Derived Category - Projective and Injective Resolutions

Projective and Injective Resolutions

One can easily show that a homotopy equivalence is a quasi-isomorphism, so the second step in the above construction may be omitted. The definition is usually given in this way because it reveals the existence of a canonical functor

In concrete situations, it is very difficult or impossible to handle morphisms in the derived category directly. Therefore one looks for a more manageable category which is equivalent to the derived category. Classically, there are two (dual) approaches to this: projective and injective resolutions. In both cases, the restriction of the above canonical functor to an appropriate subcategory will be an equivalence of categories.

In the following we will describe the role of injective resolutions in the context of the derived category, which is the basis for defining right derived functors, which in turn have important applications in cohomology of sheaves on topological spaces or more advanced cohomologies like étale cohomology or group cohomology.

In order to apply this technique, one has to assume that the abelian category in question has enough injectives which means that every object A of the category admits a monomorphism to an injective object I. (Neither the map nor the injective object has to be uniquely specified). This assumption is often satisfied. For example, it is true for the abelian category of R-modules over a fixed ring R or for sheaves of abelian groups on a topological space. Embedding A into some injective object I0, the cokernel of this map into some injective I1 etc., one constructs an injective resolution of A, i.e. an exact (in general infinite) complex

where the I* are injective objects. This idea generalizes to give resolutions of bounded-below complexes A, i.e. An = 0 for sufficiently small n. As remarked above, injective resolutions are not uniquely defined, but it is a fact that any two resolutions are homotopy equivalent to each other, i.e. isomorphic in the homotopy category. Moreover, morphisms of complexes extend uniquely to a morphism of two given injective resolutions.

This is the point where the homotopy category comes into play again: mapping an object A of to (any) injective resolution of A extends to a functor

from the bounded below derived category to the bounded below homotopy category of complexes whose terms are injective objects in .

It is not difficult to see that this functor is actually inverse to the restriction of the canonical localization functor mentioned in the beginning. In other words, morphisms Hom(A,B) in the derived category may be computed by resolving both A and B and computing the morphisms in the homotopy category, which is at least theoretically easier.

Dually, assuming that has enough projectives, i.e. for every object A there is a epimorphism map from a projective object P to A, one can use projective resolutions instead of injective ones.

In addition to these resolution techniques there are similar ones which apply to special cases, and which elegantly avoid the problem with bounded-above or -below restrictions: Spaltenstein (1988) uses so-called K-injective and K-projective resolutions, May (2006) and (in a slightly different language) Keller (1994) introduced so called cell-modules and semi-free modules, respectively.

More generally, carefully adapting the definitions, it is possible to define the derived category of an exact category (Keller 1996).