## Abstract

In analogy with Hochschild-Mitchell homology for linear categories topological Hochschild and cyclic homology (THH and TC) are defined for ring functors on a category script c sign. Fundamental properties of THH and TC are proven and some examples are analyzed. A special case of a ring functor on an exact category C is treated separately, and is compared with algebraic K-theory via a Dennis-Bökstedt trace map. Calling THH and TC applied to these ring functors simply THH(C) and TC(C), we get that the iteration of Waldhausen's S construction yields spectra {THH(S^{(n)}C)} and {TC(S^{(n)}C)}, and the maps from K-theory become maps of spectra. If C is split exact, the THH and TC spectra are Ω-spectra. The inclusion by degeneracies THH_{0}(S^{(n)}C) ⊆ THH(S^{(n)}C) is a stable equivalence, and it is shown how this leads to a weak resolution theorem for THH. If ℘_{A} is the category of finitely generated projective modules over a unital and associative ring A, we get that THH(A) ^{≃}→ THH(℘_{A}) and TC(A) ^{≃}→ TC(℘_{A}).

Original language | English (US) |
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Pages (from-to) | 231-294 |

Number of pages | 64 |

Journal | Journal of Pure and Applied Algebra |

Volume | 109 |

Issue number | 3 |

DOIs | |

State | Published - Jun 24 1996 |

## ASJC Scopus subject areas

- Algebra and Number Theory