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A sequence
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$#C+1 = 17 : ~/encyclopedia/old_files/data/E036/E.0306750 Exact sequence
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e0367501.png" /></td> </tr></table>
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of objects of an Abelian category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e0367502.png" /> and of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e0367503.png" /> such that
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A sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e0367504.png" /></td> </tr></table>
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$$
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\dots \rightarrow  A _ {n}  \rightarrow ^ { {\alpha _ n} } \
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A _ {n+} 1  \rightarrow ^ { {\alpha _ n+} 1 } \
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A _ {n+} 2  \rightarrow \dots
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$$
  
An exact sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e0367505.png" /> is called short, and consists of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e0367506.png" />, a subobject <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e0367507.png" /> of it and the corresponding quotient object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e0367508.png" />.
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of objects of an Abelian category  $  \mathfrak A $
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and of morphisms  $  \alpha _ {i} $
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such that
  
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$$
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\mathop{\rm Ker}  \alpha _ {n+} 1  = \
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\mathop{\rm Im}  \alpha _ {n} .
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$$
  
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An exact sequence  $  0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 $
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is called short, and consists of an object  $  B $,
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a subobject  $  A $
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of it and the corresponding quotient object  $  C $.
  
 
====Comments====
 
====Comments====
 
Exact sequences often occur and are often used in (co)homological considerations. There are, e.g., the long homology exact sequence
 
Exact sequences often occur and are often used in (co)homological considerations. There are, e.g., the long homology exact sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e0367509.png" /></td> </tr></table>
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$$
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\dots \rightarrow  H _ {r} ( A)  \rightarrow  H _ {r} ( X)  \rightarrow  H _ {r} ( X , A )
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\rightarrow  H _ {r-} 1 ( A)  \rightarrow \dots
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$$
  
of a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e03675010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e03675011.png" /> a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e03675012.png" />, and the long cohomology exact sequence
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of a pair $  ( X , A ) $,
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$  A $
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a subspace of $  X $,  
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and the long cohomology exact sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e03675013.png" /></td> </tr></table>
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$$
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\dots \rightarrow  H  ^ {r-} 1 ( A)  \rightarrow  H  ^ {r} ( X , A )  \rightarrow  H  ^ {r} ( X)  \rightarrow  H  ^ {r} ( X , A )  \rightarrow \dots .
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$$
  
 
Analogous long exact sequences occur in a variety of other homology and cohomology theories. Cf. [[Homology theory|Homology theory]]; [[Cohomology|Cohomology]]; [[Cohomology sequence|Cohomology sequence]]; [[Homology sequence|Homology sequence]], and various articles on the (co)homology of various kinds of objects, such as [[Cohomology of algebras|Cohomology of algebras]]; [[Cohomology of groups|Cohomology of groups]]; [[Cohomology of Lie algebras|Cohomology of Lie algebras]].
 
Analogous long exact sequences occur in a variety of other homology and cohomology theories. Cf. [[Homology theory|Homology theory]]; [[Cohomology|Cohomology]]; [[Cohomology sequence|Cohomology sequence]]; [[Homology sequence|Homology sequence]], and various articles on the (co)homology of various kinds of objects, such as [[Cohomology of algebras|Cohomology of algebras]]; [[Cohomology of groups|Cohomology of groups]]; [[Cohomology of Lie algebras|Cohomology of Lie algebras]].
  
An exact sequence of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e03675014.png" /> is sometimes called a left short exact sequence and one of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e03675015.png" /> a right short exact sequence. The exact sequence of a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e03675016.png" /> in an Abelian category is the exact sequence
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An exact sequence of the form 0 \rightarrow A _ {1} \rightarrow A \rightarrow A _ {2} $
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is sometimes called a left short exact sequence and one of the form $  A _ {1} \rightarrow A \rightarrow A _ {2} \rightarrow 0 $
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a right short exact sequence. The exact sequence of a morphism $  \alpha : X \rightarrow Y $
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in an Abelian category is the exact sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036750/e03675017.png" /></td> </tr></table>
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$$
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0  \rightarrow  \mathop{\rm Ker} \
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\alpha  \rightarrow  X  \rightarrow  Y  \rightarrow  \mathop{\rm Coker}  \alpha  \rightarrow  0 .
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$$

Latest revision as of 19:38, 5 June 2020


A sequence

$$ \dots \rightarrow A _ {n} \rightarrow ^ { {\alpha _ n} } \ A _ {n+} 1 \rightarrow ^ { {\alpha _ n+} 1 } \ A _ {n+} 2 \rightarrow \dots $$

of objects of an Abelian category $ \mathfrak A $ and of morphisms $ \alpha _ {i} $ such that

$$ \mathop{\rm Ker} \alpha _ {n+} 1 = \ \mathop{\rm Im} \alpha _ {n} . $$

An exact sequence $ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 $ is called short, and consists of an object $ B $, a subobject $ A $ of it and the corresponding quotient object $ C $.

Comments

Exact sequences often occur and are often used in (co)homological considerations. There are, e.g., the long homology exact sequence

$$ \dots \rightarrow H _ {r} ( A) \rightarrow H _ {r} ( X) \rightarrow H _ {r} ( X , A ) \rightarrow H _ {r-} 1 ( A) \rightarrow \dots $$

of a pair $ ( X , A ) $, $ A $ a subspace of $ X $, and the long cohomology exact sequence

$$ \dots \rightarrow H ^ {r-} 1 ( A) \rightarrow H ^ {r} ( X , A ) \rightarrow H ^ {r} ( X) \rightarrow H ^ {r} ( X , A ) \rightarrow \dots . $$

Analogous long exact sequences occur in a variety of other homology and cohomology theories. Cf. Homology theory; Cohomology; Cohomology sequence; Homology sequence, and various articles on the (co)homology of various kinds of objects, such as Cohomology of algebras; Cohomology of groups; Cohomology of Lie algebras.

An exact sequence of the form $ 0 \rightarrow A _ {1} \rightarrow A \rightarrow A _ {2} $ is sometimes called a left short exact sequence and one of the form $ A _ {1} \rightarrow A \rightarrow A _ {2} \rightarrow 0 $ a right short exact sequence. The exact sequence of a morphism $ \alpha : X \rightarrow Y $ in an Abelian category is the exact sequence

$$ 0 \rightarrow \mathop{\rm Ker} \ \alpha \rightarrow X \rightarrow Y \rightarrow \mathop{\rm Coker} \alpha \rightarrow 0 . $$

How to Cite This Entry:
Exact sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exact_sequence&oldid=17886
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article