This entry discusses the general notion of derived functor specified to the special context of homological algebra, hence to 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.
(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
Theorems
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.
Let be an abelian category. Without essential loss of generality, we may assume that is the category of modules over some ring, and we will often speak in terms of this case.
The category embeds into its derived category, the category of degreewise injective cochain complexes
or degreewise projective chain complexes
modulo chain homotopy. This construction of the derived category naturally gives rise to the following notion of derived functors, which we now discuss.
For two abelian categories (e.g. Mod and Mod), a functor
is called an additive functor if
maps the zero object to the zero object, ;
given any two objects , there is an isomorphism , and this respects the inclusion and projection maps of the direct sum:
Given an additive functor , it canonically induces a functor
between categories of chain complexes (its “prolongation”) by applying it to each chain complex and to all the diagrams in the definition of a chain map. Similarly it preserves chain homotopies and hence it passes to the quotient given by the strong homotopy category of chain complexes
If and have enough projectives, then their derived categories are
and
etc. One wants to accordingly derive from a functor between these derived categories. It is immediate to achieve this on the domain category, there we can simply precompose and form
But the resulting composite lands in and in general does not factor through the inclusion .
In a more general abstract discussion than we present here, one finds that by applying a projective resolution functor on chain complexes, one can enforce this factorization. However, by definition of resolution, the resulting chain complex is quasi-isomorphic to the one obtained by the above composite.
This means that if one is only interested in the “weak chain homology type” of the chain complex in the image of a derived functor, then forming chain homology groups of the chain complexes in the images of the above composite gives the desired information. This is what def. and def. below do.
Let be two abelian categories, for instance Mod and Mod. Then a functor which preserves direct sums (and hence in particular the zero object) is called
a left exact functor if it preserves kernels;
a right exact functor if it preserves cokernels;
an exact functor if it is both left and right exact.
Here to “preserve kernels” means that for every morphism in we have an isomorphism on the left of the following commuting diagram
hence that both rows are exact. And dually for right exact functors.
We record the following immediate consequence of this definition (which in the literature is often taken to be the definition).
If is a left exact functor, then for every exact sequence of the form
also
is an exact sequence. Dually, if is a right exact functor, then for every exact sequence of the form
also
is an exact sequence.
If is exact then is a monomorphism. But then the statement that is exact at says precisely that is the kernel of . So if is left exact then by definition also is the kernel of and so is in particular also a monomorphism. Dually for right exact functors.
Proposition is clearly the motivation for the terminology in def. : a functor is left exact if it preserves short exact sequences to the left, and right exact if it preserves them to the right.
Now we can state the main two definitions of this section.
Let
be a left exact functor between abelian categories such that has enough injectives. For the th right derived functor of is the composite
where
is an injective resolution functor;
is the -chain homology functor. Hence
Dually:
Let
be a right exact functor between abelian categories such that has enough projectives. For the th left derived functor of is the composite
where
is a projective resolution functor;
is the -chain homology functor. Hence
The following proposition says that in degree 0 these derived functors coincide with the original functors.
Let a left exact functor, def. in the presence of enough injectives. Then for all there is a natural isomorphism
Dually, if is a right exact functor in the presence of enough projectives, then
We discuss the first statement, the second is formally dual.
By this remark an injective resolution is equivalently an exact sequence of the form
If is left exact then it preserves this exact sequence by definition of left exactness, and hence
is an exact sequence. But this means that
The following immediate consequence of the definition is worth recording:
Let be an additive functor.
If is right exact and is a projective object, then
If is left exact and is a injective object, then
If is projective then the chain complex is already a projective resolution and hence by definition for . Dually if is an injective object.
For proving the basic property of derived functors below in prop. which continues these basis statements to higher degree, in a certain way, we need the following technical lemma.
For a short exact sequence in an abelian category with enough projectives, there exists a commuting diagram of chain complexes
where
and in addition
By prop. we can choose and . The task is now to construct the third resolution such as to obtain a short exact sequence of chain complexes, hence degreewise a short exact sequence, in the two row.
To construct this, let for each
be the direct sum and let the top horizontal morphisms be the canonical inclusion and projection maps of the direct sum.
Let then furthermore (in matrix calculus notation)
be given in the first component by the given composite
and in the second component we take
to be given by a lift in
which exists by the left lifting property of the projective object (since is a projective resolution) against the epimorphism of the short exact sequence.
In total this gives in degree 0
Let then the differentials of be given by
where the are constructed by induction as follows. Let be a lift in
which exists since is a projective object and is an epimorphism by being a projective resolution. Here we are using that by exactness the bottom morphism indeed factors through as indicated, because the definition of and the chain complex property of gives
Now in the induction step, assuming that has been been found satisfying the chain complex property, let be a lift in
which again exists since is projective. That the bottom morphism factors as indicated is the chain complex property of inside .
To see that the defines this way indeed squares to 0 notice that
This vanishes by the very commutativity of the above diagram.
This establishes such that the above diagram commutes and the bottom row is degreewise a short exact sequence, in fact a split exact sequence, by construction.
