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 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 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 to be , where the dimension is as K-vector spaces. The divisor of f/g is given by . We say these divisors are rationally equivalent to 0, and define the Chow group 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 . 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
Set . We can give a graded ring structure by defining , where the sum is over all irreducible components Z of .
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 , we add the relation . We endow K(X) with a ring structure via and which one verifies is well-defined. Similarly, we can form the K-theory 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
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 such vector bundles).
In general, we can define the topological filtration of by letting be the subgroup generated by coherent sheaves whose support has dimension at most k. Let . Given a coherent sheaf and a subvariety V of X, we can define the multiplicity as follows: on an affine open set U = Spec(A) which intersects V, V corresponds to a prime ideal P of A, corresponds to a finitely generated A-module M. So we can localize M at P to get a module over the local ring , and is the length of as an -module. Now for , we can define
There is a unique homomorphism such that and , and it factors through rational equivalence to give a map .
In the case that X is nonsingular, we can tensor this map with the rational numbers to get an isomorphism . 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 with the Chow ring , the map 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.