---
date: 2022-01-15 21:49
modification date: Monday 24th January 2022 10:08:42
title: Brauer group
aliases: ["Brauer group", "Brauer groups", "Brauer", formal Brauer group]

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- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [Azumaya algebra](Azumaya%20algebra)
	- [torsor](Unsorted/torsor.md)
	- [example computations of Brauer groups](example%20computations%20of%20Brauer%20groups.md)

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# Brauer group

Definition: $(\Br(k), \tensor_k)$ is the group of [central simple algebras](Unsorted/central%20simple%20algebra.md) algebras over $k$, up to Morita equivalence.

![](attachments/Pasted%20image%2020220806225912.png)

A higher invariant that extends the Picard group?
Computed via descent techniques?

Measures complexity of a field by asking how many iso classes of finite dimensional [central simple algebra](central%20simple%20algebra.md) exist over $k$.

For fields, equivalent to [Galois cohomology](Galois%20cohomology):
![](attachments/Pasted%20image%2020220319201026.png)

More generally, $H^2(X;\GG_m)$ for $X\in \Sch$ classifies [gerbes](Unsorted/gerbe.md) over $X$ with structure group $\GG_m$.

Finiteness of the Brauer group for scheme-theoretic surfaces is equivalent to the [Tate conjecture](Tate%20conjecture) for [divisors](Unsorted/divisor.md) on $X$, i.e. finiteness of [Sha](Unsorted/Tate-Shafarevich%20group.md) of the [Jacobian](Unsorted/Jacobian.md) variety of the [general fiber](general%20fiber).

Used in the definition of the [Brauer-Manin obstruction](Brauer-Manin%20obstruction) which obstructs lifting [adelic](adelic) points to [Unsorted/rational points](Unsorted/rational%20points.md).

![](attachments/Pasted%20image%2020220408125016.png)


See [spectral Brauer group](spectral%20Brauer%20group.md)
![](attachments/Pasted%20image%2020220505164502.png)

![](attachments/Pasted%20image%2020220510161509.png)
![](attachments/Pasted%20image%2020220510161548.png)