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exact sequence
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(Definition)
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Let $\A$ be an abelian category. We begin with a preliminary definition.
Definition 1 For any morphism $f: A \longrightarrow B$ in $\A$ let $m: X \longrightarrow B$ be the morphism equal to $\ker(\cok(f))$ Then the object $X$ is called the image of $f$ and denoted $\Im(f)$ The morphism $m$ is called the image morphism of $f$ and denoted $\im(f)$
Note that $\Im(f)$ is not the same as $\im(f)$ the former is an object of $\A$ while the latter is a morphism of $\A$ We note that $f$ factors through $\im(f)$ $$ \xymatrix{ A \ar[r]^-{e} \ar@/_1pc/[rr]_{f} & \Im(f) \ar[r]^-{m} & B } $$ The proof is as follows: by definition of cokernel, $\cok(f) f = 0$ therefore by definition of kernel, the morphism $f$ factors through $\ker(\cok(f)) = \im(f) = m$ and this factor is the morphism $e$ above. Furthermore $m$ is a monomorphism and $e$ is an epimorphism, although we do not prove these facts.
Definition 2 A sequence $$ \xymatrix{ \cdots \ar[r] & A \ar[r]^-f & B \ar[r]^-g & C \ar[r] & \cdots } $$ of morphisms in $\A$ is exact at $B$ if $\ker(g) = \im(f)$
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"exact sequence" is owned by djao. [ full author list (2) ]
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Cross-references: sequence, epimorphism, monomorphism, kernel, cokernel, proof, factors, image, object, morphism, abelian category
There are 21 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 6143 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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