Vol. 298, No. 2, 2019

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Resolutions for twisted tensor products

Anne Shepler and Sarah Witherspoon

Vol. 298 (2019), No. 2, 445–469
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.

Keywords
twisted tensor product, projective resolutions, Hochschild (co)homology, augmented algebras
Mathematical Subject Classification 2010
Primary: 16E05, 16E30, 16E40
Secondary: 16S30, 57T10
Milestones
Received: 10 January 2017
Revised: 24 July 2018
Accepted: 25 July 2018
Published: 8 March 2019
Authors
Anne Shepler
Department of Mathematics
University of North Texas
Denton, TX
United States
Sarah Witherspoon
Department of Mathematics
Texas A&M University
College Station, TX
United States