locally free

A sheaf of $\mathcal{O}_{X}$ -modules $\mathcal{F}$ on a ringed space $X$ is called locally free if for each point $p\in X$ , there is an open neighborhood (http://planetmath.org/Neighborhood) $U$ of $x$ such that $\mathcal{F}|_{U}$ is free (http://planetmath.org/FreeModule) as an $\mathcal{O}_{X}|_{U}$ -module, or equivalently, $\mathcal{F}_{p}$ , the stalk of $\mathcal{F}$ at $p$ , is free as a $(\mathcal{O}_{X})_{p}$ -module. If $\mathcal{F}_{p}$ is of finite rank (http://planetmath.org/ModuleOfFiniteRank) $n$ , then $\mathcal{F}$ is said to be of rank $n$ .

Title	locally free
Canonical name	LocallyFree
Date of creation	2013-03-22 13:52:31
Last modified on	2013-03-22 13:52:31
Owner	mps (409)
Last modified by	mps (409)
Numerical id	13
Author	mps (409)
Entry type	Definition
Classification	msc 14A99