Dilatations

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:

  1. We may assume A is henselian, see Lemma Tag 0AE4
  2. It holds when A is a G-ring, see Lemma Tag 0AE5
  3. 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?

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?

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).

Anyway, any suggestions, ideas, references, etc are very welcome. Thanks!

Bibliography + references

As I’ve said before I want to add more outside references everywhere locally in the Stacks project. Here is a way you can help: follow this link to find a listing of all results in the Stacks project whose LaTeX environment is coded as a theorem or a proposition (many important results from the literature are called lemmas in the Stacks project — if you want you can complain about this too). Then look through the list to see if you know a precise reference for one of these results (what I mean is a precise theorem/proposition/lemma in a paper which is mathematically very closely related, e.g., logically equivalent, a special case, or a generalization, or overlapping in many instances, etc). Then either leave a comment on the corresponding tag’s page or send an email to stacks dot project at google mail. Thanks!

Chapter of the day

There is a new chapter entitled “Pushouts of Algebraic Spaces”. Here is a link to the introduction. It contains a section on something that is often called formal glueing of quasi-coherent modules. The name presumably comes from the fact that this can be used and is often used to glue coherent sheaves on Noetherian schemes given on an open and on the formal completion along the closed. However, the mathematics is more like what one does when studying elementary distinguished squares in the Nishnevich topology — in fact, I wonder what topology we get if we replace elementary distinguished squares by the notion in Situation Tag 0AEV? Hmm…

Lemma of the day

Let (A,I) be a henselian pair with A Noetherian. Let A^* be the I-adic completion of A. Assume at least one of the following conditions holds

  1. A → A^* is a regular ring map,
  2. A is a Noetherian G-ring, or
  3. (A,I) is the henselization (More on Algebra, Lemma 15.7.10) of a pair (B,J) where B is a Noetherian G-ring.

Given f_1, …, f_m ∈ A[x_1, ..., x_n] and a_1, …, a_n ∈ A^* such that f_j(a_1, …, a_n) = 0 for j = 1, …, m, for every N ≥ 1 there exist b_1, …, b_n ∈ A such that a_i − b_i ∈ I^N and such that f_j(b_1, …, b_n) = 0 for j = 1, …, m. See Lemma Tag 0AH5.

Slogan: Approximation for henselian pairs.

Lemma of the day

Let A be a ring and let I be a finitely generated ideal. Let M and N be I-power torsion modules.

  1. Hom_{D(A)}(M, N) = Hom_{D(I^∞-torsion)}(M, N),
  2. Ext^1_{D(A)}(M, N) = Ext^1_{D(I^∞-torsion)}(M, N),
  3. Ext^2_{D(I^∞-torsion)}(M, N) → Ext^2_{D(A)}(M, N) is not surjective in general,
  4. (0A6N) is not an equivalence in general.

See Lemma Tag 0592.

Discussion: Let A be a ring and let I be an ideal. The derived category of complexes of A-modules with I-power torsion cohomology modules is not the same as the derived category of the category of I-power torsion modules in general, even if I is finitely generated. However, if the ring is Noetherian then it is true, see Lemma Tag 0955.

Lemma of the day

Let F be a predeformation category which has a versal formal object. Then

  1. F has a minimal versal formal object,
  2. minimal versal objects are unique up to isomorphism, and
  3. any versal object is the pushforward of a minimal versal object along a power series ring extension.

See Lemma Tag 06T5.

What is fun about this lemma is that it produces a minimal versal object (as defined in Definition Tag 06T4) from a versal one without assuming Schlessinger’s axioms. If Schlessinger’s axioms are satisfied and one is in the classical case (see Definition Tag 06GC), then a minimal versal formal object is a versal formal object defined over a ring with minimal tangent space. This is discussed in Section Tag 06IL.