Galois lattices and toric varieties

V.E.Voskresenskii^{1}

Various questions lead to consideration of algebraic varieties defined over a non-closed field k or over a ring. Let X be such variety, k_{s} a separable closure of the field k, G = Gal (k_{s}/k) the Galois group of the k_{s}/k. Then we have the object which contains very rich information — the variety X̄ = X ⊗_{k} k_{s} defined over the field k_{s} with the action of the group G on X̄ through the second factor. So we get a large collection of associated G-sets and G-Galois modules: points X(k_{s}), differential forms, divisors, Pic X̄, Br X̄, etc. So the investigation of the variety X can be divided into two steps: the geometric one — the investigation X̄ over the field k_{s}, and the algebraic one — the investigation of Galois modules associated to the scheme X̄ which is considered over k. This survey contains a detailed account of the second — algebraic — aspect of the investigation of the variety X. We suppose that the geometrical part is known.

The main part of the text is devoted to the theory of algebraic tori which are considered from the point of view of birational geometry.

During the past thirty years a new branch of mathematics was created — the theory of toric varieties which allows one to reduce a number of nonlinear problems to problems on integer points in some polyhedra. We can note such important problems as resolution of singularities, estimating the number of solutions of systems of equations, birational geometry of linear algebraic groups, the theory of invariants of transformation groups. Retrospectively, we see that the arithmetic of algebraic groups was missing more profound information about birational nature of group varieties at that time.

In the present text we give a comprehensive survey of results on birational geometry of algebraic tori. The author tried to expose some interesting results of Russian mathematicians published in hardly accessible journals. We also present here some new results on the rationality problem and on the arithmetic of integer models of tori defined over a p-adic field.

1. Diagonalizable groups and algebraic tori. First recall some facts and definitions. Let G be a linear algebraic group over a field k, G = Spec A, where A is a Hopf algebra over k, that is A is a commutative associative k-algebra endowed with additional homomorphisms:

\begin{matrix}m^{∗} : & A → & A ⊗_{k} A & (comultiplication), \\ i^{∗} : & A → & A & (coinversion), \\ m^{∗} : & A → & k & (counit), \end{matrix}

which are subject to standard conditions of co-associativity, co-inversion, and co-unit.

It is well known that group structures on the affine scheme Spec A are in one-to-one correspondence with Hopf algebra structures on the algebra A. We note that this definition of group structure can be used not only for an algebra A over a field k but also for an arbitrary commutative algebra A over a commutative ring k with unity. If necessary, we shall use such a possibility. The group G = Spec A defined over a field k is a linear algebraic group in the case when A is of finite type over k and has no nilpotents.

Let us consider the structure of algebraic k-tori. Let D_{k}(n) be a diagonal k-group, D_{k}(n) = Spec A, A = k[x_{1}, .., x_{n}, x_{1}^{−1}, .., x_{n}^{−1}], x_{i} are independent variables. Using the diagonal multiplication we can easily see that

m^{∗}(x_{i}) = x_{i} ⊗ x_{i}, i^{∗}(x_{i}) = x_{i}^{−1}, e^{∗}(x_{i}) = 1.

1Partially supported by a grant of RFFI Grant 02-01-00017

Let A^{∗} be the group of invertible elements of the ring A. In our case the group A^{∗} is the direct product A^{∗} = k^{∗}× < x_{1}, .., x_{n} > where < x_{1}, .., x_{n} > is the free Abelian group multiplicatively generated by elements x_{i}, the function x_{i} determines the regular map: x_{i} : D_{k}(n) → D_{k}(1) = G_{m,k} which is a group homomorphism. We shall call the function x_{i} a rational character of the group D_{k}(n).

We shall denote the group < x_{1}, .., x_{n} > by D̂(n) and call it the group of rational characters of the diagonal group D(n). Every character is of the form x^{a} = x_{1}^{a_{1}}x_{2}^{a_{2}}...x_{n}^{a_{n}}, where a_{i} ∈ Z. The correspondence

x^{a} → (a_{1}, a_{2}, .., a_{n})

defines an isomorphism between the multiplicative group D̂(n) and the additive group Z^{n}. Working with characters we shall freely go over from the multiplicative writing to the additive one and vice versa. We have A^{∗} = k^{∗} × D̂(n), and the ring A is the group ring k[D̂(n)] of the multiplicative group D̂(n).

Definition. A group T defined over a field k is called an algebraic torus if we have the following group isomorphism over the algebraic closure

T ⊗_{k} k̄ ≅ Dk̄(n).  (1)

In other words, T is a k-form of the group D_{k}(n) .

It is easy to show that in relation (1) we can replace the field k̄ by k_{s}. We have an isomorphism of Hopf algebras k_{s}[T] and k_{s}[D̂(n)] which shows that k_{s}[T]^{∗} is the direct product of the groups k_{s}^{∗} × T̂, where T̂ is the group of rational characters of the torus T ⊗ k_{s}. The Galois group G takes invertible elements to invertible ones, and since the unit of the group T is defined over the field k, the group G maps the group T̂ into itself. So the group T̂ is a G-module. Let x_{1}, .., x_{n} be a basis of the group T̂, σ ∈ G, then

σ(x_{j}) = ∏_{i=1}^{n} x_{i}^{a_{ij}}.

The matrix (a_{ij}) lies in the group GL(n, Z). We have a representation φ : G → GL(n, Z) with finite image. Let L = k_{s}^{ker φ}, (L : k) = m. Then L/k is the minimal splitting field of the torus T : T ⊗_{k} L ≅ D_{L}(n), the group Π = Gal (L/k) is isomorphic to the group φ(G). The torus T can be explicitly described if the Π-module T̂ and the extension L/k are known, where Π = Gal (L/k). Let us consider the group ring L[T̂], let A = (L[T̂])^{Π} be the ring of invariants. Then Spec A = T. Let B be an arbitrary commutative k-algebra. It is easy to describe the group of points T (B). In fact,

T (B) = Hom_{k−ring}(k[T], B) = Hom_{L−ring}(L[T], B ⊗_{k} L)^{Π} =
Hom_{gr}(T̂, (B ⊗_{k} L)^{∗})^{Π} = Hom_{Π}(T̂, (B ⊗_{k} L)^{∗}).

In particular, T (k) = Hom_{Π}(T̂, L^{∗}). So the torus T represents the functor

B → Hom_{Π}(T̂, (B ⊗_{k} L)^{∗}).

Examples. 1). The group Z has one generator Z =< e >, either Galois group G acts trivially: σ(e) = e, ∀σ ∈ G or there is an element σ ∈ G such that σ(e) = −e. In the first case T = D(1), in the second case the splitting field of a one dimensional torus T is a quadratic extension L, (L : k) = 2, Π = Gal (L/k) =< σ >, σ^{2} = 1, and

T (k) = Hom_{Π}(T̂, L^{∗}) = {α ∈ L | σ(α) = α^{−1}} = {α ∈ L | N_{L/k}(α) = 1}.

If L = k(\sqrt{D}) then the torus T is defined by the equation x^{2} − Dy^{2} = 1.

2). It is necessary to find all finite subgroups in GL(2, Z) up to conjugacy. There are two maximal non-conjugate groups: W_{1} and W_{2}. The group W_{1} is the group of all integer automorphisms of the quadratic form x_{1}^{2} + x_{2}^{2}, the group W_{2} is group of all integer automorphisms of the form x_{1}^{2} + x_{1}x_{2} + x_{2}^{2}. The group W_{1} = D_{4} is a dihedral group, |W_{1}| = 8. The group W_{2} = D_{6} is a dihedral group, |W_{2}| = 12.

Explicit geometric constructions describing the groups W_{1} and W_{2} allow one to find affine and projective realizations of the corresponding algebraic tori. We shall return to this question later. It should be noted that the tori corresponding to the subgroups of the group W_{1} are realized as open subsets of two-dimensional quadrics; the tori corresponding to the subgroups of W_{2} are realized either as open subsets of the projective plane P_{k}^{2} or as open subsets of a Del Pezzo surface of degree six. Here is an example. Suppose U ⊂ W_{2}, |U| = 6, F = Q( ^{3} \sqrt{D}), (F: Q) = 3, then the norm torus is determined by the following equation

T: x^{3} + Dy^{3} + D^{2}z^{3} − 3Dxyz = 1.

Tori of the form R_{F/k}(G_{m}), which are elementary examples after trivial tori D_{k}(n) = G_{m,k}^{n}, play the most important role in problems of affine realization of algebraic tori.

The group R_{F/k}(G_{m}). We know that the coordinate ring of the diagonal torus D_{k}(n) is the ring of Laurent polynomials.

k[D(n)] = k[x_{1}, .., x_{n}, x_{1}^{−1}, .., x_{n}^{−1}]

The embedding of the ring of polynomials of k[A^{n}] = k[x_{1}, .., x_{n}] in k[D_{n}] and the diagonal map : k[A^{n}] → k[A^{n}] ⊗_{k} k[D(n)] determine both the embedding of corresponding spectra D(n) → A^{n} and the action of the group D(n) on A^{n}. Let a G-module T̂ of finite Z-rank (torsion-free) be given by generators x_{1}, .., x_{n} on which the group G acts transitively by permutations. Then we have a G-invariant embedding

k_{s}[x_{1}, .., x_{n}] → k_{s}[x_{1}, .., x_{n}, x_{1}^{−1}, .., x_{n}^{−1}] = k_{s}[T̂]

Going over to G-invariants and the corresponding spectra we get an open k-embedding of the torus T into A_{k}^{n} with a natural action of the torus T on A_{k}^{n}. Let G_{1} be a subgroup in G consisting of all elements which leave elements of module T̂ fixed. Then T̂ is a Πmodule where Π = G/G_{1}. Let L = k_{s}^{G_{1}} is the minimal splitting field of T, Π = Gal (L/k), π is the stabilizer of the element x_{1} in T̂, F = L^{Π}, (F : k) = n. Then the Π-module T̂ is isomorphic to the Π-module Z[Π/π] = ZΠ ⊗_{π} Z spanned by the left cosets of Π with respect to the subgroup π. The torus T with the module T̂ = Z[Π/π] is denoted R_{F/k}(G_{m}). The group of k-points T (k) is of the form

T (k) = R_{F/k}(G_{m})(k) = Hom_{Π}(ZΠ ⊗_{π} Z, L^{∗}) = Hom_{π}(Z, L^{∗}) = F^{∗}.

