Topological Hochschild homology of ring functors and exact categories

Bjørn Ian Dundas, Randy McCarthy

Research output: Contribution to journalArticlepeer-review


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 THH0(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 languageEnglish (US)
Pages (from-to)231-294
Number of pages64
JournalJournal of Pure and Applied Algebra
Issue number3
StatePublished - Jun 24 1996

ASJC Scopus subject areas

  • Algebra and Number Theory


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