Wikipedia / Algebraic torus
Wikipedia / Algebraic torus
In mathematics, an algebraic torus, is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. These groups were named by analogy with the theory of tori in Lie group theory (see Cartan subgroup is isomorphic to the group scheme-action on a complex vector space as real manifolds.
Tori are of fundamental importance in the theory of algebraic groups and Lie groups and in the study of the geometric objects associated to them such as symmetric spaces and buildings.
Algebraic tori over fields
In most places we suppose that the base field is perfect (for example finite or characteristic zero). This hypothesis is required to have a smooth group schemepg 64.
In general one has to use separable closures instead of algebraic closures.
Multiplicative group of a field
is a field then the multiplicative group such that for any field extension. To define it properly as an algebraic group one can take the affine variety.
Definition
be a field with algebraic closure. Then a -torus to a finite product of copies of the multiplicative group.
-group it is a torus if and only if. The basic terminology associated to tori is as follows.
is called the rank or absolute rank.
The torus is said to be split is split, which is called the splitting field.
The -rank-rank equals its absolute rank.
A torus is said to be anisotropic-rank is zero.
Isogenies
An isogeny between algebraic groups is a surjective morphism with finite kernel; two tori are said to be isogenous is a power map. In particular being isogenous is an equivalence relation between tori.
Examples
Over an algebraically closed field
pg 230.
Over the real numbers
there are exactly (up to isomorphism) two tori of rank 1:
the compact form, which can be realised as the unitary group or as the special orthogonal group. It is an anisotropic torus. As a Lie group, it is also isomorphic to the 1-torus, which explains the picture of diagonalisable algebraic groups as tori.
. This gives an example of isogenous, non-isomorphic tori.
Over a finite field
Over the finite field
then the Weil restriction of its field norm-torus of rank one is either split or isomorphic to the kernel of the norm of a quadratic extension..
Weights and coweights
Over a separably closed field, a torus T admits two primary invariants. The weight lattice is the group of algebraic homomorphisms T→Gm is the group of algebraic homomorphismsGm→T, where degree is the number n such that the composition is equal to the nth power map on the multiplicative group. The functor given by taking weights is an antiequivalence of categories between tori and free abelian groups, and the coweight functor is an equivalence. In particular, maps of tori are characterized by linear transformations on weights or coweights, and the automorphism group of a torus is a general linear group overZ. The quasi-inverse of the weights functor is given by a dualization functor from free abelian groups to tori, defined by its functor of points as:
This equivalence can be generalized to pass between groups of multiplicative type (a distinguished class of formal groups) and arbitrary abelian groups, and such a generalization can be convenient if one wants to work in a well-behaved category, since the category of tori doesn't have kernels or filtered colimits.
When a field K is not separably closed, the weight and coweight lattices of a torus over K are defined as the respective lattices over the separable closure. 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 overK. 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. Tori whose weight lattices are permutation modules for the Galois group are called quasi-split, and all quasi-split tori are finite products of restrictions of scalars.
Tori in semisimple groups
Linear representations of tori
As seen in the examples above tori can be represented as linear groups. An alternative definition for tori is:
A linear algebraic group is a torus if and only if it is diagonalisable over an algebraic closure.
The torus is split over a field if and only if it is diagonalisable over this field.
Split rank of a semisimple group
then:
its rank (or absolute rank) is the rank of a maximal torus so the rank is well-defined);
its -rank (sometimes called -split rank.
-rank; the group is called split). The group is called anisotropic-rank is zero).
Classification of semisimple groups
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. In fact the classification carries over to the case of an arbitrary base field under the assumption that there exists a split maximal torus (which is automatically satisfied over an algebraically closed field). 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; obviously, only finitely many Tits indices can correspond to a given Dynkin diagram.
is the anisotropic kernel.
The first step towards a classification is then the following theorem
-algebraic groups are isomorphic if and only if they have the same Tits indices and isomorphic anisotropic kernels.
This reduces the classification problem to anisotropic groups, and to determining which Tits indices can occur for a given Dynkin diagram. The latter problem has been solved in . The former is related to the Galois cohomology. More precisely to each Tits index there is associated a unique quasi-split group-group with the same index is an inner form with coefficients in the adjoint group.
Tori and geometry
Flat subspaces and rank of symmetric spaces
is a semisimple Lie group then its real rank.
is the symmetric space.
Q-rank of lattices
has also a geometric significance. To get to it one has to introduce an arithmetic group, which is a Riemannian orbifold and hence a metric space. Then any asymptotic cone is homeomorphic to a finite simplicial complex is anisotropic.
-rank of any lattice in a semisimple Lie group, as the dimension of its asymptotic cone.
Buildings
.
Algebraic tori over an arbitrary base scheme
Definition
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. 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.
Examples
One common example of an algebraic torus is to consider the affine cone of a projective scheme.
Weights
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. If the torus is locally trivializable with respect to a weaker topology such as the etale topology, then the sheaves of groups descend to the same topologies and these representations factor through the respective quotient groupoids. In particular, an etale sheaf gives rise to a quasi-isotrivial torus, and if S is locally noetherian and normal (more generally, geometrically unibranched), the torus is isotrivial. 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.
Given a rank n torus T over S, a twisted form is a torus over S for which there exists a fpqc covering of S, where the coefficient group forms a constant sheaf. In particular, twisted forms of a split torus T over a field K with trivial Galois action on the coefficients. 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 ofK.
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. In particular, extensions of tori are classified by Ext1. Over a field, the extensions are parametrized by elements of the corresponding Galois cohomology group.
Arithmetic invariants
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:
Multiplicativity: Given two tori T1 and T2 over K, fK(T1 T2) = fK(T1) fK(T2)
Restriction: For a finite separable extension L/K, fL evaluated on an L torus is equal to fK evaluated on its restriction of scalars to K.
Projective triviality: If T is a torus over K whose weight lattice is a projective Galois module, then fK(T) = 1.
(sometimes mistakenly called the Picard group of T, although it doesn't classify Gm torsors over T), and the order of the Tate–Shafarevich group.
The notion of invariant given above generalizes naturally to tori over arbitrary base schemes, with functions taking values in more general rings. While the order of the extension group is a general invariant, the other two invariants above do not seem to have interesting analogues outside the realm of fraction fields of one-dimensional domains and their completions.
See also
Notes
References
A. Grothendieck, SGA 3 Exp. VIIIX
T. Ono, On Tamagawa Numbers
T. Ono, On the Tamagawa number of algebraic tori Annals of Mathematics 78 (1) 1963.