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locally free (Definition)

A sheaf of $ \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 $ x$ such that $ \mathcal{F}\vert _U$ is free as an $ \O _X\vert _U$-module, or equivalently, $ \mathcal{F}_p$, the stalk of $ \mathcal{F}$ at $ p$, is free as a $ (\O _X)_p$-module. If $ \mathcal{F}_p$ is of finite rank $ n$, then $ \mathcal{F}$ is said to be of rank $ n$.



"locally free" is owned by mps. [ full author list (3) | owner history (2) ]
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Cross-references: stalk, open, point, ringed space, sheaf
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This is version 10 of locally free, born on 2003-08-19, modified 2005-10-20.
Object id is 4618, canonical name is LocallyFree.
Accessed 2433 times total.

Classification:
AMS MSC: 14A99 (Algebraic geometry :: Foundations :: Miscellaneous)
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