The last relation allows one to describe the torus T (k) = R_{F/k}(G_{m}) in the most simple and natural way. Let w_{1}, ..., w_{n} be a basis of the extension F/k. We have the regular faithful representation of the elements of the field F by square matrices of order n with entries from the field k.

(αw_{1}, ..., αw_{n}) = (w_{1}, ..., w_{n})T_{α}, α → T_{α}, T_{α} ∈ GL(n, k), α ≠ 0.

The matrices T_{α}, α ∈ F^{∗}, generate a maximal k-torus in GL_{k}(n) which is R_{F/k}(G_{m}). Let L/k be a Galois extension, Π = Gal (L/k). We denote the category of algebraic

tori defined over k and split over L by C(L/k), let C(Π) be the category of torsion-free Π-modules of finite Z-rank. The correspondence T → T̂ defines a duality between the categories C(L/k) and C(Π). Let Ŝ be a permutation Π-module of the category C(Π), i.e. Ŝ has a basis on which the group Π acts by permutations. Dividing the permutation basis into orbits of the group Π we get a representation of the Π-module Ŝ as a direct sum of indecomposable permutation modules Ŝ_{i}

Ŝ = Ŝ_{1} ⊕ ... ⊕ Ŝ_{t}

. This construction gives a representation of the torus S as a direct product

S = R_{F_{1}/k}(G_{m}) × ... × R_{F_{t}/k}(G_{m})

. Tori of such form are called quasi-split. The torus S is a maximal torus in the general linear group GL_{k}(n), n = dim S, and

S(k) = F_{1}^{∗} × ... × F_{t}^{∗}

Let T be an arbitrary torus of the category C(L/k), T̂ the corresponding Π-module of rational characters. The module T̂ is the quotient Ŝ/R̂, where Ŝ is some permutation Π-module (for example, a free Π-module). An epimorphism Ŝ → T̂ defines the embedding T → S, and since S can be realized as a maximal torus of the general linear group GL_{k}(n), n = dim S, we get a faithful linear representation of our torus T. It is clear how to get a linear representation of minimal degree. Let us consider one more application of quasisplit tori. Let T̂^{0} = Hom (T̂, Z) be the module dual to the module T̂. For σ ∈ Π and f ∈ T̂^{0} let (σf)(m) = σ[f(σ^{−1}m], m ∈ T̂. Let us write an exact sequence of Π-modules

0 → R̂ → Ŝ → T̂^{0} → 0,

where Ŝ is a permutation module. Going over to dual modules we get the exact sequence

0 → T̂ → Ŝ^{0} → R̂^{0} → 0,

We note that Ŝ^{0} ≅ Ŝ, therefore we have an epimorphism of tori S → T and see that the function field k(T) is a subfield of the k(S) which is a purely transcendental extension of k. So every k-torus T is a unirational variety.

2. Picard group and Brauer group of a linear algebraic group. Let k be a field of characteristic 0, X an algebraic variety defined over k, X̄ = X ⊗_{k} k̄, G = Gal (k̄/k). There is the Hochschild–Serre spectral sequence

H^{p}(G, H^{q}(X̄, G_{m})) ⇒ H^{n}(X, G_{m})

in étale cohomology which yields the eight-term exact sequence of abelian groups

0 → H^{1}(k, k̄[X]^{∗}) → Pic X → (Pic X̄)^{G} → H^{2}(k, k[X]^{∗}) →
→ ker(Br X → Br X̄) → H^{1}(k, Pic X̄) → H^{3}(k, k̄[X]^{∗}),
Pic X = H^{1}(X, G_{m}), Br X = H^{2}(X, G_{m}).

If the set of rational points X(k) is not empty, we have a retraction Spec k → X → Spec k, and get a decomposition into direct sum

H^{q}(k, k̄[X]^{∗}) = H^{q}(k, k^{∗}) ⊕ H^{q}(k, U(X̄)) where U(X) = k[X]^{∗}/k^{∗}

In this case the image of the homomorphism H^{1}(k, Pic X̄) → H^{3}(k, k̄[X]^{∗}) lies in H^{3}(k, U(X̄). These facts can be applied in the case when X is the underlying variety of a connected linear algebraic k-group G. In this case k[G]^{∗} = k̄^{∗} × Ĝ, and sequence (1) is of the form

0 → H^{1}(k, Ĝ) → Pic G → (Pic Ḡ)^{G} → H^{2}(k, Ĝ) →
→ Br G/Br k → H^{1}(k, Pic Ḡ) → H^{3}(k, Ĝ)

In particular, if G = T is an algebraic torus then Pic T̄ = 0, and we have

Pic T = H^{1}(k, T̂), Br T = Br k ⊕ H^{2}(k, T̂)

If G is a connected semisimple group, then Ĝ = 0 and this sequence is divided into two sequences:

Pic G = (Pic G)^{G}, Br G = Br k ⊕ H^{1}(k, Pic Ḡ)

The group Ḡ contains a Borel subgroup B which is a semidirect product of a unipotent group N and a maximal torus T. The group G contains an open subset U which is isomorphic to the direct product N × T × N, and the complement Ḡ − Ū is a union ⋃_{i=1}^{r} F_{i}, where F_{i} is a prime divisor of the variety Ḡ, r is the rank of the group Ḡ i.e. r = dim T. Since all varieties are considered over the field k̄, we have Pic U = 0 and Pic Ḡ is a finite group. Since Pic G = (Pic Ḡ)^{G}, Pic G is a finite group for any connected semisimple group, taking into account an analogous result for tori we can see that Pic G is a finite group for any connected algebraic group. If G̃ is a connected simply connected semisimple group then, comparing étale and analytic cohomology, we get Pic G̃ = 0 and, more generally, Pic G = π̂(k̄)^{G}, where π is the fundamental group of a connected semisimple k- group G, π̂(k̄) = Hom (π(k̄), k̄^{∗}). The articles [Po, Sa, V1] are dedicated to the calculation of the group Pic G.

3. Birational invariants of algebraic varieties. Let G be a connected linear algebraic group defined over a field k, and let k(G) be the field of rational functions of G. If k is a perfect field, G is a group variety unirational over k. This was proved by Chevalley for groups defined over a field of characteristic zero [Che], and then generalized by Rosenlicht to the case of an arbitrary perfect field of definition [Ro]. If k is algebraically closed, then k(G) is purely transcendental over k, but for non algebraically closed fields Chevalley and Rosenlicht gave examples of groups G which are not rational over k. Further success has been reached using projective embeddings of a linear group. Methods using on this way are based, on the one hand, on the fundamental result of Hironaka about possibility of a resolution of singularities of rational maps of algebraic varieties, and on the other hand, on the theory of birational invariants developed in works of Yu.I. Manin and I.R. Shafarevich [Ma]. Since Hironaka’s theorems are proved in general only in characteristic zero, we shall assume that k is of characteristic 0.

Thus, let X and Y be nonsingular projective irreducible varieties defined over a field k of characteristic 0, and assume that there is a birational morphism φ : X → Y. We have the direct image

φ_{∗} : Pic X → Pic Y

which is an epimorphism and Ker φ_{∗} = S is a free abelian group generated by finite number of prime divisors of variety X with supports in a closed subset F ⊂ X, where F = φ^{−1}(W), W is a closed subset in Y consisting of all points in which the rational map φ^{−1} is not defined. Since we have the inverse image

φ^{∗} : Pic Y → Pic X

where φ_{∗}φ^{∗} = Id, the group Pic X is isomorphic to the direct sum:

Pic X ≅ Pic Y ⊕ S

If L is a normal extension of k, Π = Gal (L/k), then the morphism φ defines a birational L-morphism

φ ⊗ 1 : X_{L} = X ⊗_{k} L → Y ⊗_{k} L = Y_{L}

, and we get a Π-isomorphism

Pic X_{L} ≅ Pic Y_{L} ⊕ M,  (1)

where M is a torsion-free permutation Π-module of finite type. Let X and Y be birationally equivalent over k. Hironaka showed that there exist a nonsingular projective variety Z over the field k and birational morphisms Z → X and Z → Y. Then from (1) it follows that there are permutation Π- modules S_{1} and S_{2} such that the direct sums Pic X_{L} ⨁ S_{1} and Pic Y_{L} ⨁ S_{2} are Π-isomorphic.

Definition. Let Π be a group, A and B arbitrary Π-modules. We call modules A and B similar if they differ only by permutation direct summands. The similarity class of a Π-module A is denoted by [A].

Theorem. (Necessary condition of birational equivalence.)

Let X and Y be nonsingular projective algebraic varieties over a field k of characteristic 0, L/k a normal extension with Galois group Π. If X and Y are birationally equivalent over k, then Π-modules Pic X_{L} and Pic Y_{L} are similar.

Corollary 1. Under the hypotheses of the theorem let F be an intermediate field, k ⊂ F ⊂ L, π = Gal (L/F). Then H^{1}(L/F, Pic X_{L}) = H^{1}(L/F, Pic Y_{L}), the groups H^{1}(L/F, Pic X_{L}) are birational invariants of the variety X. For a finite extension L/k the groups H^{−1}(L/F, Pic X_{L}) are also birational invariants.

Let now a variety X be rational over k̄. Then X̄ = X ⊗_{k} k̄ is birationally equivalent to the projective space P^{n} ⊗_{k} k̄. Since Pic P^{n} = Z, we have Pic (X̄) ⊕ S ≅ Z ⊕ S_{1}.

Corollary 2. Let X be a smooth projective variety defined over a field k of characteristic 0. If X is rational over k̄ then the Π-module Pic X_{L} is torsion-free of finite Z-rank. If the variety X is rational over the base field k, then [Pic X_{L}] = [0] and H^{±1}(L/F, Pic X_{L}) = 0 for any normal extension L/k, k ⊂ F ⊂ L.

