Hochschild (co)homology of exterior algebras using algebraic Morse theory

In 1 Han, Y., Xu, Y. (2007). Hochschild (co)homology of exterior algebras. Comm. Algebra 35(1):115–131.Web of Science ®] , Google Scholar]], the authors computed the additive and multiplicative structure of HH*(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 HH_k(A;A) and HH^k(A;A) over ?. We provide concise presentations of algebras HH_?(A;A) and HH*(A;A) as well as determine their generators in the Hochschild complex. Finally, we compute an explicit free resolution (spanned by multisets) of the A^e-module A and describe the homotopy equivalence to its bar resolution.