Hochschild (Co)Homology of Exterior Algebras using AMT

TL;DR

This paper presents a new method to compute Hochschild (co)homology of exterior algebras, providing explicit algebra structures, generators, and resolutions over integers, enhancing understanding of their algebraic properties.

Contribution

It introduces an alternative approach (AMT) to compute Hochschild (co)homology of exterior algebras, including additive structures, generators, and explicit resolutions over integers.

Findings

Derived all Hochschild (co)homology results using AMT

Calculated additive structures over z

Provided explicit free resolutions and generators

Abstract

In 'Hochschild (co)homology of exterior algebras' (Han, Xu, 2007), the authors computed the additive and multiplicative structure of $H\!H^\ast\!(A;\!A)$, where $A$ is the $n$-th exterior algebra over a field. In this paper, we derive all their results using a different method (AMT), as well as calculate the additive structure of $H\!H_k\!(A;\!A)$ and $H\!H^k\!(A;\!A)$ over $\mathbb{Z}$. We provide concise presentations of algebras $H\!H_\ast\!(A;\!A)$ and $H\!H^\ast\!(A;\!A)$, as well as determine their generators in the Hochschild complex. Lastly, we compute an explicit free resolution (spanned by multisets) of the $A^e$-module $A$ and describe the homotopy equivalence to its bar resolution.