HYPERBOLICITY OF ORTHOGONAL INVOLUTIONS

NIKITA A. KARPENKO

Abstract. We show that a non-hyperbolic orthogonal involution on a central simple algebra over a field of characteristic ≠ 2 remains non-hyperbolic over some splitting field of the algebra.

1. Introduction

Throughout this note (besides of §3 and §4) F is a field of characteristic ≠ 2. The basic reference for the material related to involutions on central simple algebras is [12]. The degree deg A of a (finite-dimensional) central simple F -algebra A is the integer \sqrt{dim_{F} A}; the index ind A of A is the degree of a central division algebra Brauer-equivalent to A.

The main result of this paper is as follows (the proof is given in §7):

Theorem 1.1 (Main theorem). A non-hyperbolic orthogonal involution σ on a central simple F -algebra A remains non-hyperbolic over the function field of the Severi-Brauer variety of A.

To explain the statement of Abstract, let us note that the function field L of the Severi-Brauer variety of a central simple algebra A is a splitting field of A, that is, the L-algebra A_{L} is Brauer-trivial.

A stronger version of Theorem 1.1, where the word “non-hyperbolic” (in each of two appearances) is replaced by “anisotropic”, is, in general, an open conjecture, cf. [11,

Conjecture 5.2].

Let us recall that the index of a central simple algebra possessing an orthogonal involution is a power of 2. Here is the complete list of indices ind A and coindices coind A = deg A/ ind A of A for which Theorem 1.1 is known (over arbitrary fields of characteristic ≠ 2), given in the chronological order:

• ind A = 1 — trivial;

• coind A = 1 (the stronger version) — [11, Theorem 5.3];

• ind A = 2 (the stronger version) — [15, Corollary 3.4];

• coind A odd — [6, appendix by Zainoulline] and independently [8, Theorem 3.3];

• ind A = 4 and coind A = 2 — [18, Proposition 3];

• ind A = 4 — [7, Theorem 1.2].

Let us note that Theorem 1.1 for any given (A, σ) with coind A = 2 implies the stronger version version of Theorem 1.1 for this (A, σ): indeed, by [8, Theorem 3.3], if coind A = 2 and σ becomes isotropic over the function field of the Severi-Brauer variety, then σ becomes hyperbolic over this function field and the weaker version applies. Therefore we get

Date: 18 April 2009.

Key words and phrases. Algebraic groups, projective homogeneous varieties, Chow groups.

2000 Mathematical Subject Classifications: 14L17; 14C25.

Supported by the Collaborative Research Centre 701 of the Bielefeld University.

Theorem 1.2. An anisotropic orthogonal involution on a central simple F -algebra of coindex 2 remains anisotropic over the function field of the Severi-Brauer variety of the algebra. □

Sivatski’s proof of the case with deg A = 8 and ind A = 4, mentioned above, is based on the following theorem, due to Laghribi:

Theorem 1.3 ([13, Théorème 4]). Let φ be an anisotropic quadratic form of dimension 8 and of trivial discriminant. Assume that the index of the Clifford algebra C of φ is 4. Then φ remains anisotropic over the function field F (X_{1}) of the Severi-Brauer variety X_{1} of C.

The following alternate proof of Theorem 1.3, given by Vishik, is a prototype of our proof of Main theorem (Theorem 1.1). Let Y be the projective quadric of φ and let X_{2} be the Albert quadric of a biquaternion division algebra Brauer-equivalent to C. Assume that φ_{F (X_{1})} is isotropic. Then for any field extension E/F, the Witt index of φ_{E} is at least 2 if and only if X_{2}(E) ≠ ∅. By [20, Theorem 4.15] and since the Chow motive M(X_{2}) of X_{2} is indecomposable, it follows that the motive M(X_{2})(1) is a summand of the motive of Y. The complement summand of M(Y) is then given by a Rost projector on Y in the sense of Definition 5.1. Since dim Y + 1 is not a power of 2, it follows that Y is isotropic

(cf. [5, Corollary 80.11]).

