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# invertible sheaf

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

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

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## Mathematics Subject Classification

14A99*no label found*

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