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exact sequence

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Conntents

Definition

Let π’œ\mathcal{A} be an additive category (often assumed to be an abelian category, for instance π’œ=R\mathcal{A} = RMod for RR some ring).

Definition

An exact sequence in π’œ\mathcal{A} is a chain complex C β€’C_\bullet in π’œ\mathcal{A} with vanishing chain homology in each degree:

βˆ€nβˆˆβ„•.H n(C)=0. \forall n \in \mathbb{N} . H_n(C) = 0 \,.
Definition

A short exact sequence is an exact sequence, def. 1 of the form

⋯→0→0→A→B→C→0→0→⋯. \cdots \to 0 \to 0 \to A \to B \to C \to 0 \to 0 \to \cdots \,.

One usually writes this just β€œ0β†’Aβ†’Bβ†’Cβ†’00 \to A \to B \to C \to 0” or even just β€œAβ†’Bβ†’CA \to B \to C”.

Remark

A general exact sequence is sometimes called a long exact sequence, to distinguish from the special case of a short exact sequence.

Proposition

Explicitly, a sequence of morphisms

0→A→iB→pC→0 0 \to A \stackrel{i}\to B \stackrel{p}\to C \to 0

is short exact, def. 2, precisely if

  1. ii is a monomorphism,

  2. pp is an epimorphism,

  3. and the image of ii equals the kernel of pp (equivalently, the coimage of pp equals the cokernel of ii).

Proof

The third condition is the definition of exactness at BB. So we need to show that the first two conditions are equivalent to exactness at AA and at CC.

This is easy to see by looking at elements when π’œβ‰ƒR\mathcal{A} \simeq RMod, for some ring RR (and the general case can be reduced to this one using one of the embedding theorems):

The sequence being exact exact at

0→A→B 0 \to A \to B

means, since the image of 0β†’A0 \to A is just the element 0∈A0 \in A, that the kernel of Aβ†’BA \to B consists of just this element. But since Aβ†’BA \to B is a group homomorphism, this means equivalently that Aβ†’BA \to B is an injection.

Dually, the sequence being exact at

B→C→0 B \to C \to 0

means, since the kernel of C→0C \to 0 is all of CC, that also the image of B→CB \to C is all of CC, hence equivalently that B→CB \to C is a surjection.

Definition

A split exact sequence is a short exact sequence as above in which ii is a split monomorphism, or (equivalently) in which pp is a split epimorphism.

In this case, BB may be decomposed as the biproduct AβŠ•CA \oplus C (with ii and pp the usual biproduct inclusion and projection); this sense in which BB is β€˜split’ into AA and CC is the origin of the general terms β€˜split (mono/epi)morphism’.

Properties

Computing terms in an exact sequence

A typical use of a long exact sequence, notably of the homology long exact sequence, is that it allows to determine some of its entries in terms of others.

The characterization of short exact sequences in prop. 1 is one example for this: whenever in a long exact sequence one entry vanishes as in ⋅→0→C n→⋅\cdot \to 0 \to C_n \to \cdot or ⋅→C n→0→⋯\cdot \to C_n \to 0 \to \cdots, it follows that the next morphism out of or into the vanishing entry is a monomorphism or epimorphism, respectively.

In particular:

Proposition

If part of an exact sequence looks like

β‹―β†’0β†’C n+1β†’βˆ‚ nC nβ†’0β†’β‹―, \cdots \to 0 \to C_{n+1} \stackrel{\partial_n}{\to} C_n \to 0 \to \cdots \,,

then βˆ‚ n\partial_n is an isomorphism and hence

C n+1≃C n. C_{n+1} \simeq C_n \,.

Exactness and quasi-isomorphisms

Proposition

A chain complex C β€’C_\bullet is exact (is a long exact sequence), precisely if the unique chain map from the initial object, the 0-complex

0β†’C β€’ 0 \to C_\bullet

is a quasi-isomorphism.

Short exact sequences and quotients

The following are some basic lemmas that show how given a short exact sequence one obtains new short exact sequences from forming quotients/cokernels (see Wise).

Let π’œ\mathcal{A} be an abelian category.

Lemma

For

A→B→C→0 A \to B \to C \to 0

an exact sequence in π’œ\mathcal{A} and for Xβ†’BX \to B any morphism in π’œ\mathcal{A}, also

A→B/X→C/X→0 A \to B/X \to C/X \to 0

is a short exaxt sequence.

Proof

We have an exact sequence of complexes of lenght 2

0 β†’ X β†’id X β†’ 0 ↓ ↓ ↓ ↓ A β†’ B β†’ C β†’ 0 \array{ 0 &\to& X &\stackrel{id}{\to}& X &\to& 0 \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ A &\to& B &\to& C &\to& 0 }

and the exact sequence to be demonstrated is degreewise the cokernel of this sequence. So the statement reduces to the fact that forming cokernels is a right exact functor.

Lemma

For

0→A→B→C 0 \to A \to B \to C

an exact sequence and X→AX \to A any morphism, also

0→A/X→B/X→C 0 \to A/X \to B/X \to C

is exact.

Examples

Specific examples

Example

Let π’œ=β„€\mathcal{A} = \mathbb{Z}Mod ≃\simeq Ab. For nβˆˆβ„•n \in \mathbb{N} with nβ‰₯1n \geq 1 let β„€β†’β‹…nβ„€\mathbb{Z} \stackrel{\cdot n}{\to} \mathbb{Z} be the linear map/homomorphism of abelian groups which acts by the ordinary multiplication of integers by nn. This is clearly an injection. The cokernel of this morphism is the projection to the quotient group, which is the cyclic group β„€ n≔℀/nβ„€\mathbb{Z}_n \coloneqq \mathbb{Z}/n\mathbb{Z}. Hence we have a short exact sequence

0β†’β„€β†’β‹…nβ„€β†’β„€ n. 0 \to \mathbb{Z} \stackrel{\cdot n}{\to} \mathbb{Z} \to \mathbb{Z}_n \,.
Remark

The connecting homomorphism of the long exact sequence in homology induced from short exact sequences of the form in example 1 is called a Bockstein homomorphism.

Classes of examples

References

A standard introduction is for instance in section 1.1 of

The quotient lemmas from above are discussed in

in the context of the salamander lemma and the snake lemma.

Revised on May 26, 2014 03:13:34 by Urs Schreiber (31.55.33.159)