In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Ring theory and related areas of Mathematics a central simple algebra ( CSA) over a field K (also called a Brauer algebra In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where R is a commutative local ring. Goro Azumaya (1920- was a Japanese Mathematician who introduced the notion of Azumaya algebra in 1951 In Mathematics, more particularly in Abstract algebra, local rings are certain rings that are comparatively simple and serve to describe what is called 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. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany In Mathematics, the Brauer group arose out of an attempt to classify Division algebras over a given field K. 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 There are now several points of access to the basic definitions.
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 Ao (the opposite ring) on A by right-multiplication, makes their tensor product isomorphic to the r×r matrix algebra over R. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, the Tensor product of two ''R''-algebras is also an R -algebra in a natural way
For the scheme theory definition, on a scheme X with structure sheaf OX the definition as in the original Grothendieck seminar is of a sheaf of OX-algebras A that is locally isomorphic to a matrix algebra sheaf. In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on Milne, Étale Cohomology, starts instead from the definition that the stalks Ax are Azumaya algebras over the local rings OX,x at each point, in the sense given above. The Brauer group under this definition is defined as eqivalence classes of Azumaya algebras, where two algebras A1 and A2 are equivalent if there exist finite rank locally free sheaves E1 and E2 such that
Here End(Ei) denotes the endomorphism sheaf of Ei, which is a global matrix algebra. In Mathematics, the Brauer group arose out of an attempt to classify Division algebras over a given field K. In Sheaf theory, a field of mathematics a sheaf of \mathcal{O} _X-modules \mathcal{F} on a Ringed space X is called locally free The group operation is given by tensor product, and the inverse by the opposite algebra.
There have been substantive applications of these global Azumaya algebras in diophantine geometry, following work of Yuri Manin. In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only Yuri Ivanovitch Manin (Юрий Иванович Манин born 1937 Simferopol) is a Russian / German mathematician known for work in Algebraic This has helped to clarify the area of obstructions to the Hasse principle. In Mathematics, Helmut Hasse 's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation
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