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)