4. Projective models of linear algebraic groups. Let G be a connected linear algebraic group defined over a field k of characteristic zero. Consider a nonsingular projective variety X over k which contains G as an open subset. The variety X is called a projective model of the group G. The variety Ḡ = G ⊗_{k} k̄ and, hence, X̄ are rational over k̄. So, G- module Pic X̄ is torsion-free of finite Z-rank. Although the model X is not uniquely determined if dim G > 1, the similarity class of the G-module Pic X̄ is determined uniquely. We shall denote it by p_{k}(G) or simply p(G) and call it the Picard class of the group G. It is easy to show that p(G_{1} × G_{2}) = p(G_{1}) ⨁ p(G_{2}), where the addition in the right-hand side is defined by direct sum. From the embedding G → X we get an exact sequence of modules

0 → Ĝ → Ŝ → Pic X̄ → Pic Ḡ → 0

where Ŝ is a permutation G-module generated by the components of the closed subset F̄ = X̄\Ḡ. Since the variety G is normal, F̄ is an unmixed variety of codimension one. Let us consider two extreme cases:

a) G = T is an algebraic torus. Then Pic T̄ = 0 and we have an exact sequence of torsion-free modules

0 → T̂ → Ŝ → Pic X̄ → 0;  (1)

b) G is semisimple. Then Ĝ = 0, we have

0 → Ŝ → Pic X̄ → Pic Ḡ → 0.

If Ḡ is simply connected, then Pic Ḡ = 0, in this case Pic X̄ is a permutation G-module. Let us consider the case a) in detail at finite level. Let L/k be a finite Galois extension with group Π, T ∈ C(L/k). Similarly to (1) we can write the exact sequence (1) at finite level

0 → T̂ → Ŝ → Pic X̄_{L} → 0  (2)

Sequence (2) is a sequence of Π-modules, Ŝ is a permutation module. Let T_{0} be the maximal subtorus in T split over k, T_{1} = T/T_{0} is anisotropic, i.e. (T̂_{1})^{Π} = (0). The variety T is birationally equivalent to the product T_{0} × T_{1} whence

p(T) = p(T_{0} × T_{1}) = p(T_{1}) and H^{−1}(L/k, p(T)) = H^{−1}(L/k, p((T_{1})).  (3)

From sequence (2) we get an exact sequence

0 → H^{−1}(L/k, Pic X_{L}) → Ĥ^{0}(L/k, T̂).

If the torus T is anisotropic, then H^{−1}(L/k, Pic X_{L}) = 0. Formula (3) shows that for any torus T ∈ G(L/k) the birational invariant H^{−1}(L/k, p(T)) equals 0. But then this is true for every intermediate field F, k ⊂ F ⊂ L.

Theorem 1. Let T ∈ C(L/k), k ⊂ F ⊂ L. Then

H^{−1}(L/F, p_{k}(T)) = H^{−1}(L/F, p_{F} (T ⊗_{k} F)) = 0.

This is an unexpected result which is one of the most important in birational classification of algebraic tori. Now we can give a purely algebraic description of the birational invariant [Pic X_{L}] = p_{L/k}(T). Let N̂ be a Π-module of the category C(Π) with properties: H^{−1}(π, N̂) = 0 for all subgroups π of Π. Following Colliot-Thélène and Sansuc we shall call such modules flasque modules. We shall call the exact sequence of Π-modules of the category C(Π)

0 → T̂ → Ŝ → N̂ → 0,  (4)

where Ŝ is a permutation module, N̂ is a flasque one, a flasque resolution of the module T̂. One of flasque resolutions is sequence (2) obtained by geometric arguments. We have an important statement

Theorem 2. The similarity class [N̂] of the module N̂ in the sequence (4) does not depend on the choice of a flasque resolution of the module T̂. Therefore,

p_{L/k}(T) = [Pic X_{L}] = [N̂].

Since the algebraic definition of the birational invariant p(T) is absolutely independent on the characteristic of the ground field, we want to extend these concepts to the case of characteristic p > 0. It can be done in the following way.

Let X be a smooth variety (not necessary projective) over a field k of arbitrary characteristic, and let L be a finite Galois extension of k with group Π. We assume that Pic X_{L} is of finite type. Then there is a nonempty open subset U in X such that Pic (U_{L}) = 0.

The module L[U]^{∗}/L^{∗} is torsion-free of finite type. Let p(L[U]^{∗}/L^{∗}) be its Picard class in C(Π). This class does not depend on the choice of a neighbourhood U ⊂ X and is a birational invariant of the k-variety X. If X is a smooth projective variety, then p(L[U]^{∗}/L^{∗}) = [Pic X_{L}]. Since a k-torus T always has a smooth projective model X in arbitrary characteristic (it will be mentioned later), the class p(T) is a birational invariant in arbitrary characteristic.

5. The semigroup of stable equivalence. We shall call two k-varieties X_{1} and X_{2} stably equivalent if there is a birational isomorphism

X_{1} ×_{k} A^{m} ≅ X_{2} ×_{k} A^{n}

for some m and n.

In the case if X ×_{k} A^{m} ≅ A^{p} we shall call X stably rational.

Clearly, if algebraic tori T_{1} and T_{2} are stably equivalent then p(T_{1}) = p(T_{2}); if a torus T is stably rational then p(T) = [0].

Let {T } be class of stable equivalence of the torus T in the category C(L/k). On the set of classes {T } we shall define a structure of commutative semigroup setting

{T_{1}} ∘ {T_{2}} = {T_{1} ×_{k} T_{2}}.

The class of the tori which are stably rational over the field k plays the role of unity. We shall denote this semigroup by the symbol Z(L/k). Let us introduce some notation. Let S(Π) be the class of all permutation Π-modules of finite type,

H^{q}(Π) = {M ∈ C(Π) | H^{q}(π, M) = 0 ∀π ⊂ Π},

where H^{q} is the Tate cohomology. Clearly,

S(Π) ⊂ H^{−1}(Π) ∩ H^{1}(Π) = H̃(Π)

Let

D(Π) = {M ∈ C(Π) | M ⊕ M′ ∈ S(Π)}

We have the following filtration:

S(Π) ⊂ D(Π) ⊂ H̃(Π) ⊂ H^{±1}(Π) ⊂ C(Π)

The operation of direct sum of modules defines the structure of commutative semigroup on the set of similarity classes C(Π)/S(Π), and the zero element is the class S(Π). The classes D(Π)/S(Π) form the maximal subgroup in the semigroup C(Π)/S(Π). From the results of Jacobinski in the theory of lattices [Ja] it follows that this group is finitely generated. The assumption that H̃(Π) = D(Π) is not true, we shall give an example later. The map p_{L/k} : Z(L/k) → H^{−1}(Π)/S(Π) was studied in the work of the author [V3].

Theorem 1. The map p_{L/k} is an isomorphism of semigroups.

We are especially interested in tori from the kernel of the map p.

Theorem 2. The following conditions are equivalent:

a) a torus T is stably rational over the field k;

b) the Picard class p(T) is equal to zero;

c) the module of rational characters T̂ of the torus T can be embedded into an exact sequence of the form

0 → T̂ → Ŝ → Ŝ_{1} → 0,

where Ŝ and Ŝ_{1} ∈ S(Π).

6. Chevalley modules. Let us give now some important examples illustrating efficeince of the criteria mentioned in the previous paragraphs. Let Π be a finite group, π its subgroup, Z(Π) the group ring, Z[Π/π] the module spanned by cosets e_{1}, ..., e_{n} on which the group Π acts by permutations. Let us consider the map ε : Z[Π/π] → Z, defined by the equality

ε(∑_{i=1}^{n} a_{i}e_{i}) = ∑_{i=1}^{n} a_{i}.

Clearly ε is a Π-homomorphism. Let I_{Π/π} = Ker ε. We have

0 → I_{Π/π} → Z[Π/π] ⟶^{ε} Z → 0.  (1)

Going over to dual modules we get the dual sequence

0 → Z ⟶^{ε_{0}} Z[Π/π] → J_{Π/π} → 0,  (2)

where J_{Π/π} = I_{Π/π}^{0}, ε_{0}(1) = e_{1} + ... + e_{n}. Let L/k be a Galois extension with group Π, F an intermediate field such that Gal (L/F) = π. Sequences (1) and (2) define exact sequences of k-tori

1 → G_{m,k} → R_{F/k}(G_{m}) → T_{1} → 1,
1 → T_{2} → R_{F/k}(G_{m}) → G_{m,k} → 1.

Since R_{F/k}(G_{m}) can be identified with an open subset in R_{F/k}(A^{1}) = A_{k}^{n}, the torus T_{1} admits an open embedding into the projective space P_{k}^{n−1}, hence, T_{1} is rational over k, p(T_{1}) = [0]. From this point of view the torus T_{2} is more interesting. The variety T_{2} can be represented as a norm hypersurface N(x_{1}, ..., x_{n}) = 1 in the space A_{k}^{n}, where x_{1}, ..., x_{n} are the coordinates of an element of the field F in some basis. The torus T_{2} has a standard notation R_{F/k}^{(1)}(G_{m}), its module of rational characters J_{Π/π} is called Chevalley module. Chevalley was the first mathematician who discovered nonrationality of tori of the form R_{F/k}^{(1)}(G_{m}) for some extensions F/k [Che]. We consider a special case. When π = (1), the sequence (2) takes the form

0 → I_{Π} → Z[Π] ⟶^{ε} Z → 0  (3)

An arbitrary module is a quotient of a free module, hence we have an exact sequence

0 → N̂^{0} → P̂^{0} → I_{Π} → 0,  (4)

where P is a free Π-module. From (3) and (4) it follows that

H^{1}(π, N̂^{0}) = Ĥ^{0}(π, I_{Π}) = H^{−1}(π, Z) = 0

for any subgroup π of Π. So

N̂ = Hom (N̂^{0}, Z) ∈ H^{−1}(Π).

Going over in (4) to dual modules, we get a flasque resolution of the Chevalley module

0 → J_{Π} → P̂ → N̂ → 0

From (3) and (4) we conclude that H^{1}(π, N̂) = H^{3}(π, Z) is a birational invariant of the torus R_{L/k}^{(1)}(G_{m}) = T. This fact allos one to find non-rational tori. For example, if Π contains a subgroup π of the form Z/pZ × Z/pZ, where p is prime, then the torus R_{L/k}^{(1)}(G_{m}) is not rational over the field L^{Π} and moreover, over the field k, because

H^{3}(Z/pZ × Z/pZ, Z) = Z/pZ ≠ 0.

