In most places we suppose that the base field is perfect (for example finite or characteristic zero).
-rank 1 (note that restriction of scalars over an inseparable field extension will yield a commutative algebraic group that is not a torus).
-torus of rank one is either split or isomorphic to the kernel of the norm of a quadratic extension.
[2] The two examples above are special cases of this: the compact real torus is the kernel of the field norm of
Over a separably closed field, a torus T admits two primary invariants.
, where degree is the number n such that the composition is equal to the nth power map on the multiplicative group.
This induces canonical continuous actions of the absolute Galois group of K on the lattices.
The weights and coweights that are fixed by this action are precisely the maps that are defined over K. The functor of taking weights is an antiequivalence between the category of tori over K with algebraic homomorphisms and the category of finitely generated torsion free abelian groups with an action of the absolute Galois group of K. Given a finite separable field extension L/K and a torus T over L, we have a Galois module isomorphism If T is the multiplicative group, then this gives the restriction of scalars a permutation module structure.
-rank; the group is called split if and only if equality holds (that is, there is a maximal torus in
In the classical theory of semisimple Lie algebras over the complex field the Cartan subalgebras play a fundamental rôle in the classification via root systems and Dynkin diagrams.
This classification is equivalent to that of connected algebraic groups over the complex field, and Cartan subalgebras correspond to maximal tori in these.
Without the splitness assumption things become much more complicated and a more detailed theory has to be developed, which is still based in part on the study of adjoint actions of tori.
and the Tits index is a way to encode the properties of this map and of the action of the Galois group of
The Tits index is a "relative" version of the "absolute" Dynkin diagram associated to
is the anisotropic kernel: this is the semisimple algebraic group obtained as the derived subgroup of the centraliser of
As its name indicates it is an anisotropic group, and its absolute type is uniquely determined by
Thus there is a geometric definition of the real rank, as the maximal dimension of a flat subspace in
is homeomorphic to a finite simplicial complex with top-dimensional simplices of dimension equal to the
-rank of any lattice in a semisimple Lie group, as the dimension of its asymptotic cone.
Given a base scheme S, an algebraic torus over S is defined to be a group scheme over S that is fpqc locally isomorphic to a finite product of copies of the multiplicative group scheme Gm/S over S. In other words, there exists a faithfully flat map X → S such that any point in X has a quasi-compact open neighborhood U whose image is an open affine subscheme of S, such that base change to U yields a finite product of copies of GL1,U = Gm/U.
[clarification needed] One particularly important case is when S is the spectrum of a field K, making a torus over S an algebraic group whose extension to some finite separable extension L is a finite product of copies of Gm/L.
In general, the multiplicity of this product (i.e., the dimension of the scheme) is called the rank of the torus, and it is a locally constant function on S. Most notions defined for tori over fields carry to this more general setting.
For a general base scheme S, weights and coweights are defined as fpqc sheaves of free abelian groups on S. These provide representations of fundamental groupoids of the base with respect the fpqc topology.
As a partial converse, a theorem of Grothendieck asserts that any torus of finite type is quasi-isotrivial, i.e., split by an etale surjection.
Isomorphism classes of twisted forms of a split torus are parametrized by nonabelian flat cohomology
In particular, twisted forms of a split torus T over a field K are parametrized by elements of the Galois cohomology pointed set
In the one-dimensional case, the coefficients form a group of order two, and isomorphism classes of twisted forms of Gm are in natural bijection with separable quadratic extensions of K. Since taking a weight lattice is an equivalence of categories, short exact sequences of tori correspond to short exact sequences of the corresponding weight lattices.
Over a field, the extensions are parametrized by elements of the corresponding Galois cohomology group.
In his work on Tamagawa numbers, T. Ono introduced a type of functorial invariants of tori over finite separable extensions of a chosen field k. Such an invariant is a collection of positive real-valued functions fK on isomorphism classes of tori over K, as K runs over finite separable extensions of k, satisfying three properties: T. Ono showed that the Tamagawa number of a torus over a number field is such an invariant.
Furthermore, he showed that it is a quotient of two cohomological invariants, namely the order of the group