On the cohomology of ring extensions

In this article we are primarily concerned with the relativized cohomology theory of ring extensions, as initiated by G. Hochschild [ 141. Our main objective is to lay the foundation for a general vanishing theory of relativized extension functors. The significance of such a theory rests on the unification it provides for the various classical results from the cohomology theories of groups, associative algebras, and Lie algebras. The approach chosen here heavily emphasizes locally nilpotent operators as an underlying principle for vanishing and isomorphism theorems. Although in algebraic homology theory these concepts have appeared only recently (cf. [5]), they occur implicitly in singular homology theory of topological spaces. The conventional proof of the excision theorem, for instance, resorts, by means of barycentric subdivision, to the construction of a locally nilpotent operator that is chain homotopic to the identity. In Section 1 we study the question to what an extent properties of operators are inherited by resolutions. It is shown that resolutions defined by injective envelopes and projective covers are suitable tools for ordinary homology. In the framework of relative homological algebra of a given ring extension R:S the availability of standard resolutions allows the application of more computational techniques. We show that the commutator algebra of the centralizer C,(S) of S in R operates on these resolutions in a natural fashion. This result in conjunction with a certain homotopy is subsequently utilized to show that the associated action on the spaces Ext;,, S) (M, N) is trivial. The succeeding section combines these techniques with basic facts concerning locally finite operators to establish various vanishing theorems for extension functors. The third section deals with the study of the cohomology theory of Frobenius extensions. We address questions emerging from recent developments in the cohomology theories of finite dimensional modular [9] and finite dimensional restricted Lie algebras [7-93. Our results are closely