For a cyclic group Π the modules I_{Π} and J_{Π} are isomorphic, therefore for a cyclic extension L/k the torus R_{L/k}^{(1)}(G_{m}) is rational over k.

As it has been shown in investigations of, mainly, S. Endo and T. Miyata [E-M1,E-M2], the condition of invertibility of the element p(J_{Π}) imposes very serious restrictions on the structure of Π. Let us state the results in the order of strengthening of requirements.

Theorem 1. The following conditions are equivalent:

1) every Sylow subgroup of Π is either cyclic or a generalized quaternion group;

2) all abelian subgroups of Π are cyclic;

3) H^{3}(π, Z) = 0 for all subgroups π ∈ Π;

4) p(J_{Π}) ∈ H^{1}(Π)/S(Π).

We shall call a finite group metacyclic if all its Sylow subgroups are cyclic.

Theorem 2. The following conditions are equivalent:

1) the group Π is metacyclic;

2) the class p(J_{Π}) is invertible;

3) H^{1}(Π) = H^{−1}(Π) = D(Π).

Example. Let Π be the quaternion group of the order 8. Then by Theorem 1, p(J_{Π}) ∈ H^{1}(Π)/S(Π), but p(J_{Π}) is not invertible by Theorem 2. Hence,H̃(Π) ≠ D(Π).

The following subtle theorem [E-M1, E-M2] completely describes stably rational tori of the form R_{L/k}^{1}(G_{m}) in the case when L/k is a Galois extension.

Theorem 3. Let Π be a finite group. The following conditions are equivalent:

1) Π =< s, t >, t^{m} = s^{2^{n}} = 1, sts^{−1} = t^{2}, m – odd, r^{2} ≡ 1(mod m);

2) p(J_{Π}) = 0 ;

3) p(J_{Π}) has a finite order.

The rank of the group D(Π)/S(Π) was calculated by A. Dress [Dr], the torsion in some cases was studied by Endo and Miyata [E-M1, E-M2]. For example, if Π is a finite p-group, then the group D(Π)/S(Π) is a factor of the projective class group of the ring Z(Π) and hence is finite.

Let us consider one more important example, which was obtained recently by Colliot-Thélène and Kunyavskii (independently). Let F/k be a separable extension of prime degree p, T = R_{F/k}^{(1)}(G_{m}). If the extension F/k is normal then the torus T is rational over k; since F/k is cyclic, J_{Π} = I_{Π}. Let the extension F/k be not normal, let L be the normal closure of the field F in k_{s}, Π = Gal (L/k), π = Gal (L/F), (F : k) = p, (L : F) = m, (m, p) = 1. Let us choose a Π-module N̂ from the class p_{L/k}(T). For an arbitrary Sylow subgroup Π_{q} of Π, the Π_{q}-module N̂ lies in D(Π_{q}). But then N̂ ∈ D(Π). Thus there is N̂′ ∈ C(Π_{q}) such that N̂ ⨁ N̂′ ∈ S(Π). It has been shown that the product of the tori T ×_{k} N is k-rational variety. The simplest nontrivial example is given by a nonnormal extension F of degree five over k such that (L : k) = 20. In this case p(R_{F/k}^{(1)}(G_{m})) is invertible but has infinite order in the group D(Π)/S(Π).

7. Tori of small dimension. All one-dimensional tori are k-rational. As it was mentioned at the beginning, two-dimensional tori are determined by subgroups of maximal finite subgroups W_{1} and W_{2} in the group GL(2, Z). The group W_{1} of order 8 is the group of motions of a square, it acts biregularly on G_{m} × G_{m}, and using the embedding G_{m} × G_{m} → P^{1} × P^{1} its action can be extended to a biregular action on P^{1} × P^{1}. Therefore the tori of type W_{1} can be embedded into projective models which are k-forms of the variety P^{1} ×P^{1}. All such k-forms are rational over k if they have a rational k-point. Thus the tori of this type are k-rational.

The group W_{2} is the group of motions of a regular hexagon. It acts regularly on G_{m} ×G_{m} and birationally on A^{2} or P^{2}. In the homogeneous coordinates (x : y : z) of the plane P^{2} the action of the group W_{2} =< σ, τ, ω > can be given as follows: σ cyclically permutes the coordinates (x : y : z); τ(a, b) = (a^{−1}, b^{−1}), where a = x/z, b = y/z; ω is a the transposition (x : y : z) → (x : z : y). Let us choose the following rational functions on P^{2}

u = \frac{y − z}{x − y} , v = au = \frac{x(y − z)}{z(x − y)}

The fields k(u, v) and k(a, b) are isomorphic, and explicit calculations show that in the basis (u, v) the group W_{2} acts by the rule

σ(u, v) = (−\frac{u + 1}{u}, −\frac{v + 1}{v}) ,  τ(u, v) = (v, u),  ω(u, v) = (−\frac{u}{u + 1}, −\frac{v}{v + 1}) .

So the group W_{2} is conjugate in the group of birational k-transformations of the plane to a subgroup of the group Aut_{k}(P^{1} × P^{1}). Therefore, as in the previous case, the corresponding torus is birationally equivalent over k to a form of the variety P^{1} × P^{1} and, hence, is rational over k. Thus all tori of dimension 2 are rational over the ground field.

The classification of 3-dimensional tori over a ground field was studied by B. Kunyavskii [Ku1, Ku2]. Most tori are no longer rational over k but every stably rational k-torus of dimension 3 is in fact k-rational. He also studied tori with biquadratic splitting field [Ku1]. It has been shown that an arbitrary torus T ∈ C(L/k), Π = Gal (L/k) = Z/2Z × Z/2Z has a permutation resolution:

0 → T̂ → Ŝ → Ŝ_{1} → Ŝ_{2} → 0,

where Ŝ, Ŝ_{1}, Ŝ_{2} ∈ S(Π). It turned out that the semigroup Z(L/k) has one generator and is isomorphic to the additivr semigroup Z_{+} of natural numbers (with zero). The generator is the class of the 3-dimensional torus R_{L/k}^{(1)}(G_{m}). Moreover B. Kunyavskii obtained a full classification of the tori of the category C(L/k) up to birational equivalence. It turned out that any torus T ∈ C(L/k) is birationally equivalent to a direct product

T ≅ (R_{L/k}^{(1)}(G_{m})^{n} × G_{m,k}^{d}

The tori R_{L/k}^{(1)}(G_{m})^{n} × G_{m,k}^{d} and R_{L/k}^{(1)}(G_{m})^{m} × G_{m,k}^{r} are birationally equivalent iff n = m, d = r. Thus every stably rational torus T ∈ C(L/k) is rational.

The question about the number of generators of the semigroup Z(L/k) was studied in the work of A.L. Chistov [Ch2]. He has shown that as a rule the semigroup Z(L/k) has infinite number of generators. Thus, if Π = Gal (L/k) is a p-group and p > 2 then Z(L/k) has a finite number of generators only in the case of a cyclic group. For a 2-group Π it has been shown that Z(L/k) is not finitely generated if the minimal number of generators of the group Π is more than two. Then, if the order of Π is equal to 2^{n} and Π is not isomorphic to a cyclic group, a dihedral group D_{2^{n}}, a generalized quaternion group H_{2^{n}} or the group Q_{n} with generators σ, τ and relations

σ^{2^{n−1}} = τ^{2} = 1, τσ = σ^{m}τ, m = 2^{n−2} − 1,

then the semigroup Z(L/k) in not finitely generated. The cases D_{2^{n}} H_{2^{n}} Q_{n}, n ≥ 4, remain open. If Π = D_{8} then Z(L/k) ≅ Z_{+}^{11}; if Π = H_{8} then Z(L/k) = Z_{+}^{6} , if Π = Q_{3} = Z/2Z × Z/4Z, then Z(L/k) is not a finitely generated.

8. Tori with a cyclic splitting field. Let Π be the cyclic group of order n with generator σ, Π = Gal (L/k). The group ring Λ = Z[Π] = Z[σ] is isomorphic to the ring Z[x]/(x^{n} − 1). Let ζ_{n} be a primitive root of the polynomial x^{n} − 1, Φ_{n}(x) a cyclotomic polynomial, i.e. the minimal polynomial of the ring Z[x] for which ζ_{n} is a root. We have a decomposition

x^{n} − 1 = ∏_{d|n} Φ_{d}(x).

The ring Z[x]/(Φ_{n}(x)) is isomorphic to the ring of integers Z[ζ_{d}] of the cyclotomic field Q(ζ_{d}). For any monic polynomial f ∈ Z[x] dividing x^{n} − 1, let R(f) be the torus from C(L/k) with Π-module of rational characters Z[σ]/(f(σ)). Clearly R(x^{n}−1) = R_{L/k}(G_{m}). Let h_{d}(x) = x^{d}−1, d | n, and f ∈ Z[x] such that fh_{d} | h_{n}. Then we have an exact sequence of k-tori

1 → R(h_{d}) →^{α} R(fh_{d}) →^{β} R(f) → 1

and the morphism B has a rational k-section. So we have a birational isomorphism

R(fh_{d}) ≅ R(h_{d}) ×_{k} R(h_{d}),  (1)

which allows us to carry out inductive arguments. Using such “dévissage” we get the following representation.

Theorem 1. For a cyclic extension L/k we have a birational isomorphism of varieties

R_{L/k}(G_{m}) ≅ ∏_{d|n} R(Φ_{d}).

The torus R(Φ_{d}) has as the module of rational characters the ring of integers Z[ζ_{d}]. Corollary. The torus R(Φ_{n}) is stably rational over k for any n. Thus, over a number field we have infinitely many stably rational tori of rather simple

construction. The question about rationality of these tori as we shall see later is very basic but unfortunately the answer has been obtained only in very particular cases. We can simplify the problem. Let n = p_{1}^{d_{1}}, ...p_{s}^{d_{s}} be the canonical decomposition of the number n, d_{i} ≥ 1, n_{0} = p_{1}...p_{s} the square-free kernel of the number n, Π_{0} - a subgroup of order n_{0} of the group Π, L_{0} the field of Π_{0}-invariants of the field L. The formula Φ_{n}(x) = Φ_{n_{0}}(y), y = x^{n/n_{0}} shows that

R(Φ_{n}) = R_{L_{0}/k}(R(Φ_{n_{0}})).

