Exact sequence

Examples

Integers modulo two

Consider the following sequence of abelian groups: $${\displaystyle \mathbb{Z}\stackrel{2\times }{\hookrightarrow }\mathbb{Z}\twoheadrightarrow \mathbb{Z}/2\mathbb{Z}}$$

The first homomorphism maps each element ${\displaystyle i}$ in the set of integers ${\displaystyle \mathbb{Z}}$ to the element ${\displaystyle 2i}$ in ${\displaystyle \mathbb{Z}}$. The second homomorphism maps each element ${\displaystyle i}$ in ${\displaystyle \mathbb{Z}}$ to an element ${\displaystyle j}$ in the quotient group; that is, ${\displaystyle j=imod2}$. Here the hook arrow ${\displaystyle \hookrightarrow }$ indicates that the map ${\displaystyle 2\times }$ from ${\displaystyle \mathbb{Z}}$ to ${\displaystyle \mathbb{Z}}$ is a monomorphism, and the two-headed arrow ${\displaystyle \twoheadrightarrow }$ indicates an epimorphism (the map${\displaystyle mod2}$). This is an exact sequence because the image ${\displaystyle 2\mathbb{Z}}$ of the monomorphism is the kernel of the epimorphism. Essentially "the same" sequence can also be written as

$${\displaystyle 2\mathbb{Z}\hookrightarrow \mathbb{Z}\twoheadrightarrow \mathbb{Z}/2\mathbb{Z}}$$

In this case the monomorphism is ${\displaystyle 2n\mapsto 2n}$ and although it looks like an identity function, it is not onto (that is, not an epimorphism) because the odd numbers don't belong to ${\displaystyle 2\mathbb{Z}}$. The image of ${\displaystyle 2\mathbb{Z}}$ through this monomorphism is however exactly the same subset of ${\displaystyle \mathbb{Z}}$ as the image of ${\displaystyle \mathbb{Z}}$ through ${\displaystyle n\mapsto 2n}$ used in the previous sequence. This latter sequence does differ in the concrete nature of its first object from the previous one as ${\displaystyle 2\mathbb{Z}}$ is not the same set as ${\displaystyle \mathbb{Z}}$ even though the two are isomorphic as groups.

The first sequence may also be written without using special symbols for monomorphism and epimorphism: $${\displaystyle 0\to \mathbb{Z}\stackrel{2\times }{⟶}\mathbb{Z}⟶\mathbb{Z}/2\mathbb{Z}\to 0}$$

Here 0 denotes the trivial group, the map from ${\displaystyle \mathbb{Z}}$ to ${\displaystyle \mathbb{Z}}$ is multiplication by 2, and the map from ${\displaystyle \mathbb{Z}}$ to the quotient group ${\displaystyle \mathbb{Z}/2\mathbb{Z}}$ is given by reducing integers modulo 2. This is indeed an exact sequence:

• the image of the map ${\displaystyle 0\to \mathbb{Z}}$ is ${\displaystyle \{0\}}$, and the kernel of multiplication by 2 is also ${\displaystyle \{0\}}$, so the sequence is exact at the first ${\displaystyle \mathbb{Z}}$.
• the image of multiplication by 2 is ${\displaystyle 2\mathbb{Z}}$, and the kernel of reducing modulo 2 is also ${\displaystyle 2\mathbb{Z}}$, so the sequence is exact at the second ${\displaystyle \mathbb{Z}}$.
• the image of reducing modulo 2 is ${\displaystyle \mathbb{Z}/2\mathbb{Z}}$, and the kernel of the zero map is also ${\displaystyle \mathbb{Z}/2\mathbb{Z}}$, so the sequence is exact at the position ${\displaystyle \mathbb{Z}/2\mathbb{Z}}$.

The first and third sequences are somewhat of a special case owing to the infinite nature of ${\displaystyle \mathbb{Z}}$. It is not possible for a finite group to be mapped by inclusion (that is, by a monomorphism) as a proper subgroup of itself. Instead the sequence that emerges from the first isomorphism theorem is

$${\displaystyle 1\to N\to G\to G/N\to 1}$$ (here the trivial group is denoted ${\displaystyle 1,}$ as these groups are not supposed to be abelian).

