Topological Hochschild homology of ring functors and exact categories

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 β. Fundamental properties of THH and TC are proven and some examples are analyzed. A special case of a ring functor on an exact category ▪ 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( ▪) and TC( ▪), we get that the iteration of Waldhausen's S construction yields spectra { THH(S (n) ▪) } and { TC(S (n) ▪)}, and the maps from K-theory become maps of spectra. If ▪ is split exact, the THH and TC spectra are Ω-spectra. The inclusion by degeneracies THH 0 (S (n) ▪) ⊆ THH(S (n) ▪) 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) .