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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 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

B ( R ) = { ( a b 0 d ) : a , d ∈ R ∗ ,   b ∈ R } . {\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 .}

Then B(R) is a subgroup of GL2(R).

Nagao's theorem states that in the case that R is the ring K of polynomials in one variable over a field K, the group GL2(R) is the amalgamated product of GL2(K) and B(K) 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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