Relative homological algebra

Abstract

Introduction. The main purpose of this paper is to draw attention to certain functors, exactly analogous to the functors and of Cartan-Eilenberg [2], but applicable to a module theory that is relativized with respect to a given subring of the basic ring of operators. In particular, we shall show how certain cohomology theories for groups, rings, and Lie algebras can be subsumed under the theory of the Ext functor, just as (in [2]) the ordinary cohomology theories have been subsumed under the theory of the ordinary Ext functor. Among the various cohomology groups that have been considered so far, some can be expressed in terms of the ordinary Ext functor; these have been studied systematically within the framework of general homological algebra by M. Auslander (to appear). A typical feature of these groups is that they appear naturally as terms of exact sequences whose other terms are the ordinary cohomology groups of the algebraic system in question, and of its given subsystem. There is, however, another type of cohomology theory whose groups are not so intimately linked to the ordinary cohomology groups and exhibit a more individualized behaviour. Specifically, the cohomology groups for Lie algebras, as defined (in [3]) by Chevalley and Eilenberg, and the cohomology groups of groups, defined and investigated by I. T. Adamson [1], are of this second type. It is these more genuinely relative cohomology theories that fall in our present framework of homological algebra. Our plan here is to sketch the general features of the Tor and Ext functors (?2) and to illustrate some of their possible uses or interpretations by a selection of unelaborated examples. Thus, ?3 illustrates the use of the Ext functor in extending the cohomology theory for algebras. ?4 deals with homology and cohomology of groups, and involves both the Tor functor and the Ext functor. ?5 discusses the role played by the Ext functor in the cohomology theory for Lie algebras. Since this paper is intended to serve as a preliminary survey, and since the topics dealt with are supplementary to the corresponding topics of the nonrelative theory (contained in [2]), we feel justified in presupposing that the