It follows that the rationality of the form R(Φ_{n}) will be proved if it will be done for square-free numbers n.

Let now T be an arbitrary torus of the category C(L/k) where L/k is a cyclic extension with group Π. Consider a flasque resolution of the Π-module of rational characters T̂

0 ⟶ T̂ ⟶ Ŝ ⟶ N̂ ⟶ 0  (2)

All properties of the Picard class considered above used the invertibility of the class p(T) for the tori with a cyclic splitting field. This is a really basic fact, and it implies excellent algebraic-arithmetic properties of cyclic extensions. To prove invertibility of the class [N̂], consider a flasque resolution of the module N̂

0 ⟶ N̂ ⟶ Ŝ_{1} ⟶ M̂ ⟶ 0  (3)

We have Ext_{Π}(M̂, N̂) = H^{1}(Π, Hom (M̂, N̂)). Furthermore,

M̂, N̂ ∈ H^{−1}(Π) = H^{1}(Π),

because Π is a cyclic group. Let Π_{p} be the p-Sylow subgroup of Π. By induction we can check that the Π_{p}-modules M̂ and N̂ lie in D(Π_{p}). But then H^{1}(Π_{p}, Hom (M̂, N̂)) = 0 for all Sylow subgroups Π_{p} of Π, whence Ext_{Π}(M̂, N̂) = 0. Hence N̂ ⨁ M̂ ≅ Ŝ_{1} and H^{1}(Π) = D(Π).

Using invertibility of the classes [N̂] of a flasque module N̂ in the cyclic case, A. Chistov introduced new multiplication on the category C(Π) as follows. First, since the group Π is commutative, any left Π-module A can be also viewed as a right Π-module. Then the tensor product A ⊗_{Π} B is also equipped with the structure of Π-module. If one of the modules T̂_{1}, T̂_{2} lies in D(Π), set

T̂_{1} ∘ T̂_{2} = (T̂_{1} ⊗_{Π} T̂_{2})/t(T̂_{1} ⊗_{Π} T̂_{2})

where t(A) is the torsion submodule of the module A. If T̂_{1} and T̂_{2} are arbitrary modules from C(Π) and

0 ⟶ T̂_{1} ⟶ Ŝ_{1} ⟶^{p_{1}} N̂_{1} ⟶ 0

0 ⟶ T̂_{2} ⟶ Ŝ_{2} ⟶^{p_{2}} N̂_{2} ⟶ 0

are their flasque resolutions, we have an epimorphism

p̃_{2}(T̂_{1}) : T̂_{1} ∘ Ŝ_{2} → T̂_{1} ∘ N̂_{2},

induced by the projection p_{2}. We set

T̂_{1} ∘ T̂_{2} = ker p̃_{2}(T̂_{1})

One can check that the product T̂_{1}∘T̂_{2} does not depend on the choice of flasque resolutions and there are canonical isomorphisms:

T̂_{1} ∘ T̂_{2} = T̂_{2} ∘ T̂_{1},  T̂_{1} ∘ (T̂_{2} ∘ T̂_{3}) = (T̂_{1} ∘ T̂_{2}) ∘ T̂_{3}

. Note that T̂ ∘ ZΠ = T̂. Let T_{1} and T_{2} be algebraic tori of the category C(L/k). Define T_{1} ∘ T_{2} ∈ C(L/k) as the torus dual to the Π-module T̂_{1} ∘ T̂_{2}. For an arbitrary torus T ∈ C(L/k) and a monic polynomial f ∈ Z[X], f | (x^{n} − 1), we set T (f) = T ∘ R(f). Note that the Π-module T̂(Φ_{d}) can be regarded as a Z[ζ_{d}]-module, and since it is torsionfree, T̂(Φ_{d}) is a projective Z[ζ_{d}]-module, d | n. The following important lemma is a generalization of the relation (1).

Lemma. Let f be a monic polynomial from Z[X] such that fh_{d} | h_{n}. Then for any torus T ∈ C(L/k) we have a birational isomorphism of varieties

T (fh_{d}) ≅ T (f) ×_{k} T (h_{d}).

Now we can make the “dévissage” of an arbitrary torus T ∈ C(L/k) by similar arguments. We get the following remarkable theorem of Chistov [Chi1].

Theorem 2. Let L/k be a cyclic extension of degree n. Any torus T ∈ C(L/k) is birationally equivalent over k to the product

∏_{d|n} T (Φ_{d}).

For any d | n the character module T̂(Φ_{d}) is a projective Z[ζ_{d}]-module. A torus T is stably rational over k iff for any d, d | n, the module T̂(Φ_{d}) is Z[ζ_{d}]-free.

Corollary 1. In the case when Π is a cyclic p-group, every stably rational k-torus T is rational over k, T ∈ C(L/k), Π = Gal (L/k).

In this case the theorem reduces to checking the tori T (Φ_{d}) for which T (Φ_{d}) = Z[ζ_{d}], d = p^{α}. Further reduction reduces the problem to the module Z[ζ_{p}] for which the torus R(Φ_{p}) is rational.

Corollary 2. Let L/k be a cyclic extension. Then the group Z(L/k) is a finite group isomorphic to the direct product over all d | n of the class groups of the rings Z[ζ_{d}].

Corollary 3. Under the hypotheses of the theorem, let T̂ be a projective Π-module. Then the following conditions are equivalent:

1) p(T) = 0,

2) T̂(Φ_{d}) is a free Z[ζ_{d}]-module for d | n,

3) the torus T is rational over k.

Why from projectivity of T̂ and stable rationality of T does k-rationality follow? The answer is as follows. Every projective Π -module is a direct sum of a free Π-module T̂_{1} and a projective ideal T̂_{2} ∈ Z[Π]. The torus T_{1} is k-rational and the Z-rank of the module T̂_{2} is equal to n. Hence,

T_{2} ≅ ∏_{d|n} T_{2}(Φ_{d}) ≅ ∏_{d|n} R(Φ_{d}) ≅ R_{L/k}(G_{m}).

9. Projective invariant Demazure models. As we have seen problems of birational classification of varieties of linear algebraic groups necessarily lead to consideration of smooth projective varieties containing a given group as an open set. The existence of such projective models follows (in characteristic zero) from Hironaka’s theorem on the resolution of singularities. In many cases we manage to get valuable information from the very fact of the existence of such smooth models but more delicate questions require an explicit construction of the model, or least information about its Picard group. For algebraic tori we have a rather simple (we can say, canonical) construction of complete models suggested by Demazure. Since we are mainly interested in models defined over non-closed fields, our ficus is on toric varieties with maximal symmetry groups.

Let T be an algebraic k-torus, T̂ its group of rational characters. First consider the case of an algebraically closed field k. The group T̂ is a free abelian group of rank n = dim T, let T_{Q} = Q ⊗ T̂ be the corresponding linear space over Q with natural embedding T̂ ⊂ T̂_{Q}. We call a cone σ in T̂_{Q} a polyhedral cone generated by a finite system of elements m_{1}, ...m_{N} from T̂:

σ = {∑_{i=1}^{N} λ_{i}m_{i} | λ_{i} ≥ 0} .

If the cone σ contains no subspaces from T̂_{Q} besides (0), then we say that the cone has a vertex.

It is necessary for us to have two different realizations of the group T̂: additive and multiplicative. The multiplicative writing is necessary for construction of subrings of the group ring k[T̂], the additive one is suitable in geometric constructions. The group T̂ with additive structure will be often called a lattice. Let m_{1}, ..., m_{n} be a basis of the lattice T̂, and let x_{i} = e^{m_{i}} be symbols generating the multiplicative group T̂. Let us denote by A_{σ} the semigroup k-algebra k[σ ⋂ T̂] which consists of the Laurent polynomials ∑ s_{m}x^{m} where a_{m} ∈ k, m ∈ σ ⋂ T̂ and almost all coefficients a_{m} are equal to zero. We call the scheme X_{σ} an affine toric variety. The diagonal embedding A_{σ} → A_{σ} ⊗_{k} k[T̂] defines an action of the torus T on the scheme X_{σ}. It is convenient to use dual objects. Let T̂^{0} = Hom (T̂, Z) be the dual lattice and

σ^{0} = {r ∈ T̂_{Q}^{0} | r(m) ≥ 0 ∀m ∈ σ}

be the dual cone in the space T̂_{Q}^{0}. The scheme X_{σ} is smooth iff the dual cone σ^{0} is generated by a part of a basis of the lattice T̂^{0} [Da, De, Ke].

A complete toric variety is constricted by gluing affine toric varieties, and the scheme of gluing is determined by some set of cones called a fan. It is convenient to work with fans in the space T̂_{Q}^{0}.

