Derived Category - The Relation To Derived Functors

The Relation To Derived Functors

The derived category is a natural framework to define and study derived functors. In the following, let be a functor of abelian categories. There are two dual concepts:

• right derived functors are "deriving" 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, or 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 . It is the following composition:, where the first equivalence of categories is described above. The classical derived functors are related to the total one via . One might say that the RnF forget the chain complex and keep only the cohomologies, whereas R F 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. Verdier showed how derived functors associated with an abelian category A can be viewed as Kan extensions along embeddings of A into suitable derived categories .