en>Magioladitis: WP:CHECKWIKI error fixes using AWB (10515)

2014-12-14T10:05:16Z

WP:CHECKWIKI error fixes using AWB (10515)

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	~~In algebraic geometry, a~~ [[~~local ring~~]~~] ''A''~~ is ~~said~~ to ~~be '''unibranch''' if~~ the ~~[[reduced ring]] ''A''~~<~~sub~~>~~red~~<~~/sub~~> ~~(obtained by quotienting ''A'' by its~~ [~~[nilradical of a ring\|nilradical]~~]) is ~~an [[integral domain]]~~, and ~~the [[integral closure]] ''B~~'' of ''A''<~~sub~~>~~red~~<~~/sub~~> ~~is also~~ a ~~local ring~~.~~{{fact\|date=August 2012}} A unibranch local ring is said~~ to ~~be '''geometrically unibranch''' if~~ the ~~[[residue~~ field~~]] of ''B''~~ is a ~~[[purely inseparable extension]] of~~ the ~~residue field~~ of ~~''A''~~<~~sub~~>~~red~~</~~sub>~~. ~~A complex variety ''X''~~ is ~~called~~ '~~''topologically unibranch''' at~~ a ~~point ''x'' if for all complements ''Y''~~ of ~~closed algebraic subsets~~ of ~~''X'' there is~~ a ~~fundamental system~~ of ~~neighborhoods~~ (in the ~~classical topology) of ''x'' whose intersection~~ with ~~''Y''~~ is ~~connected~~.		The Benchmade HUNT Knives are right here. 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	~~In particular,~~ a ~~[[normal ring]]~~ is ~~unibranch. The notions~~ of ~~unibranch and geometrically unibranch points are used~~ in ~~some theorems in algebraic geometry~~. ~~For example~~, ~~there~~ is the ~~following result:~~

	'~~''Theorem''' {{harv\|EGA\|loc=4~~.3.~~7}} Let ''X'' and ''Y''~~ be ~~two integral locally noetherian schemes~~ and <~~math~~>~~f \colon X \to Y~~<~~/math~~> a ~~[[proper morphism\|proper]] [[dominant morphism]]~~. ~~Denote their function fields by~~ '~~'K(X~~)'' and ''~~K(Y)'', respectively~~. ~~Suppose that~~ the ~~algebraic closure~~ of ~~''K(Y)'' in ''K(X)'' has separable degree ''n''~~ and ~~that~~ <~~math~~>~~y \in Y~~<~~/math~~> is ~~unibranch. Then~~ the ~~fiber <math>f^{-~~1~~}(y)</math> has at most ''n'' connected components~~. ~~In particular~~, ~~if ''f'' is [[birational]]~~, ~~then the fibers~~ of ~~unibranch points are connected.~~

	~~In EGA~~, ~~the theorem is obtained~~ as a ~~corollary~~ of ~~[[Zariski's main theorem]]~~.

	~~==References==~~
	*{{EGA \|book=III-1}}

	~~[[Category:Algebraic geometry\|*]]~~
	~~[[Category:Commutative algebra\| ]]~~

en>Rjwilmsi: /* References */Journal cites, added 1 DOI using AWB (9888)

2014-01-25T23:53:58Z

References: Journal cites, added 1 DOI using AWB (9888)

← Older revision		Revision as of 01:53, 26 January 2014
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	~~Hello~~, I'~~m Alton~~, a ~~22 year old from Nijmegen, Netherlands~~.<br>~~My hobbies include~~ (~~but~~ are ~~not limited~~ to) ~~Shortwave listening~~, ~~Golfing~~ and ~~watching Family Guy~~.<br><br>~~Feel free to surf to my blog post :~~: [~~http~~:~~//originsofttech.net/AgaArcade/profile/librain Womens mountain bike sizing.~~]		In algebraic geometry, a [[local ring]] ''A'' is said to be '''unibranch''' if the [[reduced ring]] ''A''<sub>red</sub> (obtained by quotienting ''A'' by its [[nilradical of a ring\|nilradical]]) is an [[integral domain]], and the [[integral closure]] ''B'' of ''A''<sub>red</sub> is also a local ring.{{fact\|date=August 2012}} A unibranch local ring is said to be '''geometrically unibranch''' if the [[residue field]] of ''B'' is a [[purely inseparable extension]] of the residue field of ''A''<sub>red</sub>. A complex variety ''X'' is called '''topologically unibranch''' at a point ''x'' if for all complements ''Y'' of closed algebraic subsets of ''X'' there is a fundamental system of neighborhoods (in the classical topology) of ''x'' whose intersection with ''Y'' is connected.

			In particular, a [[normal ring]] is unibranch. The notions of unibranch and geometrically unibranch points are used in some theorems in algebraic geometry. For example, there is the following result:

			'''Theorem''' {{harv\|EGA\|loc=4.3.7}} Let ''X'' and ''Y'' be two integral locally noetherian schemes and <math>f \colon X \to Y</math> a [[proper morphism\|proper]] [[dominant morphism]]. Denote their function fields by ''K(X)'' and ''K(Y)'', respectively. Suppose that the algebraic closure of ''K(Y)'' in ''K(X)'' has separable degree ''n'' and that <math>y \in Y</math> is unibranch. Then the fiber <math>f^{-1}(y)</math> has at most ''n'' connected components. In particular, if ''f'' is [[birational]], then the fibers of unibranch points are connected.

			In EGA, the theorem is obtained as a corollary of [[Zariski's main theorem]].

			==References==
			*{{EGA \|book=III-1}}

			[[Category:Algebraic geometry\|*]]
			[[Category:Commutative algebra\| ]]

en>Melcombe: split lead and simplify, add wikilinks

2012-01-15T16:38:55Z

split lead and simplify, add wikilinks

New page

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en>Magioladitis: WP:CHECKWIKI error fixes using AWB (10515)

en>Rjwilmsi: /* References */Journal cites, added 1 DOI using AWB (9888)

en>Melcombe: split lead and simplify, add wikilinks