Since the last update of October 2014 we have added the following material:

  1. Structure modules over PIDs following Warfield Tag 0ASL
  2. Correct proof of Lemma Tag 05U9 thanks to Ofer Gabber
  3. A flat ring map which is not a directed colimit of flat finitely presented ring maps Tag 0ATE
  4. Glueing dualizing complexes Tag 0AU5
  5. Trace maps Tag 0AWG
  6. Duality for a finite morphism Tag 0AWZ
  7. Grauert-Riemenschneider for surfaces Tag 0AX7
  8. Torsion free modules Tag 0AVQ
  9. Reflexive modules Tag 0AVT
  10. Finiteness theorem for f_* Tag 0AW7
  11. Fix definition of depth (thanks to Burt) Tag 00LE and Tag 0AVY
  12. Characterizing universally catenary rings Tag 0AW1
  13. Improved section on Jacobson spaces thanks to Juan Pablo Acosta Lopez Tag 005T
  14. Faithfully flat descent of ML modules thanks to Juan Pablo Acosta Lopez Tag 05A5
  15. Improvements to the chapter on Chow homology discussed here
  16. Degrees of vector bundles on curves Tag 0AYQ
  17. Degrees of zero cycles Tag 0AZ0 and how this relates to degrees of vector bundles and with numerical intersections
  18. Quotient by category of torsion modules thanks to Ingo Blchschmidt Tag 0B0J
  19. New chapter on intersection theory discussed here
  20. Example of different colimit topologies Tag 0B2Y
  21. Section on topological groups, rings, modules Tag 0B1Y
  22. Section on tangent spaces Tag 0B28
  23. A bunch of material on (quasi-)projectivity, for example Tag 0B41 and Tag 0B44
  24. Glueing in a modification Tag 0B3W at a point of a scheme
  25. Improved material on sober spaces thanks to Fred Rohrer Tag 004U
  26. Riemann-Roch and duality for curves Tag 0B5B
  27. Fix idiotic mistake about graded projective modules, thanks to Rishi Vyas read his explanation on github
  28. Base change map in duality Tag 0AA5 is often an isomorphism (Tag 0AA8) and commutes with base change Tag 0AWG
  29. Bunch of changes thanks to Darij Grinberg
  30. Material on group schemes over fields Tag 047J
  31. Material on (locally) algebraic group schemes over fields Tag 0BF6
  32. Thickenings of quasi-affine schemes are quasi-affine Tag 0B7L
  33. Minimal closed subspaces which aren’t schemes Tag 0B7X
  34. Monomorphisms of algebraic spaces Tag 0B89
  35. Change of base field and schematic locus Tag 0B82
  36. Separated group algebraic spaces over fields are schemes Tag 0B8G
  37. Picard scheme of smooth projective curves over algebraically closed fields Tag 0B92
  38. Improved discussion of invertible modules… (too ashamed to put a link here)
  39. Jacobson algebraic spaces Tag 0BA2
  40. Nagata spaces Tag 0BAT
  41. For an algebraic space: locally Noetherian + decent => quasi-separated Tag 0BB6
  42. Various improvements on rational and birational maps Tag 01RR, Tag 01RN, and Tag 0BAJ
  43. Dimension formula for algebraic spaces Tag 0BAW
  44. Generically finite morphisms of algebraic spaces Tag 0BBA
  45. Birational morphisms of algebraic spaces Tag 0ACU
  46. Elementary etale neighbourhoods on algebraic spaces Tag 03IG
  47. Complements of affine opens have codimension 1 Tag 0BCQ
  48. Norms of invertible modules Tag 0BCX which allows us to descend ample invertible modules
  49. Descending (quasi-)projectivity through field extensions Tag 0BDB
  50. Section on splitting complexes Tag 0BCF for better handling of local structure of perfect complexes
  51. Section on stably free modules Tag 0BC2
  52. Jumping loci for perfect complexes on schemes Tag 0BDH
  53. Applications of cohomology and base change Tag 0BDM
  54. Theorem of the cube Tag 0BEZ
  55. Weil divisors on locally Noetherian schemes Tag 0BE0
  56. The Weil divisor class associated to an invertible module Tag 02SE
  57. K\”unneth formula for schemes over a field Tag 0BEC
  58. Algebraic group schemes are quasi-projective Tag 0BF7
  59. Numerical intersections Tag 0BEL
  60. Section on abelian varieties Tag 0BF9 containing just enough for our use later
  61. Tried to improve the exposition of convergence for spectral sequences using terminology mostly as in Weibel; still very far from perfect
  62. Long overdue characterization of algebraic spaces Tag 0BGQ
  63. Chapters on resolution of surface singularities one for schemes and one for algebraic spaces


Updated yet again

Just finished working through your comments once more. Please continue to leave these comments; I find them always to be helpful. In particular let us know if there is an argument that is just too quick and not clear, etc. Please also let us know what you tried…

2 and 3

Here is a question I asked myself yesterday: Suppose that X is an algebraic space which has degree 2 finite etale covering X_2 —> X and a degree 3 finite etale covering X_3 —> X such that both X_2 and X_3 are schemes. Is X a scheme?

I thought there might be a chance that the answer is yes, but just now in the common room, Philip, Anand, Davesh, and me proved that the answer is: no!

