Our goal in this paper is to define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called “module algebras” (Definition 2.1). Our motivation lies in the following problem: for an algebra A which admits a module structure over an arbitrary bialgebra B compatible with its product structure, the Hochschild or the cyclic ...

Recall that deformations of Hochschild cochains are controlled by a homotopy Baues algebra (B∞-algebra) structure, and therefore a homotopy Gerstenhaber algebra structure on the Hochschild complex. The purpose of the present article is twofold. We first answer one of the classical questions addressed to algebraic topology. Namely, we construct a B∞-algebra structure on singular cochain complexe...

We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an extension of the algebraic theory to include the case when M is not necessarily a progenerator. Our approach is complementary to recent work of Dwyer and Greenl...

The tensor product $$R\otimes S$$ of two algebras can have its multiplication deformed by a bicharacter to yield twisted ^t . We completely describe the Hochschild cohomology in terms components R and S, including full Gerstenhaber algebra structure. This description generalizes result Bergh Oppermann. A number interesting classes noncommutative arise as products, sometimes non-obvious ways. ma...