PlanetMath: exact sequence

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exact sequence

(Definition)

If we have two homomorphisms $f : A \to B$ and $g : B \to C$ in some category of modules, then we say that and are exact at if the image of is equal to the kernel of .

A sequence of homomorphisms

$\displaystyle \cdots \rightarrow A_{n+1} \buildrel {f_{n+1}} \over \longrightarrow A_n \buildrel {f_n} \over \longrightarrow A_{n-1} \rightarrow \cdots$

is said to be exact if each pair of adjacent homomorphisms $(f_{n+1}, f_n)$ is exact - in other words if ${\rm im} f_{n+1} = {\rm ker} f_n$ for all

Compare this to the notion of a chain complex.

Remark. The notion of exact sequences can be generalized to any abelian category $\mathcal{A}$ , where and above are objects and morphisms in $\mathcal{A}$ .

"exact sequence" is owned by antizeus. [ full author list (2) ]

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Cross-references: morphisms, objects, abelian category, chain complex, adjacent, sequence, kernel, image, modules, category, homomorphisms
There are 16 references to this entry.

This is version 3 of exact sequence, born on 2002-01-05, modified 2008-06-29.
Object id is 1354, canonical name is ExactSequence.
Accessed 5462 times total.

Classification:

AMS MSC:

16-00 (Associative rings and algebras :: General reference works )

Pending Errata and Addenda

None.[ View all 2 ]

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