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 GL_{2}(*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

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

Then *B*(*R*) is a subgroup of GL_{2}(*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_{2}(*R*) is the amalgamated product of GL_{2}(*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 GL_{2}(*R*) is isomorphic to the fundamental group π_{1}(*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. Zbl 0964.20027.

Nagao, Hirosi (1959). "On GL(2, K[x])". J. Inst. Polytechn., Osaka City Univ., Ser. A. 10: 117–121. MR 0114866. Zbl 0092.02504.

Serre, Jean-Pierre (2003). Trees. Springer. ISBN 3-540-44237-5.

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