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!

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_*.”

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.