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line bundle (Definition)

In algebraic geometry, the term line bundle refers to a locally free coherent sheaf of rank 1, also called an invertible sheaf. In manifold theory, it refers to a real or complex one dimensional vector bundle. These notions are equivalent on a non-singular complex algebraic variety $ X$: given a one dimensional vector bundle, its sheaf of holomorphic sections is locally free and of rank 1. Similarly, given a locally free sheaf $ \mathcal{F}$ of rank one, the space

$\displaystyle \mathcal{L}=\cup_{x\in X}\mathcal{F}_x/\mathfrak{m}_x\mathcal{F}_x,$
given the coarsest topology for which sections of $ \mathcal{F}$ define continuous functions in a vector bundle of complex dimension 1 over $ X$, with the obvious map taking the stalk over a point to that point.



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Cross-references: point, stalk, map, obvious, dimension, continuous functions, topology, sections, holomorphic, sheaf, variety, algebraic, non-singular, equivalent, vector bundle, complex, real, theory, manifold, invertible sheaf, rank, coherent sheaf, locally free, term, algebraic geometry
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This is version 2 of line bundle, born on 2003-03-13, modified 2003-08-22.
Object id is 4104, canonical name is LineBundle.
Accessed 3826 times total.

Classification:
AMS MSC14-00 (Algebraic geometry :: General reference works )

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