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exact sequence
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(Definition)
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Let
 be an abelian category. We begin with a preliminary definition.
Definition 1 For any morphism
 in
 , let
 be the morphism equal to
 . Then the object  is called the image of  , and denoted
 . The morphism  is called the image morphism of  , and denoted
 .
Note that
 is not the same as
 : the former is an object of
 , while the latter is a morphism of
 . We note that  factors through
 :
The proof is as follows: by definition of cokernel,
 ; therefore by definition of kernel, the morphism  factors through
 , and this factor is the morphism  above. Furthermore  is a monomorphism and  is an epimorphism, although we do not prove these facts.
Definition 2 A sequence
of morphisms in
 is exact at  if
 .
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"exact sequence" is owned by djao. [ full author list (2) ]
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(view preamble)
Cross-references: sequence, epimorphism, monomorphism, kernel, cokernel, proof, factors, image, object, morphism, abelian category
There are 22 references to this entry.
This is version 7 of exact sequence, born on 2002-04-24, modified 2008-06-09.
Object id is 2872, canonical name is ExactSequence2.
Accessed 4494 times total.
Classification:
| AMS MSC: | 18E10 (Category theory; homological algebra :: Abelian categories :: Exact categories, abelian categories) |
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Pending Errata and Addenda
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![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ A \ar[r]^-{e} \ar@/_1pc/[rr]_{f} & \operatorname{Im}(f) \ar[r]^-{m} & B } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/2872/l2h/img18.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ A \ar[r]^-{e} \ar@/_1pc/[rr]_{f} & \operatorname{Im}(f) \ar[r]^-{m} & B } } \end{xy}$")
![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ \cdots \ar[r] & A \ar[r]^-f & B \ar[r]^-g & C \ar[r] & \cdots } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/2872/l2h/img25.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ \cdots \ar[r] & A \ar[r]^-f & B \ar[r]^-g & C \ar[r] & \cdots } } \end{xy}$")