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- Azumaya_algebra abstract "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.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 (where Ao is the opposite algebra) is isomorphic to the matrix algebra EndR(A) ≈ Mr(R) via the map sending to the endomorphism x → axb of A.An Azumaya algebra on a scheme X with structure sheaf OX, according to the original Grothendieck seminar, is a sheaf A of OX-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. Milne, Étale Cohomology, starts instead from the definition that it is a sheaf A of OX-algebras whose stalk Ax at each point x is an Azumaya algebra over the local ring OX,x in the sense given above. Two Azumaya algebras A1 and A2 are equivalent if there exist locally free sheaves E1 and E2 of finite positive rank at every point such that where End(Ei) is the endomorphism sheaf of Ei. The Brauer group 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. 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.".
- Azumaya_algebra wikiPageID "3125089".
- Azumaya_algebra wikiPageRevisionID "580414355".
- Azumaya_algebra hasPhotoCollection Azumaya_algebra.
- Azumaya_algebra subject Category:Algebras.
- Azumaya_algebra subject Category:Ring_theory.
- Azumaya_algebra subject Category:Scheme_theory.
- Azumaya_algebra type Abstraction100002137.
- Azumaya_algebra type Algebra106012726.
- Azumaya_algebra type Algebras.
- Azumaya_algebra type Cognition100023271.
- Azumaya_algebra type Content105809192.
- Azumaya_algebra type Discipline105996646.
- Azumaya_algebra type KnowledgeDomain105999266.
- Azumaya_algebra type Mathematics106000644.
- Azumaya_algebra type PsychologicalFeature100023100.
- Azumaya_algebra type PureMathematics106003682.
- Azumaya_algebra type Science105999797.
- Azumaya_algebra comment "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.".
- Azumaya_algebra label "Algèbre d'Azumaya".
- Azumaya_algebra label "Azumaya algebra".
- Azumaya_algebra sameAs Algèbre_d'Azumaya.
- Azumaya_algebra sameAs m.08t1jt.
- Azumaya_algebra sameAs Q2835923.
- Azumaya_algebra sameAs Q2835923.
- Azumaya_algebra sameAs Azumaya_algebra.
- Azumaya_algebra wasDerivedFrom Azumaya_algebra?oldid=580414355.
- Azumaya_algebra isPrimaryTopicOf Azumaya_algebra.