Topological Hochschild homology of ring functors and exact categories Original Research Article

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 image 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(image) and TC(image), we get that the iteration of Waldhausenʹs S construction yields spectra {THH(S(n)image)} and {TC(S(n)image)}, and the maps from K-theory become maps of spectra. If image is split exact, the THH and TC spectra are Ω-spectra. The inclusion by degeneracies THH0(S(n)image) subset of or equal to THH(S(n)image) is a stable equivalence, and it is shown how this leads to a weak resolution theorem for THH. If Weierstrass pA is the category of finitely generated projective modules over a unital and associative ring A, we get that THH(A) image THH(Weierstrass pA) and TC(A) image TC(Weierstrass pA).