nLab exact sequence

Contents

category theory

Applications

Homological algebra

homological algebra

Introduction

diagram chasing

Contents

Idea

An exact sequence may be defined in a semi-abelian category, and more generally in a homological category. It is a sequential diagram in which the image of each morphism is equal to the kernel of the next morphism.

Definition

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

Definition

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

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

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

$\cdots \to 0 \to 0 \to A \to B \to C \to 0 \to 0 \to \cdots \,.$

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

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

$0 \to A \stackrel{i}\to B \stackrel{p}\to C \to 0$

is short exact, def. , precisely if

1. $i$ is a monomorphism,

2. $p$ is an epimorphism,

3. and the image of $i$ equals the kernel of $p$ (equivalently, the coimage of $p$ equals the cokernel of $i$).

Proof

The third condition is the definition of exactness at $B$. So we need to show that the first two conditions are equivalent to exactness at $A$ and at $C$.

This is easy to see by looking at elements when $\mathcal{A} \simeq R$Mod, for some ring $R$ (and the general case can be reduced to this one using one of the embedding theorems):

The sequence being exact at

$0 \to A \to B$

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

Dually, the sequence being exact at

$B \to C \to 0$

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

Definition

A split exact sequence is a short exact sequence as above in which $i$ is a split monomorphism, or (equivalently) in which $p$ is a split epimorphism.

In this case, $B$ may be decomposed as the biproduct $A \oplus C$ (with $i$ and $p$ the usual biproduct inclusion and projection); this sense in which $B$ is ‘split’ into $A$ and $C$ is the origin of the general terms ‘split (mono/epi)morphism’.

Definition in pointed sets

It is also helpful to consider a similar notion in the case of a pointed set.

Definition

In the category $Set_*$ of pointed sets, a sequence

$\array{ (A, a) & \overset{f}{\to} & (B, b) & \overset{g}{\to} & (C, c) }$

is said to be exact at $(B, b)$ if $im f = g^{-1}(c)$.

For concrete pointed categories (ie. a category $\mathcal{C}$ with a faithful functor $F: \mathcal{C} \to Set_*$), a sequence is exact if the image under $F$ is exact.

In the case of (abelian) categories like $Ab$ and $R-Mod$, the two notions of exactness coincide if we pick the point of each group/module to be $0$. Such a general notion is useful in cases such as the long exact sequence of homotopy groups where the homotopy “groups” for small $n$ are just pointed sets without a group structure.

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. is one example for this: whenever in a long exact sequence one entry vanishes as in $\cdot \to 0 \to C_n \to \cdot$ or $\cdot \to C_n \to 0 \to \cdots$, 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

$\cdots \to 0 \to C_{n+1} \stackrel{\partial_n}{\to} C_n \to 0 \to \cdots \,,$

then $\partial_n$ is an isomorphism and hence

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

Exactness and quasi-isomorphisms

Proposition

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

$0 \to C_\bullet$

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 $\mathcal{A}$ be an abelian category.

Lemma

For

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

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

$A \to B/X \to C/X \to 0$

is a short exact sequence.

Proof

We have an exact sequence of complexes of length 2

$\array{ 0 &\to& X &\stackrel{id}{\to}& X &\to& 0 \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ A &\to& B &\to& C &\to& 0 }$

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

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

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

$0 \to A/X \to B/X \to C$

is exact.

Examples

Specific examples

Example

Let $\mathcal{A} = \mathbb{Z}$Mod $\simeq$ Ab. For $n \in \mathbb{N}$ with $n \geq 1$ let $\mathbb{Z} \stackrel{\cdot n}{\to} \mathbb{Z}$ be the linear map/homomorphism of abelian groups which acts by the ordinary multiplication of integers by $n$. This is clearly an injection. The cokernel of this morphism is the projection to the quotient group, which is the cyclic group $\mathbb{Z}_n \coloneqq \mathbb{Z}/n\mathbb{Z}$. Hence we have a short exact sequence

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

The connecting homomorphism of the long exact sequence in homology induced from short exact sequences of the form in example 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.