Resolutions for Twisted Tensor Products

TL;DR

This paper develops a unified approach to resolutions for twisted tensor products of algebras, enabling efficient computation of Hochschild (co)homology and Ext/Tor, with applications to various algebraic structures.

Contribution

It introduces general bimodule and module resolutions for twisted tensor products, unifying many existing constructions and extending to important algebraic cases like Ore extensions and universal enveloping algebras.

Findings

Constructed resolutions for twisted tensor products applicable to diverse algebraic structures.

Derived Chevalley-Eilenberg resolutions for universal enveloping algebras.

Provided tools for computing Hochschild (co)homology and Ext/Tor in complex algebraic contexts.

Abstract

We build resolutions for general twisted tensor products of algebras. These bimodule and module resolutions unify many constructions in the literature and are suitable for computing Hochschild (co)homology and more generally Ext and Tor for (bi)modules. We analyze in detail the case of Ore extensions, consequently obtaining Chevalley-Eilenberg resolutions for universal enveloping algebras of Lie algebras (defining the cohomology of Lie groups and Lie algebras). Other examples include semidirect products, crossed products, Weyl algebras, Sridharan enveloping algebras, and Koszul pairs.