Definition. We call a fan in the space T̂_{Q}^{0} a finite set ∑ of cones satisfying the conditions

a) all cones from ∑ have vertices;

b) if τ is a face of a cone σ ∈ ∑, then τ ∈ ∑;

c) for σ and σ′ ∈ ∑ the intersection is a face both in σ and in σ′ Let σ ∈ ∑, and let σ^{0} be the dual cone in T̂_{Q}. The symbol X_{σ} now stands for

Spec k[σ^{0} ⋂ T̂]. Note that every X_{σ} contains an open set on which the torus T acts regularly dim X_{σ} = n. If τ is a face of a cone σ ∈ ∑, then we have a canonical morphism X_{τ} → X_{σ} which is an open embedding. Gluing together the affine parts X_{σ} with respect to the open embeddings

X_{σ ⋂ σ′} → X_{σ}, X_{σ ⋂ σ′} → X_{σ′}

we get a normal algebraic variety X_{Σ} defined over k with an action of the torus T on X_{Σ}. A fan is called complete if ⋃_{σ∈Σ} σ = T̂_{Q}^{0} It is a necessary and sufficient condition of for X_{Σ} to be a complete variety. Smoothness of the variety X_{Σ} means that all cones σ ∈ ∑ are simplicial and spanned by parts of a basis of the lattice T̂^{0}. In dimensions greater than 2 complete toric varieties may be non-projective. A criterion of projectivity says that the fan consists of cones generated by faces of a convex polyhedron D ⊂ T̂_{Q}^{0} containing 0 as an inner point. The fan ∑ admits another interpretation in the language of orbits of the torus T acting on X_{Σ}. For an open affine T -invariant subset U ⊂ X_{Σ} we denote by T̂(U) the set of characters which can be continued to regular functions on U. Let

σ(U) = {r ∈ T̂_{Q}^{0} | r(m) ≥ 0 ∀m ∈ T̂(U)}

. Then σ(U) ∈ ∑. On the other hand, every such U contains a unique closed in U orbit (the orbit of minimal dimension). This gives a one-to-one correspondence σ → O_{σ} between cones from ∑ and orbits O_{σ} of the torus T in X_{Σ}. In this case σ ⊂ τ ⇔ O_{τ} ⊂ O_{σ} and dim σ = codim O_{σ}. For example, orbits of codimension 1 are in one-to-one correspondence to one-dimensional cones (rays) from ∑. Every such ray σ ∈ ∑ is generated by a unique primitive vector r ∈ T̂^{0}, let | ∑ | be the collection of these primitive vectors, and let O_{r}, r ∈ | ∑ | be the corresponding orbit of codimension 1. The closure O_{r} = D_{r} of an orbit O_{r} in X_{Σ} is a divisor of the variety X_{Σ}. Let X_{Σ} be a smooth complete toric variety. The embedding T ⊂ X_{Σ} defines an exact sequence of lattices

0 → T̂ →^{α} Ŝ → Pic X_{Σ} → 0,  (1)

where Ŝ is a free Z-module generated by the prime divisors from the complement X \ T. We see that the divisors D_{r}, r ∈ | ∑ | form a basis of the group Ŝ so that the rank of Ŝ is equal to the cardinality of the set | ∑ |. The map α is defined by the rule:

α(m) = ∑_{r} r(m)D_{r}, r ∈ |Σ|, m ∈ T̂

The anticanonical divisor of a complete smooth variety X_{Σ} is of the form

−K = ∑ D_{r}, r ∈ |Σ|.

Let now k be a non-closed field, G = Gal (k_{s}/k), T an algebraic n-dimensional torus defined over k. We have an integral representation h : G → GL(n, Z) defined by the action of the group G on the characters T̂ of the torus T. The splitting group h(G) is a finite subgroup in GL(n, Z). The torus T can be reconstructed using h as follows. The group G_{m}^{n}⊗_{k} k_{s} carries two representations of the group G: the geometric one g → h(g)⊗1 and the arithmetic one g → 1 ⊗ g, g ∈ G. They commute. Consider the representation u(g) = h(g)⊗g of G in Aut_{k}(G_{m}^{n}⊗k_{s}). Then T is isomorphic to the factor (G_{m}^{n}⊗_{k}k_{s})/u(G). Let us choose in the lattice T̂^{0} a smooth projective fan ∑ invariant under the finite group h(G). We can always do it by a suitable partition starting with any complete fan [Br]. Then the group h(G) acts naturally by automorphisms of X_{Σ}, and we have an equivariant embedding G_{m}^{n} ⊂ X_{Σ}. The factor X = (X_{Σ} ⊗_{k} k_{s})/u(G) is a k-form of the variety X_{Σ} ⊗ k containing the torus T as an open subset. This proves the existence of a smooth projective model of the torus T over an arbitrary field. The scheme X will be called a Demazure model of the torus T. It is very convenient for the calculation of the G-module Pic (X⊗_{k}k_{s}) because the G-module Pic (X ⊗_{k} k_{s}) is isomorphic to the G-module Pic X_{Σ} on which the group G acts geometrically. The sequence (1) allows one to calculate the action of the group G on Pic X_{Σ} if we know the action of the group G on T̂ and on the edges of the fan ∑. Now let us go over to the question of practical construction of invariant projective fans and Demazure varieties of dimension n. Jordan’s theorem says that there is a finite number of non-conjugate finite subgroups in the group GL(n, Z). For every such group W we can construct a projective invariant fan in the lattice M^{0} = Z^{n} and the model X_{Σ}. Certainly, here the question about the most economic construction with minimal number of simplices arises. Of course, for distinct groups W_{1} ⊂ W_{2} ⊂ GL(n, Z) their minimal fan can be the same. If we shall not take care about minimality of the model, it will be sufficient to construct projective invariant fans for maximal finite subgroups of the group GL(n, Z). The number of maximal subgroups is less than the number of all subgroups. For example, for n = 2 there are 17 subgroups and 2 maximal subgroups; for n = 3 there are 73 subgroups and 4 maximal ones; for n = 4 there are 710 groups and 9 maximal ones.

Example. 1) There is only one smooth projective one-dimensional toric model, the projective line P^{1}.

2) For n = 2 there are two maximal subgroups W_{1} and W_{2}. The group W_{1} is of order 8, it is the group of automorphisms of the root system of type B_{2}. The root lattice B_{2} is a usual square lattice. The fan consists of 4 basic quadrants and all its parts. We have X_{Σ} = P^{1} × P^{1}.

The group W_{2} is of order 12, it is the group of automorphisms of the root system A_{2}. The fan contains 6 basic cones. The scheme X_{Σ} ⊗ k is a Del Pezzo surface of degree 6 obtained from P^{2} by blowing up three points.

3) In dimension 3 we have 4 maximal finite subgroups in the group GL(3, Z). They are, respectively, the groups of automorphisms of the cubic lattice L_{0}, the cubic lattice with the centers of cubes added L_{1}, the cubic lattice the centers of faces added L_{2}, and the group of automorphisms of the lattice which is a direct sum of the one-dimensional lattice and the two-dimensional hexagonal lattice.

Using these explicit geometric constructions, B. Kunyavskii obtained a complete birational classification of three-dimensional tori [Ku2]. From 73 three-dimensional lattices he selected 15 lattices for which the corresponding tori are not rational and are not pairwise stably equivalent. All tori of dimension 3 except these 15 tori are k-rational. In particular, all stably rational three-dimensional tori are in fact rational.

Let us consider the birational invariant H^{1}(k, p(T)) = H^{1}(G, Pic X_{Σ}) more carefully. The sequence (1) shows that for its calculation we only need information on the action of the finite group Π = h(G) on the lattice Z^{n}. The flasque resolution (1) defines a commutative diagram with exact rows

0 ⟶ H^{1}(Π, N̂) ⟶ H^{2}(Π, T̂) ⟶ H^{2}(Π, Ŝ)
↓ res ↓ res ↓ res
0 ⟶ H^{1}(< g >, N̂) ⟶ H^{2}(< g >, T̂) ⟶ H^{2}(< g >, Ŝ)

where N̂ = Pic X_{Σ}, g ∈ Π. This diagram gives a natural monomorphism

0 → H^{1}(Π, N̂) → ker[H^{2}(Π, T̂) →^{res} ⨁_{g∈Π} H^{2}(< g >, T̂)].

From the injectivity of the map

H^{2}(Π, Ŝ) → ⨁_{g∈Π} H^{2}(< g >, Ŝ)

it follows that

H^{1}(Π, p(T)) = ker[H^{2}(Π, T̂) → ⨁_{g∈Π} H^{2}(< g >, T̂)].

This formula can be found in [Sa]. An important class of tori is that of maximal tori in semisimple groups. B. Kunyavskii studying three-dimensional tori found the following phenomenon [Ku2]. Although there are finite subgroups Π ⊂ GL(3, Z) for which H^{1}(Π, Pic X_{Σ}) ≠ 0 but for all maximal finite subgroups Π ⊂ GL(3, Z) and the corresponding invariant fans ∑ we have the equality H^{1}(Π, Pic X_{Σ}) = 0. The lattices L_{0}, L_{1}, L_{2} are lattices of roots or weights of semisimple groups of rank 3. Further investigations of Kunyavskii in collaboration with the author [V-Ku] show that for a maximal k-torus T without affect in a connected adjoint semisimple group of classical type we have H^{1}(k, p(T)) = 0. We say that a maximal k-torus T in a semisimple group G has no affect if the splitting group of the torus T is isomorphic to the splitting group of the generic maximal torus of the group G [V5]. The situation with maximal tori was clarified by the following result of A. Klyachko [Kl].

Theorem. Let W (R) and A(R) be the Weyl group and the group of automorphisms of the root system R of a semisimple k-group G, Q(R) and P (R) be the root lattices and the weight lattice, respectively. Then

1) H^{1}(W, p(T)) = 0 for any W -lattice T̂, Q(R) ⊆ T̂ ⊆ P (R);

2) H^{1}(Π, p(P (R)) = H^{1}(Π, p(Q(R)) = 0, W (R) ⊆ Π ⊆ A(R);

3) if the system R is irreducible then

H^{1}(Π, p(T)) = 0 W (R) ⊆ Π ⊆ A(R), Q(R) ⊆ T̂ ⊆ P (R).

This theorem has interesting applications to the arithmetic of linear algebraic groups. Let us consider maximal tori in adjoint groups of type A_{n} in more detail. The cases n = 1, 2 will be omitted because in these cases all tori are rational. The root lattice Q(R) is constructed as follows: we choose the sublattice I_{n+1} in Z^{n+1} consisting of the points with zero sum of coordinates, then Q(R) = I_{n+1}. The symmetric group S_{n+1} = W (R) acts naturally on Q(R). The group of automorphisms A(R) is a direct product W (R)×S_{2} where the second factor acts on Q(R) by multiplication on (-1). Let us consider the A(R)- module T̂ = I_{n+1} ⊗ I_{2}. T is a maximal k-torus without affect in the adjoint group G of type A_{n}, and the Galois group acts on T̂ via the group A(R). Consider two cases:

1) G is an inner form of type A_{n}. In this case the Galois group acts via W (R), the factor I_{2} is not needed, we have an open embedding of the torus T in P^{n} over k, all maximal tori and the group G are k-rational.

2) G is an outer form of type A_{n}, T̂ = I_{n+1} ⊗ I_{2}. In this case one can prove that for even n the torus T_{n} and the group G are k-rational [V-Kl]. The torus T_{3} is not rational for the splitting group S_{4} × S_{2}.

