This entry discusses the general notion of derived functor in the context of homological algebra, hence for functors between categories of chain complexes. In the literature this is often understood to be the default case of derived functors. For more discussion of how the following relates to the more general concepts of derived functors see at derived functor – In homological algebra.
The general concept of derived functor is in homological algebra usually called the total hyper-derived functor, with just “derived functor” being reserved for a more restrictive case. In this tradition, we consider the special case first and then generalize it in stages. The relation between all these notions is discussed below.
Throughout, let and be abelian categories.
Left derived functor
Assume that the abelian category has enough projectives.
Evaluated on an object
For a right exact functor and , its left derived functor is the functor
which sends an object with (any) projective resolution to the th chain homology of the chain complex :
This definition is indeed independent, up to natural isomorphism, of the choice of resolution, see the discussion below.
Evaluated on a chain complex: left hyper-derived functor
Right derived functor
For a left exact functor and , its right derived functor is the functor
which sends an object with (any) injective resolution to the th chain homology of the chain complex
This definition is independent, up to natural isomorphism, of the choice of .
Left hyper-derived functor
Right hyper-derived functor
Total and hyper-derived functor
In degree 0
We discuss the first statement, the second is formally dual.
By the discussion there, an injective resolution is equivalently an exact sequence of the form
If is left exact then it preserves this excact sequence by definition of left exactness, and hence
is an exact sequence. But this means that
Derived adjoint functors
is a pair of additive adjoint functors, then
A left derived functor is a universal homological delta-functor.
Long exact sequence of a derived functor
Let be abelian categories and assume that has enough injectives.
Let be a left exact functor and let
be a short exact sequence in .
Then there is a long exact sequence of images of these objects under the right derived functors of def. 1
By this lemma at injective resolution we can find an injective resolution
of the given exact sequence which is itself again an exact sequence of cochain complexes.
Since is an injective object for all , its component sequences are indeed split exact sequences (see the discussion there). Splitness is preserved by a functor and so it follows that
is a again short exact sequence of cochain complexes, now in . Hence we have the corresponding homology long exact sequence
But by construction of the resolutions and by def. 1 this is equal to
In fact we even have the following.
Because an exact functor preserves all exact sequences. If is a projective resolution then also is exact in all positive degrees, and hence . Dually for .
Relation to derived categories
(…) derived category (…)
Preservation of limits and colimits
A standard textbook introduction is chapter 2 of
A systematic discussion from the point of view of homotopy theory and derived categories is in chapter 7 of
and section 13 of