---
date: 2021-10-21 18:42
modification date: Friday 22nd October 2021 15:23:59
title: Picard group
aliases: [Picard group, PIcard groupoid, Picard stack, Picard, Picard category]

---

Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #AG #NT 
- Refs:
	- #todo/add-references
- Links:
	- [examples of Picard group computations](examples%20of%20Picard%20group%20computations.md)

---

# Picard group


## Symmetric monoidal categories

![](attachments/Pasted%20image%2020220505162503.png)

## Presentable infty categories

![](attachments/Pasted%20image%2020220505164000.png)

## Classical

See [invertible module](Unsorted/invertible%20module.md)

![](attachments/Pasted%20image%2020220126101253.png)
![](attachments/Pasted%20image%2020220126101220.png)
![](attachments/Pasted%20image%2020220214091859.png)

- For a [line bundle](line%20bundle.md) on $X$ with the tensor product.
	- $\Pic \spec R = \Cl(R)$ is the class group for a [Dedekind domain](Dedekind%20domain.md)
	- Globalizes the notion of a [number field](number%20field.md).?

- Alternatively, the Picard group can be defined as the [sheaf cohomology](sheaf%20cohomology.md) 
$$H^{1} (X,{\mathcal {O}}_{X}\units).$$

- Fits into a SES
$$
0\to \Pic^0(V) \to\Pic(V) \to \NS(V) \to 0
$$
	where $\NS$ is the [[Neron Severi]] group.

	- This may require that $V$ is a [[Jacobian]]? 
	Or something special happens when it is? #unanswered_questions 

# Picard group of a manifold

![attachments/Pasted image 20210510011342.png](attachments/Pasted%20image%2020210510011342.png)

![attachments/Pasted image 20210603195814.png](attachments/Pasted%20image%2020210603195814.png)
![attachments/Pasted image 20210603195858.png](attachments/Pasted%20image%2020210603195858.png)

![attachments/Pasted image 20210626203400.png](attachments/Pasted%20image%2020210626203400.png)

# Pic 0

![](attachments/Pasted%20image%2020220214093435.png)

# Picard stacks

![](attachments/Pasted%20image%2020220516183841.png)
![](attachments/Pasted%20image%2020220516183856.png)