K₀ of Rings

Abstract

K-theory as an independent discipline is a fairly new subject, only about 35 years old. (See [Bak] for a brief history, including an explanation of the choice of the letter K to stand for the German word Klasse.) However, special cases of K-groups occur in almost all areas of mathematics, and particular examples of what we now call K₀ were among the earliest studied examples of abelian groups. More sophisticated examples of the idea of the definition of K₀ underlie the Euler-Poincaré characteristic in topology and the Riemann-Roch theorem in algebraic geometry. (The latter, which motivated Grothendieck’s first work on K-theory, will be briefly described below in §3.1.) The Euler characteristic of a space Xis the alternating sum of the Betti numbers; in other words, the alternating sum of the dimensions of certain vector spaces or free R-modules H _i (X;R) (the homology groups with coefficients in a ring R). Similarly, when expressed in modern language, the Riemann-Roch theorem gives a formula for the difference of the dimensions of two vector spaces (cohomology spaces) attached to an algebraic line bundle over a non-singular projective curve. Thus both involve a formal difference of two free modules (over a ring R which can be taken to be ℂ). The group K₁(R) makes it possible to define a similar formal difference of two finitely generated projective modules over any ring R