10. Arithmetic of linear algebraic groups. Let k be a number field, i.e. a finite extension of the field of rational numbers Q, let n = [k : Q] be the degree of this extension. For any valuation v of the field k we denote by k_{v} the completion of k with respect to v. It is known that there are finitely many r_{1} + r_{2} archimedian normalized valuations and a countable set of non-archimedian ones for every prime ideal ℘ of the field k, and r_{1} + 2r_{2} = n. In the case of a non-archimedian valuation we shall often write k_{℘} instead of k_{v}. There is an important product formula for normalized valuations

∏_{v} v(a) = 1, a ≠ 0, a ∈ k

Instead v(a) we often write |a|_{v}. The symbol O will denote the ring of integers of the field k, O_{℘} is the ring of integers of the field k_{℘}. Let now W be a vector space over the field k of finite dimension m, M a lattice in W, i.e. a finitely generated O-module in W containing a basis of W. In this case k ⊗_{O} M ≅ kM = W. For every valuation v of k we set W_{v} = k_{v} ⊗_{k} W and identify W with the k-space in W_{v} consisting of the elements 1 ⊗ x, x ∈ W. Let G be a linear algebraic group defined over k. We call an integer form of the group G a group O-scheme X such that there is a group isomorphism of k-groups X ⊗_{v} k ≅ G. The existence of an integer form X for a group G is checked very easily. One of the methods of its construction is the following one [V-F]. We have a faithful linear representation of the group G in the general linear group φ : G → GL_{k}(n) that gives an epimorphism φ^{∗} : k[GL(n) → k[G] of Hopf algebras. The ring k[GL(n)] is of the form k[x_{11}, ...x_{nn}, y^{−1}], y = det(x_{ij}), the O-algebra O[x_{11}, ...x_{nn}, y^{−1}] defines in GL_{k}(n) the integer group structure GL_{O}(n) and φ^{∗}(O[x_{11}, ...x_{nn}, y^{−1}]) = A is an O-Hopf algebra in k[G]. Note that the choice of coordinates x_{ij} in GL_{k}(n) is the group of automorphisms of V. Let M =< e_{1}, ..., e_{n} > be a free O-lattice in V. Thus, every faithful linear representation φ of G defines a group O-form G_{φ,M} = Spec A of G. Although the structure of Hopf algebra k[G] does not depend on the choice of a faithful representation φ, but a priori this cannot be said about the groups G_{φ,M}. Note that all O-forms X of the form G_{φ,M} are of finite type over O, they are reduced and faithfully flat over O. The ring A is a subring in k[G] with the conditions :

1) A is an O-Hopf algebra of finite type over O;

2) A ⋂ k = O , kA = k[G];

3) the structure of Hopf algebra in the ring A is induced by the structure of Hopf algebra k[G].

Definition. A subring A of the ring k[G] satisfying conditions 1) - 3) is called an O-Hopf order in k[G], X = Spec A is a group O-form of finite type of the group G.

Having an O-form X we can consider the scheme X ⊗_{O} O/℘, which is a group scheme over the finite field F_{℘} = O/℘, and it is natural to call it the reduction of the scheme X modulo prime ℘. Certainly, this reduction depends on the choice of an integer model X, but in the case of algebraic tori we can choose a canonical model.

Let now A be the adele ring of k,

A^{∞} = ∏_{v|∞} k_{v} ∏_{p} O_{p}

the subring of integer adeles of the ring A. We shall call the class number h(X) of the group X the number of double cosets

G(k) \ G(A)/X(A^{∞}) = H(X)

Problem. For a given k-group G, to decribe O-forms X with minimal class number h(X).

If G is commutative, the set H(X) is a finite abelian group, it will be called the class group of the scheme X.

Let us remind two important arithmetic characteristics of linear algebraic groups G defined over a number field k. The monomorphism G(k) → G(k_{v}) defines the diagonal embedding G(k) → ∏_{v} G(k_{v}). The closure \overline{G(k)} of the group G(k) in the topology of direct product is a normal subgroup and the factor

A(G) = (∏_{v} G(k_{v}))/\overline{G(k)}

is a finite commutative group. If A(G) = 0 then we shall say that the group G has weak approximation property. The group A(G) is a birational invariant in the category of connected linear algebraic groups over k, A(G) = 0 for a stably rational group G. The second invariant appears when studying the set H^{1}(k, G) of principal homogeneous spaces of the group G. We have a natural map

φ : H^{1}(k, G) → ∏_{v} H^{1}(k_{v}, G).

The kernel of the map φ is denoted by Sh(G), it is finite and consists of all principal homogeneous space of G having rational points in all fields k_{v}. As it was shown by Sansuc [Sa] the set Sh(G) has a natural structure of an abelian group, it is called the Shafarevich-Tate group and is also a birational invariant of the group G. In the work [V2, V4] the author shows that these two arithmetic invariants A(G) and Sh(G) are closely related to the birational invariant H^{1}(k, Pic X̄) in the case when G = T is a k-torus. Namely, there is an exact sequence of abelian groups

0 → A(T) → H^{1}(k, Pic X̄)^{∨} → Sh(T) → 0,  (1)

where B^{∨} = Hom (B, Q/Z). Hence, A(T) = Sh(T) = 0 if T has a metacyclic splitting field. We were sure that formula (1) is valid for every connected linear algebraic group G but realization of this idea required some efforts. First, formula (1) was not proved for semisimple connected simply connected groups, and it was an important problem. The point is that the module Pic X̄ is a permutation module for a simply connected group G, hence H^{1}(k, Pic X̄) = 0 and, having (1), we would easily get the relation A(G) = Sh(G) = 0 for simply connected groups. In fact, all terms of sequence (1) for simply connected groups are zero, but we have to calculate them independently from each other. Verification of the equality Sh(G) = 0 for simply connected groups required exhaustive work. The complete calculation can be found in the book [P-R]. Having equality (1) for tori and using triviality of A(G), Sh(G), H^{1}(k, Pic X̄) for simply connected groups, Sansuc established the formula

0 → A(G) → H^{1}(k, Pic X̄)^{∨} → Sh(G) → 0,

which is true for all connected linear algebraic groups [Sa]. Recently B. Kunyavskii and A. Skorobogatov gave an elegant proof of weak approximation property which includes the results of Kneser–Harder–Platonov–Sansuc [Ku-S]. They proved the following fact.

Theorem. Let G be a connected semisimple linear algebraic group defined over a number field k, let Y be the variety of maximal tori of the group G, F = k(Y), and let H be a generic torus of the group G over F. If H^{1}(F, p(H)) = 0 then A(G) = 0.

The proof is based on the existence of dense set of points in Y (k) over which the corresponding tori have no affect and on rationality of the variety Y.

Using the results of A. Klyachko from Section 9 we get A(G) = 0 for the following connected semisimple groups G:

1) G is simply connected,

2) G is adjoint,

3) G is absolutely simple.

We want to have something similar for calculation of the group Sh(G) for semisimple simply connected group G.

Let us come back to algebraic tori. Let k be the field of ℘-adic numbers, and let T be a torus over k. The group T (k) is locally compact. Let U be the maximal compact subgroup of T (k). Consider the O-algebra

B = {f ∈ k[T] | f(u) ∈ O ∀u ∈ U}

Since the group U is Zariski dense in T (k), the restriction of Hopf operators of the algebra k[T] to B defines the structure of Hopf O-algebra on B. Then Y = Spec B is a group scheme over O and Y ⊗_{O} k ≅ T. We shall call the scheme Y = Spec B the minimal integer model of the torus T. The definition of Y is very easy but in general the model Y is not of finite type. (This fact was pointed out by B. Moroz.) Now we study another integral model of T which is also determined only by the parameters of T, see [P-V] and [VKM]. Let L/k be the minimal splitting field of T, Π = Gal (L/k). Let T̂ be the Πmodule of rational characters of T. The ring k[T] is the ring of invariants of the group Π. This group acts on the field L and on the group T̂. The Hopf algebra k[T] of the torus T is of the form (L[T̂])^{Π} where L[T̂] is the group ring of the multiplicative group T̂. The torus T is an L/k-form of the trivial torus G_{m,k}^{d}, d = dim T, i.e. the embedding k[T] → L[T̂] can be extended to an isomorphism L ⊗_{k} C ≅ LC = L[T̂], here C = k[T]. Let us choose a basis ω_{1}, ..., ω_{n} of L/k, then we have the decomposition into a direct sum

L[T̂] = Cω_{1} ⊕ ⋯ ⊕ Cω_{n}.  (2)

Let χ ∈ T̂, then (2) implies the equality

χ = f_{1}^{(χ)}ω_{1} + ⋯ + f_{n}^{(χ)}ω_{n},  (3)

where f_{i}^{(χ)} ∈ C. Clearly, the algebra C = k[T] is generated by the coordinate functions f_{i}^{(χ)}. A finite system of generators of the algebra C can be obtained by decomposing basic characters χ_{1}, .., χ_{d} and inverse elements χ_{1}^{−1}, .., χ_{d}^{−1}, d = dim T.

Choose an integral basis ω_{1}, ..., ω_{n} of L/k and consider the ring A generated by the functions f_{i}^{(χ)} with integral coefficients. The algebra A does not depend on the choice of an integral basis L/k, it is of finite type over O, A ∩ k = O, kA = k[T]. We claim that A is a Hopf algebra compatible with the operators of k[T]. The Hopf operators act on the basic characters by

m^{∗}(χ_{i}) = χ_{i} ⊗ χ_{i}, i^{∗}(χ_{i}) = χ_{i}^{−1}, e^{∗}(χ_{i}) = 1,  (4)

where χ_{1}, .., χ_{d} is a basis of the group T̂. Let

χ_{m} = ∑_{k=1}^{n} x_{k}^{(m)}ω_{k}

be the decomposition of a basic character according to (3), and let y_{m} ∈ k[T] be the product of all distinct characters conjugate to χ_{m},,

ω_{i}ω_{j} = ∑_{k=1}^{n} c_{ij}^{k}ω_{k},  (5)

where c_{ij}^{k} ∈ O are the structure constants of the extension L/k. Easy calculations show that (4) can be written in the form

m^{∗}(x_{k}^{(m)}) = ∑ c_{ij}^{k}x_{i}^{(m)} ⊗ x_{j}^{(m)};

i^{∗}(x_{k}^{(m)}) = \frac{F_{k}^{(m)}}{y_{m}} , χ_{m}^{−1} = \frac{F_{1}^{(m)}}{y_{m}} ω_{1} + ... + \frac{F_{n}^{(m)}}{y_{m}} ω_{n};  (6)

e^{∗}(x_{k}^{(m)}) = c_{k}, 1 = ∑_{j=1}^{n} c_{j}ω_{j}.

