Cohomology and obstructions: Commutative algebras

Abstract

Associated with each of the classical cohomology theories in algebra has been a theory relating H (H as classically numbered) to obstructions to non-singular extensions and H with coefficients in a “center” to the non-singular extension theory (see [Eilenberg & MacLane (1947), Hochschild (1947), Hochschild (1954), MacLane (1958), Shukla (1961), Harrison (1962)]). In this paper we carry out the entire process using triple cohomology. Because of the special constructions which arise, we do not know how to do this in any generality. Here we restrict attention to the category of commutative (associative) algebras. It will be clear how to make the theory work for groups, associative algebras and Lie algebras. My student, Grace Orzech, is studying more general situations at present. I would like to thank her for her careful reading of the first draft of this paper. The triple cohomology is described at length elsewhere in this volume [Barr & Beck (1969)]. We use the adjoint pair