# invertible sheaf

A sheaf $\mathfrak{L}$ of $\mathcal{O}_{X}$ modules on a ringed space $\mathcal{O}_{X}$ is called 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 map.

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

Title invertible sheaf InvertibleSheaf 2013-03-22 13:52:34 2013-03-22 13:52:34 Mathprof (13753) Mathprof (13753) 8 Mathprof (13753) Definition msc 14A99