---
date: 2022-01-15 21:49
modification date: Friday 28th January 2022 22:17:31
title: invertible sheaves
aliases: [invertible sheaf]

---

Tags: 
#AG 
Refs:
[Picard group](Unsorted/Picard%20group.md)

# invertible sheaves

- Definition: note that $(\Sh(X), \tensor_{\OO_X}, \id_\tensor = \OO_X)$ forms a [symmetric monoidal category](symmetric%20monoidal%20category), and an object $A$ is *invertible* if there exists an object $B$ with $A\tensor B \cong \id_\tensor$ -- so in this case, $B\tensor_{\OO_X} A \cong \OO_X$. This an **invertible sheaf** is an invertible object in this framework. 

- Theorem: $\mcf \in \Sh(X)$ is invertible iff $\mcf$ is [locally free](locally%20free) of rank 1, so equivalently a [line bundle](line%20bundle.md) on $X$.

- $\Pic(X) \cong H^1(X; \OO_X\units)$.

![](attachments/Pasted%20image%2020210626203302.png)