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invertible sheaf (Definition)

A sheaf $ \L$ of $ \O _X$ modules on a ringed space $ \O _X$ is called invertible if there is another sheaf of $ \O _X$-modules $ \L '$ such that $ \L\otimes\L '\cong\O _X$. A sheaf is invertible if and only if it is locally free of rank 1, and its inverse is the sheaf $ \L ^{\vee}\cong\mathcal{H}om(\L ,\O _X)$, by the obvious map.

The set of invertible sheaves obviously form an abelian group under tensor multiplication, called the Picard group of $ X$.



"invertible sheaf" is owned by Mathprof. [ full author list (2) | owner history (1) ]
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Cross-references: Picard group, multiplication, tensor, abelian group, map, inverse, rank, locally free, ringed space, modules, sheaf
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This is version 5 of invertible sheaf, born on 2003-08-19, modified 2006-10-06.
Object id is 4619, canonical name is InvertibleSheaf.
Accessed 1974 times total.

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