The Grothendieck Group

In the red corner, topological K-theory! The study of stable equivalences of vector bundles over a topological space, and the engine behind Bott periodicity, a result whose reverberations are felt throughout algebraic topology. In the blue corner, modular representation theory! The story of representations in positive characteristic, where the CDE triangle powers applications in group theory and number theory. What do these subjects have to say about each other? Are they just two examples of the same construction? And. . . why is representation theory like Game of Thrones? Tune in this Friday at 5 p.m. CDT to find out, only on Sophex. 1. DEFINITION AND FIRST EXAMPLES There are many times in mathematics where some class of objects has a nice additive structure, except that one can’t always invert elements. One approach is to say, “oh well,” and develop the theory of monoids; another, which we’ll discuss today, adds in formal inverses, in effect pretending that they exist, and uses them to recover something useful. Definition. Recall that a commutative monoid (M ,+) is a set M with an associative, commutative binary operator + : M ×M → M that has an identity: e ∈ M such that e+ x = x + e = x for all x ∈ M . That is, it is an abelian group, except (possibly) for inverses. The prototypical example is N= {0,1,2,3, . . .}, the nonnegative integers under addition, and of course any abelian group satisfies the definition too. We want a way to turn a monoid into an abelian group, and we’d like to make it as well-behaved as possible. It would be nice if we could make the following work. (1) If we start with an abelian group A, we should end up with something isomorphic to A. (2) If we start with N, we should get Z (after all, this is what happened in set theory). (3) We’d like this to be well-behaved under mapping: monoid homomorphisms should correspond to group homomorphisms. It turns out we can do this. Definition. If M is a commutative monoid, the Grothendieck group of M , denoted K(M), is the abelian group satisfying the following universal property: there is a monoid homomorphism i : M → K(M) such that if A is an abelian group and f : M → A is a monoid homomorphism, there is a unique group homomorphism g : K(M)→ A making the following diagram commute. M i //