The ring A is of the form A = O[x_{j}^{(i)}, y_{i}^{−1}], 1 ≤ i ≤ d, 1 ≤ j ≤ n. From (5), (6) it follows that the Hopf operators act on the ring and define the structure of O-Hopf algebra on A. The group scheme X = Spec A is of finite type over O, it is reduced and faithfully flat over O. This form does not depend on the choice of an integral basis of L/k, nor on the choice of a basis of the group T̂. We call this model the canonical minimal model of T. The scheme Y constructed at the beginning of this section will be called the maximal integral model of T.

We have the embedding O_{L}[T̂] ⊂ A ⊗_{O} O_{L}, where O_{L} is the ring of integers of the field L. An element χ ∈ T̂ defines a homomorphism χ : X(L) = T (L) → L^{∗}. Consider the restriction of the map χ to the maximal compact subgroup U of T (k) ⊂ T (L). The group χ(U) is a compact subgroup of L^{∗}, i.e. χ maps the group U into O_{L}^{∗}. Formulas (3) show that the values of any function f(u), u ∈ U, are from O. In other words, we have an inclusion of Hopf algebras A ⊂ B and a homomorphism of group schemes Y → X. Hence, we can formulate the following statements.

Theorem 1. Let k be a field of ℘-adic numbers with ring of integers O, T an algebraic k-torus. Then there exist two Hopf O-algebras A and B in k[T] which are uniquely determined by the Π-module T̂ according to the rules mentioned above. The schemes X = Spec A and Y = Spec B are O-forms of k-torus T. They have the following properties:

1. B = {f ∈ k[T] | f(u) ∈ O ∀u ∈ U};

2. A ⊂ B ⊂ k[T], A ∩ k = B ∩ k = O, k ⊗_{O} A = k ⊗_{O} B ≅ kA = kB = k[T];

3. the algebra A is of finite type over O;

4. X(O) = U = Hom_{Π}(T̂, O_{L}^{∗});

5. O_{L}[T̂] ⊂ O_{L}A;

Proof. Clearly we only have to prove property 4). Let V be a vector space over k of dimension m and T ⊂ GL(V). Choose a lattice M in V which is invariant under the action of the group U and kM = V. The elements of U can be represented by matrices from GL(m, O) in a basis of M. Thus

X(O) = X(k) ∩ GL(m, O) = U,

and

T (k) = Hom_{Π}(T̂, L^{∗}) ⊂ Hom (T̂, L^{∗}) = T (L) ≃ (L^{∗})^{d}, d = dim T.

The group Hom (T̂, O_{L}^{∗}) ≅ (O_{L}^{∗})^{d} is the maximal compact subgroup U_{L} of the group T (L). The equality T (k) = (T (L))^{Π} implies

X(O) = U = U_{L} ∩ T (k) = U_{L}^{Π} = Hom_{Π}(T̂, O_{L}^{∗}).

It should be mentioned that the isomorphism U_{L} ≅ (O_{L}^{∗})^{d} is not a Π-isomorphism. Definition 1. We call the reduction T_{℘} of the torus T modulo ℘ the reduction

T_{℘} = X ⊗_{O} O/℘ = Spec (O/℘ ⊗_{O} A).

of its canonical integral model X.

The reduction is always a group scheme over the finite field r_{k} = O/℘. We call the reduction very nice if T_{℘} is a torus over r_{k}; nice if T_{℘} is an algebraic group over r_{k} (reduced); bad if the group scheme T_{℘} is not reduced.

Theorem 2. If L/k is unramified then the minimal integral model of a k-torus T split over L is of the form X = Spec A, where A = (O_{L}[T̂])^{Π}. The scheme X is a torus over the ring O, the reduction T_{℘} is very nice.

Proof. We have an embedding O_{L}[T̂] ⊂ O_{L}A and the decomposition χ = f_{1}ω_{1} + ... + f_{n}ω_{n}, f_{i} ∈ A, χ ∈ T̂, where ω_{1}, .., ω_{n} is an integral basis of L/k. For all σ ∈ Π we have

σχ = f_{1}σ(ω_{1}) + ... + f_{n}σ(ω_{n}).

Since L/k is unramified, the determinant of the matrix (σ_{i}(ω_{j})) is invertible in the ring O_{L}, so f_{i} ∈ O_{L}[T̂] and A = (O_{L}[T̂])^{Π}.

The reduction of a quasi-split torus. One of important objects in the theory of algebraic tori is a so-called quasi-split torus which can be represented as a product of tori of the form R_{F/k}(G_{m}). Recall the structure of the minimal model of the quasi-split torus described in [V-F]. Let ω_{1}, .., ω_{n} be an integral basis of F/k, L the normal closure of F in k_{s}, Π = Gal (L/k), Π_{1} = Gal (L/F), S = R_{F/k}(G_{m}), Ŝ = Z[Π/Π_{1}] = Z[Π] ⊗_{Π_{1}} Z. Choose a basis χ_{1}, .., χ_{n} of the group Ŝ such that the group Π acts on it by permutations, Π_{1} is the stabilizer of the element χ_{1}. We have decompositions

χ_{1} = x_{1}ω_{1} + ... + x_{n}ω_{n},
χ_{i} = σ_{i}χ_{1} = x_{1}σ_{i}(ω_{1}) + ... + x_{n}σ_{i}(ω_{n}).

The Hopf algebra A of the minimal integral model of the torus S is of the form: A = O[x_{1}, .., x_{n}, y^{−1}], here y = χ_{1}..χ_{n} is a norm form of degree n in n variables x_{1}, .., x_{n}. The scheme X = Spec A is an O-form of the torus S. Since O[x_{1}, .., x_{n}, y^{−1}] is a localization of the polynomial ring O[x_{1}, .., x_{n}], the scheme X is smooth and connected. Consequently, X_{℘} is a connected algebraic group over r_{k}. Let us study the reduction in detail.

Let E be the maximal unramified extension of k in F. Then F/E is a totally ramified extension, k ⊂ E ⊂ F, (E : k) = f, (F : E) = e, ef = n, O_{E} and O_{F} are the rings of integers of the fields E and F, respectively. We can choose an integral basis in O_{F} in the following way. Let ω_{1}, ..., ω_{f} be a basis of the unramified extension O_{E}/O and 1, π, ..., π^{e−1} an integral basis of O_{F} /O_{E}. Then ω_{i}π^{j} is an integral basis of the extension F/k. Note that

S = R_{F/k}(G_{m}) = R_{E/k}(R_{F/E}(G_{m})).

Let Y = Spec C be the canonical integral model of E-torus R_{F/E}(G_{m}). The algebra C is of the form

C = O_{E}[z_{0}, ..., z_{e−1}, u^{−1}],

where z_{i} are the coefficients of the decomposition

χ_{1} = z_{0} + z_{1}π + ... + z_{e−1}π^{e−1}, z_{i} ∈ E[R_{F/E}(G_{m})],

and u is the product of e distinct characters conjugated to χ_{1} with respect to E. Since E/k is unramified, the discriminant of the basis ω_{1}, ..., ω_{f} is a unit of O, so we can introduce new coordinates x_{k}^{(i)} by the rule z_{i} = ∑_{k=1}^{f} x_{k}^{(i)}ω_{k}. In other words, X = R_{O_{E}/O_{k}}(Y). Let us study the structure of the reduction Y_{℘_{E}} of the scheme Y. It is well known that the residue fields r_{F} and r_{E} are equal and u ≡ z_{0}^{e} (mod ℘_{E}). So

r_{E} ⊗_{O_{E}} C = r_{E}[z_{0}, z_{1}, ..., z_{e−1}, z_{0}^{−1}] = r_{E}[z_{0}, z_{0}^{−1}] ⊗ r_{E}[z_{1}, ..., z_{e−1}],

i.e. the group r_{E}-scheme Y is a direct product of the torus G_{m,r_{E}} and the commutative r_{E}-group U_{E}. The group U_{E} is isomorphic to the affine r_{E}-group A_{r_{E}}^{e−1}. Consider the regular representation of the group Y in the basis 1, π, ..., π^{e−1}. One can see that the group Y (r_{E}) can be represented by triangular matrices of the form

( \begin{matrix}z_{0} & 0 & … & 0 & 0 \\ z_{1} & z_{0} & … & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ z_{e−2} & z_{e−3} & … & z_{0} & 0 \\ z_{e−1} & z_{e−2} & … & z_{1} & z_{0} \end{matrix} ) .

Hence, the group U_{E} is triangular and the eigenvalues of U_{E} are equal to 1, i.e. U_{E} is unipotent. The reader can find a detailed description of connected unipotent groups over r_{k}, char r_{k} = p > 0 in [Se]. In particular, it is well known that any commutative unipotent r_{k}-group of period p is isomorphic to the product of groups G_{a,r_{k}}. Our group U_{E} is of period p if e ≤ p. Thus we have proved

Theorem. The reduction X_{℘} = R_{r_{E}/r_{k}}(Y_{℘_{E}}) of the torus R_{F/k}(G_{m}) can be represented as a direct product of the maximal torus T_{℘} = R_{r_{E}/r_{k}}(G_{m}) of dimension f and a connected unipotent group U_{℘} = R_{r_{E}/r_{k}}(U_{E}) of dimension (e − 1)f. If e ≤ p (where p = char r_{k}) then U_{℘} ≅ G_{a,k}^{(e−1)f}.

If S is a quasi-split k-torus, i.e. S is a direct product of tori R_{F/k}(G_{m}), then the canonical integral model X of S is a product of canonical integral models of the tori R_{F/k}(G_{m}). Hence, the reduction X_{℘} is the product of the reductions of factors.

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