Inmathematics, anexact sequence is a sequence ofmorphisms between objects (for example,groups,rings,modules, and, more generally, objects of anabelian category) such that theimage of one morphism equals thekernel of the next.

In the context of group theory, a sequence

${\displaystyle G_0\stackrel{f_1}{\to }{G}_1\stackrel{f_2}{\to }{G}_2\stackrel{f_3}{\to }\cdots \stackrel{f_n}{\to }{G}_n}$

of groups andgroup homomorphisms is said to beexactat${\displaystyle G_i}$ if${\displaystyle \mathrm{im}(f_i)=\ker (f_{i+1})}$. The sequence is calledexact if it is exact at each${\displaystyle G_i}$ for all, i.e., if the image of each homomorphism is equal to the kernel of the next.

The sequence of groups and homomorphisms may be either finite or infinite.

A similar definition can be made for otheralgebraic structures. For example, one could have an exact sequence ofvector spaces andlinear maps, or of modules andmodule homomorphisms. More generally, the notion of an exact sequence makes sense in anycategory withkernels andcokernels, and more specially inabelian categories, where it is widely used.

To understand the definition, it is helpful to consider relatively simple cases where the sequence is of group homomorphisms, is finite, and begins or ends with thetrivial group. Traditionally, this, along with the single identity element, is denoted 0 (additive notation, usually when the groups are abelian), or denoted 1 (multiplicative notation).

• Consider the sequence 0 →A →B. The image of the leftmost map is 0. Therefore the sequence is exact if and only if the rightmost map (fromA toB) has kernel {0}; that is, if and only if that map is amonomorphism (injective, or one-to-one).
• Consider the dual sequenceB →C → 0. The kernel of the rightmost map isC. Therefore the sequence is exact if and only if the image of the leftmost map (fromB toC) is all ofC; that is, if and only if that map is anepimorphism (surjective, or onto).
• Therefore, the sequence 0 →X →Y → 0 is exact if and only if the map fromX toY is both a monomorphism and epimorphism (that is, abimorphism), and so usually anisomorphism fromX toY (this always holds inexact categories likeSet).

Short exact sequences are exact sequences of the form

${\displaystyle 0\to A\stackrel{f}{\to }B\stackrel{g}{\to }C\to 0.}$

As established above, for any such short exact sequence,f is a monomorphism andg is an epimorphism. Furthermore, the image off is equal to the kernel ofg. It is helpful to think ofA as asubobject ofB withf embeddingA intoB, and ofC as the corresponding factor object (orquotient),B/A, withg inducing an isomorphism

${\displaystyle C\cong B/\mathrm{im}(f)=B/\ker (g)}$

The short exact sequence

${\displaystyle 0\to A\stackrel{f}{\to }B\stackrel{g}{\to }C\to 0}$

is calledsplit if there exists a homomorphismh :C →B such that the compositiong ∘h is the identity map onC. It follows that if these areabelian groups,B is isomorphic to thedirect sum ofA andC:

${\displaystyle B\cong A\oplus C.}$

A general exact sequence is sometimes called along exact sequence, to distinguish from the special case of a short exact sequence.^[1]

A long exact sequence is equivalent to a family of short exact sequences in the following sense: Given a long sequence

${\displaystyle A_0\stackrel{f_1}{\to }{A}_1\stackrel{f_2}{\to }{A}_2\stackrel{f_3}{\to }\cdots \stackrel{f_n}{\to }{A}_n,}$(1)

withn ≥ 2, we can split it up into the short sequences

(2)

where${\displaystyle K_i=\mathrm{im}(f_i)}$ for every${\displaystyle i}$. By construction, the sequences(2) are exact at the${\displaystyle K_i}$'s (regardless of the exactness of(1)). Furthermore,(1) is a long exact sequence if and only if(2) are all short exact sequences.

Seeweaving lemma for details on how to re-form the long exact sequence from the short exact sequences.

