PlanetMath: chain complex

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chain complex

(Definition)

Let be a ring. A sequence of -modules and homomorphisms

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

is said to be a chain complex (or

-complex, or just complex) if each pair of adjacent homomorphisms $(d_{n+1}, d_n)$ satisfies the relation $d_n\circ d_{n+1} = 0$ . This is equivalent to saying that $\operatorname{im}d_{n+1} \subseteq \operatorname{ker}d_n$ . We often denote such a complex by $({\bold A}, d)$ , or simply ${\bold A}$ .

Compare this to the notion of an exact sequence, which requires $\operatorname{im}d_{n+1} = \operatorname{ker}d_n$ .

The homomorphisms in the chain complex are called boundary operators, or boundary maps.

"chain complex" is owned by yark. [ full author list (2) | owner history (1) ]

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See Also: homology of a chain complex

Other names:

R-complex

Also defines:

boundary operator, boundary map

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Cross-references: exact sequence, homomorphisms, sequence, ring
There are 30 references to this entry.

This is version 9 of chain complex, born on 2002-01-05, modified 2008-03-18.
Object id is 1353, canonical name is ChainComplex.
Accessed 7898 times total.

Classification:

AMS MSC:	16E05 (Associative rings and algebras :: Homological methods :: Syzygies, resolutions, complexes)
	18G35 (Category theory; homological algebra :: Homological algebra :: Chain complexes)

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