ON INTEGRAL MODELS OF AFFINE TORIC VARIETIES

B.E. Kunyavskii^{1} and B.Z. Moroz

1. The goal of this paper is to further investigate the class of (the systems of) Diophantine equations studied by one of us in a recent series of papers, [8-11]. Each of these systems of equations defines an integral model of an affine toric variety. To be precise, given an affine toric variety X defined over ℚ, these equations define an affine ℤ-scheme X whose generic fibre X ⊗_{ℤ} ℚ is isomorphic to X. After formalising this construction in n^{∘}2, we go on to generalise it. In n^{∘}3 a class of integral models of an affine toric variety defined over a number field is described; in the last section (n^{∘}4) we establish some finiteness properties of these integral models. It seems to be the first attempt to study such models systematically (cf., however, [14]). On the other hand, integral models of algebraic tori have been discussed in the literature, [1], [3], [4], [7], [12], [13], and we comment on the relation of our “standard model” of an algebraic torus to other models, leaving the detailed discussion of this point to a subsequent publication.

In what follows ℚ, ℤ, and ℚ_{p} stand for the field of rational numbers, the ring of rational integers, and the p-adic number field respectively; G_{m,L} denotes the multiplicative group defined over a field L; let A(𝔳) be the category of commutative 𝔳-algebras for a given commutative ring 𝔳.

2. Let k be a field of characteristic zero. An algebraic torus T of dimension d defined over k splits over a finite normal extension K | k, so that T ×_{k} K ≅ G_{m,K}^{d}. Let 𝔊 be the Galois group of K | k, and let M = Hom(T ×_{k}K, G_{m}) be the 𝔊-module of characters of T. Let ρ : 𝔊 → GL(d, ℤ) be the integral representation afforded by M; the dual module M̌ = Hom(M, ℤ) affords the contragredient representation ρ̌. We fix a ℤ-basis {e_{1}, …, e_{d}} of M̌, a basis {ω_{1}, …, ω_{n}} of K | k, introduce a system of independent variables x = (…, x_{ij}, …), 0 ≤ i ≤ d, 1 ≤ j ≤ n, and write

gt_{i} = ∑_{j=1}^{n} x_{ij}gω_{j}, t_{i}^{g} = ∏_{j=1}^{d} t_{j}^{r_{ji}(g)}, 1 ≤ i ≤ d, g ∈ 𝔊,

where ρ̌(g)e_{i} = ∑_{j=1}^{d} e_{j}r_{ji}(g). The torus T can be defined by the following system of equations in k[x] :

gt = t^{g}, t_{0} ∏_{i=1}^{d} t_{i} = 1, g ∈ 𝔊,  (1)

Typeset by AMS-TEX

1Dept. of Mathematics and Computer Science, Bar-Ilan University, 52900 Ramat Gan, Israel.

Research of the first author is partially supported by the Ministry of Absorption (Israel), and by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities

where t := (t_{1}, …, t_{d}), t_{0} = ∑_{j=1}^{n} x_{oj}ω_{j}. Given a 𝔊-invariant strongly convex rational polyhedral cone σ_{ℚ} in the ℚ-vector space V = M ⊗_{ℤ} ℚ, one defines the dual cone

σ̌_{ℚ} = {v | v ∈ V̌ , (v | u) ≥ 0  for  u ∈ σ_{ℚ}}

in V̌ = M̌ ⊗_{ℤ} ℚ, and the semigroup σ̌ = σ̌_{ℚ} ∩ M̌; let {u_{1}, …, u_{l}} be the minimal set of generators of σ̌ (cf. [6]). We write

u_{i} = ∑_{j=1}^{d} e_{j}c_{ji}, 1 ≤ i ≤ ℓ; e_{j} = ∑_{i=1}^{ℓ} u_{i}c̃_{ij}, 1 ≤ j ≤ d.

Let r̃(g) = c̃r(g)c, c̃ = (c̃_{ij})_{1≤i≤ℓ,1≤j≤d}, c = (c_{ij})_{1≤i≤d,1≤j≤ℓ}; clearly, ρ̌(g)u_{i} = ∑_{j=1}^{ℓ} u_{j}r̃_{ji}(g), g ∈ 𝔊, 1 ≤ i ≤ ℓ. In the new variables y = (…, y_{ih}, …), 1 ≤ i ≤ ℓ, 1 ≤ h ≤ n, equations

∑_{h=1}^{n} y_{ih}gω_{h} = ∏_{j=1}^{ℓ}(∑_{h=1}^{n} y_{jh}ω_{h})^{r̃_{ji}(g)}, g ∈ 𝔊, 1 ≤ i ≤ ℓ,  (2)

define an algebraic variety X_{T,σ} over k. It is a toric variety, containing the torus T as a dense open orbit.

