Algebraic torus

In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by formula_1, formula_2, or formula_3, is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled as a product of algebraic groups formula_1. These groups were named by analogy with the theory of "tori" in Lie group theory (see Cartan subgroup). For example, over the complex numbers formula_5 the algebraic torus formula_1 is isomorphic to the group scheme formula_7, which is the scheme theoretic analogue of the Lie group formula_8. In fact, any formula_1-action on a complex vector space can be pulled back to a formula_10-action from the inclusion formula_11 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, since for an algebraic group formula_12 to be smooth over characteristic formula_13, the mapsformula_14 must be geometrically reduced for large enough formula_15, meaning the image of the corresponding map on formula_12 is smooth for large enough formula_15. 
In general one has to use separable closures instead of algebraic closures.
Multiplicative group of a field.
If formula_18 is a field then the "multiplicative group" over formula_18 is the algebraic group formula_1 such that for any field extension formula_21 the formula_22-points are isomorphic to the group formula_23. To define it properly as an algebraic group one can take the affine variety defined by the equation formula_24 in the affine plane over formula_18 with coordinates formula_26. The multiplication is then given by restricting the regular rational map formula_27 defined by formula_28 and the inverse is the restriction of the regular rational map formula_29.
Definition.
Let formula_18 be a field with algebraic closure formula_31. Then a "formula_18-torus" is an algebraic group defined over formula_18 which is isomorphic over formula_31 to a finite product of copies of the multiplicative group.
In other words, if formula_35 is an formula_18-group it is a torus if and only if formula_37 for some formula_38. The basic terminology associated to tori is as follows.
Isogenies.
An isogeny between algebraic groups is a surjective morphism with finite kernel; two tori are said to be "isogenous" if there exists an isogeny from the first to the second. Isogenies between tori are particularly well-behaved: for any isogeny formula_51 there exists a "dual" isogeny formula_52 such that formula_53 is a power map. In particular being isogenous is an equivalence relation between tori.
Examples.
Over an algebraically closed field.
Over any algebraically closed field formula_54 there is up to isomorphism a unique torus of any given rank. For a rank formula_55 algebraic torus over formula_56 this is given by the group scheme formula_57pg 230.
Over the real numbers.
Over the field of real numbers formula_58 there are exactly (up to isomorphism) two tori of rank 1:
Any real torus is isogenous to a finite sum of those two; for example the real torus formula_63 is doubly covered by (but not isomorphic to) formula_64. This gives an example of isogenous, non-isomorphic tori.
Over a finite field.
Over the finite field formula_65 there are two rank-1 tori: the split one, of cardinality formula_66, and the anisotropic one of cardinality formula_67. The latter can be realised as the matrix group
formula_68
More generally, if formula_21 is a finite field extension of degree formula_70 then the Weil restriction from formula_22 to formula_18 of the multiplicative group of formula_22 is an formula_18-torus of rank formula_70 and formula_18-rank 1 (note that restriction of scalars over an inseparable field extension will yield a commutative algebraic group that is not a torus). The kernel formula_77 of its field norm is also a torus, which is anisotropic and of rank formula_78. Any formula_18-torus of rank one is either split or isomorphic to the kernel of the norm of a quadratic extension. The two examples above are special cases of this: the compact real torus is the kernel of the field norm of formula_80 and the anisotropic torus over formula_65 is the kernel of the field norm of formula_82.
Weights and coweights.
Over a separably closed field, a torus "T" admits two primary invariants. The weight lattice formula_83 is the group of algebraic homomorphisms "T" → Gm, and the coweight lattice formula_84 is the group of algebraic homomorphisms Gm → "T". These are both free abelian groups whose rank is that of the torus, and they have a canonical nondegenerate pairing formula_85 given by formula_86, where degree is the number "n" such that the composition is equal to the "n"th 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 over Z. 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 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. 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:
The torus is split over a field if and only if it is diagonalisable over this field.
Split rank of a semisimple group.
