Nagao's theorem

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In mathematics, Nagao's theorem, named after Hirosi Nagao, is a result about the structure of the group of 2-by-2 invertible matrices over the ring of polynomials over a field. It has been extended by Jean-Pierre 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 general ring R let GL₂(R) denote the group of invertible 2-by-2 matrices with entries in R, and let R^* denote the group of units of R, and let

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

Then B(R) is a subgroup of GL₂(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 GL₂(R) is the amalgamated product of GL₂(K) and B(K[t]) over their intersection B(K).

Serre's extension

In this setting, C is a 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 with at most one non-terminal vertex, such that GL₂(R) is isomorphic to the fundamental group π₁(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 | doi-access=free }}
{{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:Theorems in group theory