Hochschild (Co)Homology of Exterior Algebras using AMT

In ‘Hochschild (co)homology of exterior algebras’ (Han, Xu, 2007), the authors computed the additive and multiplicative structure of $H\text{\hspace{0.17em}⁣}{H}^{\ast }\text{\hspace{0.17em}⁣}(A;\text{\hspace{0.17em}⁣}A)$, 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 $H\text{\hspace{0.17em}⁣}{H}_k\text{\hspace{0.17em}⁣}(A;\text{\hspace{0.17em}⁣}A)$ and $H\text{\hspace{0.17em}⁣}{H}^k\text{\hspace{0.17em}⁣}(A;\text{\hspace{0.17em}⁣}A)$ over $\mathbb{Z}$. We provide concise presentations of algebras $H\text{\hspace{0.17em}⁣}{H}_{\ast }\text{\hspace{0.17em}⁣}(A;\text{\hspace{0.17em}⁣}A)$ and $H\text{\hspace{0.17em}⁣}{H}^{\ast }\text{\hspace{0.17em}⁣}(A;\text{\hspace{0.17em}⁣}A)$, 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.