Editing Exact sequence

{{short description|Sequence of homomorphisms such that each kernel equals the preceding image}}
[[File:Illustration of an Exact Sequence of Groups.svg|thumb|Illustration of an exact sequence of [[Group (mathematics)|groups]] <math>G_i</math> using [[Euler diagram|Euler diagrams]].|alt=Illustration of an exact sequence of groups using Euler diagrams. Each group is represented by a circle, within which there is a subgroup that is simultaneously the range of the previous homomorphism and the kernel of the next one, because of the exact sequence condition.]]
In [[mathematics]], an '''exact sequence''' is a sequence of [[morphisms]] between objects (for example, [[Group (mathematics)|groups]], [[Ring (mathematics)|rings]], [[Module (mathematics)|modules]], and, more generally, objects of an [[abelian category]]) such that the [[Image (mathematics)|image]] of one morphism equals the [[kernel (algebra)|kernel]] of the next.

==Definition==
In the context of group theory, a sequence
:<math>G_0\;\xrightarrow{\ f_1\ }\; G_1 \;\xrightarrow{\ f_2\ }\; G_2 \;\xrightarrow{\ f_3\ }\; \cdots \;\xrightarrow{\ f_n\ }\; G_n</math>
of groups and [[group homomorphism]]s is said to be '''exact''' '''at''' <math>G_i</math> if <math>\operatorname{im}(f_i)=\ker(f_{i+1})</math>. The sequence is called '''exact''' if it is exact at each <math>G_i</math> for all <math>1\leq i<n</math>, 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 other [[algebraic structure]]s.  For example, one could have an exact sequence of [[vector space]]s and [[linear map]]s, or of modules and [[module homomorphism]]s.  More generally, the notion of an exact sequence makes sense in any [[category (mathematics)|category]] with [[kernel (category theory)|kernel]]s and [[cokernel]]s, and more specially in [[abelian categories]], where it is widely used. 

===Simple cases===
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 the [[trivial 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 (from ''A'' to ''B'') has kernel {0}; that is, if and only if that map is a [[monomorphism]] (injective, or one-to-one).
* Consider the dual sequence ''B'' → ''C'' → 0. The kernel of the rightmost map is ''C''. Therefore the sequence is exact if and only if the image of the leftmost map (from ''B'' to ''C'') is all of ''C''; that is, if and only if that map is an [[epimorphism]] (surjective, or onto).
* Therefore, the sequence 0 → ''X'' → ''Y'' → 0 is exact if and only if the map from ''X'' to ''Y'' is both a monomorphism and epimorphism (that is, a [[bimorphism]]), and so usually an [[isomorphism]] from ''X'' to ''Y'' (this always holds in [[exact categories]] like '''Set''').

===Short exact sequence===
Short exact sequences are exact sequences of the form
:<math>0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0.</math>
As established above, for any such short exact sequence, ''f'' is a monomorphism and ''g'' is an epimorphism. Furthermore, the image of ''f'' is equal to the kernel of ''g''. It is helpful to think of ''A'' as a [[subobject]] of ''B'' with ''f'' embedding ''A'' into ''B'', and of ''C'' as the corresponding factor object (or [[Quotient object|quotient]]), ''B''/''A'', with ''g'' inducing an isomorphism
:<math>C \cong B/\operatorname{im}(f) = B/\operatorname{ker}(g)</math>

The short exact sequence
:<math>0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0\,</math>
is called '''[[split exact sequence|split]]''' if there exists a homomorphism ''h'' : ''C'' → ''B'' such that the composition ''g'' ∘ ''h'' is the identity map on ''C''. It follows that if these are [[abelian group]]s, ''B'' is isomorphic to the [[direct sum]] of ''A'' and ''C'':
:<math>B \cong A \oplus C.</math>

===Long exact sequence===
A general exact sequence is sometimes called a '''long exact sequence''', to distinguish from the special case of a short exact sequence.<ref>{{Cite web|title=exact sequence in nLab, Remark 2.3|url=https://ncatlab.org/nlab/show/exact+sequence#Definition|access-date=2021-09-05|website=ncatlab.org}}</ref>

