Feigenbaum function

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 $A{\otimes }_R{A}^{\circ }$ (where A^o is the opposite algebra) is isomorphic to the matrix algebra End_R(A) ≈ M_r(R) via the map sending $a\otimes b$ to the endomorphism x → axb of A.

An Azumaya algebra on a scheme X with structure sheaf O_X, according to the original Grothendieck seminar, is a sheaf A of O_X-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_X-algebras whose stalk A_x at each point x is an Azumaya algebra over the local ring O_X,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

$A_1\otimes \mathrm{E}\mathrm{n}\mathrm{d}(E_1)\simeq A_2\otimes \mathrm{E}\mathrm{n}\mathrm{d}(E_2),$

where End(E_i) is the endomorphism sheaf of E_i. 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.

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
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