Relative homological algebra

A homological algebra associated with a pair of Abelian categories $ ( \mathfrak A , \mathfrak M ) $ and a fixed functor $ \Delta : \mathfrak A \rightarrow \mathfrak M $( cf. Abelian category). The functor $ \Delta $ is taken to be additive, exact and faithful. A short exact sequence of objects of $ \mathfrak A $,

$$ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 $$

is said to be admissible if the exact sequence

$$ 0 \rightarrow \Delta A \rightarrow \Delta B \rightarrow \Delta C \rightarrow 0 $$

splits in $ \mathfrak M $( cf. Split sequence). By means of the class $ {\mathcal E} $ of admissible exact sequences, the class of $ {\mathcal E} $- projective (respectively, $ {\mathcal E} $- injective) objects is defined as the class of those objects $ P $( respectively, $ Q $) for which the functor $ \mathop{\rm Hom} _ {\mathfrak A} ( P, -) $( respectively, $ \mathop{\rm Hom} _ {\mathfrak A} ( - , Q) $) is exact on the admissible short exact sequences.

Any projective object $ P $ of $ \mathfrak A $ is $ {\mathcal E} $- projective, although this does not mean that in $ \mathfrak A $ there are enough relative projective objects (i.e. that for any object $ A $ from $ \mathfrak A $, an admissible epimorphism $ P \rightarrow A $ of a certain $ {\mathcal E} $- projective object of $ \mathfrak A $ exists). If $ \mathfrak A $ contains enough $ {\mathcal E} $- projective or $ {\mathcal E} $- injective objects, then the usual constructions of homological algebra make it possible to construct derived functors in this category, which are called relative derived functors.

Examples. Let $ \mathfrak A $ be the category of $ R $- modules over an associative ring $ R $ with a unit, let $ \mathfrak M $ be the category of Abelian groups and let $ \Delta : \mathfrak A \rightarrow \mathfrak M $ be the functor which "forgets" the module structure. In this case all exact sequences are admissible, and as a result the "absolute" (i.e. usual) homological algebra is obtained.

If $ G $ is a group, then every $ G $- module is, in particular, an Abelian group. If $ R $ is an algebra over a commutative ring $ k $, then every $ R $- module is a $ k $- module. If $ R $ and $ S $ are rings and $ R \supset S $, then every $ R $- module is an $ S $- module. In all these cases there is a functor from one Abelian category into the other defining the relative derived functors.

References

[1]	S. MacLane, "Homology" , Springer (1963)
[2]	J.C. Moore, S. Eilenberg, "Foundations of relative homological algebra" , Amer. Math. Soc. (1965)