Given a subring 𝔳 of the field k, an 𝔳-scheme X is said to be an integral model of X_{T,σ} if X ×_{𝔳} k ≅ X_{T,σ}. Suppose now that k is the quotient field of a ring 𝔳 and K is the quotient field of the ring 𝔒 = ∑_{i=1}^{n} ⊕ω_{i}𝔳; let, moreover, gω_{i} ∈ 𝔒 for g ∈ 𝔊, 1 ≤ i ≤ n. Then equations (2) define a standard integral model X_{0} of X_{T;σ}. Clearly,

X_{0}(A) = {α | α ∈ B^{l}, gα = α^{g} for g ∈ 𝔊}

for A ∈ A(𝔳), where B = A ⊗_{𝔳} O, and α_{i}^{g} := ∏_{j=1}^{ℓ} α_{j}^{r̃_{ji}(g)}, g ∈ 𝔊, 1 ≤ i ≤ ℓ. For a standard model T_{0} of the k-torus T, defined by (1), this gives

T_{0}(A) = {α | α ∈ (B^{∗})^{d}, gα = α^{g} for g ∈ 𝔊}  (3)

with α_{i}^{g} := ∏_{j=1}^{d} α_{j}^{r_{ji}(g)}, 1 ≤ i ≤ d, g ∈ 𝔊, in this case. Note that T_{0} is a group 𝔳-scheme since T_{0}(A) is an Abelian group for A ∈ A(𝔳). An assiduous reader will note that equations t^{g} = gt, g ∈ 𝔊, define T as a closed subset in the torus S = R_{K/k}G_{m,K}^{d}, while the condition ∏_{i=1}^{d} t_{i} ≠ 0 (equivalent to the equation t_{0} ∏_{i=1}^{d} t_{i} = 1) defines S as an open subset in 𝔸_{k}^{N}, N = dn. In other words, equations (1) define a natural embedding of the torus T into the quasi-trivial torus S = R_{K/k}T_{K}. Since S can be regarded as a maximal k-torus of GL_{k}(V), where V is an Ndimensional linear k-space, one obtains a faithful linear k-representation φ : T ↣ GL_{k}(V);let φ^{∗} : k[GL(V)] → k[T] be the corresponding homomorphism of the function algebras. The choice of an integer basis of K | k induces an integral structure in GL(V), or in other terms, defines a Hopf algebra H := 𝔳[GL(V)] in k[GL(V)]. The group 𝔳-scheme T_{0} = Spec 𝔳[x]/J, where J denotes the ideal defined by the equations (1), coincides with Spec φ^{∗}(H). This construction is reminiscent of the considerations of V.E. Voskresenskii and T.V. Fomina, [3], [4] (cf. also [15], [7]).

3. Let now k be a number field (that is, a finite extension of ℚ), and let 𝔳 and 𝔒 denote the rings of integers of k and K respectively. We fix a sequence 𝔅 = (𝔅_{1}, …, 𝔅_{l}) of fractional ideals of K satisfying the conditions

g𝔅_{i} = ∏_{j=1}^{ℓ} 𝔅_{j}^{r̃_{ji}(g)} for g ∈ 𝔊, 1 ≤ i ≤ ℓ,  (4)

and define a functor 𝔉_{𝔅} : A(𝔳) → Sgr, where A(𝔳) is the category of commutative 𝔳-algebras, and Sgr is the category of commutative semigroups, as follows. Let A ∈ A(𝔳), and let B_{i} = A ⊗_{𝔳} 𝔅_{i}, B = (B_{1}, …, B_{ℓ}). By definition,

𝔉_{𝔅}(A) = {α | α ∈ B, gα = α^{g} for g ∈ 𝔊},

where (gα)_{i} = gα_{i}, and (α^{g})_{i} = ∏_{j=1}^{ℓ} α_{j}^{r̃_{ji}(g)}.

Theorem 1. There is an integral model X_{𝔅} of X_{T,σ} representing the functor 𝔉_{𝔅} (in other words, X_{𝔅} is an 𝔳-scheme, X_{𝔅}(A) = 𝔉_{𝔅}(A) for A ∈ A(𝔳), and X_{𝔅} ⊗_{𝔳} k ≅ X_{T,σ}). The scheme X_{𝔅} can be covered by two open affine subsets.

