TOPOLOGICAL HOCHSCHILD HOMOLOGY OF RING FUNCTORS AND EXACT CATEGORIES

In analogy with Hochschild-Mitchell homology for linear categories top ological Hochschild and cyclic homology (THH and TC) are defined for r ing functors on a category C. Fundamental properties of THH and TC are proven and some examples are analyzed. A special case of a ring funct or on an exact category C is treated separately, and is compared with algebraic K-theory via a Dennis-Bokstedt trace map. Calling THH and TC applied to these ring functors simply THH(C) and TC(C), we get that t he 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 a is split exact, the THH and TC spectra are Omega-spectra. The i nclusion by degeneracies THH0(S-(n)C) subset of or equal to THH(S-(n)C ) is a stable equivalence, and it is shown how this leads to a weak re solution theorem for THH. If P-A is the category of finitely generated projective modules over a unital and associative ring A, we get that THH(A)-->(THH)-T-similar or equal to(P-A) and TC(A)-->(TC)-T-similar o r equal to(P-A).