Intertwiners and the K-theory of commutative rings

Abstract

Since around 1970, the main approach to the K-theory of a ring A has been by means of the homotopy of the plus-construction applied to the classifying space of the general linear group of A. In the case of a commutative ring A, we show how to capture K 0A information that is neglected by this definition, while retaining the higher K-theory. To accomplish this, we expand the algebraic focus from invertible matrices to what we call intertwining matrices. S in M nA is an intertwining matrix if it is not a zero divisor and satisfies (M nA)S = S(M nA). We establish a number of properties of intertwiners in abstract monoids, and in particular of interwining matrices, so as to make the classifying space and its plus-construction more accessible. This ultimately leads to new insights on the action of K 0A on the higher K-groups, and traditional matters like the Rosenberg-Zelinsky theorem. The theory attains greatest power when A is a domain of dimension 1, where it provides a new description of torsion in the Picard group of A. Number fields are an abundant source of examples.