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

{{Equation|1=A_0\;\xrightarrow{\ f_1\ }\; A_1 \;\xrightarrow{\ f_2\ }\; A_2 \;\xrightarrow{\ f_3\ }\; \cdots \;\xrightarrow{\ f_n\ }\; A_n,|2=1}}

with ''n ≥'' 2, we can split it up into the short sequences

{{Equation|1=\begin{align}
      0 \rightarrow K_1 \rightarrow {} & A_1 \rightarrow  K_2 \rightarrow  0 ,\\
      0 \rightarrow K_2 \rightarrow {} & A_2 \rightarrow  K_3 \rightarrow  0 ,\\
                                       & \ \,\vdots \\
  0 \rightarrow K_{n-1} \rightarrow {} & A_{n-1} \rightarrow  K_n \rightarrow  0 ,\\
\end{align}|2=2}}
where <math>K_i = \operatorname{im}(f_i)</math> for every <math>i</math>. By construction, the sequences ''(2)'' are exact at the <math>K_i</math>'s (regardless of the exactness of ''(1)''). Furthermore, ''(1)'' is a long exact sequence if and only if ''(2)'' are all short exact sequences.

See [[#Weaving lemma|weaving lemma]] for details on how to re-form the long exact sequence from the short exact sequences.

== Examples ==

=== Integers modulo two ===
Consider the following sequence of abelian groups:
:<math>\mathbf{Z} \mathrel{\overset{2\times}{\,\hookrightarrow}} \mathbf{Z} \twoheadrightarrow \mathbf{Z}/2\mathbf{Z}</math>

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

:<math>2\mathbf{Z} \mathrel{\,\hookrightarrow} \mathbf{Z} \twoheadrightarrow \mathbf{Z}/2\mathbf{Z}</math>

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

The first sequence may also be written without using special symbols for monomorphism and epimorphism:
:<math>0 \to \mathbf{Z} \mathrel{\overset{2\times}{\longrightarrow}} \mathbf{Z} \longrightarrow \mathbf{Z}/2\mathbf{Z} \to 0</math>

Here 0 denotes the trivial group, the map from '''Z''' to '''Z''' is multiplication by 2, and the map from '''Z''' to the [[factor group]] '''Z'''/2'''Z''' is given by reducing integers [[modular arithmetic|modulo]] 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 first '''Z'''.
* the image of multiplication by 2 is 2'''Z''', and the kernel of reducing modulo 2 is also 2'''Z''', so the sequence is exact at the second '''Z'''.
* the image of reducing modulo 2 is '''Z'''/2'''Z''', and the kernel of the zero map is also '''Z'''/2'''Z''', so the sequence is exact at the position '''Z'''/2'''Z'''.

The first and third sequences are somewhat of a special case owing to the infinite nature of '''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 

:<math>1 \to N \to G \to G/N \to 1</math>
(here the trivial group is denoted <math>1,</math> as these groups are not supposed to be [[abelian group|abelian]]).

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

:<math>1 \to C_n \to D_{2n} \to C_2 \to 1</math>

where <math>C_n</math> is the [[cyclic group]] of order ''n'' and <math>D_{2n}</math> is the [[dihedral group]] of order 2''n'', which is a non-abelian group.

=== Intersection and sum of modules ===
Let {{math|''I''}} and {{math|''J''}} be two [[Ideal (ring theory)|ideal]]s of a ring {{math|''R''}}.
Then
:<math>0 \to I\cap J \to I\oplus J \to I + J \to 0 </math>
is an exact sequence of {{math|''R''}}-modules, where the module homomorphism <math>I\cap J \to I\oplus J</math> maps each element {{math|''x''}} of <math>I\cap J</math> to the element {{tmath|(x,x)}} of the [[direct sum]] <math>I\oplus J</math>, and the homomorphism <math>I\oplus J \to I+J</math> maps each element {{tmath|(x,y)}} of <math>I\oplus J</math> to {{tmath|x-y}}.