As a more concrete example of an exact sequence on finite groups:

${\displaystyle 1\to C_n\to D_{2n}\to C_2\to 1}$

where ${\displaystyle C_n}$ is the cyclic group of order n and ${\displaystyle D_{2n}}$ is the dihedral group of order 2n, which is a non-abelian group.

Intersection and sum of modules

Let ${\displaystyle I}$ and ${\displaystyle J}$ be two ideals of a ring ${\displaystyle R}$. Then $${\displaystyle 0\to I\cap J\to I\oplus J\to I+J\to 0}$$ is an exact sequence of ${\displaystyle R}$-modules, where the module homomorphism ${\displaystyle I\cap J\to I\oplus J}$ maps each element ${\displaystyle x}$ of ${\displaystyle I\cap J}$ to the element ⁠${\displaystyle (x,x)}$⁠ of the direct sum ${\displaystyle I\oplus J}$, and the homomorphism ${\displaystyle I\oplus J\to I+J}$ maps each element ⁠${\displaystyle (x,y)}$⁠ of ${\displaystyle I\oplus J}$ to ⁠${\displaystyle x-y}$⁠.

These homomorphisms are restrictions of similarly defined homomorphisms that form the short exact sequence

$${\displaystyle 0\to R\to R\oplus R\to R\to 0}$$

Passing to quotient modules yields another exact sequence

$${\displaystyle 0\to R/(I\cap J)\to R/I\oplus R/J\to R/(I+J)\to 0}$$

Properties

The splitting lemma states that, for a short exact sequence $${\displaystyle 0\to A\stackrel{f}{\to }B\stackrel{g}{\to }C\to 0,}$$ the following conditions are equivalent.

• There exists a morphism ${\displaystyle t:B\to A}$ such that ${\displaystyle t\circ f}$ is the identity on ${\displaystyle A}$.
• There exists a morphism ${\displaystyle u:C\to B}$ such that ${\displaystyle g\circ u}$ is the identity on ${\displaystyle C}$.
• There exists a morphism ${\displaystyle u:C\to B}$ such that ${\displaystyle B}$ is the direct sum of ${\displaystyle f(A)}$ and ${\displaystyle u(C)}$.

For non-commutative groups, the splitting lemma does not apply, and one has only the equivalence between the two last conditions, with "the direct sum" replaced with "a semidirect product".

In both cases, one says that such a short exact sequence splits.

The snake lemma shows how a commutative diagram with two exact rows gives rise to a longer exact sequence. The nine lemma is a special case.

The five lemma gives conditions under which the middle map in a commutative diagram with exact rows of length 5 is an isomorphism; the short five lemma is a special case thereof applying to short exact sequences.

Weaving lemma

The importance of short exact sequences is underlined by the fact that every exact sequence results from "weaving together" several overlapping short exact sequences. Consider for instance the exact sequence

$${\displaystyle A_1\to A_2\to A_3\to A_4\to A_5\to A_6}$$

which implies that there exist objects C_k in the category such that

$${\displaystyle C_k\cong \ker (A_k\to A_{k+1})\cong \mathrm{im}(A_{k-1}\to A_k)}$$.

Suppose in addition that the cokernel of each morphism exists, and is isomorphic to the image of the next morphism in the sequence:

$${\displaystyle C_k\cong \mathrm{coker}(A_{k-2}\to A_{k-1})}$$

(This is true for a number of interesting categories, including any abelian category such as the abelian groups; but it is not true for all categories that allow exact sequences, and in particular is not true for the category of groups, in which ${\displaystyle \mathrm{coker}(f):G\to H}$ is not ${\displaystyle H/\mathrm{im}(f)}$ but ${\displaystyle H/{⟨\mathrm{im}f⟩}^H}$, the quotient of ${\displaystyle H}$ by the conjugate closure of ${\displaystyle \mathrm{im}(f)}$.) Then we obtain a commutative diagram in which all the diagonals are short exact sequences:

The only portion of this diagram that depends on the cokernel condition is the object $C_7$ and the final pair of morphisms $A_6\to C_7\to 0$. If there exists any object ${\displaystyle A_{k+1}}$ and morphism ${\displaystyle A_k\to A_{k+1}}$ such that ${\displaystyle A_{k-1}\to A_k\to A_{k+1}}$ is exact, then the exactness of ${\displaystyle 0\to C_k\to A_k\to C_{k+1}\to 0}$ is ensured. Again taking the example of the category of groups, the fact that ${\displaystyle \mathrm{im}(f)}$ is the kernel of some homomorphism on ${\displaystyle H}$ implies that it is a normal subgroup, which coincides with its conjugate closure; thus ${\displaystyle \mathrm{coker}(f)}$ is isomorphic to the image ${\displaystyle H/\mathrm{im}(f)}$ of the next morphism.

