# general linear group scheme

###### Definition 1

Fix a positive integer $n$. We define the *general linear group scheme* ${\mathrm{GL}}_{n}$ as the affine scheme^{} defined by

$$ |

Observe that if $R$ is any commutative ring, as usual (http://planetmath.org/ExampleOfFunctorOfPointsOfAScheme) with schemes, an $R$-point of ${\mathrm{GL}}_{n}$ is given by specifying, for each $i$ and $j$, an element ${r}_{ij}$ that is the image of ${X}_{ij}$, and by specifying one other element $r$ such that

$$rdet\left(\begin{array}{ccc}\hfill {r}_{11}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {r}_{1n}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {r}_{n1}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {r}_{nn}\hfill \end{array}\right)=1.$$ |

In other words, an $R$-point of ${\mathrm{GL}}_{n}$ is an invertible matrix with entries in $R$.

As usual with schemes, we denote the $R$-points of ${\mathrm{GL}}_{n}$ by ${\mathrm{GL}}_{n}(R)$; we see that this notion does not lead to confusion, since it is exactly what is meant by the usual usage of this notation (see entry General Linear Group^{}).

Title | general linear group scheme |
---|---|

Canonical name | GeneralLinearGroupScheme |

Date of creation | 2013-03-22 14:11:16 |

Last modified on | 2013-03-22 14:11:16 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 7 |

Author | alozano (2414) |

Entry type | Example |

Classification | msc 14K99 |

Classification | msc 14A15 |

Classification | msc 14L10 |

Classification | msc 20G15 |

Related topic | GeneralLinearGroup |