nLab
derived functor in homological algebra

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.

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Definition

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 $𝒜$ and $ℬ$ be abelian categories.

Left derived functor

Assume that the abelian category $𝒜$ has enough projectives.

Evaluated on an object

Definition

For $F:𝒜\to ℬ$ a right exact functor and $n\in \mathbb{N}$, its left derived functor is the functor

$$L_{n}F:𝒜\to ℬ$$L_n F : \mathcal{A} \to \mathcal{B}

which sends an object $X\in 𝒜$ with (any) projective resolution $((QX{)}_{•}\to X)\in {\mathrm{Ch}}_{•\ge 0}(𝒜)$ to the $n$th chain homology of the chain complex $F((QX{)}_{•})\in {\mathrm{Ch}}_{•}(ℬ)$:

$$(L_{n}F)(X)≔H_n(F((QX{)}_{•})).$$(L_n F)(X) \coloneqq H_n(F((Q X)_\bullet)) \,.

This definition is indeed independent, up to natural isomorphism, of the choice of resolution, see the discussion below.

Remark

That $q:(QX)\to X$ is a projective resolution means that there is a chain map

$$\begin{array}{ccc}⋮& & ⋮\\ \downarrow & & \downarrow \\ (QX{)}_2& \stackrel{q_2}{\to }& 0\\ {\downarrow }^{{\partial }_1^{QX}}& & \downarrow \\ (QX{)}_1& \stackrel{q_1}{\to }& 0\\ {\downarrow }^{{\partial }_0^{QX}}& & \downarrow \\ (QX{)}_0& \stackrel{q_0}{\to }& X\end{array}$$\array{ \vdots && \vdots \\ \downarrow && \downarrow \\ (Q X)_2 &\stackrel{q_2}{\to}& 0 \\ \downarrow^{\mathrlap{\partial^{Q X}_1}} && \downarrow \\ (Q X)_1 &\stackrel{q_1}{\to}& 0 \\ \downarrow^{\mathrlap{\partial^{Q X}_0}} && \downarrow \\ (Q X)_0 &\stackrel{q_0}{\to}& X }

which is a quasi-isomorphism and such that $(QX{)}_n\in 𝒜$ is a projective object for all $n\in \mathbb{N}$.

This being a quasi-isomorphism, in turn, is equivalent to

1. $q_0$ inducing an isomorphism $(QX{)}_0/\mathrm{im}({\partial }_0^{QX})=H_0(QX)\to X$;

2. and isomorphisms $H_n(QX)\to 0$ for all $n\ge 1$.

This happens to be equivalent to the statement that the augmented complex

$$\cdots \stackrel{{\partial }_1^{QX}}{\to }(QX{)}_2\stackrel{{\partial }_1^{QX}}{\to }(QX{)}_1\stackrel{{\partial }_0^{QX}}{\to }(QX{)}_0\stackrel{q_0}{\to }X$$\cdots \stackrel{\partial_1^{Q X}}{\to} (Q X)_2 \stackrel{\partial_1^{Q X}}{\to} (Q X)_1 \stackrel{\partial_0^{Q X}}{\to} (Q X)_0 \stackrel{q_0}{\to} X

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 $(QX{)}_0$ 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).

Evaluated on a chain complex: left hyper-derived functor

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Right derived functor

Definition

For $F:𝒜\to ℬ$ a left exact functor and $n\in \mathbb{N}$, its right derived functor is the functor

$$R_{n}F:𝒜\to ℬ$$R_n F : \mathcal{A} \to \mathcal{B}

which sends an object $X\in 𝒜$ with (any) injective resolution $Q_{•}\in {\mathrm{Ch}}_{•\ge 0}(𝒜)$ to the $n$th chain homology of the chain complex $F(Q_{•})\in {\mathrm{Ch}}_{•}(ℬ)$

$$(R_{n}F)(X)≔H_n(F(Q_{•})).$$(R_n F)(X) \coloneqq H_n(F(Q_\bullet)) \,.

This definition is independent, up to natural isomorphism, of the choice of $P_{•}$.

Left hyper-derived functor

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Right hyper-derived functor

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Total and hyper-derived functor

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Properties

In degree 0

Proposition

Let $F:𝒜\to ℬ$ a left exact functor in the presence of enough injectives. Then for all $X\in 𝒜$ there is a natural isomorphism

$$R^{0}F(X)\simeq F(X).$$R^0F(X) \simeq F(X) \,.

Dually, of $F$ is a right exact functor in the presence of enough projectives, then

$$L_{0}F(X)\simeq F(X).$$L_0 F(X) \simeq F(X) \,.

Proof

We discuss the first statement, the second is formally dual.

By the discussion there, an injective resolution $X\stackrel{{\simeq }_{\mathrm{qi}}}{\to }{X}^{•}$ is equivalently an exact sequence of the form

$$0\to X\hookrightarrow X^0\to X^1\to \cdots .$$0 \to X \hookrightarrow X^0 \to X^1 \to \cdots \,.

If $F$ is left exact then it preserves this excact sequence by definition of left exactness, and hence

$$0\to F(X)\hookrightarrow F(X^0)\to F(X^1)\to \cdots$$0 \to F(X) \hookrightarrow F(X^0) \to F(X^1) \to \cdots

is an exact sequence. But this means that

$$R^{0}F(X)≔\ker (F(X^0)\to F(X^1))\simeq F(X).$$R^0 F(X) \coloneqq ker(F(X^0) \to F(X^1)) \simeq F(X) \,.

