Hochschild (Co)Homology of Exterior Algebras using AMT

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.