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!
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.
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.
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!
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.
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!
Since the last update of October 2014 we have added the following material:
Enjoy!
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…
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!):
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!