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 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

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

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

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