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'chain complex'
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| Title of object: |
chain complex |
| Canonical Name: |
ChainComplex |
| Type: |
Definition |
| Created on: |
2002-01-05 14:58:54 |
| Modified on: |
2007-01-08 11:12:26 |
| Classification: |
msc:16E05, msc:18G35 |
| Synonyms: |
chain complex=R-complex |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\def\im{\operatorname{im}}
\def\ker{\operatorname{ker}} |
Content:
\PMlinkescapeword{adjacent}
\PMlinkescapeword{complex}
\PMlinkescapeword{equivalent}
\PMlinkescapeword{relation}
\PMlinkescapeword{satisfies}
Let $R$ be a ring.
A sequence of \PMlinkname{$R$-modules}{Module} and homomorphisms
\[
\cdots \rightarrow
A_{n+1} \buildrel {d_{n+1}} \over \longrightarrow
A_n \buildrel {d_n} \over \longrightarrow
A_{n-1} \rightarrow
\cdots
\]
is said to be a \emph{chain complex}
(or \emph{$R$-complex}, or just \emph{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
$\im d_{n+1} \subseteq \ker d_n$.
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 $\im d_{n+1} = \ker d_n$.
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