# Up to date with comments

Those of you who have left comments: OK, I worked through all your comments and I fixed almost all of them. For many of your comments I left a corresponding message on the same webpage as where you left your comment, but not for all of them.

Please continue to leave comments, suggestions, etc. I always learn things when I work through these. Thanks!

# Different ideal

This is just to record some thoughts on the different ideal or equivalently the ramification divisor in the case of quasi-finite morphisms f : X —> Y of locally Noetherian schemes.

The model for the construction is the case where (a) f is finite flat, (b) f is generically etale, and (c) X and Y are Gorenstein. In this case we let ω = Hom(f_*O_X, O_Y) viewed as an O_X-module. By property (c) ω is an invertible O_X-module. By property (a) the trace map Tr_{X/Y} defines a global section τ : O_X —> ω. By property (b) this section is nonzero in all the generic points of X. Since X is Gorenstein we conclude that τ is a regular section. Hence the scheme of zeros of τ is an effective Cartier divisor R ⊂ X. This is the ramification divisor. In this situation it follows from the definitions that the norm of R is the discriminant of f (defined as the determinant of the trace pairing).

Easy generalizations: (1) By suitable localizing and glueing we can replace the assumption that f is finite flat by the assumption that f is quasi-finite and flat. (2) Instead of assuming that X and Y are Gorenstein it suffices to assume that the fibres of f are Gorenstein.

To deal with nonflat cases, the construction works whenever f is quasi-finite, generically etale (i.e., etale at all the generic points of X), the relative dualizing sheaf ω is invertible, and there is a global section τ of ω whose restriction to the etale locus is as above. To make τ unique let’s assume X —Y is etale also at all the embedded points of X.

The trickiest part to verify is the existence of the section τ. If X is S_2, then it suffices to check in codimension 1. Beyond the usual case where X and Y are regular in codimension 1, it works also if the map X —> Y looks like a Harris-Mumford type admissible cover in codimension 1: for example consider the nonflat morphism corresponding to the ring map A = R[x, y]/(xy) —> R[u, v]/(uv) = B sending x, y to u^n, v^n where n is a nonzerodivisor in the Noetherian ring R. Then the ramification divisor is given by the ideal generated by n in the ring B!

In this way we obtain the well known observation that admissible coverings in characteristic zero are not ramified at the nodes.

PS: From the point of view above, the problem with nonbalanced maps, such as the map R[x, y]/(xy) —> R[u, v]/(uv) sending x to u^2 and y to v^3, is that τ is not even defined. So you cannot really even begin to say that it is (un)ramified…

[Edit a bit later] and in fact you can compose with the map R[u, v]/(uv) —> R[a, b]/(ab) sending u to a^3 and v to b^2 to get the map R[x, y]/(xy) —> R[a, b]/(ab) sending x, y to a^6, b^6 whose ramification divisor is empty (provided 6 is invertible in R)…

[Edit on Sept 18] The morphism given by A = R[x, y]/(xy) —> R[u, v]/(uv) = B sending x, y to u^n, v^n is a morphism which is both “not ramified” in the sense above and “not unramified” in the sense of Tag 02G3.

# Apologies for backlog

Just a quick message to apologize in the delay in working through the comments on the Stacks project webpage and the comments sent to the email address as well as the pull requests at github. Eventually I will get to them.

Currently I am very excited about the topics course I am teaching about \’etale fundamental groups. The topic was chosen on the one hand because it fits well with the remynar on \’etale cohomology organized by Remy and on the other hand because it fits with recent work on the Stacks project: see this chapter on fundamental groups.

Another thing I am very excited about is our graduate student seminar this semester. Here we will be working through Milne’s amazing paper on a conjecture of Artin and Tate. We also intend to try and make a concept map along the way. Not sure if this will work out but see this page of Daniel Halpern-Leistner example.

# Pardon the interruption

Dear visitors of the Stacks project website. Tomorrow Tuesday August 18 after 3:30 PM the Stacks project site and this blog will be down for a while. Our administrator is switching server hardware… But we should be back up by Wednesday morning at the latest.

[Edit: Aug 19, 2015.] OK and we’re back. Let us know if there is something screwy with the stacks website. Thanks!

# Tags in pdfs

With a few changes to one of our scripts and using the marginnotes latex package, we can now show the tags and references in the pdfs. Here is an example of what this looks like. (The hyperlinks in this file do not work but this will get fixed if we put it on the server.)

I am not completely convinced that this is a good idea, so I need a couple of you guys to tell me you think it is a good idea. Thanks!

[Edit Aug 14, 2015] OK, this is now on the server.

# Evolution of a lemma

Pieter Belmans just put another feature of the Stacks project website online: a way to browse the edits done over time to a given result and its proof in the Stacks project. This is a new and somewhat experimental feature, but it already works quite well in my opinion. Really the only way to understand what it does is to try some of the links below and do some clicking around.

To see the history of a given Tag just go to the page of the tag and look in the sidebar on the right for a link entitled “history”; we’ve not implemented this for chapters or sections. Here is a list of examples:

The motivation for having this in place is that it is technologically possible and that it provides detailed information about when and how the material evolved over time. This is all part of the whole idea that development on the Stacks project is completely open and accessible to all.

Enjoy!

# Update

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

Enjoy!

# 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!):

# 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!