under construction
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.
and
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.
Throughout, let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories.
Assume that the abelian category $\mathcal{A}$ has enough projectives.
For $F : \mathcal{A} \to \mathcal{B}$ a right exact functor and $n \in \mathbb{N}$, its left derived functor is the functor
which sends an object $X \in \mathcal{A}$ with (any) projective resolution $((Q X)_\bullet \to X) \in Ch_{\bullet \geq 0}(\mathcal{A})$ to the $n$th chain homology of the chain complex $F((Q X)_\bullet) \in Ch_{\bullet}(\mathcal{B})$:
This definition is indeed independent, up to natural isomorphism, of the choice of resolution, see the discussion below.
That $q : (Q X) \to X$ is a projective resolution means that there is a chain map
which is a quasi-isomorphism and such that $(Q X)_n \in \mathcal{A}$ is a projective object for all $n \in \mathbb{N}$.
This being a quasi-isomorphism, in turn, is equivalent to
$q_0$ inducing an isomorphism $(Q X)_0/im(\partial_0^{Q X}) = H_0(Q X) \to X$;
and isomorphisms $H_n(Q X) \to 0$ for all $n \geq 1$.
This happens to be equivalent to the statement that the augmented complex
is an exact sequence, and traditionally the condition is often stated this way. But it somewhat hides the true meaning of the resolution and notably the fact that $(Q X)_0$ and $X$ are to be regared as being in the same degree. It also does not naturally generalize to the case where $X$ itself is already a chain complex not concentrated in lowest degree (discussed below).
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For $F : \mathcal{A} \to \mathcal{B}$ a left exact functor and $n \in \mathbb{N}$, its right derived functor is the functor
which sends an object $X \in \mathcal{A}$ with (any) injective resolution $Q_\bullet \in Ch_{\bullet \geq 0}(\mathcal{A})$ to the $n$th chain homology of the chain complex $F(Q_\bullet) \in Ch_{\bullet}(\mathcal{B})$
This definition is independent, up to natural isomorphism, of the choice of $P_\bullet$.
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Let $F \colon \mathcal{A} \to \mathcal{B}$ a left exact functor in the presence of enough injectives. Then for all $X \in \mathcal{A}$ there is a natural isomorphism
Dually, of $F$ is a right exact functor in the presence of enough projectives, then
We discuss the first statement, the second is formally dual.
By the discussion there, 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 excact sequence by definition of left exactness, and hence
is an exact sequence. But this means that
If
is a pair of additive adjoint functors, then
the left adjoint $F$ is right exact;
the right adjoint $G$ is left exact;
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A left derived functor $L_\bullet F$ is a universal homological delta-functor.
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. 1
in $\mathcal{B}$.
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 $A^n$ 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 a functor $F$ 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
But by construction of the resolutions and by def. 1 this is equal to
Prop. 2 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 $0 \to F(A) \to F(B) \to F(C) \to$ is a short exact sequence itself precisely if $R^1 F(A) \simeq 0$.
In fact we even have the following.
Let $F$ be an additive functor which is an exact functor. Then
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 $R^n F$.
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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.
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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