# 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 LocallyFree 2013-03-22 13:52:31 2013-03-22 13:52:31 mps (409) mps (409) 13 mps (409) Definition msc 14A99