Comments and fixes

OK, I worked through your online comments as well as those emailed to me privately. This time I decided not to answer all the comments individually one by one as I’ve done in the past (it takes a fair amount of time and is kind of boring). The downside of this is that some of the comments are unanswered but already fixed in the text, so a casual visitor of the site may be confused.

If you are wondering how we dealt with a comment, or what the state of affairs was before the fix, you’ll have to look in the commit log for the Stacks project. For example, Kiran’s comment was addressed in this commit. If you look closely, you’ll see that I made a typo in the fix, which was then fixed here.

Unfortunately, there isn’t a good way to find my commit responding to Kiran’s comment unless you are comfortable using the command line and git. However, in most cases the comment will be about a lemma, proposition, remark, or theorem and then there is an easy way to do so. For example consider this comment by Keenan. To see how I addressed his comment, surf to the page, click on “history” in the right panel, and click on the diff corresponding to the edit on February 4, 2016.

Enjoy!

Worked through your comments

Thanks for all the comments on the Stacks project (both online and via email). I think I am up to date again. If there is something I missed, please let me know. Also, please continue to leave comments and help out. Thanks!

Notes by Pak-Hin Lee

On the webpage of Pak-Hin Lee you can find some live-texed notes of some courses here at Columbia. Truly amazing!

Including among these is Pak-Hin Lee’s notes of my course on \’etale fundamental groups. The idiotic thing is that looking at these notes, it appears I did a smashing job of giving these lectures. But then, when you stare harder and more locally at these notes, then you may start to wonder…

In any case, enjoy this over the winter break!

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:

  1. Nakayama’s lemma
  2. Lemma on generically finite morphisms
  3. A lemma of Serp’e
  4. Lemma proven with help from David Rydh
  5. Topological invariance of the etale site

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!