Loop spaces for the Q-construction

Abstract

The algebraic K-groups of an exact category M are defined by Quillen as K i M=л i+1(Q M) , i≥0, where Q M is a category known as the Q-construction on M . For a ring R, K i P R = K iR (the usual algebraic K-groups of R) where P R is the category of finitely generated projective right R-modules. Previous study of K ∗ R has required not only the Qconstruction, but also a model for the loop space of Q P R, known as the S −1 S-construction. Unfortunately, the S −1 S-construction does not yield a loop space for Q M when M is arbitrary. In this paper, two useful models of a loop space for Q M , with no restriction on the exact category M , are descibed. Moreover, these construction are shown to be directly related to the S −1 S-construction. The simpler of the two constructions fails to have a certain symmetry property with respect to dualization of the exact category M . This deficiency is eliminated in the second construction, which is somewhat more complicated. Applications are given to the relative algebraic K-theory of an exact functor of exact categories, with special attention given to the case when the exact functor is cofinal.