---
tag: VKZTT484WQ
kind: concept
name: Azumaya algebra
msc: [16H05, 16K50]
specializes: [3CTW1P4HBP]
sources:
  - url: https://en.wikipedia.org/wiki/Azumaya_algebra
same_as:
  wikidata: Q2835923
  wikipedia: Azumaya algebra
---

## Summary

An Azumaya algebra over a [[commutative ring|0310REJJY8]] $R$ is an [[associative algebra|3CTW1P4HBP]] that behaves like a [[central simple algebra|5HRKJBAY2V]] over every point of $R$.
It is a faithful, finitely generated projective $R$-module $A$ for which $A \otimes_R A^{\mathrm{op}} \to \End_R(A)$ is an isomorphism.
Over a field the Azumaya algebras are exactly the central simple algebras.
Azumaya algebras define the [[Brauer group|BEW48TA9WZ]] of a commutative ring and, in geometry, of a scheme.

## Explanations

### L1 human

A central simple algebra over a field looks like a matrix algebra after a change of field.
An Azumaya algebra is the same idea with the field replaced by a commutative ring.
Its fibre at each maximal ideal $\mathfrak{m}$ of $R$ is a central simple algebra over the field $R/\mathfrak{m}$, and these fibres vary nicely.
The simplest example is the ring $M_n(R)$ of $n \times n$ matrices over $R$.