# Azumaya algebra

In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where R is a commutative local ring. The notion was developed further in ring theory, and in algebraic geometry, where Alexander Grothendieck made it the basis for his geometric theory of the Brauer group in Bourbaki seminars from 1964-5. There are now several points of access to the basic definitions.

## Over a ring

An Azumaya algebra over a commutative local ring R is an R-algebra A that is free and of finite rank r≥1 as an R-module, such that the tensor product ${\displaystyle A\otimes _{R}A^{\circ }}$ (where Ao is the opposite algebra) is isomorphic to the matrix algebra EndR(A) ≈ Mr(R) via the map sending ${\displaystyle a\otimes b}$ to the endomorphism ${\displaystyle x\mapsto axb}$ of A.

## On a scheme

An Azumaya algebra on a scheme X with structure sheaf ${\displaystyle {\mathcal {O}}_{X}}$, according to the original Grothendieck seminar, is a sheaf ${\displaystyle {\mathcal {A}}}$ of ${\displaystyle {\mathcal {O}}_{X}}$-algebras that is étale locally isomorphic to a matrix algebra sheaf; one should, however, add the condition that each matrix algebra sheaf is of positive rank. This definition makes an Azumaya algebra on ${\displaystyle (X,{\mathcal {O}}_{X})}$ into a 'twisted-form' of the sheaf ${\displaystyle M_{n}({\mathcal {O}}_{X})}$. Milne, Étale Cohomology, starts instead from the definition that it is a sheaf ${\displaystyle {\mathcal {A}}}$ of ${\displaystyle {\mathcal {O}}_{X}}$-algebras whose stalk ${\displaystyle {\mathcal {A}}_{x}}$ at each point ${\displaystyle x}$ is an Azumaya algebra over the local ring ${\displaystyle {\mathcal {O}}_{X,x}}$ in the sense given above. Two Azumaya algebras ${\displaystyle {\mathcal {A}}_{1}}$ and ${\displaystyle {\mathcal {A}}_{2}}$ are equivalent if there exist locally free sheaves ${\displaystyle {\mathcal {E}}_{1}}$ and ${\displaystyle {\mathcal {E}}_{2}}$ of finite positive rank at every point such that

${\displaystyle A_{1}\otimes \mathrm {End} _{{\mathcal {O}}_{X}}({\mathcal {E}}_{1})\simeq A_{2}\otimes \mathrm {End} _{{\mathcal {O}}_{X}}({\mathcal {E}}_{2}),}$

where ${\displaystyle \mathrm {End} _{{\mathcal {O}}_{X}}({\mathcal {E}}_{i})}$ is the endomorphism sheaf of ${\displaystyle {\mathcal {E}}_{i}}$. The Brauer group ${\displaystyle B(X)}$ of X (an analogue of the Brauer group of a field) is the set of equivalence classes of Azumaya algebras. The group operation is given by tensor product, and the inverse is given by the opposite algebra. Note that this is distinct from the cohomological Brauer group which is defined as ${\displaystyle H^{2}(X_{\text{et}},\mathbb {G} _{m})}$.

## Applications

There have been significant applications of Azumaya algebras in diophantine geometry, following work of Yuri Manin. The Manin obstruction to the Hasse principle is defined using the Brauer group of schemes.

## References

• Knus, Max-Albert (1991), Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften, 294, Berlin etc.: Springer-Verlag, ISBN 3-540-52117-8, Zbl 0756.11008
• Knus, Max-Albert; Ojanguren, Manuel (1974), Théorie de la descente et algèbres d'Azumaya, Lecture Notes in Mathematics, 389, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0057799, MR 0417149, Zbl 0284.13002
• Saltman, David J. (1999). Lectures on division algebras. Regional Conference Series in Mathematics. 94. Providence, RI: American Mathematical Society. ISBN 0-8218-0979-2. Zbl 0934.16013.