Localization of Monoid Objects and Hochschild Homology

Abstract

Let A be a (not necessarily commutative) monoid object in an abelian symmetric monoidal category (C, ⊗,1) satisfying certain conditions. In this paper, we continue our study of the localization M S of any A-module M with respect to a subset S ⊆ Hom A−Bimod (A, A) that is closed under composition. In particular, we prove the following theorem: if P is an A-bimodule such that P is symmetric as a bimodule over the center Z(A) of A, we have isomorphisms HH *(A, P) S ≅ HH *(A, P S ) ≅ HH *(A S , P S ) of Hochschild homology groups.