1 results
On the algebraic K-theory of formal power series
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- Journal:
- Journal of K-Theory / Volume 10 / Issue 1 / August 2012
- Published online by Cambridge University Press:
- 04 April 2012, pp. 165-189
- Print publication:
- August 2012
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- Article
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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 πR denote the ring of formal (derived) power series in M over R. We define a natural transformation of functors of simplicial R-bimodules Φ: 
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 
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 
