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'locally free'
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| Title of object: |
locally free |
| Canonical Name: |
LocallyFree |
| Type: |
Definition |
| Created on: |
2003-08-19 05:51:58 |
| Modified on: |
2004-02-19 13:48:42 |
| Classification: |
msc:14A99 |
Revision comment (for changes between this and next version):
| Revise according to altomani's comment ($\O_X$ -> $(\O_X)_p$). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\newcommand{\F}{\mathcal{F}}
\renewcommand{\O}{\mathcal{O}} |
Content:
\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 $x\in X$, 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, $\F_x$, the stalk of $\F$ at $x$, is free as a $\O_X$-module. If $\F_x$ is of \PMlinkname{finite rank}{ModuleOfFiniteRank} $n$, then $\F$ is said to be of rank $n$. |
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