If formula_89 is a semisimple algebraic group over a field formula_18 then:
Obviously the rank is greater than or equal the formula_18-rank; the group is called "split" if and only if equality holds (that is, there is a maximal torus in formula_89 which is split over formula_18). The group is called "anisotropic" if it contains no split tori (i.e. its formula_18-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.
If formula_35 is a maximal torus in a semisimple algebraic group formula_89 then over the algebraic closure it gives rise to a root system formula_103 in the vector space formula_104. On the other hand, if formula_105 is a maximal formula_18-split torus its action on the formula_18-Lie algebra of formula_89 gives rise to another root system formula_109. The restriction map formula_110 induces a map formula_111 and the Tits index is a way to encode the properties of this map and of the action of the Galois group of formula_112 on formula_103. The Tits index is a "relative" version of the "absolute" Dynkin diagram associated to formula_103; obviously, only finitely many Tits indices can correspond to a given Dynkin diagram.
Another invariant associated to the split torus formula_115 is the "anisotropic kernel": this is the semisimple algebraic group obtained as the derived subgroup of the centraliser of formula_115 in formula_89 (the latter is only a reductive group). As its name indicates it is an anisotropic group, and its absolute type is uniquely determined by formula_109.
The first step towards a classification is then the following theorem
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 groups of formula_18. More precisely to each Tits index there is associated a unique quasi-split group over formula_18; then every formula_18-group with the same index is an inner form of this quasi-split group, and those are classified by the Galois cohomology of formula_18 with coefficients in the adjoint group.
Tori and geometry.
Flat subspaces and rank of symmetric spaces.
If formula_12 is a semisimple Lie group then its real rank is the formula_58-rank as defined above (for any formula_58-algebraic group whose group of real points is isomorphic to formula_12), in other words the maximal formula_15 such that there exists an embedding formula_129. For example, the real rank of formula_130 is equal to formula_131, and the real rank of formula_132 is equal to formula_133.
If formula_134 is the symmetric space associated to formula_12 and formula_136 is a maximal split torus then there exists a unique orbit of formula_137 in formula_134 which is a totally geodesic flat subspace in formula_134. It is in fact a maximal flat subspace and all maximal such are obtained as orbits of split tori in this way. Thus there is a geometric definition of the real rank, as the maximal dimension of a flat subspace in formula_134.
Q-rank of lattices.
If the Lie group formula_12 is obtained as the real points of an algebraic group formula_89 over the rational field formula_143 then the formula_143-rank of formula_89 has also a geometric significance. To get to it one has to introduce an arithmetic group formula_146 associated to formula_147, which roughly is the group of integer points of formula_89, and the quotient space formula_149, which is a Riemannian orbifold and hence a metric space. Then any asymptotic cone of formula_150 is homeomorphic to a finite simplicial complex with top-dimensional simplices of dimension equal to the formula_143-rank of formula_89. In particular, formula_150 is compact if and only if formula_89 is anisotropic.
Note that this allows to define the formula_155-rank of any lattice in a semisimple Lie group, as the dimension of its asymptotic cone.
Buildings.
If formula_89 is a semisimple group over formula_157 the maximal split tori in formula_89 correspond to the apartments of the Bruhat-Tits building formula_134 associated to formula_89. In particular the dimension of formula_134 is equal to the formula_162-rank of formula_89.
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 G"m"/"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 "GL"1,"U" = G"m"/"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 G"m"/"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 formula_164 of a projective scheme formula_165. Then, with the origin removed, the induced projection map formula_166 gives the structure of an algebraic torus over formula_134.
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" for which their base extensions are isomorphic, i.e., it is a torus of the same rank. Isomorphism classes of twisted forms of a split torus are parametrized by nonabelian flat cohomology formula_168, where the coefficient group forms a constant sheaf. In particular, twisted forms of a split torus "T" over a field "K" are parametrized by elements of the Galois cohomology pointed set formula_169 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 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. In particular, extensions of tori are classified by Ext1 sheaves. These are naturally isomorphic to the flat cohomology groups formula_170. 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 "f"K 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 formula_171 (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.