LocallyFree 2003-08-19 05:51:58 2005-10-20 23:48:55 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\F}{\mathcal{F}} \renewcommand{\O}{\mathcal{O}} \PMlinkescapeword{free} \PMlinkescapeword{rank} A sheaf of $\O_X$-modules $\F$ on a ringed space $X$ is called {\em locally free} if for each point $p\in X$, there is an open \PMlinkname{neighborhood}{Neighborhood} $U$ of $x$ such that $\F|_U$ is \PMlinkname{free}{FreeModule} as an $\O_X|_U$-module, or equivalently, $\F_p$, the stalk of $\F$ at $p$, is free as a $(\O_X)_p$-module. If $\F_p$ is of \PMlinkname{finite rank}{ModuleOfFiniteRank} $n$, then $\F$ is said to be of rank $n$.