ChainMap 2002-01-23 12:21:51 2002-02-12 08:07:08 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics \usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $(A,d)$ and $(A^{'},d^{'})$ be chain complexes. A \emph{chain map} $f:A \to A^{'}$ is a sequence of homomorphisms $\{f_n\}$ such that $d_{n}^{'} \circ f_{n} = f_{n-1} \circ d_{n}$ for each $n$. Diagramatically, this says that the following diagram commutes: $$ \xymatrix{ & A_{n} \ar[d]^{f_n} \ar[r]^{d_{n}} & A_{n-1} \ar[d]^{f_{n-1}} \\ & A_{n}^{'} \ar[r]^{d_{n}^{'}} & A_{n-1}^{'} } $$