# 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 |