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[parent] general linear group scheme (Example)
Definition 1   Fix a positive integer $ n$. We define the general linear group scheme $ {\mathrm{GL}}_n$ as the affine scheme defined by
$\displaystyle {\mathbb{Z}[Y,X_{11},\ldots,X_{1n},\ldots,X_{n1},\ldots,X_{nn}]} ... ...X_{1n}\ \vdots&\ddots&\vdots\ X_{n1}&\cdots&X_{nn} \end{pmatrix}-1\right>} $

Observe that if $ R$ is any commutative ring, as usual 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

$\displaystyle 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 $ {\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).



"general linear group scheme" is owned by alozano. [ full author list (2) | owner history (1) ]
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See Also: general linear group


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Cross-references: general linear group, matrix, invertible, words, image, schemes, commutative ring, affine scheme, integer, positive
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This is version 4 of general linear group scheme, born on 2004-02-24, modified 2004-08-09.
Object id is 5617, canonical name is GeneralLinearGroupScheme.
Accessed 2023 times total.

Classification:
AMS MSC: 14K99 (Algebraic geometry :: Abelian varieties and schemes :: Miscellaneous)
 14A15 (Algebraic geometry :: Foundations :: Schemes and morphisms)
 14L10 (Algebraic geometry :: Algebraic groups :: Group varieties)
 20G15 (Group theory and generalizations :: Linear algebraic groups :: Linear algebraic groups over arbitrary fields)
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