On the algebraic K-theory of formal power series

Abstract

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 $\LARGE{\tau}\normalsize{_{R}}$-bimodule, we let $\LARGE{\tau}\normalsize{_{R}\big(M\big)}$ denote the (derived) tensor algebra of $M$ over $R$, and $\LARGE{\tau}\;\normalsize{^{\pi}_{R}}$ denote the ring of formal (derived) power series in $M$ over $R$. We define a natural transformation of functors of simplicial $R$-bimodules $\phi: \sum\tilde{K}\big(R;\; \big)\rightarrow\tilde{K}\big(\LARGE{\tau}\normalsize{_{R}\big(\; \big)\big)}$.which is closely related to Waldhausen's equivalence $\sum\tilde{K}\big(\text{Nil}\big(R; \;\big)\big)\rightarrow\tilde{K}\big(\LARGE{\tau}\normalsize{^{\pi}_{R}\big( \big)\big)}$. 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 $\sum W\big(R; \;\big)\rightarrow^{\simeq}\text{holim}_{n}\tilde{K}\big(\LARGE{\tau}\normalsize{_{R}\big(\; \big)/I^{n+1}\big)}$, and for connected bimodules, also an equivalence $\sum\tilde{K}\big(R; \;\big)\rightarrow^{\simeq}\tilde{K}\big(\LARGE{\tau}\normalsize{_{R}\big( \;\big)\big)}$.