On the algebraic K-theory of formal power series

摘要

In this paper we extend the computation of the the typical curves of algebraic K-theory done by Lars Hesselholt and Ib Madsen to general tensor algebras. The models used allow us to determine the stages of the Taylor tower of algebraic K-theory as a functor of augmented algebras, as defined by Tom Goodwillie, when evaluated on derived tensor algebras. For R a discrete ring, and M a simplicial R-bimodule, we let R(M) denote the (derived) tensor algebra of M over R, and _R ~π(M) denote the ring of formal (derived) power series in M over R. We define a natural transformation of functors of simplicial R-bimodules ? which is closely related to Waldhausen's equivalence We show that φ induces an equivalence on any finite stage of Goodwillie's Taylor towers of the functors at any simplicial bimodule. This is used to show that there is an equivalence of functors, and for connected bimodules, also an equivalence