Namely, following Hironaka’s example, we made a smooth proper 3-fold X_6 with an action of a cyclic group G of order 6 such that X = X_6/G is not a scheme, but the two intermediate quotients X_2 and X_3 are schemes. Namely, start with a smooth projective 3-fold Y with an action of G and a 6-gon of smooth rational curves C_0, C_1, C_2, C_3, C_4, C_5 which are cyclically permuted by G. In other words, I mean that C_i ∩ C_{i + 1} is exactly one point P_i and the intersection is transversal. Then you do as in the Hironaka example: blow up all these curves but at P_i analytic locally blow up C_i first and C_{i + 1} second. The result is a proper scheme X_6 with an action of G. Over the P_i there are two curves E_i, E’_i such that E_0 + E_1 + … + E_5 is zero in the Chow group (see picture of Philip below). Hence orbit of a point on E_0 cannot be contained in an affine open of X_6, which proves that X_6/G is not a scheme. However, the morphism

X_6 – E_i ∪ E’_i —-> Y – P_i

is a usual blow up, hence the source is a quasi-projective variety and any finite collection of points is contained in an affine open. This quickly gives that X_3 and X_2 are schemes.
To finish, here is a picture X_6 (thanks Philip!):IMG_20150422_153508_334

Lecture notes on etale cohomology

In 2009 I gave a course on etale cohomology here at Columbia University. Notes were taken by Thibaut Pugin, Zachary Maddock and Min Lee (in reverse alphabetical ordering). There were the basis for the chapter on etale cohomology of the Stacks project. Since the Stacks project is all about creating one consistent whole with lemmas proved at the correct level of generality, the chapter no longer has the flavor of lecture notes. Since some of you may prefer the original exposition and since the link to the lecture notes on Pugin’s home page no longer works, here are the original lecture notes by Thibaut Pugin, Zachary Maddock and Min Lee with all the mistakes and imperfections, etc, etc, etc. Enjoy!

Updated once more

Allright, I worked through your comments (as well as the suggestions we get via our email address) once more. Thanks to all. Go back there and leave some more comments.

Just a reminder: Once we reach 5000 pages we’re going to have a party — you’ll be invited via this blog. Currently we’re at 4665 pages. You can help make the party happen!

Intersection Theory

Today I finished the first complete version of a chapter on intersection theory. As usual comments and suggestion are very welcome.

The chapter uses Serre’s Tor formula and moving lemmas to define an intersection product on the Chow groups of nonsingular projective varieties over an algebraically closed ground field and that is all it does. You can read the introduction for a little bit more information.

There are some improvements that can be made to this chapter. The first is that some of the material on Serre’s Tor formula belongs properly in one of the chapters on commutative algebra. Of course, there is a lot more one can say about regular local rings and the Tor formula, leading up to recent work on homological conjectures in commutative algebra. Also, some of the arguments in the moving lemmas use geometric arguments on varieties over algebraically closed fields and we need to write more of the API to easily translate these into scheme theoretic language. Finally, the chapter is missing examples and more references to the literature.

What often happens with new chapters is that a few years down the road, we take a second look and make substantial improvements.

One aspect of the material in the new chapter is that it was not as straightforward to write as the material on constructible sheaves which was like butter. The conclusion must therefore be that intersection theory is not like butter!

Nonetheless: Enjoy!

Updated again

As you may have noticed I do not always immediately respond to the comments on the stacks project. It turns out to be more efficient to wait till there are quite a few of them and then take out a block of time to deal with them. I just finished with this once more and this time it took a bit more time. Thanks especially to Juan Pablo Acosta López.

Chow homology

Last week I started looking at the chapter on chow homology. When I first wrote it in 2009, it was a bit of an experiment. As a consequence, various proofs used three different approaches: one using K-groups, one using blow-up lemmas as in Fulton, and one via the key lemma (see below). Today I rearranged the whole thing to simplify the exposition and stick with the approach using the key lemma.

How much intersection theory does the chapter cover? We work consistently with schemes locally of finite type over a fixed universally catenary, locally Noetherian base scheme. We introduce cycle groups, flat pullback, proper pushforward, and rational equivalence. After proving some basic properties, we introduce the operation c_1(L) ∩ – where L is an invertible sheaf and the Gysin map for an effective Cartier divisor. Having proved the basic properties of these operations, we have enough to introduce chern classes of locally free sheaves and prove their basic properties. The chapter ends with stating the Grothendieck-Riemann-Roch theorem (without proof).

The key lemma is a statement about tame symbols over a Noetherian local domain of dimension 2. I think of it as the statement that the secondary ramifications add up to zero. I’d love it if you could tell me a reference for this lemma in the literature (I assume there is some paper on K-theory that contains this result).

The key lemma implies our key formula. I am sure I discussed this statement with somebody in my office at some point, but I cannot remember who; if it was you, please email me! Anyway, the key formula quickly implies that c_1(L) ∩ – passes through rational equivalence (Section Tag 02TG), the fact that c_1(L) ∩ c_1(N) ∩ – = c_1(N) ∩ c_1(L) ∩ -, and that the Gysin map for an effective Cartier divisor passes through rational equivalence (Section Tag 02TK).


Up to date again

With a very stuffy nose and lot’s of sneezing, today I worked through the comments that were left on the Stacks project. Thanks to all of you!

Here is what is very helpful (especially on those days were my head is all stuffed up): when you find a mistake, please also point out how to fix it (if you know how). Even just making a guess at what went wrong is helpful. Thanks!

Up to date with comments

Hello again! This is just to let you know that I’ve just finished working through the comments on the Stacks project and all (known) mistakes, typos etc are fixed. Thanks for all your help! Please go back there and find more.

Also, please let me know if you or somebody else have helped out in the past and we’ve forgotten to add your or their name to the contributors list. It seems idiotic, but it turns out it is actually kind of hard to keep track of everybody who is helping out (and in some sense we still do not have a really good way to do this). Any suggestions on how to manage this are very welcome too.