Geometric equivalence relations on modules

Abstract

In [4] Gersten developed the notion of homotopy for ring homomorphisms. A simple homotopy of two ring homomorphisms f,g : A +B can be viewed as a deformation over the parameter space Spec Z[t]. This is a homomorphism A+B[t] which restricts to f when t = 0 and to g when t = 1. Two homomorphisms f and g are homotopic if there is a chain of homomorphisms starting with f and ending with g such that each term is simple homotopic to the next. Let A be a k-algebra and k a field. Consider the finite dimensional representations of A and require that a simple homotopy of representations be given by a deformation over Speck[t] = A:. Using direct sum we can make the homotopy classes of representations into an abelian monoid. Now it is more useful to use any nonsingular, rational affine curve as well as A: for the parameter space of a homotopy. (This becomes apparent when A is commutative; see Section 1.3.) The abelian monoid of homotopy classes is denoted by N(A) and its associated group by R(A). Two modules (representations) whose classes in R(A) are the same are said to be ‘rationally equivalent’. In Section 1.3 we show that when A is commutative R(A) is isomorphic to the Chow group of O-cycles of SpecA modulo rational equivalence. In Section 1.2 we prove basic structure theorems about the functor H from kalgebras (finitely generated) to abelian monoids. Now consider more general deformations using any connected affine k-scheme as the parameter space. The result is a coarser equivalence relation on the finite dimensional A-modules and a corresponding abelian monoid which we denote C(A). In Section 1 .I we determine C(A) for some algebras: finite dimensional algebras, enveloping algebras, commutative algebras. We prove basic structure theorems about C(A *kB), C(A Ok@, C(A x B). Let D(A) be the associated abelian group. Two finite dimensional modules M and N are said to be ‘algebraically