Volodin K-theory of A∞-ring spaces

Abstract

IN [IS], Suslin showed that a certain simplicial set Y(G, R), constucted by Volodin, was weakly equivalent to R(BG(R)+), where G(R) = CL(R), E(R) or St(R), and R is a discrete ring with unit. This model for the loop space of K-theory has had a number of important applications, ranging from the stability problem in algebraic K-theory Cl63 to the construction of invariants for stable pseudio-isotopy theory [ZO]. More recently, it has been used in determining the homotopy-type of n-relative K-theory ([6], [9]), and in showing that the Waldhausen K-theory of various diagrams of simplicial rings can be computed degreewise [6]. The existence of a Volodin model for A, rings is presupposed in a crucial reduction step required to prove the equivalence of Waldhausen’s stable K-theory and Biikstedt’s topological Hochschild homology [lo]. In this paper, we construct a Volodin model for the loop space of K(R), where R is an A, ring. It is worth noting that even in the case of simplicial rings, the proof that the Volodin model has the right homotopy type does not follow directly by the arguments of [lS]. and as far as we are aware of there is no rigorous proof of this fact in the existing literature. A key point in Suslin’s argument for discrete rings is that the fundamental group of the Volodin space acts trivially on the higher homotopy groups. This is not true for simplicial rings. For consider the natural map R + q,R for a simplicial ring. The induced map on Volodin spaces V-Y(R) -+ ^YY(noR) (VU( ) is denoted X( ) in [15]) is an isomorphism in homology (both spaces are acyclic) and on fundamental groups (Sr(q,R) in both cases). St(n,R) acts trivially on the higher homotopy groups of ,V9’(noR). If it acts trivially on the higher homotopy groups of YY(R) too, then 3/Y(R) + YY(noR) would be a nilpotent homology equivalence and hence a homotopy equivalence, which is not true in general. In the ratioal case as well as for simplicial rings with torsion-free no, Song [14] proved that the Volodin model has the right homotopy type using methods of homological algebra to reduce the proof to Suslin’s arguments. For general A, rings, the proof is much harder. For this reason, we have devoted this paper to proving the existence and the relevant properties of our Volodin model, deferring to other papers the applications which provided the initial motivation for the construction of such a model for RK(R). After the introduction of A, rings with involution in Section 1 (which we will need for the proof of our main result) we introduce the Volodin construction in Section 2 and prove our main theorem based on three lemmas. In Section 3 we give alternative models for this construction which we will need for the proofs of the three lemmas in Sections 4, 5, and 6. Finaly, $7 is devoted to generalizing the group completion theorem of [4] and casting it in homotopy invariant form. In the process we also simplify a number of arguments in that