Feigenbaum function - Revision history

140.103.56.231: /* Functional equation */

2014-09-15T21:57:11Z

Functional equation

← Older revision		Revision as of 23:57, 15 September 2014
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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~~.		Greetings! I am [http://Www.Myelectronicmd.com/get_reference.php?Id=1296&condition=ANAL%20HERPES%20SIMPLEX%20VIRUS&symname=A&typ=3 Marvella] and I really feel comfortable when people use the complete name. What I love performing is to collect std testing at home badges but I've been using on new things recently. Puerto Rico is exactly where he's always been [http://Rhrealitycheck.org/article/2013/02/22/trich-the-most-common-sexually-transmitted-infection-you-may-never-have-heard-of/ residing] but she requirements to [http://www.revleft.com/vb/member.php?u=160656 at home std testing] move simply because of her family members. Since she was eighteen she's been operating as a receptionist but her promotion by [http://www.alemcheap.fi/people/anhipkiss http://www.alemcheap.fi] no means comes.<br><br>Take a look at my webpage ... over the counter home std test kit std test ([http://xrambo.com/user/NEme Link Home Page])

	~~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''.~~

	~~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.~~
	~~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~~
	:~~<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~~.

	~~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==~~
	*{{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 }}
	*{{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}}~~
	* {{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 }}

	~~[[Category:Ring theory]]~~
	~~[[Category:Scheme theory]]~~
	~~[[Category:Algebras]~~]

en>Bgwhite: WP:CHECKWIKI error fix. Broken bracket problem. Do general fixes and cleanup if needed. - using AWB (9449)

2013-08-29T09:05:53Z

WP:CHECKWIKI error fix. Broken bracket problem. Do general fixes and cleanup if needed. - using AWB (9449)

← Older revision		Revision as of 11:05, 29 August 2013
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	[~~http://www.webmdbook.com/index.php?do=/profile~~-~~11685/info/ at home std test~~] ~~Greetings~~. ~~Allow me begin by telling you~~ the ~~writer~~'~~s title - Phebe. He utilized to be unemployed but now he~~ is a ~~pc operator but his promotion by no means comes~~. ~~My home std test kit family life~~ in ~~Minnesota~~ and ~~my family loves~~ it. ~~One~~ of the ~~extremely best issues in~~ the ~~world for me~~ is ~~home std test~~ to do [~~http:~~//~~water.Me.vccs.edu~~/~~courses/env108/lesson5_2~~.~~htm aerobics~~] ~~and I~~'~~ve been doing it for quite [http:~~//~~www~~.~~hotporn123.com~~/~~user~~/~~RPocock~~ at ~~home std testing~~] ~~a over~~ the ~~counter std test whilst~~.<br><br>~~Here is my webpage ::~~ [~~https~~://~~alphacomms~~.~~zendesk~~.~~com/entries/53455304~~-~~Curing~~-~~Your~~-~~Candida~~-~~Albicans~~-~~How~~-To-~~Make~~-It-~~Happen~~-~~Easily alphacomms~~.~~zendesk.com~~]		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.

			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''.

			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.
			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
			:<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.

			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==
			*{{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 }}
			*{{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}}
			* {{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 }}

			[[Category:Ring theory]]
			[[Category:Scheme theory]]
			[[Category:Algebras]]

en>R. J. Mathar: /* References */ added Varin preprint

2011-09-08T11:18:48Z

References: added Varin preprint

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