After introducing some notation in §2 and discussing some important general principles concerning Chow motives in §3, we produce in §4 a replacement of [20, Theorem 4.15] (used right above to split off the summand M(X_{2})(1) from the motive of Y) valid for more general (as projective quadrics) algebraic varieties. In §5 we reproduce some recent results due to Rost concerning the modulo 2 Rost correspondences and Rost projectors on more general (as projective quadrics) varieties. In §6 we apply some standard motivic decompositions of projective homogeneous varieties to certain varieties related to a central simple algebra with an isotropic orthogonal involution. We also reproduce (see Theorem 6.1) some results of [9] which contain the needed generalization of indecomposability of the motive of an Albert quadric used in the previous paragraph. Finally, in §7 we prove Main theorem (Theorem 1.1) following the strategy of [7] and using results of [9] which were not available at the time of [7].

Acknowledgements. Thanks to Anne Quéguiner for asking me the question and to Alexander Vishik for telling me the alternate proof of Theorem 1.3.

2. Notation

We understand under a variety a separated scheme of finite type over a field.

Let D be a central simple F -algebra. The F -dimension of any right ideal in D is divisible by deg D; the quotient is the reduced dimension of the ideal. For any integer i, we write X(i; D) for the generalized Severi-Brauer variety of the right ideals in D of reduced dimension i. In particular, X(0; D) = Spec F = X(deg D; D) and X(i, D) = ∅ for i < 0 and for i > deg D.

More generally, let V be a right D-module. The F -dimension of V is then divisible by deg D and the quotient rdim V = dim_{F} V/ deg D is called the reduced dimension of V. For any integer i, we write X(i; V) for the projective homogeneous variety of the D-submodules in V of reduced dimension i (non-empty iff 0 ≤ i ≤ rdim V). For a finite sequence of integers i_{1}, …, i_{r}, we write X(i_{1} ⊂ ⋯ ⊂ i_{r}; V) for the projective homogeneous variety of flags of the D-submodules in V of reduced dimensions i_{1}, …, i_{r} (non-empty iff 0 ≤ i_{1} ≤ ⋯ ≤ i_{r} ≤ rdim V).

Now we additionally assume that D is endowed with an orthogonal involution τ. Then we write X(i; (D, τ)) for the variety of the totally isotropic right ideals in D of reduced dimension i (non-empty iff 0 ≤ i ≤ deg D/2).

If moreover V is endowed with a hermitian (with respect to τ) form h, we write X(i; (V, h)) for the variety of the totally isotropic D-submodules in V of reduced dimension i.

We refer to [10] for a detailed construction and basic properties of the above varieties.

3. Krull-Schmidt principle

The characteristic of the base field F is arbitrary in this section.

Our basic reference for Chow groups and Chow motives (including notation) is [5]. We fix an associative unital commutative ring Λ (we shall take Λ = 𝔽_{2} in the application) and for a variety X we write CH(X; Λ) for its Chow group with coefficients in Λ. Our category of motives is the category CM(F, Λ) of graded Chow motives with coefficients in Λ, [5, definition of §64]. By a sum of motives we always mean the direct sum.

We shall often assume that our coefficient ring Λ is finite. This simplifies significantly the situation (and is sufficient for our application). For instance, for a finite Λ, the endomorphism rings of finite sums of Tate motives are also finite and the following easy statement applies:

Lemma 3.1. An appropriate power of any element of any finite associative (not necessarily commutative) ring is idempotent.

Proof. Since the ring is finite, any its element x satisfies x^{a} = x^{a+b} for some a ≥ 1 and b ≥ 1. It follows that x^{ab} is an idempotent. □

Let X be a smooth complete variety over F. We call X split, if its integral motive M(X) ∈ CM(F, ℤ) (and therefore its motive with any coefficients) is a finite sum of Tate motives. We call X geometrically split, if it splits over a field extension of F. We say that X satisfies the nilpotence principle, if for any field extension E/F and any coefficient ring Λ, the kernel of the change of field homomorphism End(M(X)) → End(M(X)_{E}) consists of nilpotents. Any projective homogeneous variety is geometrically split and satisfies the nilpotence principle, [3, Theorem 8.2].

Corollary 3.2 ([9, Corollary 2.2]). Assume that the coefficient ring Λ is finite. Let X be a geometrically split variety satisfying the nilpotence principle. Then an appropriate power of any endomorphism of the motive of X is a projector.

