On the algebraic K-theory of formal power series

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 T-R(M) denote the (derived) tensor algebra of M over R, and T-R(M) denote the ring of formal (derived) power series in M over R. We define a natural transformation of functors of simplicial R-bimodules Phi : Sigma(K) over tilde (R; ) -> (k) over tilde (T-R(pi)()) which is closely related to Waldhausen's equivalence Sigma(K) over tilde (Nil(R; )) -> similar or equal to (K) over tilde (T-R()). We show that Phi 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 Sigma W(R; ) -> similar or equal to holim(n)(K) over tilde (T-R()/In+1) and for connected bimodules, also an equivalence Sigma(K) over tilde (R; ) -> similar or equal to (k) over tilde (T-R()).