On the spectrum of algebraic 𝐾-theory

Abstract

The groups Kt(A) of Bass for i < 0 are identified as homotopy groups of the spectrum of algebraic A -theory. The spectrum itself is identified. Applications to Laurent polynomials and to -theory exact sequences are given. Quillen has recently proposed a K-theory for unital rings [12], [13]. He associates to a ring A a space BG1(A) whose homology is that of the group G\(A) and whose homotopy groups nt BG1(A) + he defines as Kt(A i ^ 1. The space BG1(A) is known to be an H-space, and indeed an infinite loop space. Hence one is motivated to define Kt(A), for i e Z, as 7Zi(E(A)) where E(A) is the associated Q-spectrum. This note describes E(A) and identifies the groups Kt(A i < 0. In fact, we show that the groups Kt(A) are exactly the groups L~K0(A) discussed in Bass' book [3, p. 664] for i < 0. Recall from the work of Karoubi and Villamayor [10] the cone CA and suspension SA of a ring A. An infinite matrix is called permutant if it is an infinite permutation matrix times a diagonal matrix of finite type. The diagonal matrix is of finite type if its diagonal entries are chosen from a finite subset of the ring. The ring CA is the ring generated by permutant matrices. The cone CA contains the two-sided ideal A = \Jn Mn(A) and the quotient ring is called the suspension of A. We can now state our main result. THEOREM A. The space Q (BG1(SA)) has the homotopy type of K0{A) x BGl(A) . COROLLARY. For all ieZ we have Kt(A) = Ki+1(SA). Since Karoubi [9] has already identified X0(<SM) with Bass' groups K-i(A% the Corollary above completes the identification of Bass' groups with the negative homotopy of the spectrum E(A). In proving Theorem A we must first analyze the cone construction. THEOREM B. The space BG1(CA) is contractible. This result generalizes work of Karoubi and Villamayor [11] who show that Ki(CA) = 0 for i ^ 2. To prove Theorem B we observe that it suffices AMS 1970 subject classifications. Primary 18F25, 55B15, 16A54, 13D15, 55F50, 18G30, 55B20, 55D35.