Consider the following sequence of abelian groups:

${\displaystyle \mathbf{Z}\stackrel{2\times }{\hookrightarrow }\mathbf{Z}\twoheadrightarrow \mathbf{Z}/2\mathbf{Z}}$

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

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

In this case the monomorphism is 2n ↦ 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 2Z. The image of 2Z through this monomorphism is however exactly the same subset ofZ as the image ofZ throughn ↦ 2n used in the previous sequence. This latter sequence does differ in the concrete nature of its first object from the previous one as 2Z is not the same set asZ even though the two are isomorphic as groups.

The first sequence may also be written without using special symbols for monomorphism and epimorphism:

${\displaystyle 0\to \mathbf{Z}\stackrel{2\times }{⟶}\mathbf{Z}⟶\mathbf{Z}/2\mathbf{Z}\to 0}$

Here 0 denotes the trivial group, the map fromZ toZ is multiplication by 2, and the map fromZ to thefactor groupZ/2Z is given by reducing integersmodulo 2. This is indeed an exact sequence:

• the image of the map 0 →Z is {0}, and the kernel of multiplication by 2 is also {0}, so the sequence is exact at the firstZ.
• the image of multiplication by 2 is 2Z, and the kernel of reducing modulo 2 is also 2Z, so the sequence is exact at the secondZ.
• the image of reducing modulo 2 isZ/2Z, and the kernel of the zero map is alsoZ/2Z, so the sequence is exact at the positionZ/2Z.

The first and third sequences are somewhat of a special case owing to the infinite nature ofZ. It is not possible for afinite group to be mapped by inclusion (that is, by a monomorphism) as a proper subgroup of itself. Instead the sequence that emerges from thefirst 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 beabelian).

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 thecyclic group of ordern and${\displaystyle D_{2n}}$ is thedihedral group of order 2n, which is a non-abelian group.

LetI andJ be twoideals of a ringR.Then

${\displaystyle 0\to I\cap J\to I\oplus J\to I+J\to 0}$

is an exact sequence ofR-modules, where the module homomorphism${\displaystyle I\cap J\to I\oplus J}$ maps each elementx of${\displaystyle I\cap J}$ to the element⁠${\displaystyle (x,x)}$⁠ of thedirect 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 toquotient modules yields another exact sequence

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

Thesplitting 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 morphismt :B →A such thatt ∘f is the identity onA.
• There exists a morphismu:C →B such thatg ∘u is the identity onC.
• There exists a morphismu:C →B such thatB is thedirect sum off(A) andu(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 "asemidirect product".

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

Thesnake lemma shows how acommutative diagram with two exact rows gives rise to a longer exact sequence. Thenine lemma is a special case.

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

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 objectsC_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 thecategory of groups, in which coker(f) :G →H is notH/im(f) but${\displaystyle H/{⟨\mathrm{im}f⟩}^H}$, the quotient ofH by theconjugate closure of 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 im(f) is the kernel of some homomorphism onH implies that it is anormal subgroup, which coincides with its conjugate closure; thus coker(f) is isomorphic to the imageH/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.

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

Theextension problem is essentially the question "Given the end termsA andC of a short exact sequence, what possibilities exist for the middle termB?" In the category of groups, this is equivalent to the question, what groupsB haveA as a normal subgroup andC as the corresponding factor group? This problem is important in theclassification of groups. See alsoOuter automorphism group.

Notice that in an exact sequence, the compositionf_i+1 ∘f_i mapsA_i to 0 inA_i+2, so every exact sequence is achain complex. Furthermore, onlyf_i-images of elements ofA_i are mapped to 0 byf_i+1, so thehomology of this chain complex is trivial. More succinctly:

Exact sequences are precisely those chain complexes which areacyclic.

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 along exact sequence (that is, an exact sequence indexed by the natural numbers) on homology by application of thezig-zag lemma. It comes up inalgebraic topology in the study ofrelative homology; theMayer–Vietoris sequence is another example. Long exact sequences induced by short exact sequences are also characteristic ofderived functors.

Exact functors arefunctors that transform exact sequences into exact sequences.

Citations

Sources

• Spanier, Edwin Henry (1995).Algebraic Topology. Berlin: Springer. p. 179.ISBN 0-387-94426-5.
• Eisenbud, David (1995).Commutative Algebra: with a View Toward Algebraic Geometry. Springer-Verlag New York. p. 785.ISBN 0-387-94269-6.

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