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en>Wavelength: inserting 1 hyphen: —> "one-dimensional"—wikt:one-dimensional

2014-02-20T17:42:13Z

inserting 1 hyphen: —> "one-dimensional"—wikt:one-dimensional

← Older revision		Revision as of 19:42, 20 February 2014
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	~~In mathematics~~, ~~'''Nagao's theorem'''~~, ~~named~~ after ~~Hirosi Nagao~~, is ~~a result about the structure~~ of ~~the [[Group (mathematics)\|group]]~~ of ~~2-by-2 [[Invertible matrix\|invertible matrices]] over~~ the ~~[[ring of polynomials]] over a [[Field (mathematics)\|field]]~~. ~~It has been extended by [[Jean-Pierre Serre\|Serre]] to give a description of~~ the ~~structure~~ of the ~~corresponding matrix group over the [[coordinate ring]]~~ of ~~a [[projective curve]]~~.

	~~==Nagao's theorem==~~

	~~For a [[Ring (mathematics)\|general ring]] ''R'' we let GL~~<~~sub~~>2</sub>(''R'') denote the group of invertible 2-by-2 matrices with entries in ''R'', and let ''R''<sup>*</~~sup~~> ~~denote the [[Group_of_units#Group_of_units\|group of units]] of ''R'', and let~~

	: <~~math~~> ~~B(R) = \left\lbrace{~~ ~~\left({\begin{array}{{20}c} a & b \\ 0 & d \end{array}}\right)~~ : ~~a,d \in R^, ~ b \in R }\right\rbrace~~. </~~math>~~

	~~Then ''B''(''R'') is a subgroup of GL<sub>2<~~/~~sub>(''R'')~~.

	~~Nagao's theorem states that in the case that ''R'' is the ring ''K''[''t''] of polynomials in one variable over a field ''K'', the group GL~~<~~sub~~>2</~~sub>(''R'') is the [[amalgamated product]] of GL<sub>2<~~/~~sub>(''K'') and ''B''(''K''[''t'']) over their intersection ''B''(''K'')~~.

	~~==Serre's extension==~~
	~~In this setting, ''C'' is a [[Singular point of an algebraic variety\|smooth]] projective curve ''C'' over a field ''K''~~. ~~For a [[closed point]] ''P'' of ''C'' let ''R'' be the corresponding coordinate ring of ''C'' with ''P'' removed~~. ~~There exists a [[graph of groups]] (''G'',''T'') where ''T'' is a [[Tree (graph theory)\|tree]] with at most one non-terminal vertex, such that GL~~<~~sub~~>2</~~sub>(''R'') is isomorphic~~ to ~~the~~ [~~[fundamental group]] π<sub>1<~~/~~sub>(''G'',''T'')~~.

	~~==References==~~
	* {{cite journal \| last=Mason \| first=A. ~~\| year=2001 \| title=Serre's generalization of Nagao's theorem: an elementary approach \| journal=Transactions of the American Mathematical Society \| volume=353 \| pages=749–767 \| doi=10.1090~~/~~S0002-9947-00-02707-0 \| zbl=0964~~.~~20027 \| issue=2 }}~~
	* {{cite journal \| zbl=0092.02504 \| mr=0114866 \| last=Nagao \| first=Hirosi \| title=On GL(2, ''K''[''x'']~~) \| journal=J~~. ~~Inst~~. ~~Polytechn., Osaka City Univ., Ser~~. ~~A \| volume~~=~~10 \| pages=117–121 \| year=1959 }}~~
	* {{cite book \| title=Trees \| first=Jean-Pierre \| last=Serre \| authorlink=Jean-Pierre Serre \| publisher=Springer \| year=2003 \| isbn=3-540-44237-5 }}

	~~[[Category:Group theory]]~~

	~~{{algebra-stub}}~~

en>Mojdadyr at 12:44, 8 October 2013

2013-10-08T12:44:58Z

← Older revision		Revision as of 14:44, 8 October 2013
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			In mathematics, '''Nagao's theorem''', named after Hirosi Nagao, is a result about the structure of the [[Group (mathematics)\|group]] of 2-by-2 [[Invertible matrix\|invertible matrices]] over the [[ring of polynomials]] over a [[Field (mathematics)\|field]]. It has been extended by [[Jean-Pierre Serre\|Serre]] to give a description of the structure of the corresponding matrix group over the [[coordinate ring]] of a [[projective curve]].

			==Nagao's theorem==

			For a [[Ring (mathematics)\|general ring]] ''R'' we let GL<sub>2</sub>(''R'') denote the group of invertible 2-by-2 matrices with entries in ''R'', and let ''R''<sup>*</sup> denote the [[Group_of_units#Group_of_units\|group of units]] of ''R'', and let

			: <math> B(R) = \left\lbrace{ \left({\begin{array}{{20}c} a & b \\ 0 & d \end{array}}\right) : a,d \in R^, ~ b \in R }\right\rbrace. </math>

			Then ''B''(''R'') is a subgroup of GL<sub>2</sub>(''R'').

			Nagao's theorem states that in the case that ''R'' is the ring ''K''[''t''] of polynomials in one variable over a field ''K'', the group GL<sub>2</sub>(''R'') is the [[amalgamated product]] of GL<sub>2</sub>(''K'') and ''B''(''K''[''t'']) over their intersection ''B''(''K'').

			==Serre's extension==
			In this setting, ''C'' is a [[Singular point of an algebraic variety\|smooth]] projective curve ''C'' over a field ''K''. For a [[closed point]] ''P'' of ''C'' let ''R'' be the corresponding coordinate ring of ''C'' with ''P'' removed. There exists a [[graph of groups]] (''G'',''T'') where ''T'' is a [[Tree (graph theory)\|tree]] with at most one non-terminal vertex, such that GL<sub>2</sub>(''R'') is isomorphic to the [[fundamental group]] π<sub>1</sub>(''G'',''T'').

			==References==
			* {{cite journal \| last=Mason \| first=A. \| year=2001 \| title=Serre's generalization of Nagao's theorem: an elementary approach \| journal=Transactions of the American Mathematical Society \| volume=353 \| pages=749–767 \| doi=10.1090/S0002-9947-00-02707-0 \| zbl=0964.20027 \| issue=2 }}
			* {{cite journal \| zbl=0092.02504 \| mr=0114866 \| last=Nagao \| first=Hirosi \| title=On GL(2, ''K''[''x'']) \| journal=J. Inst. Polytechn., Osaka City Univ., Ser. A \| volume=10 \| pages=117–121 \| year=1959 }}
			* {{cite book \| title=Trees \| first=Jean-Pierre \| last=Serre \| authorlink=Jean-Pierre Serre \| publisher=Springer \| year=2003 \| isbn=3-540-44237-5 }}

			[[Category:Group theory]]

			{{algebra-stub}}

en>SwisterTwister: Unreferenced article

2012-07-29T05:48:09Z

Unreferenced article

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