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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,&nbsp;''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}}

Latest revision as of 18:42, 20 February 2014

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