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.


Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

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\mathcal{A} and \mathcal{B} be abelian categories.

Left derived functor

Assume that the abelian category 𝒜\mathcal{A} has enough projectives.

Evaluated on an object

Definition

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

L nF:𝒜 L_n F : \mathcal{A} \to \mathcal{B}

which sends an object X𝒜X \in \mathcal{A} with (any) projective resolution ((QX) X)Ch 0(𝒜)((Q X)_\bullet \to X) \in Ch_{\bullet \geq 0}(\mathcal{A}) to the nnth chain homology of the chain complex F((QX) )Ch ()F((Q X)_\bullet) \in Ch_{\bullet}(\mathcal{B}):

(L nF)(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)Xq : (Q X) \to X is a projective resolution means that there is a chain map

(QX) 2 q 2 0 1 QX (QX) 1 q 1 0 0 QX (QX) 0 q 0 X \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) n𝒜(Q X)_n \in \mathcal{A} is a projective object for all nn \in \mathbb{N}.

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

  1. q 0q_0 inducing an isomorphism (QX) 0/im( 0 QX)=H 0(QX)X(Q X)_0/im(\partial_0^{Q X}) = H_0(Q X) \to X;

  2. and isomorphisms H n(QX)0H_n(Q X) \to 0 for all n1n \geq 1.

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

1 QX(QX) 2 1 QX(QX) 1 0 QX(QX) 0q 0X \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(Q X)_0 and XX are to be regared as being in the same degree. It also does not naturally generalize to the case where XX itself is already a chain complex not concentrated in lowest degree (discussed below).

Evaluated on a chain complex: left hyper-derived functor

(…)

Right derived functor

Definition

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

R nF:𝒜 R_n F : \mathcal{A} \to \mathcal{B}

which sends an object X𝒜X \in \mathcal{A} with (any) injective resolution Q Ch 0(𝒜)Q_\bullet \in Ch_{\bullet \geq 0}(\mathcal{A}) to the nnth chain homology of the chain complex F(Q )Ch ()F(Q_\bullet) \in Ch_{\bullet}(\mathcal{B})

(R nF)(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 P_\bullet.

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:𝒜F \colon \mathcal{A} \to \mathcal{B} a left exact functor in the presence of enough injectives. Then for all X𝒜X \in \mathcal{A} there is a natural isomorphism

R 0F(X)F(X). R^0F(X) \simeq F(X) \,.

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

L 0F(X)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 qiX X \stackrel{\simeq_{qi}}{\to} X^\bullet is equivalently an exact sequence of the form

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

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

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

is an exact sequence. But this means that

R 0F(X)ker(F(X 0)F(X 1))F(X). R^0 F(X) \coloneqq ker(F(X^0) \to F(X^1)) \simeq F(X) \,.

Derived adjoint functors

Observation

If

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

is a pair of additive adjoint functors, then

(…)

General

Theorem

A left derived functor L FL_\bullet F is a universal homological delta-functor.

Long exact sequence of a derived functor

Proposition

Let 𝒜,\mathcal{A}, \mathcal{B} be abelian categories and assume that 𝒜\mathcal{A} has enough injectives.

Let F:𝒜F : \mathcal{A} \to \mathcal{B} be a left exact functor and let

0ABC0 0 \to A \to B \to C \to 0

be a short exact sequence in 𝒜\mathcal{A}.

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

0R 0F(A)=F(A)R 0F(B)=F(B)R 0F(C)δ 0R 1F(A)R 1F(B)R 1F(C)δ 1R 2F(A) 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 \mathcal{B}.

Proof

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

0A B C 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 nA^n is an injective object for all nn, its component sequences 0A nB nC n00 \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 FF and so it follows that

0F(A˜ )F(B˜ )F(C˜ )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 \mathcal{B}. Hence we have the corresponding homology long exact sequence

H n1(F(A ))H n1(F(B ))H n1(F(C ))δH n(F(A ))H n(F(B ))H n(F(C ))δH n+1(F(A ))H n+1(F(B ))H n+1(F(C )). \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

R n1F(A)R n1F(B)R n1F(C)δR nF(A)R nF(B)R nF(C)δR n+1F(A)R n+1F(B)R n+1F(C). \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 1F(A)R^1 F(A) is as a “measure for how a left exact functor FF fails to be an exact functor”. For, with ABCA \to B \to C any short exact sequence, this proposition gives the exact sequence

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

and hence 0F(A)F(B)F(C)0 \to F(A) \to F(B) \to F(C) \to is a short exact sequence itself precisely if R 1F(A)0R^1 F(A) \simeq 0.

In fact we even have the following.

Let FF be an additive functor which is an exact functor. Then

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

and

L 1F=0. L_{\geq 1} F = 0 \,.
Proof

Because an exact functor preserves all exact sequences. If Y AY_\bullet \to A is a projective resolution then also F(Y) F(Y)_\bullet is exact in all positive degrees, and hence L n1F(A))H n(F(Y))=0L_{n\geq 1} F(A) ) H_{n \geq}(F(Y)) = 0. Dually for R nFR^n F.

Relation to derived categories

(…) derived category (…)

Preservation of limits and colimits

Proposition

Let F:𝒜F \colon \mathcal{A} \to \mathcal{B} be an additive right exact functor with codomain an AB5-category.

Then the left derived functors L nFL_n F preserves filtered colimits precisely if filtered colimits of projective objects in 𝒜\mathcal{A} are FF-acyclic objects.

See here.

Examples

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References

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

Revised on October 18, 2012 11:27:56 by Urs Schreiber (82.169.65.155)