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).