Locally free sheaf

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In sheaf theory, a field of mathematics, 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 $U$ of $p$ such that $\mathcal{F}| _U$ is free as an $\mathcal{O} _X| _U$ -module. This implies that $\mathcal{F}_p$ , the stalk of $\mathcal{F}$ at $p$ , is free as a $(\mathcal{O} _X)_p$ -module for all $p$ . The converse is true if $\mathcal{F}$ is moreover coherent. If $\mathcal{F}_p$ is of finite rank $n$ for every $p\in X$ , then $\mathcal{F}$ is said to be of rank $n.$