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Homotopy liftings and Hochschild cohomology of some twisted tensor products

Pablo S. Ocal, Tolulope Oke, Sarah Witherspoon

Abstract

The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present new proofs of these isomorphisms, using Volkov's homotopy liftings that were introduced for handling Gerstenhaber brackets expressed on arbitrary bimodule resolutions. Our results illustrate the utility of homotopy liftings for theoretical purposes.

Homotopy liftings and Hochschild cohomology of some twisted tensor products

Abstract

The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present new proofs of these isomorphisms, using Volkov's homotopy liftings that were introduced for handling Gerstenhaber brackets expressed on arbitrary bimodule resolutions. Our results illustrate the utility of homotopy liftings for theoretical purposes.

Paper Structure

This paper contains 4 sections, 6 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Twisted tensor product and Gerstenhaber bracket techniques
  3. Isomorphisms of Gerstenhaber algebras
  4. Example

Key Result

Lemma 2.3

GNW There is a chain map that is an isomorphism of $(A\otimes^t B)^e$-modules in each degree, given by on $(P_i\otimes^t Q_j)\otimes_{A\otimes^t B}(P_u\otimes^t Q_v)$.

Theorems & Definitions (12)

  • Lemma 2.3
  • Definition 2.8
  • Theorem 2.10
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.5: Grimley--Nguyen--Witherspoon GNW
  • proof
  • Lemma 3.8
  • proof
  • ...and 2 more