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 (born 1920), 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. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K. It is an abelian group with elements isomorphism classes of division algebras over K, such that the center is exactly K. The group is named for the algebraist Richard Brauer. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Alexander Grothendieck (Berlin, March 28, 1928) was one of the most important mathematicians active in the 20th century. ... In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K. It is an abelian group with elements isomorphism classes of division algebras over K, such that the center is exactly K. The group is named for the algebraist Richard Brauer. ... The Séminaire Nicolas Bourbaki (Bourbaki Seminar) is a series of seminars (in fact public lectures with printed notes distributed) that has been held in Paris since 1948. ...

For R a local ring, an Azumaya algebra is an R-algebra A which is free and of finite rank r as an R-module, and for which the natural action of A on itself by left-multiplication, and of A^o (the opposite ring) on A by right-multiplication, makes the tensor product isomorphic to the r×r matrix algebra over R. In mathematics, there is a construction in abstract algebra of the tensor product of commutative rings; which puts a ring structure on the tensor product as abelian groups of two commutative rings R and S. This structure on RZS then makes it a coproduct in the category of commutative rings. ... ...

For the scheme theory definition, on a scheme X with structure sheaf O_X the definition as in the original Grothendieck seminar is of a sheaf of O_X-algebras A that is locally isomorphic to a matrix algebra sheaf. Milne, Étale Cohomology, starts instead from the definition that the stalks A_x are Azumaya algebras over the local rings O_X,x at each point, in the sense given above. The Brauer group under this definition is defined by the tensor product operation, with classes identified by product with (global) matrix algebras, in other words endomorphism sheaves of locally free sheaves of finite rank. In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ... In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K. It is an abelian group with elements isomorphism classes of division algebras over K, such that the center is exactly K. The group is named for the algebraist Richard Brauer. ...

There have been substantive applications of these global Azumaya algebras in diophantine geometry, following work of Yuri Manin. This has helped to clarify the area of obstructions to the Hasse principle. In mathematics, a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. ... Yuri Ivanovitch Manin (born 1937) is a Russian-born mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. ... In mathematics, Helmut Hasses local-global principle, also known as the Hasse principle, is the assertion that an equation can be solved over the rational numbers if and only if it can be solved over the real numbers and over the p-adic numbers for every prime p. ...