INCOMPRESSIBILITY OF GENERIC TORSORS OF NORM TORI

NIKITA A. KARPENKO

Abstract. Let p be a prime integer, F a field of characteristic not p, T the norm torus of a degree p^{n} extension field of F, and E a T -torsor over F such that the degree of each closed point on E is divisible by p^{n} (a generic T -torsor has this property). We prove that E is p-incompressible. Moreover, all smooth compactifications of E (including those given by toric varieties) are p-incompressible. The main requisites of the proof are: (1) A. Merkurjev’s degree formula (requiring the characteristic assumption), generalizing M. Rost’s degree formula, and (2) combinatorial construction of a smooth projective fan invariant under an action of a finite group on the ambient lattice due to J.-L. Colliot-Thélène - D. Harari - A.N. Skorobogatov, produced by refinement of J.-L. Brylinski’s method with a help of an idea of K. Künnemann.

Let F be a field, p a prime integer. We say that an F -variety (by which we mean just a separated F -scheme of finite type) is p-incompressible (resp., incompressible), if its canonical p-dimension (resp., canonical dimension), defined as in [13, §4b], is equal to its usual dimension. Paraphrasing the definition, an integral F -variety X is incompressible if and only if X(L) = ∅ for any extension field L of F which is a subfield of the function field F (X) of transcendence degree < dim X; a connected smooth complete variety X is incompressible if and only if any rational map X −−→ X is dominant, [13, Corollary 4.3(2)]; p-incompressibility is a p-local version of incompressibility implying the incompressibility.

Given an arbitrary F -variety V, we write n_{V} for the greatest common divisor of the degrees of the closed points on V. Usually, we are only interested in v_{p}(n_{V}), where v_{p} is the p-adic valuation.

By a compactification of an F -variety V we mean a complete F -variety X containing a dense open subvariety isomorphic to V.

Given a finite separable extension field (or, more generally, an étale algebra) K/F, its norm torus T = T_{K/F}, also called norm one torus and denoted by R_{K/F}^{(1)} (𝔾_{m}), is the algebraic torus defined as the kernel of the norm map of algebraic tori N_{K/F} : R_{K/F} (𝔾_{m}) → 𝔾_{m}_{, F}, where R_{K/F} is the Weil transfer with respect to K/F. The group of F -points of T is the subgroup of norm 1 elements in K^{×}.

We consider T -torsors over F (i.e., the principal homogeneous spaces of T) and call them simply T -torsors. Any element a ∈ F^{×} produces a T -torsor E_{a} with the set of F - points being the set of norm a elements in K^{×}. The isomorphism class of E_{a} corresponds to the image of a under the connecting homomorphism H^{0}(F, 𝔾_{m}) → H^{1}(F, T) of the long exact sequence in galois cohomology, arising from the short exact sequence of the definition of T. This connecting homomorphism is surjective (and its kernel is the norm subgroup). Thus every T -torsor E is isomorphic to E_{a} for some element a ∈ F^{×} (whose class modulo N_{K/F} (K^{×}) is uniquely determined by E).

Date: 5 August 2012.

Key words and phrases. Algebraic tori, toric varieties, incompressibility, Chow groups and Steenrod operations. Mathematical Subject Classification (2010): 14L17; 14C25.

Note that for any E, the integer n_{E} divides dim_{F} K. Moreover, if K is the product of étale F -algebras K_{1}, …, K_{r}, then n_{E} divides dim_{F} K_{i} for each i = 1, …, r. In particular, n_{E} = dim_{F} K is possible only if K is a field.

The main result of this note is the following theorem known for cyclic K/F (see [13, §11d]):

Theorem 1. Assume that char F ≠ p. For some integer n ≥ 0, let K/F be a (separable) extension field of degree p^{n}, T its norm torus, and E a T -torsor such that n_{E} = p^{n}. Then the F -variety E is p-incompressible. Any smooth compactification of the variety E is also p-incompressible.

