Sankt Petersburg University, Bibliotechnaya pl. 2, Sankt Petersburg, Russia
Contents
The origins of relative homological algebra can be found in different branches of algebra but mainly in the theory of abelian groups and in the representation theory of finite groups. Prüfer introduced in 1923 the notion of purity which nowadays is one of the most important notions of abelian group theory []. Generalizations of purity in the category of abelian groups and in module categories have many applications and are really tools of homological algebra.
In representation theory we have also the important notions of relative projectives and relative injectives, and the analysis of their properties has led Hochschild in 1956 to the discovery of ``relative homological algebra'' []. It is worth noting that ideas of relative homological algebra were contained ``internally'' in the ``Homological algebra'' of Cartan and Eilenberg [].
Finally, Buchsbaum [] and others (see []) have given axioms for a ``proper class'' of short exact sequences in any abelian category and MacLane has rewritten in his ``Homology'' [] a part of homological algebra from the point of view of relative homological algebra.
The first years after that have yielded many interesting examples of proper classes which have been used for proving ``relative'' versions of ``absolute'' theorems. Thus in representation theory Lam and Reiner [] discovered the relative Grothendieck group, Warfield [] introduced the notion of Cohn-purity in a module category which generalized Prüfer purity and he used it for researching the interesting class of algebraically compact modules, Eilenberg and Moore [] generalized the notion of proper class starting from a triple and this allows us to develop relative homological algebra in categories more general than the abelian.
Mishina and Skornyakov in 1969 [] (see also the expanded English translation in 1976) and then Sklyarenko in 1978 [] have given good surveys of the development of relative homological algebra at that time.
In this article we do not intend to give a review of all contributions to relative homological algebra but we only present the main ideas of the theory and also some recent advances in it.
In Section we introduce the notion of a proper class of cokernels in a pre-abelian category. This generalization of the usual notion of a proper class of short exact sequences will be used in Section for defining relative derived categories. This construction was made in [] to give a unified approach to homological algebra in pre-abelian categories which allows to include the approaches in [] in the framework of a single theory.
Section is devoted to relative homological algebra in module categories and we discuss recent results on the classification of inductively closed proper classes which are closely related with algebraically compact modules. Some results concern the structure of such modules. We also discuss in this section the so-called ``group of relations'' of relative Grothendieck groups.
The language of relative homological algebra is useful in defining the cohomology of small categories (see []), and the corresponding theory is presented in Section . Moreover, we discuss there a new cohomology introduced by Baues--Wirsching cohomology Baues and Wirsching [] which generalizes the Hochschild--Mitchell cohomology [].
Section contains some applications of the results of the preceding sections to the cohomology of partially ordered sets (= posets).
Hochschild--Mitchell dimension of the poset of real numbers is equal to 3
Section contains the cohomology theory of coalgebras (including the relative case). For simplicity we restrict ourselves to the case where the base commutative ring is a field. Note that the general case has been considered in [], which uses the relative homological algebra developed in [].