On the homology of commutative algebras
The authors give a method for computing the homology and cohomology groups of a commutative \(A\)-algebra \(B\) with coefficients in a \(B\)-module \(M\). It is extended the 2-fold crossed module of the theory of groups to commutative algebra, and using 2-crossed resolutions and the homotopy operator, it is defined a homology theory and it is proved that it coincides with the André-Quillen theory for \(n=0,1,2,3\). Using this linear method it is obtained the Jacobi-Zariski sequence. The authors use these results to characterize complete intersection rings by the vanishing of the third homology group. The corresponding results for cohomology also hold.
scientific article; zbMATH DE number 89566