Posted by: Steven Sam | November 9, 2009

Chow rings and K-theory

I want to write a post about the set-valued tableaux of Buch and how they are related to Schur polynomials. But since these two things are related to the K-theory and Chow ring, respectively, of the Grassmannian, I thought I would write a post explaining some basic generalities between the Chow ring and K-theory. Let X be a variety over an algebraically closed field K.

First let’s define the Chow groups. We first form the k-cycles Z_k(X) to be the free Abelian group spanned by the k-dimensional subvarieties of X. Let [V] be the basis element corresponding to a subvariety V. Pick a subvariety W of X of dimension k+1, and a nonzero rational function f/g defined on W. If V is a codimension 1 subvariety of W, let \mathcal{O}_{W,V} be the ring obtained by taking the ring of polynomial functions on W and inverting all polynomial functions which are not identically zero on V. We define the order {\rm ord}_V(f/g) to be \dim_K \mathcal{O}_{W,V}/(f) - \dim_K \mathcal{O}_{W,V}/(g), where the dimension is as K-vector spaces. The divisor of f/g is given by {\rm div}(f/g) = \sum_{\dim V = k} {\rm ord}_V(f/g) [V]. We say these divisors are rationally equivalent to 0, and define the Chow group {\rm A}_k(X) to be the group of k-cycles modulo rational equivalence.

Now let’s assume that X is nonsingular of dimension n. Given subvarieties V and W of X, let Z be an irreducible component of the intersection V \cap W. Restrict to an open affine subset U of X, so that V and W are defined by ideals I and J, respectively. We define the intersection multiplicity to be the following alternating sum

\displaystyle \mu(Z; V,W) = \sum_{i \ge 0} (-1)^i {\rm length}_{\mathcal{O}_{X,Z}} {\rm Tor}^i_{\mathcal{O}_{Z,Z}} (\mathcal{O}_{X,Z} / I, \mathcal{O}_{X,Z} / J).

Set {\rm A}^k(X) = {\rm A}_{n-k}(X). We can give {\rm A}^*(X) a graded ring structure by defining [V] \cdot [W] = \sum \mu(Z; V,W) [Z], where the sum is over all irreducible components Z of V \cap W.

Now we discuss the K-theory K(X) of vector bundles. For now we don’t need to assume X nonsingular yet. We first consider the free Abelian group on isomorphism classes of vector bundles on X, modulo relations given by short exact sequences: for any sequence 0 \to E_1 \to E_2 \to E_3 \to 0, we add the relation [E_1] - [E_2] + [E_3] = 0. We endow K(X) with a ring structure via [E] + [F] = [E \oplus F] and [E] \cdot [F] = [E \otimes F] which one verifies is well-defined. Similarly, we can form the K-theory {\rm K}_\circ(X) of coherent sheaves on X, which doesn’t necessarily have a multiplication since tensoring with an arbitrary coherent sheaf need not preserve exact sequences. There is a map

\varepsilon \colon {\rm K}(X) \to {\rm K}_\circ(X)

obtained by sending the class of a vector bundle to itself, considered as a locally free sheaf. When X is nonsingular, this map is an isomorphism. The reason is that in this situation, one can always resolve any coherent sheaf by a finite resolution of vector bundles (one never needs more than \dim X + 1 such vector bundles).

In general, we can define the topological filtration of {\rm K}_\circ(X) by letting F_k {\rm K}_\circ(X) be the subgroup generated by coherent sheaves whose support has dimension at most k. Let {\rm gr}_k {\rm K}_\circ(X) = F_k {\rm K}_\circ(X) / F_{k-1} {\rm K}_\circ(X). Given a coherent sheaf \mathscr{F} and a subvariety V of X, we can define the multiplicity m_V(\mathscr{F}) as follows: on an affine open set U = Spec(A) which intersects V, V corresponds to a prime ideal P of A, \mathscr{F} corresponds to a finitely generated A-module M. So we can localize M at P to get a module over the local ring A_P, and m_V(\mathscr{F}) is the length of M_P as an A_P-module. Now for \mathscr{F} \in F_k {\rm K}_\circ(X), we can define

\displaystyle Z_k(\mathscr{F}) = \sum_{\dim V = k} m_V(\mathscr{F}) [V] \in {\rm A}_k(X).

There is a unique homomorphism \varphi \colon Z_k(X) \to {\rm gr}_k {\rm K}_\circ(X) such that [V] \mapsto [\mathcal{O}_V] and Z_k(\mathscr{F}) \mapsto [\mathscr{F}], and it factors through rational equivalence to give a map \varphi \colon {\rm A}_k(X) \to {\rm gr}_k {\rm K}_\circ(X).

In the case that X is nonsingular, we can tensor this map with the rational numbers to get an isomorphism \varphi_{\bf Q}. Then K-theory of coherent sheaves is the same as K-theory of vector bundles, and the topological filtration in this case is a filtration by subrings. In fact, after identifying {\rm A}_*(X) with the Chow ring {\rm A}^*(X), the map \varphi_{\mathbf{Q}} is an isomorphism of rings. This says that cohomology is a sort of approximation to K-theory.

Next time, I’ll specialize to the case when X is a Grassmannian, and explain the combinatorics involved with the Chow ring and K-theory. This will make the isomorphism above more clear. Click here for the next post.

-Steven

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