Lemma of the day

Let X —> Y —> Z be morphism of schemes. Let P be one of the following properties of morphisms of schemes: flat, locally finite type, locally finite presentation. Assume that X —> Z has P and that {X —> Y} can be refined by an fppf covering of Y. Then Y —> Z is P. See Tag 06NB.

Unobstructed in codimension 3

So this is a follow up on the post about Burch’s theorem. Namely, I’ve just learned in the last month or so that the next case of this is in Eisenbud + Buchsbaum Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. It says that the resolution of a codimension 3 Gorenstein singularity R/I with R regular has a free resolution of the form

0 —> R —> R^n —f—> R^n —> R

where f is an alternating matrix and the other arrows are given by Pfaffians of f.

Moreover, if R/J is an almost complete intersection of grade 3, then R/J is linked to a Gorenstein R/I as above and a similar type of resolution can be obtained (results of Brown, kustin, etc).

OK, this is cool, very cool.

It seems completely clear that similarly to Burch’s theorem this implies that such a singularity is unobstructed, just as in the codimension 2 Cohen-Macaulay case. To be precise, as a simple consequence of the paper we obtain:

If R = k[[x, y, z]] and R —> S is an Artinian quotient ring such that either (1) S is Gorenstein, or (2) the kernel of R —> S is generated by at most 4 elements, then the miniversal deformation space of S is a power series ring over k.

Right…?

What I’d like is a reference to articles (with page and line numbers) stating exactly the above for (1) and (2). A generic reference to unobstructedness of determinantal singularities doesn’t count. I’ve googled and binged, but no luck so far. Can you help?

Or maybe this is just one of the innumerable results in our field that are so clearly true that you cannot formulate it in a paper as your paper will be immediately rejected?

Lemma of the day

Let (An) be an inverse system of abelian groups. The following are equivalent

1. (An) is zero as a pro-object,
2. lim An = 0 and R1lim An = 0 and the same holds for ⨁ i ∈ N (An).

See Tag 091C.

Lemma of the day

Let A be a Noetherian ring and I ⊂ A an ideal. For every n let M_n be a flat A/I^n-module. Let M_{n + 1} —> M_n be a surjective A-module map. Then the inverse limit M =lim M_n is a flat A-module (see Tag 0912).

Update

Since the last update we have added the following material:

