## 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 R-bimodule, we let R(M) denote the (derived) tensor algebra of M over R, and ^{π} _{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 Î 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

Original language | English (US) |
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Pages (from-to) | 165-189 |

Number of pages | 25 |

Journal | Journal of K-Theory |

Volume | 10 |

Issue number | 1 |

DOIs | |

State | Published - Aug 2012 |

## Keywords

- Formal Power Series
- Goodwillie Calculus
- K-theory of Endomorphisms
- Key Words Algebraic K-theory
- Tensor Algebra

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology