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)
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 $\mathcal{A}$ be an abelian category. Without essential loss of generality, we may assume that $\mathcal{A} = R Mod$ is the category of modules over some ring, and we will often speak in terms of this case.
The category $\mathcal{A}$ 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 $\mathcal{A}, \mathcal{B}$ two abelian categories (e.g. $R$Mod and $R'$Mod), a functor
is called an additive functor if
$F$ maps the zero object to the zero object, $F(0) \simeq 0 \in \mathcal{B}$;
given any two objects $x, y \in \mathcal{A}$, there is an isomorphism $F(x \oplus y) \cong F(x) \oplus F(y)$, and this respects the inclusion and projection maps of the direct sum:
Given an additive functor $F : \mathcal{A} \to \mathcal{A}'$, 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 $\mathcal{A}$ and $\mathcal{A}'$ have enough projectives, then their derived categories are
and
etc. One wants to accordingly derive from $F$ a functor $\mathcal{D}_\bullet(\mathcal{A}) \to \mathcal{D}_\bullet(\mathcal{A})$ 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 $\mathcal{K}_\bullet(\mathcal{A}')$ and in general does not factor through the inclusion $\mathcal{D}_\bullet(\mathcal{A}') = \mathcal{K}_\bullet(\mathcal{P}_{\mathcal{A}'}) \hookrightarrow \mathcal{K}_\bullet(\mathcal{A}')$.
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. 4 and def. 5 below do.
Let $\mathcal{A}, \mathcal{A}'$ be two abelian categories, for instance $\mathcal{A} = R$Mod and $\mathcal{A}' = R'$Mod. Then a functor $F \colon \mathcal{A} \to \mathcal{A}'$ 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 $X \stackrel{f}{\to} Y$ in $\mathcal{A}$ 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 $F$ is a left exact functor, then for every exact sequence of the form
also
is an exact sequence. Dually, if $F$ is a right exact functor, then for every exact sequence of the form
also
is an exact sequence.
If $0 \to A \to B \to C$ is exact then $A \hookrightarrow B$ is a monomorphism. But then the statement that $A \to B \to C$ is exact at $B$ says precisely that $A$ is the kernel of $B \to C$. So if $F$ is left exact then by definition also $F(A) \to F(B)$ is the kernel of $F(B) \to F(C)$ and so is in particular also a monomorphism. Dually for right exact functors.
Proposition 1 is clearly the motivation for the terminology in def. 3: 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 $\mathcal{A}$ has enough injectives. For $n \in \mathbb{N}$ the $n$th right derived functor of $F$ is the composite
where
$P$ is an injective resolution functor;
$\mathcal{K}^\bullet(F)$ is a prolongation of $F$ analogous to def. 2;
$H^n(-)$ is the $n$-chain homology functor. Hence
Dually:
Let
be a right exact functor between abelian categories such that $\mathcal{A}$ has enough projectives. For $n \in \mathbb{N}$ the $n$th left derived functor of $F$ is the composite
where
$Q$ is a projective resolution functor;
$\mathcal{K}_\bullet(F)=\mathcal{K}(F)$ is the prolongation of $F$ according to def. 2;
$H_n(-)$ is the $n$-chain homology functor. Hence
The following proposition says that in degree 0 these derived functors coincide with the original functors.
Let $F \colon \mathcal{A} \to \mathcal{B}$ a left exact functor, def. 3 in the presence of enough injectives. Then for all $X \in \mathcal{A}$ there is a natural isomorphism
Dually, if $F$ 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 $X \stackrel{\simeq_{qi}}{\to} X^\bullet$ is equivalently an exact sequence of the form
If $F$ 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 $F$ be an additive functor.
If $F$ is right exact and $N \in \mathcal{A}$ is a projective object, then
If $F$ is left exact and $N \in \mathcal{A}$ is a injective object, then
If $N$ is projective then the chain complex $[\cdots \to 0 \to 0 \to N]$ is already a projective resolution and hence by definition $L_n F(N) \simeq H_n(0)$ for $n \geq 1$. Dually if $N$ is an injective object.
For proving the basic property of derived functors below in prop. 4 which continues these basis statements to higher degree, in a certain way, we need the following technical lemma.
For $0 \to A \stackrel{i}{\to} B \stackrel{p}{\to} C \to 0$ 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. \ref{ExistenceOfInjectiveResolutions} we can choose $f_\bullet$ and $h_\bullet$. The task is now to construct the third resolution $g_\bullet$ 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 $n \in \mathbb{N}$
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 $C_0$ (since $C_\bullet$ is a projective resolution) against the epimorphism $p : B \to C$ of the short exact sequence.
In total this gives in degree 0
Let then the differentials of $B_\bullet$ be given by
where the $\{e_k\}$ are constructed by induction as follows. Let $e_0$ be a lift in
which exists since $C_1$ is a projective object and $A_0 \to A$ is an epimorphism by $A_\bullet$ being a projective resolution. Here we are using that by exactness the bottom morphism indeed factors through $A$ as indicated, because the definition of $\zeta$ and the chain complex property of $C_\bullet$ gives
Now in the induction step, assuming that $e_{n-1}$ has been been found satisfying the chain complex property, let $e_n$ be a lift in
which again exists since $C_{n+1}$ is projective. That the bottom morphism factors as indicated is the chain complex property of $e_{n-1}$ inside $d^{B_\bullet}_{n-1}$.
To see that the $d^{B_\bullet}$ defines this way indeed squares to 0 notice that
This vanishes by the very commutativity of the above diagram.
This establishes $g_\bullet$ 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 $g_\bullet$ is indeed a quasi-isomorphism, consider the homology long exact sequence associated to the short exact sequence of cochain complexes $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$. In positive degrees it implies that the chain homology of $B_\bullet$ indeed vanishes. In degree 0 it gives the short sequence $0 \to A \to H_0(B_\bullet) \to B\to 0$ 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 1 is the following.
For $0 \to A \to B \to C \to 0$ 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 $\mathcal{A}, \mathcal{B}$ be abelian categories and assume that $\mathcal{A}$ has enough injectives.
Let $F : \mathcal{A} \to \mathcal{B}$ be a left exact functor and let
be a short exact sequence in $\mathcal{A}$.
Then there is a long exact sequence of images of these objects under the right derived functors $R^\bullet F(-)$ of def. 4
in $\mathcal{B}$.
By lemma 2 we can find an injective resolution
of the given exact sequence which is itself again an exact sequence of cochain complexes.
Since $A^n$ is an injective object for all $n$, its component sequences $0 \to A^n \to B^n \to C^n \to 0$ are indeed split exact sequences (see the discussion there). Splitness is preserved by any functor $F$ (and also since $F$ is additive it even preserves the direct sum structure that is chosen in the proof of lemma 1) and so it follows that
is a again short exact sequence of cochain complexes, now in $\mathcal{B}$. Hence we have the corresponding homology long exact sequence from prop. \ref{HomologyLongExactSequence}:
By construction of the resolutions and by def. 4, this is equal to
Finally the equivalence of the first three terms with $F(A) \to F(B) \to F(C)$ is given by prop. 2.
Prop. 4 implies that one way to interpret $R^1 F(A)$ is as a “measure for how a left exact functor $F$ fails to be an exact functor”. For, with $A \to B \to C$ any short exact sequence, this proposition gives the exact sequence
and hence $0 \to F(A) \to F(B) \to F(C) \to 0$ is a short exact sequence itself precisely if $R^1 F(A) \simeq 0$.
Dually, if $F$ is right exact functor, then $L_1 F (C)$ “measures how $F$ fails to be exact” for then
is an exact sequence and hence is a short exact sequence precisely if $L_1F(C) \simeq 0$.
Notice that in fact we even have the following statement (following directly from the definition).
Let $F$ 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 $Y_\bullet \to A$ is a projective resolution then also $F(Y)_\bullet$ is exact in all positive degrees, and hence $L_{n\geq 1} F(A) ) H_{n \geq}(F(Y)) = 0$. Dually for $R^n F$.
Conversely:
Let $F \colon \mathcal{A} \to \mathcal{B}$ be a left or right exact additive functor. An object $A \in \mathcal{A}$ is called an $F$-acyclic object if all positive-degree right/left derived functors of $F$ are zero on $A$.
We now discuss how the derived functor of an additive functor $F$ may also be computed not necessarily with genuine injective/projective resolutions as in def. 4, but with (just) “$F$-injective”/“$F$-projective resolutions”.
While projective resolutions in $\mathcal{A}$ are sufficient for computing every left derived functor on $Ch_\bullet(\mathcal{A})$ and injective resolutions are sufficient for computing every right derived functor on $Ch^\bullet(\mathcal{A})$, if one is interested just in a single functor $F$ 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 $F$-projective resolutions and $F$-injective resolutions, respectively, we discuss now. A special case of both are $F$-acyclic resolutions.
Let $\mathcal{A}, \mathcal{B}$ be abelian categories and let $F \colon \mathcal{A} \to \mathcal{B}$ be an additive functor.
Assume that $F$ is left exact. An additive full subcategory $\mathcal{I} \subset \mathcal{A}$ is called $F$-injective (or: consisting of $F$-injective objects) if
for every object $A \in \mathcal{A}$ there is a monomorphism $A \to \tilde A$ into an object $\tilde A \in \mathcal{I} \subset \mathcal{A}$;
for every short exact sequence $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ with $A, B \in \mathcal{I} \subset \mathcal{A}$ also $C \in \mathcal{I} \subset \mathcal{A}$;
for every short exact sequence $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ with $A\in \mathcal{I} \subset \mathcal{A}$ also $0 \to F(A) \to F(B) \to F(C) \to 0$ is a short exact sequence in $\mathcal{B}$.
And dually:
Assume that $F$ is right exact. An additive full subcategory $\mathcal{P} \subset \mathcal{A}$ is called $F$-projective (or: consisting of $F$-projective objects) if
for every object $A \in \mathcal{A}$ there is an epimorphism $\tilde A \to A$ from an object $\tilde A \in \mathcal{P} \subset \mathcal{A}$;
for every short exact sequence $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ with $B, C \in \mathcal{P} \subset \mathcal{A}$ also $A \in \mathcal{P} \subset \mathcal{A}$;
for every short exact sequence $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ with $C\in \mathcal{I} \subset \mathcal{A}$ also $0 \to F(A) \to F(B) \to F(C) \to 0$ is a short exact sequence in $\mathcal{B}$.
