Posted by: Steven Sam | August 3, 2009

Chern classes and Riemann–Roch formalism

Last time I gave the definition of \lambda-rings and tried to motivate them from the perspective of K-theory. I’d like to continue with how to define abstract Chern classes for \lambda-rings and explain Riemann–Roch formalism. We’ll assume all \lambda-rings R have an augmentation, are equipped with a positive structure, and have an involution. I’ll once again use K-theory as motivation for the definitions.

In topological K-theory, if we have a space X with a vector bundle E, there exists a map f \colon X' \to X such that the pullback f^*E decomposes into a direct sum of line bundles, and the map on cohomology rings f^* \colon \mathrm{H}^*(X) \to \mathrm{H}^*(X') is injective. The corresponding statement for \lambda-rings is that we can embed any \lambda-ring R into a larger one such that every element can be written as a sum of elements with augmentation \pm 1.

Let A = \bigoplus_{d \ge 0} A_d be a graded ring. We let 1+A^+ be the group under multiplication of formal power series of the form 1 + \sum_{n \ge 0} a_n t^n where a_n \in A_n. A map of Abelian groups c_t \colon R \to 1+A^+ (we use addition for R), written c_t(x) = \sum_i c^i(x) t^i is a Chern class homomorphism if

  • For each line element u, c^i(u) = 0 for i>1.
  • For two line elements u and v, c^1(uv) = c^1(u) + c^1(v).

We will also assume that each c_t(x) is a polynomial. Chern class homomorphisms are compatible with the splitting principle in the following way. If we want to split an element X in R, then there exists a \lambda-ring R’ and a graded ring A’ which extends A, together with a Chern class homomorphism c_t \colon R' \to 1+A'^+ which restricts to the one for R. Hence, Chern classes are completely determined by what they do to line elements.

So we get “factorizations” c_t(x) = \prod_i (1+a_i t). This allows us to define the Chern character \mathrm{ch} \colon R \to \mathbf{Q}[t] via \displaystyle \mathrm{ch}(x) = \sum_i \exp(a_i) =  \sum_i \sum_{n \ge 0} \frac{a_n}{n!} and the Todd class \displaystyle \mathrm{td}(x) = \prod_i \frac{a_i\exp(a_i)}{\exp(a_i) - 1}.

Now we move on to Riemann–Roch formalism. Let C be a category. A Riemann–Roch functor is a triple (K, \rho, A) such that

  • For all X in C, we have commutative rings K(X) and A(X). Let UK(X) and UA(X) be the underlying additive groups.
  • For every morphism f \colon X \to Y, we have ring homomorphisms (pullbacks) f^K \colon K(Y) \to K(X) and f^A \colon A(Y) \to A(X) which make K and A into contravariant functors from C to commutative rings.
  • For every morphism f \colon X \to Y, we have group homomorphisms (pushforwards) f_K \colon UK(X) \to UK(Y) and f_A \colon UA(X) \to UA(Y) which make UK and UA into covariant functors from C to Abelian groups.
  • The projection formula f_H(x \cdot f^H(y)) = f_H(x) \cdot y where f \colon X \to Y and x \in H(X) and y \in H(Y) holds for H=K and H=A.
  • \rho is a natural transformation from K to A.

The statement that Riemann–Roch holds for a morphism f in C means that there exists an element \tau_f \in A(X) such that \rho_Y f_K(x) = f_A(\tau_f \cdot \rho_X(x)) for all x in K(X). The element \tau_f measures the failure for \rho \colon UK \to UA to be a natural transformation of covariant functors.

If Riemann–Roch holds for two morphisms, then it also holds for their composition.

To bring in \lambda-rings, we focus more specifically on Chern class functors. This is a Riemann–Roch functor (K, c, A) where we further assume that

  • The image of K is the category of \lambda-rings with involution.
  • The image of A is the category of graded rings whose morphisms are degree 0 maps.
  • For every X in C, c \colon K(X) \to 1+A(X)^+ is a Chern class homomorphism.

Fix this Chern class functor. Since this post is already getting too technical, let me just mention that one defines what it means for a morphism in C to be an elementary embedding and elementary projection, and shows that Riemann–Roch holds for such morphisms. The elements \tau_f in both cases are defined in terms of Todd classes.

Now we relate this to the Grothendieck–Riemann–Roch theorem. For this, we fix some Noetherian commutative ring R and let C be the category whose objects are schemes over Spec R that are quasi-projective and connected. The morphisms in our category are those which are projective local complete intersection morphisms: these are all morphisms of the form X \xrightarrow{i} \mathbf{P}(E) \xrightarrow{p} Y where E is a vector bundle on Y, and \mathbf{P}(E) is the corresponding projective bundle, and i is a closed embedding whose image is locally a complete intersection (i.e., there is an open affine cover where in each open affine, the ideal of X is generated by a regular sequence). Here K is the K-theory of algebraic vector bundles over X, and A is a certain associated graded ring of K(X). We have pullbacks of vector bundles which makes K and A into contravariant functors. The pushforward f_K on K is given by f_K = \sum_{i \ge 0} (-1)^i \mathrm{R}^i f_* an alternating sum of higher direct images. In this case, i is an elementary embedding, and p is an elementary projection in the language used above.

What comes out of all of this (and the stuff I skipped on elementary embeddings and elementary projections), is that Riemann–Roch holds for all morphisms in this category, and the element \tau_f can be written as the Todd class \mathrm{td}(T_f) where T_f is the virtual tangent bundle of f defined as
T_f = [i^*\mathscr{T}_{\mathbf{P}(E) / Y}] - [\mathscr{N}_{X,\mathbf{P}(E)}] using the notation above. here \mathscr{T} denotes the relative tangent bundle, and \mathscr{N}_{X,\mathbf{P}(E)} is the normal bundle of X inside of \mathbf{P}(E) (this is a bundle because X is a local complete intersection).

We’ll do one last thing: specialize to the case that R=k is a field and we consider only connected nonsingular quasiprojective varieties over k and proper morphisms between them. Given X, we can embed it inside some projective space \mathbf{P}^n via a map i. A map is proper f \colon X \to Y if and only if the graph morphism X \to \mathbf{P}^n \times Y induced by the maps i and f is a closed embedding. Whenever a nonsingular variety is embedded as a closed subvariety of another nonsingular variety, it is automatically a local complete intersection, so we see that this case is really a special case of the preceding paragraph (here E is a trivial rank n+1 vector bundle over Y). Also, on a nonsingular variety, every coherent sheaf has a finite resolution by vector bundles, so K(X) is in fact the Grothendieck group of coherent sheaves (though the multiplication for coherent sheaves which aren’t vector bundles involves sums and Tor functors). Furthermore, we can let A be the Chow ring functor (which shows that it coincides with the associated graded of K in this case), and our natural transformation \rho is the usual Chern character ch. We’ll finish with what Grothendieck–Riemann–Roch says in this case:

Let f \colon X \to Y be a proper morphism between nonsingular quasiprojective varieties, and let [F] be an element in K(X). Then

\mathrm{ch}(f_K([F])) = f_*(\mathrm{td}([T_f]) \cdot \mathrm{ch}([F])).



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