general linear group schemeGeneralLinearGroupScheme2004-02-24 02:39:292004-08-09 14:21:10Examplegroup scheme% this is the default PlanetMath preamble. as your knowledge
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\begin{defn}
Fix a positive integer $n$. We define the \emph{general linear group scheme} $\GL_n$ as the affine scheme defined by
\[
{\mathbb{Z}[Y,X_{11},\ldots,X_{1n},\ldots,X_{n1},\ldots,X_{nn}]}
/
{\left<Y\det\begin{pmatrix}
X_{11}&\cdots&X_{1n}\\
\vdots&\ddots&\vdots\\
X_{n1}&\cdots&X_{nn}
\end{pmatrix}-1\right>}
\]
\end{defn}
Observe that if $R$ is any commutative ring, as \PMlinkname{usual}{ExampleOfFunctorOfPointsOfAScheme} with schemes, an $R$-point of $\GL_n$ 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 $\GL_n$ is an invertible matrix with entries in $R$.
As usual with schemes, we denote the $R$-points of $\GL_n$ by $\GL_n(R)$; 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).