Tags: #AG Refs: Picard group

invertible sheaves

• Definition: note that \(({\mathsf{Sh}}(X), \otimes_{{\mathcal{O}}_X}, \operatorname{id}_\otimes= {\mathcal{O}}_X)\) forms a symmetric monoidal category, and an object \(A\) is invertible if there exists an object \(B\) with \(A\otimes B \cong \operatorname{id}_\otimes\) – so in this case, \(B\otimes_{{\mathcal{O}}_X} A \cong {\mathcal{O}}_X\). This an invertible sheaf is an invertible object in this framework.

• Theorem: \({\mathcal{F}}\in {\mathsf{Sh}}(X)\) is invertible iff \({\mathcal{F}}\) is locally free of rank 1, so equivalently a line bundle on \(X\).

• \(\operatorname{Pic}(X) \cong H^1(X; {\mathcal{O}}_X^{\times})\).