- Azumaya algebra
In
mathematics , an Azumaya algebra is a generalization ofcentral simple algebra s to "R"-algebras where "R" need not be a field. Such a notion was introduced in a 1951 paper ofGoro Azumaya , for the case where "R" is acommutative local ring . The notion was developed further inring theory , and inalgebraic geometry , whereAlexander Grothendieck made it the basis for his geometric theory of theBrauer group inBourbaki seminar s from 1964-5. 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 "A"o (the
opposite ring ) on "A" by right-multiplication, makes their tensor product isomorphic to the "r"×"r"matrix algebra over "R".For the
scheme theory definition, on a scheme "X" withstructure 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. TheBrauer group under this definition is defined as eqivalence classes of Azumaya algebras, where two algebras "A"1 and "A"2 are equivalent if there exist finite ranklocally free sheaves "E"1 and "E"2 such that:A_1otimesmathrm{End}(E_1) simeq A_2otimesmathrm{End}(E_2).
Here End("E"i) denotes the endomorphism sheaf of "E"i, which is a global matrix algebra. 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 ofYuri Manin . This has helped to clarify the area of obstructions to theHasse principle .
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