general linear group scheme

Definition 1

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

{\mathbb{Z}[Y,X_{11},\ldots,X_{1n},\ldots,X_{n1},\ldots,X_{nn}]}/{\left<Y\det% \begin{pmatrix}X_{11}&\cdots&X_{1n}\\ \vdots&\ddots&\vdots\\ X_{n1}&\cdots&X_{nn}\end{pmatrix}-1\right>}

Observe that if $R$ is any commutative ring, as usual (http://planetmath.org/ExampleOfFunctorOfPointsOfAScheme) with schemes, an $R$ -point of $\operatorname{GL}_{n}$ 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

r\det\begin{pmatrix}r_{11}&\cdots&r_{1n}\\ \vdots&\ddots&\vdots\\ r_{n1}&\cdots&r_{nn}\end{pmatrix}=1.

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

As usual with schemes, we denote the $R$ -points of $\operatorname{GL}_{n}$ by $\operatorname{GL}_{n}(R)$ ; 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