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

Definition in additive categories

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).

Definition

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

Definition

A short exact sequence is an exact sequence, def. 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 “0→A→B→C→00 \to A \to B \to C \to 0” or even just “A→B→CA \to B \to C”.

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. , precisely if

ii is a monomorphism,
pp is an epimorphism,
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 𝒜≃R\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 at

0→A→B 0 \to A \to B

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

In this case, BB may be decomposed as the biproduct A⊕CA \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’.

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 *Set_* of pointed sets, a sequence

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

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

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

In the case of (abelian) categories like AbAb and R−ModR-Mod, the two notions of exactness coincide if we pick the point of each group/module to be 00. Such a general notion is useful in cases such as the long exact sequence of homotopy groups where the homotopy “groups” for small nn 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 ⋅→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

⋯→0→C n+1→∂ nC n→0→⋯, \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+1≃C n. C_{n+1} \simeq C_n \,.

Exactness and quasi-isomorphisms

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.

Lemma

For

A→B→C→0 A \to B \to C \to 0

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

A→B/X→C/X→0 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

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.

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 𝒜=ℤ\mathcal{A} = \mathbb{Z}Mod ≃\simeq Ab. For n∈ℕn \in \mathbb{N} with n≥1n \geq 1 let ℤ→⋅nℤ\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 ℤ n≔ℤ/nℤ\mathbb{Z}_n \coloneqq \mathbb{Z}/n\mathbb{Z}. Hence we have a short exact sequence

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

Example

Classes of examples

References

A standard introduction is for instance in section 1.1 of

Charles Weibel, An Introduction to Homological Algebra

The quotient lemmas from above are discussed in

Jonathan Wise, The Snake Lemma (pdf)

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

Last revised on April 1, 2021 at 11:08:27. See the history of this page for a list of all contributions to it.