Derived adjoint functors

Observation

If

$$(F⊣G):𝒜\stackrel{\stackrel{G}{\leftarrow }}{\underset{F}{\to }}ℬ$$(F \dashv G) : \mathcal{A} \stackrel{\overset{G}{\leftarrow}}{\underset{F}{\to}} \mathcal{B}

is a pair of additive adjoint functors, then

• the left adjoint $F$ is right exact;

• the right adjoint $G$ is left exact;

(…)

General

Theorem

A left derived functor $L_{•}F$ is a universal homological delta-functor.

Long exact sequence of a derived functor

Proposition

Let $𝒜,ℬ$ be abelian categories and assume that $𝒜$ has enough injectives.

Let $F:𝒜\to ℬ$ be a left exact functor and let

$$0\to A\to B\to C\to 0$$0 \to A \to B \to C \to 0

be a short exact sequence in $𝒜$.

Then there is a long exact sequence of images of these objects under the right derived functors $R^{•}F(-)$ of def. 1

$$0\to R^{0}F(A)=F(A)\to R^{0}F(B)=F(B)\to R^{0}F(C)\stackrel{{\delta }_0}{\to }{R}^{1}F(A)\to R^{1}F(B)\to R^{1}F(C)\stackrel{{\delta }_1}{\to }{R}^{2}F(A)\to \cdots$$0 \to R^0F (A) = F(A) \to R^0 F(B) = F(B) \to R^0 F(C) \stackrel{\delta_0}{\to} R^1 F(A) \to R^1 F(B) \to R^1F(C) \stackrel{\delta_1}{\to} R^2 F(A) \to \cdots

in $ℬ$.

Proof

By this lemma at injective resolution we can find an injective resolution

$$0\to A^{•}\to B^{•}\to C^{•}\to 0$$0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0

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

$$0\to F({\tilde{A}}^{•})\to F({\tilde{B}}^{•})\to F({\tilde{C}}^{•})\to 0$$0 \to F(\tilde A^\bullet) \to F(\tilde B^\bullet) \to F(\tilde C^\bullet) \to 0

is a again short exact sequence of cochain complexes, now in $ℬ$. Hence we have the corresponding homology long exact sequence

$$\cdots \to H^{n-1}(F(A^{•}))\to H^{n-1}(F(B^{•}))\to H^{n-1}(F(C^{•}))\stackrel{\delta }{\to }{H}^n(F(A^{•}))\to H^n(F(B^{•}))\to H^n(F(C^{•}))\stackrel{\delta }{\to }{H}^{n+1}(F(A^{•}))\to H^{n+1}(F(B^{•}))\to H^{n+1}(F(C^{•}))\to \cdots .$$\cdots \to H^{n-1}(F(A^\bullet)) \to H^{n-1}(F(B^\bullet)) \to H^{n-1}(F(C^\bullet)) \stackrel{\delta}{\to} H^n(F(A^\bullet)) \to H^n(F(B^\bullet)) \to H^n(F(C^\bullet)) \stackrel{\delta}{\to} H^{n+1}(F(A^\bullet)) \to H^{n+1}(F(B^\bullet)) \to H^{n+1}(F(C^\bullet)) \to \cdots \,.

But by construction of the resolutions and by def. 1 this is equal to

$$\cdots \to R^{n-1}F(A)\to R^{n-1}F(B)\to R^{n-1}F(C)\stackrel{\delta }{\to }{R}^{n}F(A)\to R^{n}F(B)\to R^{n}F(C)\stackrel{\delta }{\to }{R}^{n+1}F(A)\to R^{n+1}F(B)\to R^{n+1}F(C)\to \cdots .$$\cdots \to R^{n-1}F(A) \to R^{n-1}F(B) \to R^{n-1}F(C) \stackrel{\delta}{\to} R^{n}F(A) \to R^{n}F(B) \to R^{n}F(C) \stackrel{\delta}{\to} R^{n+1}F(A) \to R^{n+1}F(B) \to R^{n+1}F(C) \to \cdots \,.

Remark

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

$$0\to F(A)\to F(B)\to F(C)\to R^{1}F(A)$$0 \to F(A) \to F(B) \to F(C) \to R^1 F(A)

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

$$R^{\ge 1}F=0$$R^{\geq 1} F = 0

and

$$L_{\ge 1}F=0.$$L_{\geq 1} F = 0 \,.

Proof

Because an exact functor preserves all exact sequences. If $Y_{•}\to A$ is a projective resolution then also $F(Y{)}_{•}$ is exact in all positive degrees, and hence $L_{n\ge 1}F(A))H_{n\ge }(F(Y))=0$. Dually for $R^{n}F$.

Relation to derived categories

(…) derived category (…)

Preservation of limits and colimits

Proposition

Let $F:𝒜\to ℬ$ 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 $𝒜$ are $F$-acyclic objects.

See here.

Examples

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References

A standard textbook introduction is chapter 2 of

• Charles Weibel, An Introduction to Homological Algebra.

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)

and section 13 of

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006)