Nagao’s theorem

For any integral domain $k$, the group of $n\times n$ invertible matrices with coefficients in $k[t]$ is the amalgamated free product of invertible matrices over $k$ and invertible upper triangular matrices over $k[t]$, amalgamated over the upper triangular matrices of $k$. More compactly

$${\mathrm{GL}}_n(k[t])\cong {\mathrm{GL}}_n(k){*}_{B(k)}B(k[t]).$$

Title	Nagao’s theorem
Canonical name	NagaosTheorem
Date of creation	2013-03-22 13:58:31
Last modified on	2013-03-22 13:58:31
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	6
Author	bwebste (988)
Entry type	Theorem
Classification	msc 20G15