Developments from Barrʹs thesis Original Research Article

In 1962, Barr exhibited an idempotent in the rational group algebra of the symmetric group which split the Hochschild cochain complex of a commutative algebra into the direct sum of the Harrison complex and a complement. This discovery has led to significant developments both in the cohomology of commutative algebras and in understanding the relations between Hochschild cohomology and simplicial and sheaf cohomology.