1. universal property of blowing up (schemes) Tag 0806
2. admissible blowups (schemes) Tag 080J
3. strict transform (schemes) Tag 080C
4. a section on fitting ideals (algebra) Tag 07Z6
5. flattening by blowing up (schemes) Tag 080X
6. proper modifications can be dominated by blowups (schemes) Tag 081T
7. relative spectrum (spaces) Tag 03WD
8. scheme theoretic closure (spaces) Tag 0831
9. effective Cartier divisors (spaces) Tag 083A
10. relative Proj (spaces) Tag 0848
11. blowing up (spaces) Tag 085P
12. strict transforms (spaces) Tag 0861
13. admissible blowups (spaces) Tag 086A
14. generalities on limits (spaces) Tag 07SB
15. flattening by blowups (spaces) Tag 087A
16. David Rydh’s result that a decent space has a dense open subscheme Tag 086U
17. QCoh is Grothendieck (spaces and stacks) Tag 077V Tag 0781
18. proper modifications can be dominated by blowups (spaces) Tag 087G
19. multiple versions of Chow’s lemma (spaces) Tag 089J Tag 088U Tag 089L Tag 089M
20. David Rydh’s result that a locally separated algebraic space is decent Tag 088J
21. Grothendieck existence theorem (schemes) Tag 087V Tag 0886
22. Grothendieck algebraization theorem (schemes) Tag 089A
23. proper pushforward preserves coherence (spaces) Tag 08AP
24. theorem on formal functions (spaces) Tag 08AU
25. Grothendieck existence theorem (spaces) Tag 08BE Tag 08BF
26. a little bit about m-regularity Tag 08A2
27. connected spaces are nonempty (thanks to Burt) Tag 004S
28. decent group space over field is separated Tag 08BH
29. derived Mayer-Vietoris (ringed spaces) Tag 08BR
30. derived categories of modules (schemes) Tag 08CV
31. D(QCoh(O_X) = D_{QCoh}(O_X) for X quasi-compact with affine diagonal (schemes) Tag 08DB
32. Lipman and Neeman’s result on approximation by perfect complexes (schemes) Tag 08ES
33. derived categories of modules (spaces), Tag 08EZ
34. Induction principle for quasi-compact and quasi-separated algebraic spaces using distinguished squares (this is really fun!) Tag 08GL
35. derived Mayer-Vietoris using distinguished squares Tag 08GS
36. D(QCoh(O_X) = D_{QCoh}(O_X) for X quasi-compact with affine diagonal (spaces) Tag 08H1
37. approximation by perfect complexes (spaces) Tag 08HP
38. bunch of improvements to the bibliography bibliography
39. being projective is not local on the base Tag 08J0
40. descent data for schemes need not be effective, even for a projective morphism Tag 08KE
41. base change for Rf_*RHom(E, G) (schemes) Tag 08IC
42. base change for Rf_*RHom(E, G) (spaces) Tag 08JM
43. the Hom functor Tag 08JS
44. the stack of coherent sheaves Tag 08WC
45. deformation theory: rings, modules, ringed spaces, sheaves of modules on ringed spaces, ringed topoi, sheaves of modules on ringed topoi Tag 08KX
46. subtopoi Tag 08LT
47. standard simplicial resolutions Tag 08N8
48. cotangent complex Tag 08P6
49. snake lemma now has a proof without picking elements Tag 010H
50. constructing polynomial resolutions Tag 08PX
51. (trivial) Kan fibrations Tag 08NK Tag 08NT
52. Quillen’s spectral sequence Tag 08RF
53. cotangent complex and obstructions (algebra) Tag 08SP
54. cotangent complex and obstructions (ringed spaces) Tag 08UZ
55. cotangent complex and obstructions (ringed topoi) Tag 08V5
56. fixed an error in Artin’s axioms point out by David Rydh 2ccbbe3087e4dc2b1df2193c81ede7486931424c
57. skeleton chapter on dualizing complexes (algebra) Tag 08XH
58. descent for universally injective morphisms (thanks to Kiran Kedlaya) Tag 08WE

This brings us up to May 1 of this year. At that point I started to work on a chapter on pro-\’etale cohomology, in order to advertise work by Bhargav Bhatt and Peter Scholze in some lectures in Stockholm (KTH). The authors graciously send me a copy of their (for the moment) unfinished manuscript. The chapter covers only a small part of their material, leading up to the definition of constructible complexes and the proper base change theorem. All mistakes are mine. I’ve tried to put most of the background material in other chapters. As is usual for the Stacks project, whenever you try to add something new you are forced to add a lot of background material to go along with it. Here is a list of some of the things we added.

1. pro-\’etale cohomology (schemes) Tag 0966
2. Gleason’s theorem on extremally disconnected spaces (I strongly recommend the original paper) Tag 08YH
3. Hochster’s spectral spaces (I strongly recommend the original paper) Tag 08YF
4. Stone Cech compactification Tag 0908
5. Olivier’s theorem on absolutely flat extensions of strictly henselian rings (I strongly recommend the original paper) Tag 092Z
6. weakly \’etale morphisms (schemes) Tag 094N
7. derived completion (algebra; I’ve tried to give some references but I’d love to know more about the history of this topic) Tag 091N
8. constructible sheaves (etale) Tag 05BE Tag 095M
9. derived completion (ringed topoi) Tag 0995 Tag 099L Tag 099P
10. derived category D_c (etale) Tag 095V

Enjoy!