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
Canonical name	InvertibleSheaf
Date of creation	2013-03-22 13:52:34
Last modified on	2013-03-22 13:52:34
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	8
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 14A99