# Completeness

For everybody reading and using the Stacks project here is something to keep in mind:

Of course the results in a particular section or chapter do not cover all possible results about the topic discussed in that section or chapter.

First of all, taken literally, this is simply not possible. But even trying to do justice to a topic and mention all the wonderful things one can say, would slow down progress to a halt. Adding all possible deductions and recombinations of lemmas in the section and earlier ones also often takes too much work.

Instead what we try to do is as follows. Each time we broach a new topic we try to have a skeleton outline of the basic material. Often we do this when there is just one tiny result we want to use. Then over time, we come back to the section/chapter with more material as needed. Also, sometimes a result cannot be formulated or proved immediately because it needs more terminology or results proven later in the Stacks project. When this happens we try to put in a pointer to this material in the earlier section.

This has turned out to work fairly well. But if you find cases where it didn’t then please let us know. For example, if you find a case where some elementary results are being used which have no formulation in earlier chapters, then please let us know so we can fix that. On the other hand, if you have a result you would like to see mentioned, then send us a latex file with the actual math and we will consider it for inclusion. Thanks!

# Blowing down exceptional curves

Let X be a Noetherian separated scheme. Let E ⊂ X be an effective Cartier divisor such that there is an isomorphism E → P1k where k is a field. Then we say E is an exceptional curve of the first kind if the normal sheaf of E in X has degree -1 on E over k.

You can get an example of the situation above by starting with a Noetherian separated scheme Y and a closed point y such that the local ring of Y at y is a regular local ring of dimension 2 and taking the blowup b : X → Y of y and taking E to be the exceptional divisor.

Conversely, if E ⊂ X is gotten in this manner we say that E can be contracted.

The following questions have been bugging me for a while now.

Question 1: Given an exceptional curve E of the first kind on a separated Noetherian scheme X is there a contraction of E?

Question 2: Given an exceptional curve E of the first kind on a separated Noetherian scheme X is there a contraction of E but where we allow Y to be an algebraic space?

Question 3: Suppose that Y is a separated Noetherian algebraic space and that y is a closed point of Y such that the henselian local ring of Y at y is regular of dimension 2. Is there an open neighbourhood of y which is a scheme?

Question 4: With assumptions as in Question 3 assume moreover that the blow up of Y in y is a scheme. Then is Y a scheme?

In these questions the answer is positive if we assume that X or Y is of finite type over an excellent affine Noetherian scheme (and I think in the literature somewhere; I’d be thankful for references).

But… it might be interesting and fun to try and find counter examples for the general statements. Let me know if you have one!

# Canonical divisor

Let k be a field and let X be a finite type scheme over k. Let F be a coherent O_X-module which is generically invertible. This means there exists a an open dense subscheme such that F is an invertible module when restricted to that open.

Lemma: There exists an open subscheme U containing all codimension 1 points, an invertible O_U-module L, and a map a : L → F|_U which is generically an isomorphism, i.e., there exists an open dense subscheme of U such that a restricted to that open is an isomorphism.

Proof. We already have a triple (U, L, a) for some dense open U in X. To prove the lemma we can proceed by adding 1 codimension 1 point ξ at a time. To do this we may work over the 1-dimensional local ring at ξ, where the existence of the extension is more or less clear.

Now assume that X is equidimensional of dimension d. Then we have a Chow group A_{d-1}(X) of codimension 1 cycles. If X is integral this is called the Weil divisor class group. For F as above we pick (U, L, a) as in the lemma. Observe that A_{d-1}(U) = A_{d-1}(X).

Def: The divisor associated to F is c_1(L) ∩ [U]_d + [Coker(a)]_{d-1} – [Ker(a)]_{d-1}

The notation here is as in the chapter Chow Homology of the Stacks project. The first term c_1(L) ∩ [U]_d is the first chern class of L on U and the other two terms involve taking lengths at codimension 1 points. Using the lemma to compare different triples for F it is easy to verify this is well defined as an element of A_{d-1}(X).

Def: Assume in addition X is generically Gorenstein, i.e., there exists a dense open which is Gorenstein. Let ω and ω’ be the cohomology sheaves of the dualizing complex of X in degrees -d and -d+1. The canonical divisor K_X is the divisor associated to ω minus [ω’]_{d-1}.

There you go; you’re welcome!

Rmks:
1. Fulton’s “Intersection Theory” defines the todd class of X in complete generality.
2. If X is generically reduced, then X is generically regular, hence generically Gorenstein and our definition applies.
3. The term [ω’]_{d-1} is zero if X is Cohen-Macaulay in codim 1.
4. If X is Gorenstein in codimension 1, then our canonical divisor agrees with the canonical divisor you find in many papers.
5. A canonical divisor of an equidimensional X can always be defined: either by Fulton or by generalizing the definition of the divisor associated to F to the case where F and O_X define the same class in K_0(Coh(U)) for some dense open U. This will always be true for ω. Just takes a bit more work.
6. If X is proper and equidimensional of dimension 1, then χ(F) = deg(divisor asssociated to F) + χ(O_X) whenever F is generically invertible.
7. If X is proper and equidimensional of dimension 1, then deg(K_X) = – 2χ(O_X).
8. If X is a curve and f : Y → X is the normalization, then K_X = f_*(K_Y) + 2 ∑ δ_P P where δ_P is the delta invariant at the point P (Fulton, Example 18.3.4).
9. If X is equidimensional of dimension 1 and Z ⊂ X is the largest CM subscheme agreeing with X generically, then K_X = K_Z – 2 ∑ t_P P where t_P is the length of the torsion submodule in O_{X,P}.

Edit 3/1/2016: Jason Starr commented below that there is a refinement which is sometimes useful, namely, one can ask for a Todd class and Riemann-Roch in K-theory and he just added by email: “In our joint work on rational simple connectedness of low degree complete intersections, we need to know that certain (integral) Cartier divisor classes on moduli spaces are Q-linearly equivalent. It is not enough to know that the pushforward cycles classes to the (induced reduced) coarse moduli scheme are rationally equivalent. So we need the Riemann-Roch that works on K-theory. In fact, the relevant computations are in our earlier manuscript about “Virtual canonical bundle …”, and we slightly circumvent Riemann-Roch in the computation. But, morally, we are using a Todd class that lives in K-theory, not just in CH_*.”

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!

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.