71.114.94.216: Fixed incorrect formula

2014-12-27T01:39:51Z

Fixed incorrect formula

← Older revision		Revision as of 03:39, 27 December 2014
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	~~In mathematics, an '''essentially finite vector bundle''' is~~ a ~~particular type~~ of ~~[[vector bundle]] defined by Madhav Nori,<ref>M~~. V. ~~Nori~~ '~~'On~~ the ~~Representations of~~ the ~~Fundamental Group'', Compositio Mathematica, Vol~~. ~~33, Fasc~~. ~~1, (1976), p~~. ~~29–42~~<~~/ref~~><~~ref~~>T. ~~Szamuely~~ '~~'Galois Groups and Fundamental Groups~~.~~'' Cambridge Studies in Advanced Mathematics~~, ~~Vol. 117~~ (~~2009~~)~~</ref> as the main tool in the construction~~ of the ~~[[fundamental group scheme]]~~. Even if the ~~definition~~ is ~~not intuitive there is a nice characterization that makes essentially finite vector bundles quite natural objects to study in [[algebraic geometry]]~~. ~~So before recalling the definition we give this characterization:~~		Getting the very best kitchen knives can be a difficult job to achieve with so a lot of distinct brands and tends to make out there. 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	~~==Characterization==~~
	~~Let~~ <~~math~~>X<~~/math~~> be a ~~reduced and connected [[Scheme~~ (~~mathematics~~)~~\|scheme]] over~~ a ~~perfect [[field~~ (~~mathematics~~)~~\|field]]~~ <~~math~~>k<~~/math~~> ~~endowed with~~ a ~~section <math>x\in X~~(k)~~</math>~~. ~~Then~~ a ~~vector bundle~~ <~~math~~>V<~~/math~~> ~~over~~ <~~math~~>X<~~/math~~> is ~~essentially finite if~~ and ~~only if there exists a~~ [~~[Finite morphism\|finite]~~] <~~math~~>k<~~/math~~>-[~~[group scheme]] <math>G<~~/~~math> and a <math>G<~~/~~math>~~-~~[[torsor]~~] <~~math~~>~~p:P\to X~~<~~/math~~> such that <~~math~~>V<~~/math~~> ~~becomes trivial over <math>P</math> (i~~.e. ~~<math>p^*(V)\cong O_P^{\oplus r}</math>~~, ~~where <math>r=rk~~(~~V)</math>~~).

	~~==Definition==~~

	~~{{Empty section\|date=October 2012}}~~

	~~==Notes==~~
	~~<references/>~~

	~~[[Category:Scheme theory]]~~
	~~[[Category:Topological methods of algebraic geometry]]~~

198.137.20.217 at 17:05, 1 February 2014

2014-02-01T17:05:46Z

← Older revision		Revision as of 19:05, 1 February 2014
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	~~Hello from Denmark~~. I'~~m glad to came here~~. ~~My first name is Crystal~~. <br>~~I live~~ in a ~~small town called Kobenhavn K~~ in ~~nothern Denmark~~.<br>~~I was also born~~ in ~~Kobenhavn K 40 years ago~~. ~~Married in November year 2010~~. ~~I'm working at the post office~~.		In mathematics, an '''essentially finite vector bundle''' is a particular type of [[vector bundle]] defined by Madhav Nori,<ref>M. V. Nori ''On the Representations of the Fundamental Group'', Compositio Mathematica, Vol. 33, Fasc. 1, (1976), p. 29–42</ref><ref>T. Szamuely ''Galois Groups and Fundamental Groups.'' Cambridge Studies in Advanced Mathematics, Vol. 117 (2009)</ref> as the main tool in the construction of the [[fundamental group scheme]]. Even if the definition is not intuitive there is a nice characterization that makes essentially finite vector bundles quite natural objects to study in [[algebraic geometry]]. So before recalling the definition we give this characterization:

			==Characterization==
			Let <math>X</math> be a reduced and connected [[Scheme (mathematics)\|scheme]] over a perfect [[field (mathematics)\|field]] <math>k</math> endowed with a section <math>x\in X(k)</math>. Then a vector bundle <math>V</math> over <math>X</math> is essentially finite if and only if there exists a [[Finite morphism\|finite]] <math>k</math>-[[group scheme]] <math>G</math> and a <math>G</math>-[[torsor]] <math>p:P\to X</math> such that <math>V</math> becomes trivial over <math>P</math> (i.e. <math>p^*(V)\cong O_P^{\oplus r}</math>, where <math>r=rk(V)</math>).

			==Definition==

			{{Empty section\|date=October 2012}}

			==Notes==
			<references/>

			[[Category:Scheme theory]]
			[[Category:Topological methods of algebraic geometry]]

en>ZéroBot: r2.7.1) (Robot: Adding ru:Шаровой сектор

2012-04-19T11:35:04Z

r2.7.1) (Robot: Adding ru:Шаровой сектор

New page

Hello from Denmark. I'm glad to came here. My first name is Crystal. <br>I live in a small town called Kobenhavn K in nothern Denmark.<br>I was also born in Kobenhavn K 40 years ago. Married in November year 2010. I'm working at the post office.

Spherical sector - Revision history

71.114.94.216: Fixed incorrect formula

198.137.20.217 at 17:05, 1 February 2014

en>ZéroBot: r2.7.1) (Robot: Adding ru:Шаровой сектор