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_*.”

5 thoughts on “Canonical divisor

  1. I do not agree that Fulton defines the Todd class of $X$ in “complete” generality. The Todd class homomorphism is defined in quite general circumstances. However, the question of whether the Todd class is defined for all separated, integral, finite type schemes is open, if memory serves. It is very related to the question of whether every such scheme admits an immersion to a smooth scheme. It is also related to the resolution property on the scheme. I believe that Totaro has an article exploring these connections.

    • In his book in section 18.3 he defines it for all schemes locally of finite type and separated and he defines the todd class of X in that generality too. Of course he uses Q coefficients and maybe what you are thinking about is whether you can do it with Z coefficients.

      • Maybe I am remembering the problem of constructing the relative Todd class as an element in K-theory for perfect morphisms (although presumably that is in SGA 6 . . .). Apart from the Q-coefficients issue (which seems unavoidable), Grothendieck’s formulation (for perfect morphisms, etc.) takes values in a ring formed from K-theory, which could be much bigger than the corresponding Chow groups (which Fulton defines in terms of the associated reduced scheme, so definitely lose information).

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