exact sequence


Category theory

Homological algebra

homological algebra


nonabelian homological algebra


Basic definitions

Stable homotopy theory notions



diagram chasing

Homology theories




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


An exact sequence in 𝒜\mathcal{A} is a chain complex C C_\bullet in 𝒜\mathcal{A} with vanishing chain homology in each degree:

n.H n(C)=0. \forall n \in \mathbb{N} . H_n(C) = 0 \,.

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

00ABC00. \cdots \to 0 \to 0 \to A \to B \to C \to 0 \to 0 \to \cdots \,.

One usually writes this just “0ABC00 \to A \to B \to C \to 0” or even just “ABCA \to B \to C”.


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


Explicitly, a sequence of morphisms

0AiBpC0 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).


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 𝒜R\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

0AB 0 \to A \to B

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

Dually, the sequence being exact at

BC0 B \to C \to 0

means, since the kernel of C0C \to 0 is all of CC, that also the image of BCB \to C is all of CC, hence equivalently that BCB \to C is a surjection.


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 ACA \oplus C (with ii and pp the usual biproduct inclusion and projection); this sense in which BB is ‘split’ into AA and CC is the origin of the general terms ‘split (mono/epi)morphism’.


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 0C n\cdot \to 0 \to C_n \to \cdot or C n0\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:


If part of an exact sequence looks like

0C n+1 nC n0, \cdots \to 0 \to C_{n+1} \stackrel{\partial_n}{\to} C_n \to 0 \to \cdots \,,

then n\partial_n is an isomorphism and hence

C n+1C n. C_{n+1} \simeq C_n \,.

Exactness and quasi-isomorphisms


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

0C 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 𝒜\mathcal{A} be an abelian category.



ABC0 A \to B \to C \to 0

an exact sequence in 𝒜\mathcal{A} and for XBX \to B any morphism in 𝒜\mathcal{A}, also

AB/XC/X0 A \to B/X \to C/X \to 0

is a short exaxt sequence.


We have an exact sequence of complexes of lenght 2

0 X id X 0 A B C 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.



0ABC 0 \to A \to B \to C

an exact sequence and XAX \to A any morphism, also

0A/XB/XC 0 \to A/X \to B/X \to C

is exact.


Specific examples


Let 𝒜=\mathcal{A} = \mathbb{Z}Mod \simeq Ab. For nn \in \mathbb{N} with n1n \geq 1 let n\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 n/n\mathbb{Z}_n \coloneqq \mathbb{Z}/n\mathbb{Z}. Hence we have a short exact sequence

0n n. 0 \to \mathbb{Z} \stackrel{\cdot n}{\to} \mathbb{Z} \to \mathbb{Z}_n \,.

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


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 (