Example 2. Let t be an indeterminate, K/F an arbitrary finite separable field extension, and E_{t} the T_{K(t)/F (t)}-torsor of norm t elements (E_{t} is the generic principal homogeneous space of T, or generic T -torsor in the sense of [12, §3], produced out of the imbedding of T into the special algebraic group R_{K/F} (𝔾_{m})). Then the degree of every closed point on E_{t} is divisible by d := [K : F]. Proving this, one may replace the base field F (t) by F ((t)). If for a finite extension field L/F ((t)), the element t ∈ L is the norm for the d-dimensional étale L-algebra K((t)) ⊗_{F ((t))} L, then v(t) is divisible by d (see [4, Theorem in §(2.5) of Chapter 2]), where v is the extension to L of the t-adic discrete valuation on F ((t)); therefore d divides [L : F ((t))] (see [4, Exercise 1c in §2 of Chapter 2]).

Proof of Theorem 1. According to [2, Corollaire 1], there exists a smooth projective (toric) F -variety X containing E as an open subvariety. Clearly, E is p-incompressible if X is so. (Actually, by [17, Proposition 4], for any extension field L/F, one has X(L) ≠ ∅ if and only if E(L) ≠ ∅ so that E is p-incompressible if and only if X is so.) Since the property of being p-incompressible is birationally invariant on connected smooth complete varieties (see e.g. [10, Remark 4.13] or [9, Lemma 3.6]), all smooth compactifications of E are p-incompressible provided that X is so.

A connected smooth complete variety V over a field of characteristic ≠ p is called (p, n)-rigid here, if it is R^{p}-rigid in the sense of [14, §7] for the infinite sequence

R := (0, …, 0, 1, 0, …)

of 0 and 1 with precisely one 1 staying on the nth position. By definition, (p, n)-rigidity of V means that v_{p}(n_{V}) = v_{p}(deg c_{R}(−T_{V})), where T_{V} is the tangent bundle of V and c_{R} is the Chern class corresponding to the sequence R and the prime p in the sense of [14, §4].

By Theorem 3 right below, the variety X is (p, n)-rigid. A (p, n)-rigid variety is pincompressible by [14, Corollary 7.3]. This is the place where the degree formula of [14] is used and where the characteristic assumption is needed. The projectivity assumption made in [14] is superfluous because of [1, §10]. □

Theorem 3. For E as in Theorem 1, any smooth compactification of E is (p, n)-rigid.

Proof. Since the property of being (p, n)-rigid is birationally invariant [14, Remark 7.5], it suffices to construct one (p, n)-rigid smooth compactification of E. For this, let Γ be the galois group of the normalization L of K/F and let 𝔛 be the Γ-set corresponding to the étale F -algebra K in the sense of [11, §18]. The cardinality of the set 𝔛 is equal to p^{n} = [K : F].

The cocharacter lattice N of the split torus T_{L} is the lattice of the elements in the free abelian group ℤ[𝔛] on 𝔛 with the sum of coordinates = 0. There exists a smooth projective fan A of the lattice N (we do not require that A is invariant under the action of Γ on N yet), for instance, a fan producing the toric variety given by the projective space (see [6, Exercise of §1.4]).

The symmetric group S of all permutations of the set 𝔛 act on N by permutations of the coordinates. By [2, Theorem 1], there exists a smooth projective S-invariant fan B of N which is a subdivision of A. This produces a smooth projective toric variety X_{k} (over any given field k) endowed with an action of S (see [3, §5.5]) as well as with an action of the split k-torus with the cocharacter lattice N. In particular, Γ acts on X_{F}. Twisting X_{F} by the principal homogenous space Spec L of the constant group Γ (quasi-projectivity of X_{F} is needed for existence of the twisting, see [5, Proposition 2.12] or [16, V.20]) we get a smooth projective T -equivariant compactification X of T (cf. [2, Preuve du Corollaire 1 á partir du Théorème 1]). Twisting afterwards X by the T -torsor E as in [5, Proposition 2.12] (using (quasi-)projectivity once again), we get a smooth compactification Y of E. We claim that the variety Y is (p, n)-rigid.

