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.

Key Words:

• Symmetric monoidal categories
• Hochschild homology
• Central localization

2000 Mathematics Subject Classification:

• 16E30
• 16E40
• 18D10

Notes

Communicated by M. Bresar.