Algebraic Tori

An algebraic torus is a linear algebraic group that becomes isomorphic to a finite power of the multiplicative group over a separable closure of the base field. Tori are simultaneously a special case of reductive groups and the foundation on which the structure theory of reductive groups is built: maximal tori carry root data, maximal split tori carry relative root data and apartments, and the arithmetic of tori over discretely valued fields underlies parahoric subgroups and Bruhat–Tits buildings. This topic develops the theory of tori in itself, before and independently of reductive structure theory.

Conventions

Fixed for every node of this topic and for downstream consumers.

• \(k\) denotes the base field, \(k_s\) a separable closure, \(\Theta = \operatorname{Gal}(k_s/k)\). Statements are phrased over \(k_s\), never over an algebraic closure: a torus splits over a finite separable extension, so \(\bar{k}\) is an unnecessary strengthening.
• The character lattice is \(X^*(T) = \operatorname{Hom}_{k_s\text{-grp}}(T_{k_s}, \mathbb{G}_m)\) and the cocharacter lattice is \(X_*(T) = \operatorname{Hom}_{k_s\text{-grp}}(\mathbb{G}_m, T_{k_s})\), in perfect duality; the continuous \(\Theta\)-action on each is part of the data, not built into the Hom. See Character and Cocharacter Lattices.
• Anisotropic means \(X_*(T)^{\Theta} = 0\); this is equivalent to \(X^*(T)^{\Theta} = 0\) (see Anisotropic Torus), and the cocharacter form is the one used downstream.
• \(X_*\) is the preferred working lattice: valuation-theoretic invariants of \(T(k)\) and, downstream, apartments of buildings are expressed through \(X_*\).
• Primitive maps are \(\mathbb{Z}\)-valued into finitely generated abelian groups. Tensoring with \(\mathbb{Q}\) or \(\mathbb{R}\) happens at the point of use, never inside a definition.

Core nodes

• Algebraic Torus — the definition.
• Tori among Groups of Multiplicative Type — tori among groups of multiplicative type; the characteristic-\(p\) trichotomy.
• F-Tori Are Classified by Galois Modules — the anti-equivalence with Galois lattices, the complete invariant.
• The Minimal Splitting Field of a Torus — the minimal splitting field.
• Split Torus, Anisotropic Torus — the two extreme classes.
• Subtori and Quotients via the Character Lattice — the lattice dictionary for subtori and quotients.
• Split-Anisotropic Decomposition of a Torus — every torus is an almost direct product of the two extremes.
• Weil Restriction of Tori, Norm-One Torus of a Separable Extension, Classification of Rank-One Tori — the basic examples.
• Induced Torus, Resolution of a Torus by Induced Tori — the class that behaves like \(\mathbb{G}_m\), and the resolution of an arbitrary torus by it.
• Unramified Torus, Tamely Ramified Torus, Weakly Induced Torus — ramification classes over discretely valued fields.

Planned extensions

Galois cohomology of tori, the Kottwitz homomorphism and the subgroups \(T(k)_0 \subseteq T(k)_1\) of \(T(k)\), integral models and their filtrations, and the dual torus with the local Langlands correspondence for tori are tracked as later stages of this topic.