Hochschild cohomology and higher order extensions of associative algebras

The nth Hochschild cohomology group is described by (n-2)-extensions (Theorem 1). When n = 2, 3, the theorem reduces to the well-known classical results; for n = 1, we get a description of the group of derivations by extensions; and for n ≥ 4, this result was recently obtained by Baues and Pirashvili for Shukla cohomology. However, their proof is not explicit. We provide a different and explicit proof in the case of Hochschild cohomology. One can consider this theorem as an alternative definition of cohomology theory. So, one has some kind of hint to define cohomology theory for various algebraic structures. © Pleiades Publishing, Inc., 2006.