With the $\mathcal{I},\mathcal{P}\subset \mathcal{A}$ as above, we say:
For $A \in \mathcal{A}$,
an $F$-injective resolution of $A$ is a cochain complex $I^\bullet \in Ch^\bullet(\mathcal{I}) \subset Ch^\bullet(\mathcal{A})$ and a quasi-isomorphism
an $F$-projective resolution of $A$ is a chain complex $Q_\bullet \in Ch_\bullet(\mathcal{P}) \subset Ch^\bullet(\mathcal{A})$ and a quasi-isomorphism
Let now $\mathcal{A}$ have enough projectives / enough injectives, respectively.
For $F \colon \mathcal{A} \to \mathcal{B}$ an additive functor, let $Ac \subset \mathcal{A}$ be the full subcategory on the $F$-acyclic objects, def. 6. Then
if $F$ is left exact, then $\mathcal{I} \coloneqq Ac$ is a subcategory of $F$-injective objects;
if $F$ is right exact, then $\mathcal{P} \coloneqq Ac$ is a subcategory of $F$-projective objects.
Consider the case that $F$ is right exact. The other case works dually. Then the first condition of def. 7 is satisfied because every injective object is an $F$-acyclic object and by assumption there are enough of these.
For the second and third condition of def. 7 use that there is the long exact sequence of derived functors prop. 4
For the second condition, by assumption on $A$ and $B$ and definition of $F$-acyclic object we have $R^n F(A) \simeq 0$ and $R^n F(B) \simeq 0$ for $n \geq 1$ and hence short exact sequences
which imply that $R^n F(C)\simeq 0$ for all $n \geq 1$, hence that $C$ is acyclic.
Similarly, the third condition is equivalent to $R^1 F(A) \simeq 0$.
The $F$-projective/injective resolutions (def. 9) by acyclic objects as in example 1 are called $F$-acyclic resolutions.
Let $\mathcal{A}$ be an abelian category with enough injectives. Let $F \colon \mathcal{A} \to \mathcal{B}$ be an additive left exact functor with right derived functor $R_\bullet F$, def. 4. Finally let $\mathcal{I} \subset \mathcal{A}$ be a subcategory of $F$-injective objects, def. 7.
If a cochain complex $A^\bullet \in Ch^\bullet(\mathcal{I}) \subset Ch^\bullet(\mathcal{A})$ is quasi-isomorphic to 0,
then also $F(X^\bullet) \in Ch^\bullet(\mathcal{B})$ is quasi-isomorphic to 0
Consider the following collection of short exact sequences obtained from the long exact sequence $X^\bullet$:
and so on. Going by induction through this list and using the second condition in def. 7 we have that all the $im(d^n)$ are in $\mathcal{I}$. Then the third condition in def. 7 says that all the sequences
are exact. But this means that
is exact, hence that $F(X^\bullet)$ is quasi-isomorphic to 0.
For $A \in \mathcal{A}$ an object with $F$-injective resolution $A \stackrel{\simeq_{qi}}{\to} I_F^\bullet$, def. 9, we have for each $n \in \mathbb{N}$ an isomorphism
between the $n$th right derived functor, def. 4 of $F$ evaluated on $A$ and the cochain cohomology of $F$ applied to the $F$-injective resolution $I_F^\bullet$.
By this prop. we may also find an injective resolution $A \stackrel{\simeq_{qi}}{\to} I^\bullet$. By this prop there is a lift of the identity on $A$ to a chain map $I^\bullet_F \to I^\bullet$ such that the diagram
commutes in $Ch^\bullet(\mathcal{A})$. Therefore by the 2-out-of-3 property of quasi-isomorphisms it follows that $f$ is a quasi-isomorphism
Let $Cone(f) \in Ch^\bullet(\mathcal{A})$ be the mapping cone of $f$ and let $I^\bullet \to Cone(f)$ be the canonical chain map into it. By the explicit formulas for mapping cones, we have that
there is an isomorphism $F(Cone(f)) \simeq Cone(F(f))$;
$Cone(f) \in Ch^\bullet(\mathcal{I})\subset Ch^\bullet(\mathcal{A})$ (because $F$-injective objects are closed under direct sum).
The first implies that we have a homology exact sequence
Observe that with $f^\bullet$ a quasi-isomorphism $Cone(f^\bullet)$ is quasi-isomorphic to 0. Therefore the second item above implies with lemma 3 that also $F(Cone(f))$ 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 $F$ is right exact;
the right adjoint $G$ is left exact;
(…)
Let $F \colon \mathcal{A} \to \mathcal{B}$ be an additive right exact functor with codomain an AB5-category.
Then the left derived functors $L_n F$ preserves filtered colimits precisely if filtered colimits of projective objects in $\mathcal{A}$ are $F$-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