A short proof of Klyachko’s theorem about rational algebraic tori

Mathieu Florence

Abstract In this paper, we give another proof of a theorem by Klyachko ([Kl]), which asserts that Zariski’s conjecture holds for a special class of tori over an arbitrary ground field.

1 Introduction

The main purpose of this paper is to give a much simpler proof of a theorem due to Klyachko ([Kl]; see also [Vo], chap. 2, 6.3), which is here theorem 2.4. To achieve this, we first prove a generalization of a theorem due to Voskresenskii ([Vo], chap. 2, 5.1, corollary). To be more precise, we show stable rationality for a certain class of algebraic tori over a given field k, strictly containing the cyclotomic ones. What is more, we give an effective way of presenting the character module of these tori as the kernel of a surjection between permutation modules (that is, lattices that contain a basis which is permuted by the action of the absolute Galois group of k). Recall that, according to loc. cit., chap. 2, 4.7, theorem 2 , the existence of such a surjection is a necessary and sufficient condition for a torus to be stably rational. All the basic material concerning algebraic tori and rationality questions related to these is contained in loc. cit., chap.2; we shall assume that the reader is familiar with this reference. In the following section, the symbol ⊗ alone means ⊗_{ℤ}. If k is a field with separable closure k_{s}, we denote by Γ_{k} the profinite group Gal(k_{s}/k). Let Γ be a profinite group. By a Γ-lattice, we mean a free ℤ-module of finite rank, endowed with a continuous action of Γ. We will say simply ’exact sequence’ instead of ’exact sequence of Γ-lattices’.

2 Stably rational and rational algebraic tori

To begin this section, we prove an elementary but crucial lemma.

Lemma 2.1 Let Γ be a profinite group. Let A_{i}, B_{i}, C_{i}, i = 1, 2 be Γ-lattices, fitting into two exact sequences

0 ⟶ A_{i} ⟶^{j_{i}} B_{i} ⟶^{π_{i}} C_{i} ⟶ 0.

Assume we are given s_{i} : C_{i} ⟶ B_{i}, and d_{1}, d_{2} two coprime integers, such that π_{i} ∘ s_{i} = d_{i}Id, i = 1, 2. Let

A_{3} = A_{1} ⊗ A_{2},
B_{3} = (B_{1} ⊗ B_{2}) ⊕ (C_{1} ⊗ C_{2}),

and

C_{3} = (C_{1} ⊗ B_{2}) ⊕ (B_{1} ⊗ C_{2}).

Then there is an exact sequence

0 ⟶ A_{3} ⟶^{j_{3}} B_{3} ⟶^{π_{3}} C_{3} ⟶ 0,

together with a morphism s_{3} : C_{3} ⟶ B_{3}, satisfying π_{3} ∘ s_{3} = d_{1}d_{2}Id.

Proof. We have an exact sequence

0 ⟶ A_{1}⊗A_{2} ⟶ B_{1}⊗B_{2} ⟶^{(π_{1}⊗Id)⊕(Id⊗π_{2})} (C_{1}⊗B_{2})⊕(B_{1}⊗C_{2}) ⟶^{π} C_{1}⊗C_{2} ⟶ 0,

where π = Id ⊗ π_{2} − π_{1} ⊗ Id. Select integers u, v such that vd_{2} − ud_{1} = 1. Then the map

s : C_{1} ⊗ C_{2} ⟶ (C_{1} ⊗ B_{2}) ⊕ (B_{1} ⊗ C_{2}),
c_{1} ⊗ c_{2} ↦ (vc_{1} ⊗ s_{2}(c_{2}), us_{1}(c_{1}) ⊗ c_{2})

is a splitting of π. Hence we have an exact sequence

0 ⟶ A_{3} ⟶^{j_{3}} B_{3} ⟶^{π_{3}} C_{3} ⟶ 0

as stated, where

π_{3} : (B_{1} ⊗ B_{2}) ⊕ (C_{1} ⊗ C_{2}) ⟶^{((π_{1}⊗Id)⊕(Id⊗π_{2}),s)} (C_{1} ⊗ B_{2}) ⊕ (B_{1} ⊗ C_{2}).