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

:<math>0\to R \to R\oplus R \to R \to 0 </math> 

Passing to [[quotient module]]s yields another exact sequence

:<math>0\to R/(I\cap J) \to R/I \oplus R/J \to R/(I+J) \to 0 </math>

== Properties ==
The [[splitting lemma]] states that, for a  short exact sequence
:<math>0 \to A \;\xrightarrow{\ f\ }\; B \;\xrightarrow{\ g\ }\; C \to 0,</math> the following conditions are equivalent.
*There exists a morphism {{math|''t'' : ''B'' → ''A''}} such that {{math|''t'' ∘ ''f''}} is the identity on {{math|''A''}}.
*There exists a morphism {{math|''u'': ''C'' → ''B''}} such that {{math|''g'' ∘ ''u''}} is the identity on {{math|''C''}}.
*There exists a morphism {{math|''u'': ''C'' → ''B''}} such that {{math|''B''}} is the [[direct sum]] of {{math|''f''(''A'')}} and {{math|''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

:<math>A_1\to A_2\to A_3\to A_4\to A_5\to A_6</math>

which implies that there exist objects ''C<sub>k</sub>'' in the category such that

:<math>C_k \cong \ker (A_k\to A_{k+1}) \cong \operatorname{im} (A_{k-1}\to A_k)</math>.

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

:<math>C_k \cong \operatorname{coker} (A_{k-2}\to A_{k-1})</math>

(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 coker(''f'') : ''G'' → ''H'' is not ''H''/im(''f'') but <math>H / {\left\langle \operatorname{im} f \right\rangle}^H</math>, the quotient of ''H'' by the [[conjugate closure]] of im(''f'').)  Then we obtain a commutative diagram in which all the diagonals are short exact sequences:

:[[Image:long short exact sequences.png]]
The only portion of this diagram that depends on the cokernel condition is the object <math display="inline">C_7</math> and the final pair of morphisms <math display="inline">A_6 \to C_7\to 0</math>.  If there exists any object <math>A_{k+1}</math> and morphism <math>A_k \to A_{k+1}</math> such that <math>A_{k-1} \to A_k \to A_{k+1}</math> is exact, then the exactness of <math>0 \to  C_k \to A_k \to C_{k+1} \to  0</math> is ensured.  Again taking the example of the category of groups, the fact that im(''f'') is the kernel of some homomorphism on ''H'' implies that it is a [[normal subgroup]], which coincides with its conjugate closure; thus coker(''f'') is isomorphic to the image ''H''/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 [[subobject | subobjects]] and factor objects.

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

Notice that in an exact sequence, the composition ''f''<sub>''i''+1</sub> ∘ ''f''<sub>''i''</sub> maps ''A''<sub>''i''</sub> to 0 in ''A''<sub>''i''+2</sub>, so every exact sequence is a [[chain complex]]. Furthermore, only ''f''<sub>''i''</sub>-images of elements of ''A''<sub>''i''</sub> are mapped to 0 by ''f''<sub>''i''+1</sub>, so the [[homology (mathematics)|homology]] of this chain complex is trivial. More succinctly:
:Exact sequences are precisely those chain complexes which are [[acyclic complex|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 functor]]s.

[[Exact functor]]s are [[functor]]s that transform exact sequences into exact sequences.

==References==
;Citations
{{reflist}}

;Sources
*{{cite book|first=Edwin Henry|last=Spanier|author-link=Edwin Spanier|title=Algebraic Topology|url=https://archive.org/details/algebraictopolog00span|url-access=limited|publisher=Springer|location=Berlin|year=1995|page=[https://archive.org/details/algebraictopolog00span/page/n98 179]|isbn=0-387-94426-5}}
*{{cite book|first=David|last=Eisenbud|author-link=David Eisenbud|title=Commutative Algebra: with a View Toward Algebraic Geometry|url=https://archive.org/details/commutativealgeb00eise_849|url-access=limited|publisher=Springer-Verlag New York|year=1995|page=[https://archive.org/details/commutativealgeb00eise_849/page/n777 785]|isbn=0-387-94269-6}}

{{Topology}}

[[Category:Homological algebra]]
[[Category:Additive categories]]