Abstract

this paper, we construct and investigate the natural homology theory for coherently homotopy commutative dg-algebras, usually known as E1 -algebras. We call the theory \Gamma-homology for historical reasons (see, for instance, [3]). Since discrete commutative rings are E1 rings, we obtain by specialization a new homology theory for commutative rings. This special case is far from trivial. It has the following application in stable homotopy theory, which was our original motivation and which will be treated in a sequel to this paper. The obstructions to an E1 multiplicative structure on a spectrum lie (under mild hypotheses) in the \Gamma-cohomology of the corresponding dual Steenrod algebra, just as the obstructions to an A1 -structure lie in the Hochschild cohomology of that algebra [15]. The \Gamma-homology of a discrete commutative algebra B can be understood as a refinement of Harrison homology, which was originally defined as the homology of the quotient of the Hochschild complex by the subcomplex generated by nontrivial shuffle products. It is better defined as the homology of a related complex which one obtains by tensoring each term

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