In this post, I want to discuss Grothendieck’s -rings and how they provide an abstract setting for Riemann–Roch formalism. The references I’ll be using are
- Donald Knutson, Lambda-Rings and the Representation Theory of the Symmetric Group
- William Fulton and Serge Lang, Riemann–Roch Algebra
The definition of a -ring is a bit technical, but it starts with a commutative ring R together with operations for all nonnegative integers i such that and for all r in R together with some axioms. In particular, we should say what these lambda operations do to sums and products, and we might also want to know what compositions of them look like. To motivate these axioms, we’ll look at K-theory (where it originates).
Let X be a topological space, and consider the set of all vector bundles over X (topological, smooth, holomorphic, whatever you want). We define lambda operations using exterior powers: . Of course, the set of vector bundles on X isn’t a ring, but the free Abelian group of isomorphism classes of vector bundles is a ring if we use tensor product as the multiplication. But we have to define exterior products on “negatives” of isomorphism classes of vector bundles. For actual vector bundles, we have the identity
so we can use this to extend to “negatives” and we make this an axiom for a general -ring:
(L1) for all r and s in R.
In fact, the above identity holds if we pass to the Grothendieck group of vector bundles over X (add the relations whenever we have a short exact sequence of the form ) because in general, if E’ is an extension of E and F, then has a filtration whose associated graded is the direct sum on the RHS above.
What about products? i.e., what should be? The exterior power of a tensor product of two vector bundles has a rather complicated expression. Nonetheless, there exist integer valued polynomials in 2n variables such that
How do we get these polynomials? Let be algebraically independent variables, and let denote the ith elementary symmetric function in the Xs and Ys, respectively. Then we define the polynomials via the identity
So we have some complicated family of polynomials, and the axiom
(L2) for all r and s in R.
There are also some integer-valued polynomials of degree nm for expressing the compositions . I won’t get into that, but this gives the third axiom
(L3) for all r in R.
A morphism of -rings is a ring homomorphism which commutes with the lambda-operations.
For a simple combinatorial example, take X to be a point. In this case, vector bundles are just vector spaces, and . Identifying vector spaces with their dimension, the lambda operations become , which is just the binomial coefficient when n is nonnegative. A morphism is called an augmentation. For vector bundles, this map is given by sending a vector bundle to its rank and extending linearly to virtual vector bundles. We’ll assume our -rings are augmented.
We can place some further requirements and operations on our -rings. First, in , we naturally have a notion of what it means to be “positive”: any class which represents an actual vector bundle. The set of positive elements has the property that it’s closed under addition and multiplication, and every element of can be expressed as a difference of two positive elements. Furthermore, whenever x is positive, for sufficiently large i, and all positive elements of augmentation 1 (line bundles) have multiplicative inverses. We’ll take all of these features to be an axiom system for a “positive subset” of a -ring. Motivated by the K-theory, we’ll call positive elements of augmentation 1 line elements. We’ll assume that our -rings are equipped with a positive structure.
K-theory also has a nice involution: send a vector bundle to its dual bundle. In general, we’ll say that is an involution of our -ring if it satisfies
- for all x,
- for all x,
- for all line elements u.
We’ll further assume that our -rings are equipped with an involution.
Another example of -rings comes from representation rings. Given a group G and a representation V, we define to be the representation with the diagonal action of G. The augmentation here sends a representation to its dimension (over the ground field), the positive elements are the representations, and the involution sends a representation to its dual. One particular example is when G is the general linear group and we consider only rational representations, so that the representation ring is the ring of symmetric functions in n variables (together with a multiplicative inverse for the product of the n variables). In this case, the lambda operations are plethysm: where is the nth elementary symmetric function, and positive means Schur positive. [I first saw -rings in context of symmetric functions, so the definitions seemed a bit mysterious to me.]
In the next post, I’ll discuss abstract Chern classes in the context of -rings and Riemann–Roch formalism, and say how this relates to Grothendieck–Riemann–Roch for proper maps between nonsingular varieties.