The last assertion is obvious: if r_{i} : B_{i} ⟶ A_{i} (i = 1, 2) are such that r_{i} ∘ j_{i} = d_{i}Id, then

r_{3} := (r_{1} ⊗ r_{2}, 0) : B_{3} ⟶ A_{3}

satisfies r_{3} ∘ j_{3} = d_{1}d_{2}Id.□

From this we can derive the following Theorem 2.2 Let k be a field, and X_{1}, ..., X_{r} be finite Γ_{k}-sets. For i = 1, ..., r, denote by J_{i} the kernel of the canonical surjection ℤ^{X_{i}} ⟶^{π_{i}} ℤ. Let J = ⊗_{i}J_{i}. If the orders of the X_{i} are two by two coprime, then we have an exact sequence

0 ⟶ J ⟶ ⨁_{I∈ℐ_{0}} ℤ^{Π_{i∈I}X_{i}} ⟶^{π} ⨁_{I∈ℐ_{1}} ℤ^{Π_{i∈I}X_{i}} ⟶ 0,

where ℐ_{i} is the set of subsets of {1, ..., r} whose cardinality is congruent to r − i mod 2. In particular, a k-torus with character module isomorphic to J is stably rational over k. What is more, let d denote the product of the orders of the X_{i}, i = 1, ..., r. Then there exists

s : ⨁_{I∈ℐ_{1}} ℤ^{Π_{i∈I}X_{i}} ⟶ ⨁_{I∈ℐ_{0}} ℤ^{Π_{i∈I}X_{i}}

such that π ∘ s = dId.

Proof. For i = 1, ..., r, we have a canonical map

s_{i} : ℤ ⟶ ℤ^{X_{i}},

1 ↦ ∑_{x∈X_{i}} x,

which satisfies π_{i} ∘ s_{i} = d_{i}Id, where d_{i} is the order of X_{i}. The proof is then an easy induction using the previous lemma and the obvious isomorphism ℤ^{X} ⊗ ℤ^{Y} ≃ ℤ^{X×Y}, for any two finite sets X and Y. □

As a particular case of this theorem, we recover a result due to Voskresenskii ([Vo], chap. 2, 5.1 corollary).

Corollary 2.3 Let k be a field, and l/k a Galois extension with cyclic Galois group G of ordrer n = p_{1}...p_{r}, where the p_{i} are prime numbers. Let σ be a generator of this Galois group, and T/k the n^{th} cyclotomic torus, i.e. the torus with character group isomorphic to ℤ[X]/ϕ_{n}(X), where ϕ_{n}(X) is the n^{th} cyclotomic polynomial, the action of σ being given by multiplication by X (in other words, the character group of T is isomorphic to the ring of integers of the n^{th} cyclotomic extension of ℚ, with the action of σ being given by multiplication by a primitive n^{th} root of unity). Then T is stably rational over k.

Proof. For i = 1, ..., r, let X_{i} be the unique quotient of G isomorphic to ℤ/p_{i}. With the notations of the preceding theorem, the Γ_{k}-module J is isomorphic to the character module of T (this is just the fact that the ring of integers of the n^{th} cyclotomic extension of ℚ is naturally isomorphic to the tensor product of the rings of integers of the p_{i}^{th} cyclotomic extensions of ℚ), whence the claim.□

We are now able to give a simple proof of the following theorem.

Theorem 2.4 (Klyachko) Let k be a field, and X, Y two finite Γ_{k}-sets, of coprime orders p and q, respectively. Consider the two basic exact sequences

0 ⟶ J_{X} ⟶ ℤ^{X} ⟶ ℤ ⟶ 0,
0 ⟶ J_{Y} ⟶ ℤ^{Y} ⟶ ℤ ⟶ 0.

Then, a k-torus T with character module isomorphic to J := J_{X} ⊗ J_{Y} is rational over k.