Proof. It follows from the definition of X_{T,σ} that if X (A) = 𝔉_{𝔅}(A) for A ∈ A(𝔳) then X ×_{𝔳} k ≅ X_{T,σ}. To construct an 𝔳-scheme X_{𝔅} representing the functor 𝔉_{𝔅} we appeal to a theorem of Steinitz’s (see, for instance, [5, Theorem 13 on p. 95]), and denote by St(𝔅_{j}) the Steinitz class of the 𝔳-module 𝔅_{j}. By definition, St(𝔅_{j}) ∈ H_{k}, 1 ≤ j ≤ ℓ, where H_{k} is the class group of k. Since each ideal class contains infinitely many prime ideals, we may choose prime ideals 𝔭_{jν} in such a way that 𝔭_{jν} ∈ St(𝔅_{j}) and 𝔭_{jν} ≠ 𝔭_{iμ} for (j, ν) ≠ (i, μ), 1 ≤ i, j ≤ ℓ, {ν, μ} ⫅ {1, 2}. By construction,

𝔅_{j} = 𝔭_{jν}ω_{1j}^{(ν)} ⊕ ∑_{h=2}^{n} ⊕𝔳_{ν}ω_{hj}, 1 ≤ j ≤ ℓ, ν = 1, 2,

for some ω_{hj}, ω_{1j}^{(ν)} in K. Let 𝔄_{ν} = ∏_{j=1}^{ℓ} 𝔭_{jν}, ν = 1, 2. Since (𝔄_{1}, 𝔄_{2}) = 1, there are two integers b_{1}, b_{2} in 𝔳 such that (b_{1}, b_{2}) = 1 and 𝔄_{ν}|b_{ν}, ν = 1, 2. Let 𝔳_{ν} = 𝔳[1/b_{ν}], and let 𝔳_{12} = 𝔳[1/b_{1}b_{2}](= 𝔳[1/b_{1}, 1/b_{2}]). Clearly, 𝔭_{jν}𝔳_{ν} = 𝔳_{ν}, and 𝔭_{jν}𝔳_{12} = 𝔳_{12} for 1 ≤ j ≤ ℓ, ν = 1, 2. Therefore

𝔅_{j} ⊗_{𝔳} 𝔳_{ν} = 𝔳_{ν}ω_{1j}^{(ν)} ⊕ ∑_{h=2}^{n} ⊕𝔳_{ν}ω_{hj}  (5)

is a free 𝔳_{ν}-module, and

𝔅_{j} ⊗_{𝔳} 𝔳_{12} = 𝔳_{12}ω_{1j}^{(ν)} ⊕ ∑_{h=2}^{n} ⊕𝔳_{12}ω_{hj} (ν = 1, 2)  (6)

is a free 𝔳_{12}-module. Note that 𝔭_{j1} = α_{j}𝔭_{j2} and ω_{1j}^{(2)} = α_{j}ω_{1j}^{(1)} for some α_{j} in k^{∗} since 𝔭_{jν} ∈ St(𝔅_{j}), ν = 1, 2, 1 ≤ j ≤ ℓ. In view of (4), equations

gz^{(ν)} = (z^{(ν)})^{g}, g ∈ 𝔊,  (7)

where

gz_{j}^{(ν)} := y_{j1}^{(ν)}gω_{1j}^{(ν)} + ∑_{h=2}^{n} y_{jh}gω_{hj}, (z^{(ν)})_{j}^{g} = ∏_{i=1}^{ℓ}(z_{i}^{(ν)})^{r̃_{ij}(g)}, 1 ≤ j ≤ ℓ,

define an ideal 𝔄_{ν} in 𝔳_{ν}[y^{(ν]}]. After clearing the denominators, if necessary, we may (and shall) regard 𝔄_{ν} as an ideal in 𝔳[y^{(ν)}]. Let ℭ_{ν} = Spec 𝔳[y^{(ν)}]/𝔄_{ν}, and let U_{ν} = {𝔭 | 𝔭 ∈ ℭ_{ν}, 𝔭 ∤ b_{ν}}, ν = 1, 2. By definition, X_{𝔅} = W_{1} ∪ W_{2} with W_{ν} ≅ U_{ν} and W_{1} ∩ W_{2} ≅ U_{12}^{(ν)}, where U_{12}^{(ν)} = {𝔭 | 𝔭 ∈ ℭ_{ν}, 𝔭 ∤ b_{1}b_{2}}, ν = 1, 2, the gluing isomorphism φ : U_{12}^{(1)} → U_{12}^{(2)} being given by φ : y_{1j}^{(1)} ↦ α_{j}y_{1j}^{(2)}, 1 ≤ j ≤ ℓ. Identifying W_{ν} with Spec 𝔳_{ν}[y^{(ν)}]/𝔄_{ν}, and W_{12} with

Spec 𝔳_{12}[y^{(1)}]/𝔄_{1} ≅ Spec 𝔳_{12}[y^{(2)}]/𝔄_{2},

and bearing in mind the relations (5), (6), and (b_{1}, b_{2}) = 1, one can easily check that the 𝔳-scheme X_{𝔅} represents the functor F_{𝔅}, as required.