Conversely, given any list of overlapping short exact sequences, their middle terms form an exact sequence in the same manner.

Applications of exact sequences

In the theory of abelian categories, short exact sequences are often used as a convenient language to talk about subobjects and factor objects.

The extension problem is essentially the question "Given the end terms ${\displaystyle A}$ and ${\displaystyle C}$ of a short exact sequence, what possibilities exist for the middle term ${\displaystyle B}$?" In the category of groups, this is equivalent to the question, what groups ${\displaystyle B}$ have ${\displaystyle A}$ as a normal subgroup and ${\displaystyle C}$ as the corresponding factor group? This problem is important in the classification of groups. See also Outer automorphism group.

Notice that in an exact sequence, the composition ${\displaystyle f_{i+1}\circ f_i}$ maps ${\displaystyle A_i}$ to 0 in ${\displaystyle A_{i+2}}$, so every exact sequence is a chain complex. Furthermore, only ${\displaystyle f_i}$-images of elements of ${\displaystyle A_i}$ are mapped to 0 by ${\displaystyle f_{i+1}}$, so the homology of this chain complex is trivial. More succinctly:

Exact sequences are precisely those chain complexes which are acyclic.

Given any chain complex, its homology can therefore be thought of as a measure of the degree to which it fails to be exact.

If we take a series of short exact sequences linked by chain complexes (that is, a short exact sequence of chain complexes, or from another point of view, a chain complex of short exact sequences), then we can derive from this a long exact sequence (that is, an exact sequence indexed by the natural numbers) on homology by application of the zig-zag lemma. It comes up in algebraic topology in the study of relative homology; the Mayer–Vietoris sequence is another example. Long exact sequences induced by short exact sequences are also characteristic of derived functors.

Exact functors are functors that transform exact sequences into exact sequences.

de Rham complex

The term "exact" originates from exact differential forms in the context of the de Rham complex: $${\displaystyle 0\to {\mathrm{\Omega }}^0(X)\stackrel{d^0}{\to }{\mathrm{\Omega }}^1(X)\stackrel{d^1}{\to }{\mathrm{\Omega }}^2(X)\stackrel{d^2}{\to }{\mathrm{\Omega }}^3(X)\stackrel{d^3}{\to }\cdots \stackrel{d^{n-1}}{\to }{\mathrm{\Omega }}^n(X)\to 0}$$ When ${\displaystyle n=3}$ this sequence can be written as: $${\displaystyle 0\to {\mathrm{\Omega }}^0(X)\stackrel{\text{grad}}{\to }{\mathrm{\Omega }}^1(X)\stackrel{\text{curl}}{\to }{\mathrm{\Omega }}^2(X)\stackrel{\text{div}}{\to }{\mathrm{\Omega }}^3(X)\to 0}$$ Where ${\displaystyle \text{grad},\text{curl},\text{div}}$ represent the gradient, curl, and divergence. The sequence then yields the identities ${\displaystyle \text{curl}\cdot \text{grad}=0}$ and ${\displaystyle \text{div}\cdot \text{curl}=0}$ from advanced calculus.

A differential form ${\displaystyle \omega \in {\mathrm{\Omega }}^p(X)}$ is closed when ${\displaystyle \omega \in \text{im }{d}^{p-1}}$, i.e. ${\displaystyle \omega =d^{p-1}{\omega }^{\prime }}$, and is exact when ${\displaystyle \omega \in \ker d^p}$, that is, ${\displaystyle d^p\omega =0}$. The de Rham complex is an exact sequence of modules if and only if every closed form is exact, which explains why the word exact is an adjective of "exact sequence".