Nagao's theorem

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 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 we 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

[math]\displaystyle{ 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 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

• Mason, A. (2001). "Serre's generalization of Nagao's theorem: an elementary approach". Transactions of the American Mathematical Society 353 (2): 749–767. doi:10.1090/S0002-9947-00-02707-0. 
• Nagao, Hirosi (1959). "On GL(2, K[x])". J. Inst. Polytechn., Osaka City Univ., Ser. A 10: 117–121. 
• Serre, Jean-Pierre (2003). Trees. Springer. ISBN 3-540-44237-5.