, to get such an example we have to assume that A is nonexcellent since otherwise Artin’s result on dilatations kicks in to show that X does exist. In fact we have the following:
- We may assume A is henselian, see Lemma Tag 0AE4
- It holds when A is a G-ring, see Lemma Tag 0AE5
- There exists a blow up Y’ —> Spec(A*) with center supported on the closed point which dominates Y and which is the base change of some X’ —> Spec(A) as above, see Lemma Tag 0AE6 and Lemma Tag 0AFK.
I’ve tried to make a counter example for non-G-rings, but failed. So now I am beginning to wonder: maybe there isn’t one? [Edit 20 October 2014: there is none as can be seen by visiting the upgraded Lemma Tag 0AE5.]
If so, then perhaps Artin’s result on dilatations (in formal moduli II) holds for Noetherian algebraic spaces without any supplementary conditions. Yes, this is a ridiculous step to take (Artin’s result is about formal algebraic spaces and a lot stronger than the question asked above), and I say this, not because I have a good reason to think this is true, but just to make it easier for you and me to make a counter example. I don’t have one, do you? [Edit 20 October 2014: there is none as can be seen by visiting this blog post.]
You should probably stop reading here, because now things become really vague. Looking at affine schemes \’etale over Y leads to the following type of question. Suppose that f : V* —> Spec(A*) is a finite type morphism with V* affine and f^{-1}(U*) —> U* \’etale. Then we can ask whether V* is the base change of a similar type of morphism V —> Spec(A). The answer is a resounding NO because for example the morphism f could be an open immersion whose complement is a closed subscheme of Spec(A*) which is not the base change of a closed subscheme of Spec(A). But suppose we only ask for a V —> Spec(A) such that the m_A-adic formal completion of V is isomorphic to the m_{A*}-adic formal completion of V*? Namely, if this question has a positive answer, then we might be able to use this to construct an X as above whose base change is Y by glueing affine pieces. I also would dearly love a counter example to this question (again it holds if A is a G-ring so a counter example would have to involve some kind of bad ring). [Edit 20 October 2014: The existence of these algebras follows from the paper by Elkik on solutions of equations over henselian rings, see this blog post.]
Anyway, any suggestions, ideas, references, etc are very welcome. Thanks!
This is a non-comment, since I do not actually have anything to add. However, I would love to hear if there is any news on this. I saw a recent article of Abramovich and Temkin regarding birational morphisms of *quasi-excellent* schemes, so maybe that is something slightly more general than excellent schemes (although maybe their work has nothing to do with your question — I cannot tell).
What they do is a thousand times more interesting than what I talked about in the blog post. It is related in the sense that quasi-excellent is weaker than excellent and many things can be done for quasi-excellent schemes (or even G-schemes).