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Twisted tensor products: Alexander-Whitney and Eilenberg-Zilber maps

Anne V. Shepler, Sarah Witherspoon

Abstract

Alexander-Whitney and Eilenberg-Zilber maps traditionally convert between the tensor product of standard resolutions and the standard resolution of a tensor product of algebras. We examine Alexander-Whitney and Eilenberg-Zilber maps for twisted tensor products, which include skew group algebras, smash products of Hopf algebras, Ore extensions, and universal enveloping algebras. These maps convert between the twist of standard resolutions and the standard resolution of a twist. We extend these to chain maps to and from twists of other resolutions. This allows one to transfer homological information between various resolutions of algebras and to expedite results on the deformation theory of twisted tensor product algebras.

Twisted tensor products: Alexander-Whitney and Eilenberg-Zilber maps

Abstract

Alexander-Whitney and Eilenberg-Zilber maps traditionally convert between the tensor product of standard resolutions and the standard resolution of a tensor product of algebras. We examine Alexander-Whitney and Eilenberg-Zilber maps for twisted tensor products, which include skew group algebras, smash products of Hopf algebras, Ore extensions, and universal enveloping algebras. These maps convert between the twist of standard resolutions and the standard resolution of a twist. We extend these to chain maps to and from twists of other resolutions. This allows one to transfer homological information between various resolutions of algebras and to expedite results on the deformation theory of twisted tensor product algebras.

Paper Structure

This paper contains 12 sections, 14 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Twisted product resolutions
  3. Twisting bar resolutions
  4. Twisted Alexander-Whitney and Eilenberg-Zilber maps
  5. Twisted Alexander-Whitney and Eilenberg-Zilber maps as chain maps
  6. Twisted AW and EZ maps split the reduced bar resolution
  7. Compatible chain maps
  8. Conversion between Resolutions using compatible surjective and injective maps
  9. Conversion between Resolutions using only compatible injective maps
  10. Applications: Smash products
  11. Applications to Hopf actions on Koszul algebras
  12. Acknowledgments

Key Result

Lemma 3.7

Iterative twisting commutes with multiplication in the algebras $R$ and $S$: For any $\ell, j \geq 0$, the following diagram of vector spaces is commutative and a similar diagram holds for $\tau_{_{{\bf B}_S}}$ with the roles of $S$ and $R$ reversed.

Theorems & Definitions (44)

  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.8
  • Lemma 3.7
  • Lemma 3.8
  • proof
  • Example 4.4
  • Example 4.8
  • Example 4.9
  • ...and 34 more