Corollary 1.

There is an 𝔳-scheme T_{0} satisfying relation (3).

Proof. Take σ = {0}, then σ̌_{ℚ} = ℚ^{d}, ℓ = 2d, and we may let u_{i} = e_{i}, u_{i+d} = −e_{i} for 1 ≤ i ≤ d, 𝔅_{j} = O for 1 ≤ j ≤ d. In this case, r̃_{ij}(g) = r_{ij}(g) and r̃_{ij+d}(g) = −r_{ij}(g) for 1 ≤ i, j ≤ d, r̃_{ij}(g) = 0 for i > d, g ∈ 𝔊. Therefore (α^{g})_{i} = ∏_{j=1}^{d} α_{j}^{r_{ji}(g)}, α_{i}α_{i+d} = 1 for 1 ≤ i ≤ d, g ∈ 𝔊, and the assertion follows from the definition of the functor 𝔉_{𝔅} and the theorem.

The 𝔳-group scheme T_{0} is called the standard integral model of the k-torus T; if the extension K | k has an integral basis, T_{0} coincides with the affine 𝔳-scheme defined in n^{∘} 2.

Remark 1. An 𝔳-scheme X_{𝔅} may be regarded as a T_{0}-toric variety in the following sense. There is an 𝔳-scheme morphism φ_{𝔅} : T_{0} × X_{𝔅} → X_{𝔅} such that the diagram

T_{0} × X_{𝔅} ⟶^{φ_{𝔅}} X_{𝔅}
↓  ↓
T × X_{T,σ} ⟶^{ψ} X_{T,σ}

commutes (here the vertical arrows are induced by the base change 𝔳 → 𝔳⊗_{𝔳}k(= k)). Thus the action of the 𝔳-scheme T_{0} on any of the 𝔳-schemes X_{𝔅}, defined by φ_{𝔅}, extends the natural action of the k-torus T on the k-variety X_{T,σ} (defined by ψ). By definition, the morphism φ_{𝔅} induces the group T_{0}(A) action on the set X_{𝔅}(A) for A ∈ A(𝔳).

4. Let

I(σ) = {𝔅 | 𝔅 ∈ I(K)^{ℓ}, g𝔅 = 𝔅^{g} for g ∈ 𝔊},

where (𝔅^{g})_{j} = ∏_{i=1}^{ℓ} 𝔅_{i}^{r_{ij}(g)}, 1 ≤ j ≤ ℓ, and I(K) stands for the group of fractional ideals of K. In this section we prove that the class of integral models

D(T, σ) = {X_{𝔅} | 𝔅 ∈ I(σ)}

of the toric variety X_{T,σ}, constructed in n^{∘} 3, contains only finitely many pairwise non-isomorphic 𝔳-schemes. To state our result, let

I_{pr}(σ) = {(α) | α ∈ K^{∗ℓ}, gα = α^{g} for 𝔊},

where (α^{g})_{j} = ∏_{i=1}^{ℓ} α_{i}^{r̃_{ij}(g)}, and (α) := ((α_{1}), …, (α_{ℓ})) stands for a sequence of principal ideals in I(K)^{ℓ}; let H(σ) = I(σ)/I_{pr}(σ).

Proposition 1. The group H(σ) is finite. Moreover, if 𝔅^{−1}𝔅′ ∈ I_{pr}(σ) then X_{𝔅} ≅ X_{𝔅′} (as 𝔳-schemes).

Proof. Let 𝔅′ = (α)𝔅 with (α) ∈ I_{pr}(σ). Then the schemes X_{𝔅} and X_{𝔅′} are (locally) defined by the same system of equations (7) if one chooses {α_{j}ω_{1j}^{(ν)}, α_{j}ω_{hj} | 2 ≤ h ≤ n} to serve as an 𝔳_{ν}-basis of 𝔅_{j} in (5), (6). Let J(σ) = I(σ) ∩ I_{pr}(K)^{ℓ}, where I_{pr}(K) = {(α) | α ∈ K^{∗}} denotes the group of principal ideals of K. Since I(σ)/J(σ) is finite, being isomorphic to a subgroup of the finite group H(K)^{ℓ} (here H(K) denotes the ideal class group of K), it suffices to prove that J(σ)/I_{pr}(σ) is a finite group. Let (α) ∈ J(σ); then gα = ϵ(α, g)α^{g} with ϵ(α, g) ∈ (𝔒^{∗})^{ℓ}. Clearly, (α) ∈ I_{pr}(σ) if and only if

