Basic Examples
The following canonical examples appear throughout the theory; they provide the building blocks from which every classical reductive group is assembled.
The additive group \(\mathbb{G}_a\)
\(\mathbb{G}_{a,R}\) \(= \operatorname{Spec} R[T]\) with primitive Hopf structure \(\Delta(T) = T \otimes 1 + 1 \otimes T\). Its [[node:affine_group_schemes.additive_group_scheme_points|functor of points]] sends \(B \mapsto (B, +)\). One-dimensional, smooth, connected, non-reductive.
The multiplicative group \(\mathbb{G}_m\)
\(\mathbb{G}_{m,R}\) \(= \operatorname{Spec} R[T, T^{-1}]\) with group-like Hopf structure \(\Delta(T) = T \otimes T\). Its [[node:affine_group_schemes.multiplicative_group_scheme_points|functor of points]] sends \(B \mapsto B^\times\). One-dimensional, smooth, connected, reductive (a torus).
The \(n\)-th roots of unity \(\mu_n\)
\(\mu_{n,R}\) \(= \operatorname{Spec} R[T]/(T^n - 1) \cong \operatorname{Spec} R[\mathbb{Z}/n]\). Its points are \(\{x \in B^\times : x^n = 1\}\). Finite of order \(n\); étale iff \(n\) is invertible in \(R\); for \(n = p = \operatorname{char}(R)\) it acquires an infinitesimal part.
The general linear group \(\operatorname{GL}_n\)
\(\operatorname{GL}_{n,R}\) \(= \operatorname{Spec} R[X_{ij}, \det^{-1}]\) with matrix-multiplication Hopf structure \(\Delta(X_{ij}) = \sum_k X_{ik} \otimes X_{kj}\). Its [[node:affine_group_schemes.general_linear_group_scheme_points|functor of points]] sends \(B \mapsto \operatorname{GL}_n(B)\). Of dimension \(n^2\), smooth, connected, reductive.
The special linear group \(\operatorname{SL}_n\)
\(\operatorname{SL}_{n,R} = \ker(\det : \operatorname{GL}_{n,R} \to \mathbb{G}_{m,R})\), i.e. \(\operatorname{Spec} R[X_{ij}] / (\det(X_{ij}) - 1)\). Its points are matrices of determinant \(1\). Of dimension \(n^2 - 1\), smooth, connected, simply connected, semisimple.
Tori (split form)
A split torus of rank \(r\) is \(\mathbb{G}_{m,R}^r\), the \(r\)-fold product. A general torus is a group scheme becoming a split torus after base change to a finite étale cover. Tori have a character lattice \(X^*(T) = \operatorname{Hom}(T, \mathbb{G}_m) \cong \mathbb{Z}^r\) in the split case.
Finite étale group schemes
Finite groups \(\Gamma\) give rise to constant group schemes \(\underline{\Gamma}_S\); twisting by Galois action yields finite étale group schemes more generally. These appear as components of disconnected algebraic groups and as fundamental groups of tori.
Diagonal, Borel, unipotent inside \(\operatorname{GL}_n\)
Inside \(\operatorname{GL}_n\):
- the diagonal torus \(T \cong \mathbb{G}_m^n\),
- the Borel subgroup \(B\) of upper-triangular matrices,
- the unipotent radical \(U\) of strictly upper-triangular matrices (an iterated extension of \(\mathbb{G}_a\)).
These are the prototypes of maximal tori, Borel subgroups, and unipotent groups in arbitrary reductive group schemes.
Trivial group scheme
[[node:affine_group_schemes.trivial_group_scheme|The trivial group
scheme]] \(\mathrm{e}_{/S} = S\) (with the identity structure morphism)
is the terminal object of Grp(Over S) — every group scheme has a unique
morphism to it.