General Linear Group Scheme \(\operatorname{GL}_n\)

Let \(n\) be a positive integer and \(S\) a scheme. The general linear group scheme of rank \(n\) over \(S\), denoted \(\operatorname{GL}_{n,S}\), is defined by base change:

\[ \operatorname{GL}_{n,S} \;=\; S \,\times_{\operatorname{Spec} \mathbb{Z}}\, \operatorname{Spec} \mathbb{Z}[x_{ij}\colon 1 \le i,j \le n]\bigl[\det(x_{ij})^{-1}\bigr], \]

where \(\mathbb{Z}[x_{ij}]\) is the polynomial ring in the \(n^2\) indeterminates \(x_{ij}\) and the localization inverts the determinant of the formal \(n \times n\) matrix \((x_{ij})\). Its coordinate Hopf algebra is generated over the base by the entries \(x_{ij}\) and by \(\det^{-1}\), with the matrix-multiplication Hopf structure:

\[ \Delta(x_{ij}) \;=\; \sum_{k=1}^{n} x_{ik} \otimes x_{kj}, \qquad \varepsilon(x_{ij}) \;=\; \delta_{ij}, \qquad S(x_{ij}) \;=\; (\det)^{-1} \cdot C_{ji}, \]

where \(C_{ji}\) is the \((j, i)\)-cofactor of \((x_{ij})\) (so that \((S(x_{ij}))\) is the formal matrix inverse of \((x_{ij})\)).

The structure morphism \(\operatorname{GL}_{n,S} \to S\) is affine and locally of finite type. Hence \(\operatorname{GL}_{n,S}\) is an algebraic group over \(S\); it is smooth of relative dimension \(n^2\) and is, for \(n \ge 2\), non-commutative. Its functor of points is identified by the GL_n points formula.

The group scheme \(\operatorname{GL}_n\) is the target of the embedding theorem: every algebraic group over a field is a closed subgroup of some \(\operatorname{GL}_n\). The closed subgroup \(\operatorname{SL}_n = \ker(\det)\), together with subgroups of \(\operatorname{GL}_n\) cut out by symmetry conditions, gives rise to the classical group schemes.