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

In Hochschild (co)homology of exterior algebras, Y. Han and Y, Hu 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 Z. We provide concise presentations 
of algebras HH_*(A;A) and HH^*(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.