Proof. Select integers u, v such that up − vq = 1. Theorem 2.2 gives the following presentation of J:

0 ⟶ J ⟶ ℤ^{X×Y} ⊕ ℤ ⟶^{π} ℤ^{X} ⊕ ℤ^{Y} ⟶ 0,

where π(x ⊗ y, 0) = (x, y) and π(0, 1) = (u ∑_{x∈X} x, v ∑_{y∈Y} y). Let E/k (resp. F/k) be the etale extension of k corresponding to X (resp. to Y). Then, in terms of tori, this exact sequence reads as

1 ⟶ R_{E/k}(𝔾_{m}) × R_{F/k}(𝔾_{m}) ⟶^{i} R_{E⊗_{k}F/k}(𝔾_{m}) × 𝔾_{m} ⟶ T ⟶ 1,

where R denotes Weil scalar restriction. The map i is given on the k-points of the considered tori by the following formula:

i(x, y) = (x ⊗ y, N_{E/k}(x)^{u}N_{F/k}(y)^{v}), x ∈ E^{∗}, y ∈ F^{∗}.

Thus, we have a generically free action of the algebraic k-group H := R_{E/k}(𝔾_{m}) × R_{F/k}(𝔾_{m}) on the k-vector space V := (E ⊗_{k} F) ⊕ k, such that T is birational to the quotient V/H (of course, such a quotient is defined up to birational equivalence only). Assume that p < q. Let G/k be the algebraic k-group GL_{k}(E) × R_{F/k}(𝔾_{m}) (E being viewed as a k-vector space). I claim that the action of H on V can be naturally extended to an action of G on V (H being viewed as a subgroup of G the obvious way). Indeed, this new action is given on the k-points by the formula, for g = (ϕ, y) ∈ G(k), v = (e ⊗ f, λ) ∈ V:

g.v = (ϕ(e) ⊗ yf, det(ϕ)^{u}N_{F/k}(y)^{v}λ).

This action is generically free. Indeed, this is an easy consequence of the equality up − vq = 1 and of the following lemma.

Lemma 2.5 Let G act on E ⊗_{k} F the obvious way. Then the stabilizer of a generic element is the subgroup 𝔾_{m} of G given, on the level of k-points, by elements of the form (x, x^{−1}) ∈ GL_{k}(E) × F^{∗}, for x ∈ k^{∗}.

We postpone the proof until the end of this section. Assuming this lemma, we have a birational G-equivariant isomorphism V ≃ (V/G) × G, where the action of G on the right is given by translation. Indeed, this is a direct consequence of Hilbert’s theorem 90, asserting that H^{1}(l, G) = 1 for any field extension l of k. Hence we have birational isomorphisms

T ≃ V/H ≃ V/G × G/H.

It is clear that the k-variety G/H = GL_{k}(E)/R_{E/k}(𝔾_{m}) is k-rational. As in Klyachko’s original proof, the key point is here that the k-variety (defined up to birational equivalence) V/G is independent of E (seen as an etale k-algebra). Hence, the birational equivalence class of T is independent of E; we may therefore assume that E is split, i.e. that the action of Γ_{k} on X is trivial. But then J is isomorphic to J_{Y} ^{p−1}, hence T is birational to (R_{F/k}(𝔾_{m})/𝔾_{m})^{p−1}, which is a rational variety (it is an open subvariety of (ℙ_{k}^{q−1})^{p−1}). □

Proof of lemma 2.5. We may assume that F is split, i.e. F = k^{q} as an etale k-algebra. Let f_{i}, i = 1, ..., q denote the canonical k-basis of F. Consider an element w = ∑_{i} e_{i} ⊗ f_{i} ∈ E ⊗_{k} F in general position. Let g = (ϕ, (λ_{1}, ..., λ_{q})) ∈ GL_{k}E × F^{∗} be such that g.w = w. This amounts to saying that ϕ(e_{i}) = λ_{i}^{−1}e_{i} for all i. Since p < q and since w is in general position, e_{1}, ..., e_{p} form a basis of E with respect to which the i’th component of e_{p+1} is non zero for all i = 1, ..., p. This readily implies that the λ_{i} are all equal to some scalar λ and that ϕ = λ^{−1}Id, thus proving the claim. □

References

[Vo] V. E. Voskresenskii. — Algebraic Groups and their Birational Invariants, Translations of Mathematical Monographs 179 (1991), Amer. Math. Soc.

[Kl] A. A. Klyachko. — On rationality of tori with a cyclic splitting field, Arithmetic and Geometry of Varieties, Kuibyshev Univ. Press (1988), 73-78 (russian).