general linear group scheme

Definition 1

Fix a positive integer n. We define the general linear group scheme GLn as the affine schemeMathworldPlanetmath defined by


Observe that if R is any commutative ring, as usual ( with schemes, an R-point of GLn is given by specifying, for each i and j, an element rij that is the image of Xij, and by specifying one other element r such that


In other words, an R-point of GLn is an invertible matrix with entries in R.

As usual with schemes, we denote the R-points of GLn by GLn(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 GroupMathworldPlanetmath).

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