Shukla Cohomology and Triples*

Thus the comparison ends in a manner fully as satisfactory as in [I] and [2]. The proof uses essentially the same method of acyclic models as was used in those papers. A standard (i.e., functorial) complex for each theory is constructed and the hypotheses of the theorem of acyclic models of [2] are shown to be satisfied, from which chain equivalence of these complexes follows. Here the standard complexes are not generally acyclic. Thus, chain equivalence as modules is required, in contrast to the previous situations in which it would have sufficed to show that a certain complex was merely acyclic. The following notations will be used throughout. K denotes a commutative ring with unit, A is a fixed unitary K-algebra and M is a fixed A-bimodule. A @ without subscript denotes OK and if A is any K-module we let Afn) denote an nth tensor power of A over K. If B is the free R-module on the base X, for some ring R, we will let (x) denote the basis element corresponding to an x E X. We let (Alg -K, ~4) denote the category whose