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| An '''exact sequence''' is a concept in [[mathematics]], especially in [[Ring (mathematics)|ring]] and [[module (mathematics)|module]] theory, [[homological algebra]], as well as in [[differential geometry]] and [[group theory]]. An exact sequence is a [[sequence]], either finite or infinite, of objects and [[morphism]]s between them such that the [[Image (mathematics)|image]] of one morphism equals the [[kernel (algebra)#Group homomorphisms|kernel]] of the next.
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| ==Definition==
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| In the context of [[group theory]], a sequence
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| :<math>G_0 \;\xrightarrow{f_1}\; G_1 \;\xrightarrow{f_2}\; G_2 \;\xrightarrow{f_3}\; \cdots \;\xrightarrow{f_n}\; G_n</math> | |
| of [[group (mathematics)|groups]] and [[group homomorphism]]s is called '''exact''' if the [[Image (mathematics)|image]] (or [[Range (mathematics)|range]]) of each homomorphism is equal to the [[Kernel (algebra)|kernel]] of the next:
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| :<math>\mathrm{im}(f_k) = \mathrm{ker}(f_{k+1})</math> | |
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| Note that the sequence of groups and homomorphisms may be either finite or infinite.
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| A similar definition can be made for certain other [[algebraic structure]]s. For example, one could have an exact sequence of [[vector space]]s and [[linear map]]s, or of [[module (mathematics)|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.
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| ===Short exact sequence===
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| <!-- :<math>A \;\xrightarrow{f}\; B \;\twoheadrightarrow\; C</math> -->
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| The most common type of exact sequence is the '''short exact sequence'''. This is an exact sequence of the form
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| :<math>A \;\overset{f}{\hookrightarrow}\; B \;\overset{g}{\twoheadrightarrow}\; C</math> | |
| where ƒ is a [[monomorphism]] and ''g'' is an [[epimorphism]]. In this case, ''A'' is a [[subobject]] of ''B'', and the corresponding [[quotient]] is [[isomorphism|isomorphic]] to ''C'':
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| :<math>C \cong B/f(A)</math> | |
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| (where ''f(A)'' = im(''f'')).
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| A short exact sequence of [[abelian group]]s may also be written as an exact sequence with five terms:
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| :<math>0 \;\xrightarrow{}\; A \;\xrightarrow{f}\; B \;\xrightarrow{g}\; C \;\xrightarrow{}\; 0</math>
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| where 0 represents the [[Initial and terminal objects|zero object]], such as the [[trivial group]] or a zero-dimensional vector space. The placement of the 0's forces ƒ to be a monomorphism and ''g'' to be an epimorphism (see below).
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| If instead the objects are groups not known to be abelian, then multiplicative rather than additive notation is traditional, and the identity element -- as well as the trivial group -- is often written as "1" instead of "0". So in that case a short exact sequence would be written as follows:
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| :<math>1 \;\xrightarrow{}\; A \;\xrightarrow{f}\; B \;\xrightarrow{g}\; C \;\xrightarrow{}\; 1</math>
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| == Example ==
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| Consider the following sequence of [[abelian group]]s:
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| :<math>\Bbb{Z} \;\overset{2\cdot}{\hookrightarrow}\; \Bbb{Z} \twoheadrightarrow \Bbb{Z}/2\Bbb{Z}</math>
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| The first operation forms an element in the set of integers, '''Z''', using multiplication by 2 on an element from '''Z''' i.e. ''j'' = 2''i''. The second operation forms an element in the quotient space, ''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.
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| This sequence may also be written without using special symbols for monomorphism and epimorphism:
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| :<math>0\to \Bbb{Z} \;\xrightarrow{2\cdot}\; \Bbb{Z} \to \Bbb{Z}/2\Bbb{Z}\to 0</math> | |
| Here 0 denotes the trivial abelian group with a single element, the map from '''Z''' to '''Z''' is multiplication by [[two|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:
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| * 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'''.
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| * 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'''.
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| * the image of reducing modulo 2 is all of '''Z'''/2'''Z''', and the kernel of the zero map is also all of '''Z'''/2'''Z''', so the sequence is exact at the position '''Z'''/2'''Z'''
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| Another example, from [[differential geometry]], especially relevant for work on the [[Maxwell equations]]:
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| :<math>\Bbb{H}_1\ \xrightarrow{\mbox{grad}}\ \Bbb{H}_\mbox{curl}\ \xrightarrow{\mbox{curl}}\ \Bbb{H}_\mbox{div}\ \xrightarrow{\mbox{div}}\ \Bbb{L}_2</math>
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| based on the fact that on properly defined [[Hilbert space]]s,
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| : <math>\begin{align}
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| \mbox{curl}\,(\mbox{grad}\,f ) &= \nabla \times (\nabla f) = 0 \\
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| \mbox{div}\,(\mbox{curl}\,\vec v ) &= \nabla \cdot \nabla \times \vec{v} = 0
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| \end{align}</math>
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| in addition, [[curl (mathematics)|curl]]-free vector fields can always be written as a [[Conservative vector field|gradient of a scalar function]] (as soon as the space is assumed to be [[simply connected]], see '''Note 1''' below), and that a [[divergence]]less field can be written as a curl of another field.<ref>{{cite web |url=http://mathworld.wolfram.com/DivergencelessField.html |title=Divergenceless field|date=December 6, 2009}}</ref>
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| '''Note 1''': this example makes use of the fact that 3-dimensional space is topologically trivial.
