Pieter Belmans and Johan Commelin have added a bunch of new features to the Stacks project website:

  1. Nested enumerations are now displayed correctly; this lemma is an example.
  2. Outside references are more visible; this lemma has one.
  3. Footnotes are now displayed as footnotes; this section has four footnotes.
  4. They have implemented a system for historical remarks; Nakayama’s lemma is currently the only tag which has a historical remark.
  5. They have implemented a system for slogans; these lemmas are examples (look at the top underneath the header and breadcrumb).
  6. They have written the sloganerator; click or read below.

Huge thanks for all the work put into this by Pieter and Johan.

What is the sloganerator? Roughly, it gives you a random result from the Stacks project and asks you to type in a slogan describing the result. Your suggestion will become a comment on the tag’s page the Stacks project website. In due time we will then add the slogan to the actual Stacks project and you slogan will become visible as in the examples above.

The idea of having “slogans” or “human readable” descriptions of the results in the Stacks project has been around for a while. See this blog post and follow the links to the older blog posts. This will hopefully give you a better idea of what this is all about.


PS: If you’d like to suggest a slogan on any given tag of the Stacks project, then please just leave a comment on the tag’s page. Similarly with outside references and historical remarks.


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? [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!

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.

Fixed all issues

Hello again. This just a quick post to let you know that I worked through your recent comments and updated the Stacks project website with the latest version. So get back there and find more mistakes. Thank you all!