in the homotopy category, which is at least theoretically easier. In fact, it is enough to resolve Y: for any complex X and any bounded below complex Y of injectives, Dually, assuming that A has enough projectives, i.e. for every object X there is an epimorphism from a projective object P to X, 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: uses so-called K-injective and K-projective resolutions, and (in a slightly different language) 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 . The relation to derived functors The derived category is a natural framework to define and study derived functors. In the following, let F: A → B be a functor of abelian categories. There are two dual concepts: right derived functors come from left exact functors and are calculated via injective resolutions left derived functors come from right exact functors and are calculated via projective resolutions In the following we will describe right derived functors. So, assume that F is left exact. Typical examples are F: A → Ab given by X ↦ Hom(X, A) or X ↦ Hom(A, X) for some fixed object A, or the global sections functor on sheaves or the direct image functor. Their right derived functors are Extn(–,A), Extn(A,–), Hn(X, F) or Rnf∗ (F), respectively. The derived category allows us to encapsulate all derived functors RnF in one functor, namely the so-called total derived functor RF: D+(A) → D+(B). It is the following composition: D+(A) ≅ K+(Inj(A)) → K+(B) → D+(B), where the first equivalence of categories is described above. The classical derived functors are related to the total one via RnF(X) = Hn(RF(X)). One might say that the RnF forget the chain complex and keep only the cohomologies, whereas RF does keep track of the complexes. Derived categories are, in a sense, the "right" place to study these functors. For example, the Grothendieck spectral sequence of a composition of two functors such that F maps injective objects in A to G-acyclics (i.e. RiG(F(I)) = 0 for all i > 0 and injective I), is an expression of the following identity of total derived functors R(G∘F) ≅ RG∘RF. J.-L.