Delooping classifying spaces in algebraic K-theory

Abstract

FOR ANY associative ring with identity A let BGL(A)+ denote the “classifying space” for algebraic K-theory given in [I51 where a definition for the higher K-theory functors Ki, i >, 1, is proposed by Quillen as K,(A) = n,(BGL(A)+). This K, and KZ agree with the K, of Bass [2] and the K2 of Milnor [ 141 and the theory has other very pleasant properties [ 151. Now Anderson [l] and Segal [17] have shown how to associate a generalized cohomology theory K’(X; q) to any category %? with a “commutative” and “associative” internal operation ?? x %’ -+ e. If one takes the category 9’ of finitely generated projective modules over A with morphisms the isomorphisms and internal operation the direct sum, then K,(A) = K-‘(pt; 9’) for i 2 0. See [18]. Our purpose here is to show that