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| '''Note 2''': <math>\Bbb{H}_\mbox{curl}\ </math> and <math>\Bbb{H}_\mbox{div}\ </math> are the domains for the curl and div operators respectively.
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| ==Special cases==
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| To make sense of the definition, it is helpful to consider what it means in relatively simple cases where the sequence is finite and begins or ends with 0.
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| * The sequence 0 → ''A'' → ''B'' is exact at ''A'' if and only if the map from ''A'' to ''B'' has kernel {0}, i.e. if and only if that map is a [[monomorphism]] (one-to-one).
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| * Dually, the sequence ''B'' → ''C'' → 0 is exact at ''C'' if and only if the image of the map from ''B'' to ''C'' is all of ''C'', i.e. if and only if that map is an [[epimorphism]] (onto).
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| * A consequence of these last two facts is that the sequence 0 → ''X'' → ''Y'' → 0 is exact if and only if the map from ''X'' to ''Y'' is an [[Morphism#Some_specific_morphisms|isomorphism]].
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| Important are '''short exact sequences''', which are exact sequences of the form
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| : <math> 0 \rightarrow A~\overset{f}{\rightarrow}~B~\overset{g}{\rightarrow}~C \rightarrow 0</math>
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| By the above, we know that 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'' being the embedding of ''A'' into ''B'', and of ''C'' as the corresponding factor object ''B''/''A'', with the map ''g'' being the natural projection from ''B'' to ''B''/''A'' (whose kernel is exactly ''A'').
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| == Facts ==
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| The [[splitting lemma]] states that if the above short exact sequence admits a morphism ''t'': ''B'' → ''A'' such that ''t'' <small>o</small> ''f'' is the identity on ''A'' [[logical disjunction|or]] a morphism ''u'': ''C'' → ''B'' such that ''g'' <small>o</small> ''u'' is the identity on ''C'', then ''B'' is a [[twisted direct sum]] of ''A'' and ''C''. (For groups, a twisted direct sum is a [[semidirect product]]; in an abelian category, every twisted direct sum is an ordinary [[biproduct|direct sum]].) In this case, we say that the short exact sequence ''splits''.
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| 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.
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| 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.
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| 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
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| :<math>A_1\to A_2\to A_3\to A_4\to A_5\to A_6</math>
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| which implies that there exist objects ''C<sub>k</sub>'' in the category such that
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| :<math>C_k \cong \ker (A_k\to A_{k+1}) \cong \operatorname{im} (A_{k-1}\to A_k)</math>.
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| Suppose in addition that the [[cokernel]] of each morphism exists, and is isomorphic to the image of the next morphism in the sequence:
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| :<math>C_k \cong \operatorname{coker} (A_{k-2}\to A_{k-1})</math>
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| (This is true for a number of interesting categories, including any abelian category such as the [[abelian group]]s; 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:
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| :[[Image:long short exact sequences.png]]
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| Note that the only portion of this diagram that depends on the cokernel condition is the object ''C<sub>7</sub>'' and the final pair of morphisms ''A<sub>6</sub>'' → ''C<sub>7</sub>'' → 0. If there exists any object <math>A_{k+1}</math> and morphism <math>A_k \rightarrow A_{k+1}</math> such that <math>A_{k-1} \rightarrow A_k \rightarrow A_{k+1}</math> is exact, then the exactness of <math>0 \rightarrow C_k \rightarrow A_k \rightarrow C_{k+1} \rightarrow 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.
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| Conversely, given any list of overlapping short exact sequences, their middle terms form an exact sequence in the same manner.
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| ==Applications of exact sequences==
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| In the theory of abelian categories, short exact sequences are often used as a convenient language to talk about sub- and factor objects.
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| 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]].
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| Notice that in an exact sequence, the composition ''f''<sub>''i''+1</sub> <small>o</small> ''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:
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| :Exact sequences are precisely those chain complexes which are [[acyclic complex|acyclic]].
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| Given any chain complex, its homology can therefore be thought of as a measure of the degree to which it fails to be exact.
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| 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''' (i.e. 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.
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| [[Exact functor]]s are [[functor]]s that transform exact sequences into exact sequences.
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| ==References==
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| ;General
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| *{{cite book
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| |first=Edwin Henry
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| |last=Spanier
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| |title=Algebraic Topology
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| |publisher=Springer
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| |location=Berlin
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| |year=1995
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| |pages=179
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| |isbn=0-387-94426-5
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| }}
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| *{{cite book
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| |first1=M.R.
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| |author1=Adhikari
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| |first2=Avishek
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| |last2=Adhikari
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| |title=Groups, Rings and Modules with Applications
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| |publisher=Universities Press
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| |location=India
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| |year=2003
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| |pages=216
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| |isbn=81-7371-429-0
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| }}
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| ;Citations
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| {{reflist}}
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| ==External links==
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| *{{planetmath reference|id=1354|title=Exact sequence}}
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| *{{MathWorld|title=Exact Sequence|urlname=ExactSequence}}
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| *{{MathWorld|title=Short Exact Sequence|urlname=ShortExactSequence}}
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| [[Category:Homological algebra]]
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| [[Category:Additive categories]]
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