Topic Linear Algebraic Groups
topic admitted

Linear Algebraic Groups

A linear algebraic group over a base scheme \(S\) is, in the modern sense (Milne, SGA 3), an affine group scheme that is locally of finite type over \(S\). Equivalently, a closed subgroup scheme of some \(\operatorname{GL}_{n,S}\) (see the embedding theorem).

When \(S = \operatorname{Spec} k\) for a field \(k\), this recovers the classical theory of linear algebraic groups over \(k\); when \(S\) is arbitrary, base change and descent considerations enter.

Scope of this topic

The linear_algebraic_groups topic collects the foundational infrastructure used throughout the Langlands roadmap:

  1. Definitions and structure. Affine group scheme + locally of finite type; the Path-4 + Level-1 idiom records this via (G : Scheme) [G.Over S] [GrpObj (Scheme.asOver G S)] plus [IsAffineHom (G ↘ S)] and [LocallyOfFiniteType (G ↘ S)].

  2. Coordinate-ring duality. Coordinate rings and Hopf algebras establishes that affine group schemes over \(S\) correspond to commutative Hopf algebras over \(\Gamma(S, \mathcal{O}_S)\); the functor implementing this is <code>hopfSpec</code>.

  3. Canonical examples. Basic examples — \(\mathbb{G}_a\), \(\mathbb{G}_m\), \(\mu_n\), \(\operatorname{GL}_n\), \(\operatorname{SL}_n\), tori, finite étale group schemes — provide the building blocks for everything that follows.

  4. Categorical structure. Morphisms and closed subgroups, kernels and quotients, base change, automorphism groups.

  5. Geometric properties. Connected components, smoothness, tangent spaces, and Lie algebras.

  6. Representation theory. representations as comodules, faithful linear representations, the embedding into \(\operatorname{GL}_n\).

Position in the Langlands roadmap

This topic supplies the foundational layer beneath reductive structure (root data, parabolics, Borel subgroups), descent and forms (inner/outer forms, quasi-split groups), and buildings and parahorics (the local-field theory). Every later topic assumes the affine group-scheme infrastructure recorded here.

References

  • Milne, Algebraic Groups (CUP, 2017).
  • Waterhouse, Introduction to Affine Group Schemes (GTM 66, 1979).
  • SGA 3 (Demazure–Grothendieck), Schémas en Groupes.
  • Jantzen, Representations of Algebraic Groups (AMS, 2003).

Prerequisite Topics

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