ϵ(α, g) = \frac{gλ(α)}{λ(α)^{g}} with λ(α) ∈ (𝔒^{∗})^{ℓ}.  (8)

On the other hand, it follows from the identity

ϵ(α, g_{1}g_{2}) = g_{1}ϵ(α, g_{2})ϵ(α, g_{1})^{g_{2}}

that ϵ(α, g)^{n} satisfies relation (8) with λ(α) = ∏_{g∈𝔊} g^{−1}ϵ(α, g). Since the group (𝔒^{∗})^{ℓ} is finitely generated, it follows that the group J(σ)/I_{pr}(σ) is indeed finite. This proves the proposition.

Corollary 2. Suppose that σ contains an element v with (u | v) > 0 for each u in σ̌, and let I_{0}(K) be the semigroup of integral ideals in K, I_{0}(σ) = I(σ) ∩ I_{0}(K)^{ℓ}. There is a finite subset S of I_{0}(σ) such that any scheme X in D(T, σ) is isomorphic to a scheme X_{𝔅} with 𝔅 ∈ S.

Proof. Let v′ = ∑_{g∈𝔊} gv; clearly, (u_{i} | v′) > 0 for 1 ≤ i ≤ ℓ and gv′ = v′ for g ∈ 𝔊. Let X_{𝔅} ∈ D(T, σ); choose a natural number n such that n𝔅 ∈ I_{0}(K)^{ℓ}, and let α = (n^{m_{1}}, …, n^{m_{e}}) with m_{i} = (u_{i} | v′). Let 𝔅′ = (α)𝔅, then 𝔅′ ∈ I_{0}(σ) and X_{𝔅} ≅ X_{𝔅′} since gα = α^{g} = α for g ∈ 𝔊.

Remark 2. We do not know whether

X_{𝔅} ≅ X_{𝔅′} ⇒ 𝔅^{−1}𝔅′ ∈ I_{pr}(σ),

but suspect it is not the case.

Remark 3. If σ = {0} then X_{T,σ} ≅ T, and therefore X_{𝔅} is an 𝔳-integral model of the k-scheme T for 𝔅 ∈ I(σ). However, X_{𝔅} is not an 𝔳-group scheme unless 𝔅 ∈ I_{pr}(σ), in which case X_{𝔅} ≅ T_{0}.

As it has been noted at the end of n^{∘} 2, equations (1) define a closed embedding φ : T ↪ S, where S = R_{K/k}T_{K}, , T_{K} ≅ G_{m,K}^{d}. Let S be the Néron-Raynaud model of S (cf. [1, Ch. 10]). The connected component S^{0} of the identity element in S is isomorphic to R_{𝔒/𝔳}G_{m,𝔒}^{d} [12, Lemma 3.1]; therefore T_{0} coincides with the schematic closure of T in S and, moreover, T_{0} ⊆ S^{0} (in view of (3)). If the extension K | k is tame, one may deduce from the results of Edixhoven, [2], that T_{0} is a smooth 𝔳group scheme, and therefore in this case it coincides with the connected component of the identity element in the Néron-Raynaud model of the torus T, by Proposition 10.1.4 in [1]. In general, however, T_{0} may fail to be a smooth 𝔳-scheme. Let, for example, k = ℚ, K = ℚ(\sqrt{d}), d = 3(mod 4), and let T = R_{K/k}^{1}G_{m} be the norm-torus, defined as the kernel of the norm N : R_{K/k}G_{m,K} → G_{m,k}. Then

T_{0} = Spec ℤ[x, y]/(x^{2} − dy^{2} − 1),

and the fibre of T_{0} at the prime 2 is non-reduced, because f^{2} = 0 in F_{2}[x, y]/(x^{2} − dy^{2} − 1) for f = x + dy + 1 (in this case the Néron-Raynaud model can be obtained from T_{0} by blowing up two closed points, [2, Example 4.3]).

Acknowledgement. We are grateful to Professor G. Faltings for an important consultation, and to Professor V.E. Voskresenskii and Dr. T.V. Fomina for a few conversations and correspondence relating to integral models of algebraic tori. This work was done at the Max-Planck-Institut für Mathematik, whose hospitality and financial assistance are gratefully acknowledged.

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Max-Planck-Institut für Mathematik, Gottfried-Claren-Str. 26, 53225 Bonn, Germany