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| In [[mathematics]], an '''Azumaya algebra''' is a generalization of [[central simple algebra]]s to ''R''-algebras where ''R'' need not be a [[field (mathematics)|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 seminar]]s from 1964-5. There are now several points of access to the basic definitions.
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| 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]] <math>A \otimes_R A^\circ</math> (where ''A''<sup>o</sup> is the [[opposite ring|opposite algebra]]) is isomorphic to the [[matrix algebra]] End<sub>''R''</sub>(''A'') ≈ M<sub>''r''</sub>(''R'') via the map sending <math>a \otimes b</math> to the endomorphism ''x'' → ''axb'' of ''A''.
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| An Azumaya algebra on a scheme ''X'' with [[structure sheaf]] ''O''<sub>''X''</sub>, according to the original Grothendieck seminar, is a sheaf ''A'' of ''O''<sub>''X''</sub>-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 ''O''<sub>''X''</sub>-algebras whose stalk ''A''<sub>''x''</sub> at each point ''x'' is an Azumaya algebra over the [[local ring]] ''O''<sub>''X,x''</sub> in the sense given above.
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| Two Azumaya algebras ''A''<sub>1</sub> and ''A''<sub>2</sub> are ''equivalent'' if there exist [[locally free sheaves]] ''E''<sub>1</sub> and ''E''<sub>2</sub> of finite positive rank at every point such that
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| :<math>A_1\otimes\mathrm{End}(E_1) \simeq A_2\otimes\mathrm{End}(E_2),</math> | |
| where End(''E''<sub>i</sub>) is the endomorphism sheaf of ''E''<sub>''i''</sub>. 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.
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| 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.
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| ==References==
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| *{{citation | last=Knus | first=Max-Albert | title=Quadratic and Hermitian forms over rings | series=Grundlehren der Mathematischen Wissenschaften | volume=294 | location=Berlin etc. | publisher=[[Springer-Verlag]] | year=1991 | isbn=3-540-52117-8 | zbl=0756.11008 }}
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| *{{Citation | last1=Knus | first1=Max-Albert | last2=Ojanguren | first2=Manuel | title=Théorie de la descente et algèbres d'Azumaya | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | doi=10.1007/BFb0057799 | mr=0417149 | zbl=0284.13002 | year=1974 | volume=389}}
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| * {{cite book | last=Saltman | first=David J. | title=Lectures on division algebras | series=Regional Conference Series in Mathematics | volume=94 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1999 | isbn=0-8218-0979-2 | zbl=0934.16013 }}
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| [[Category:Ring theory]]
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| [[Category:Scheme theory]]
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| [[Category:Algebras]]
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