To see that is indeed a quasi-isomorphism, consider the homology long exact sequence associated to the short exact sequence of cochain complexes . In positive degrees it implies that the chain homology of indeed vanishes. In degree 0 it gives the short sequence sitting in a commuting diagram
where both rows are exact. That the middle vertical morphism is an isomorphism then follows by the five lemma.
The formally dual statement to lemma is the following.
For a short exact sequence in an abelian category with enough injectives, there exists a commuting diagram of cochain complexes
where
and in addition
The central general fact about derived functors to be discussed here is now the following.
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.
in .
By lemma 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 any functor (and also since is additive it even preserves the direct sum structure that is chosen in the proof of lemma ) 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 from prop. :
By construction of the resolutions and by def. , this is equal to
Finally the equivalence of the first three terms with is given by prop. .
Prop. implies that one way to interpret is as a “measure for how a left exact functor fails to be an exact functor”. For, with any short exact sequence, this proposition gives the exact sequence
and hence is a short exact sequence itself precisely if .
Dually, if is right exact functor, then “measures how fails to be exact” for then
is an exact sequence and hence is a short exact sequence precisely if .
Notice that in fact we even have the following statement (following directly from the definition).
Let be an additive functor which is an exact functor. Then its left and right derived functors vanish in positive degree:
and
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 .
Conversely:
Let be a left or right exact additive functor. An object is called an -acyclic object if all positive-degree right/left derived functors of are zero on .
We now discuss how the derived functor of an additive functor may also be computed not necessarily with genuine injective/projective resolutions as in def. , but with (just) “-injective”/“-projective resolutions”.
While projective resolutions in are sufficient for computing every left derived functor on and injective resolutions are sufficient for computing every right derived functor on , if one is interested just in a single functor then such resolutions may be more than necessary. A weaker kind of resolution which is still sufficient is then often more convenient for applications. These -projective resolutions and -injective resolutions, respectively, we discuss now. A special case of both are -acyclic resolutions.
Let be abelian categories and let be an additive functor.
Assume that is left exact. An additive full subcategory is called -injective (or: consisting of -injective objects) if
for every object there is a monomorphism into an object ;
for every short exact sequence in with also ;
for every short exact sequence in with also is a short exact sequence in .
And dually:
Assume that is right exact. An additive full subcategory is called -projective (or: consisting of -projective objects) if
for every object there is an epimorphism from an object ;
for every short exact sequence in with also ;
for every short exact sequence in with also is a short exact sequence in .
With the as above, we say:
For ,
an -injective resolution of is a cochain complex and a quasi-isomorphism
an -projective resolution of is a chain complex and a quasi-isomorphism
Let now have enough projectives / enough injectives, respectively.
For an additive functor, let be the full subcategory on the -acyclic objects, def. . Then
if is left exact, then is a subcategory of -injective objects;
if is right exact, then is a subcategory of -projective objects.
Consider the case that is left exact. The other case works dually. Then the first condition of def. is satisfied because every injective object is an -acyclic object and by assumption there are enough of these.
For the second and third condition of def. use that there is the long exact sequence of derived functors prop.
For the second condition, by assumption on and and definition of -acyclic object we have and for and hence short exact sequences
which imply that for all , hence that is acyclic.
Similarly, the third condition is equivalent to .
The -projective/injective resolutions (def. ) by acyclic objects as in example are called -acyclic resolutions.
Let be an abelian category with enough injectives. Let be an additive left exact functor with right derived functor , def. . Finally let be a subcategory of -injective objects, def. .
Consider the following collection of short exact sequences obtained from the long exact sequence :
and so on. Going by induction through this list and using the second condition in def. we have that all the are in . Then the third condition in def. says that all the sequences
are exact. But this means that
is exact, hence that is quasi-isomorphic to 0.
For an object with -injective resolution , def. , we have for each an isomorphism
between the th right derived functor, def. of evaluated on and the cochain cohomology of applied to the -injective resolution .
By this prop. we may also find an injective resolution . By this prop there is a lift of the identity on to a chain map such that the diagram
commutes in . Therefore by the 2-out-of-3 property of quasi-isomorphisms it follows that is a quasi-isomorphism
Let be the mapping cone of and let be the canonical chain map into it. By the explicit formulas for mapping cones, we have that
there is an isomorphism ;
(because -injective objects are closed under direct sum).
The first implies that we have a homology exact sequence
Observe that with a quasi-isomorphism is quasi-isomorphic to 0. Therefore the second item above implies with lemma that also is quasi-isomorphic to 0. This finally means that the above homology exact sequences consists of exact pieces of the form
If
is a pair of additive adjoint functors, then
the left adjoint is right exact;
the right adjoint is left exact;
(…)
Let be an additive right exact functor with codomain an AB5-category.
Then the left derived functors preserves filtered colimits precisely if filtered colimits of projective objects in are -acyclic objects.
See here.
(…)
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
Pierre Schapira, Categories and homological algebra (2011) (pdf)
Masaki Kashiwara, Pierre Schapira, section 13 of Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006)
The above text is taken from
Last revised on February 6, 2024 at 16:50:26. See the history of this page for a list of all contributions to it.