# 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

 ${\mathbb{Z}[Y,X_{11},\ldots,X_{1n},\ldots,X_{n1},\ldots,X_{nn}]}/{\left}$

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 GeneralLinearGroupScheme 2013-03-22 14:11:16 2013-03-22 14:11:16 alozano (2414) alozano (2414) 7 alozano (2414) Example msc 14K99 msc 14A15 msc 14L10 msc 20G15 GeneralLinearGroup