chain complex ChainComplex 2002-01-05 14:58:54 2008-03-18 09:20:42 Definition boundary operator boundary map \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \def\im{\operatorname{im}} \def\ker{\operatorname{ker}} \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 $\im d_{n+1} = \ker d_n$. The homomorphisms $d_n$ in the chain complex are called \emph{boundary operators}, or \emph{boundary maps}.