First of all, by [17, Proposition 4], we have v_{p}(n_{Y}) = v_{p}(n_{E}) = n. Therefore to check (p, n)-rigidity of Y we have to check that v_{p}(deg c_{R}(−T_{Y})) = n, where R is the sequence introduced above. The integer deg c_{R}(−T_{Y}) can be expressed in terms of the fan B (see [6, Propositions of §4.3 and of §5.2]) and therefore does not depend on the base field anymore so that we may replace Y by X_{k} with an arbitrary chosen field k.

Let us choose a field k (of characteristic ≠ p) possessing a degree p^{n} cyclic extension field l such that for its norm torus T′ = T_{l/k} there exists a T′-torsor E′ with v_{p}(n_{E′}) = n (we can find it using Example 2). Fixing an arbitrary bijection of the (order p^{n} cyclic) galois group Γ′ of l/k with the set 𝔛, we get an action of Γ′ on 𝔛 and therefore on X_{k}. Twisting X_{k} by the principal homogeneous space Spec l of Γ′ and then by the T′-torsor E′, we get a smooth compactification Y′ of E′. Another smooth compactification of E′ is given by a Severi-Brauer variety of certain central division k-algebra of degree p^{n} (cf. [13]). Concretely, if E′ is given by some a ∈ k^{×}, the Severi-Brauer variety of the cyclic division algebra (l/k, a) can be taken. The Severi-Brauer variety is (p, n)-rigid by [14, §7.2], therefore Y′ is (p, n)-rigid. Since v_{p}(n_{Y′}) = n, it follows that v_{p}(deg c_{R}(−T_{Y′})) = n. □

A (p, n)-rigid variety is actually strongly p-incompressible in the sense of [8, §2]. Therefore for E as in Theorem 1, any smooth compactification of E is strongly p-incompressible. This statement is stronger than the part of the statement of Theorem 1 saying that any smooth compactification of E is p-incompressible. It can be formulated in terms of E alone (without mentioning its compactification) as follows:

Corollary 4. Let E be as in Theorem 1 and let Y be an integral complete (not necessarily smooth) F -variety such that v_{p}(n_{Y}) ≥ n (= v_{p}(n_{E})) and v_{p}(n_{Y_{F (E)}}) = 0 (i.e., Y_{F (E)} has a closed point of a prime to p degree). Then

(1) dim Y ≥ dim E;

(2) if dim Y = dim E then v_{p}(Y) = n and v_{p}(n_{E_{F (Y )}}) = 0.

Proof. Apply the strong incompressibility of a smooth compactification of E given by a toric variety X, taking into account [17, Proposition 4] saying that for any extension field L/F, X(L) = ∅ provided that E(L) = ∅. □

Here is an application of Theorem 1 suggested in [13, §11d]:

Corollary 5. For any p-primary (separable) field extension K/F in characteristic ≠ p, the essential p-dimension as well as the essential dimension of the functor of non-zero norms of K/F (defined as in [13, Example 11.11]) is equal to the degree [K : F].

Proof. A proof for the case of cyclic K/F is given in [13, §11d]. The only missing point to make it work in the general case was absence of Theorem 1 (for non-cyclic K/F). □

Remark 6. In the case of cyclic K/F, the statement of Corollary 5 as well as the statement of Theorem 1 hold also in characteristic p due to existence of a proof of pincompressibility for Severi-Brauer varieties avoiding a use of the degree formula (see [8, Examples 2.4 and 3.3]). On the other hand, neither Theorem 3 nor Corollary 4 are known in characteristic p even for cyclic field extensions. One may expect that Theorems 1, 3 and Corollaries 4, 5 hold in characteristic p for general K/F. This is so in the case of [K : F] = p (i.e., in the case of n = 1) due to results of O. Haution, [7, Corollary 10.2] (for Theorem 1 and Corollary 5 alone it suffices to use [15, Proposition 1.5(2)]=[13, Proposition 2.4(2)]).

Acknowledgements. I am grateful to Alexander Sergeevich Merkurjev for involving me into the subject.

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Université Pierre et Marie Curie, Institut de Mathématiques de Jussieu, Paris, FRANCE

E-mail address: karpenko at math.jussieu.fr, web page:  www.math.jussieu.fr/~karpenko