Satellites and Homology

In Chapters VI and VII we gave “concrete” applications of the theory of derived functors established in Chapter IV, namely to the category of groups and the category of Lie algebras over a field K. In this chapter our first purpose is to broaden the setting in which a theory of derived functors may be developed. This more general theory is called relative homological algebra, the relativization consisting of replacing the class of all epimorphisms (monomorphisms) by a suitable subclass in defining the notion of projective (injective) object. An important example of such a relativization, which we discuss explicitly, consists in taking, as our projective class of epimorphisms in the category ###m_Λ of Λ-modules, those epimorphisms which split as abelian group homomorphisms.