We say that the Krull-Schmidt principle holds for a given pseudo-abelian category, if every object of the category has one and unique decomposition in a finite direct sum of indecomposable objects. In the sequel, we are constantly using the following statement:

Corollary 3.3 ([4, Corollary 35], see also [9, Corollary 2.6]). Assume that the coefficient ring Λ is finite. The Krull-Schmidt principle holds for the pseudo-abelian Tate subcategory in CM(F, Λ) generated by the motives of the geometrically split F -varieties satisfying the nilpotence principle. □

Remark 3.4. Replacing the Chow groups CH(−; Λ) by the reduced Chow groups \overline{CH}(−; Λ) (cf. [5, §72]) in the definition of the category CM(F, Λ), we get a “simplified” motivic category \overline{CM}(F, Λ) (which is still sufficient for the main purpose of this paper). Working within this category, we do not need the nilpotence principle any more. In particular, the Krull-Schmidt principle holds (with a simpler proof) for the pseudo-abelian Tate subcategory in \overline{CM}(F, Λ) generated by the motives of the geometrically split F -varieties.

4. Splitting off a motivic summand

The characteristic of the base field F is still arbitrary in this section.

In this section we assume that the coefficient ring Λ is connected. We shall often assume that Λ is finite.

Before climbing to the main result of this section (which is Proposition 4.6), let us do some warm up.

The following definition of [9] extends some terminology of [19]:

Definition 4.1. Let M ∈ CM(F, Λ) be a summand of the motive of a smooth complete irreducible variety of dimension d. The summand M is called upper, if CH^{0}(M; Λ) ≠ 0. The summand M is called lower, if CH_{d}(M; Λ) ≠ 0. The summand M is called outer, if it is simultaneously upper and lower.

For instance, the whole motive of a smooth complete irreducible variety is an outer summand of itself. Another example of an outer summand is the motive given by a Rost projector (see Definition 5.1).

Given a correspondence α ∈ CH_{dim X}(X × Y; Λ) between some smooth complete irreducible varieties X and Y, we write mult α ∈ Λ for the multiplicity of α, [5, definition of §75]. Multiplicity of a composition of two correspondences is the product of multiplicities of the composed correspondences (cf. [11, Corollary 1.7]). In particular, multiplicity of a projector is idempotent and therefore ∈ {0, 1} because the coefficient ring Λ is connected.

Characterizations of outer summands given in the two following Lemmas are easily obtained:

Lemma 4.2 (cf. [9, Lemmas 2.8 and 2.9]). Let X be a smooth complete irreducible variety. The motive (X, p) given by a projector p ∈ CH_{dim X}(X × X; Λ) is upper if and only if mult p = 1. The motive (X, p) is lower if and only if mult p^{t} = 1, where p^{t} is the transpose of p.

Lemma 4.3 (cf. [9, Lemma 2.12]). Assume that a summand M of the motive of a smooth complete irreducible variety of dimension d decomposes into a sum of Tate motives. Then M is upper if and only if the Tate motive Λ is present in the decomposition; it is lower if and only if the Tate motive Λ(d) is present in the decomposition.

The following statement generalizes (the finite coefficient version of) [20, Corollary 3.9]:

Lemma 4.4. Assume that the coefficient ring Λ is finite. Let X and Y be smooth complete irreducible varieties such that there exist multiplicity 1 correspondences

α ∈ CH_{dim X}(X × Y; Λ) and β ∈ CH_{dim Y} (Y × X; Λ).

Assume that X is geometrically split and satisfies the nilpotence principle. Then there is an upper summand of M(X) isomorphic to an upper summand of M(Y). Moreover, for any upper summand M_{X} of M(X) and any upper summand M_{Y} of M(Y), there is an upper summand of M_{X} isomorphic to an upper summand of M_{Y}.

Proof. By Corollary 3.2, the composition p := (β ∘ α)^{∘n} for some n ≥ 1 is a projector. Therefore q := (α∘β)^{∘2n} is also a projector and the summand (X, p) of M(X) is isomorphic to the summand (Y, q) of M(Y): mutually inverse isomorphisms are, say,

α ∘ (β ∘ α)^{∘(2n)} : (X, p) → (Y, q) and β ∘ (α ∘ β)^{∘(4n−1)} : (Y, q) → (X, p) .

Since mult p = (mult β · mult α)^{n} = 1 and similarly mult q = 1, the summand (X, p) of M(X) and the summand (Y, q) of M(Y) are upper by Lemma 4.2.

We have proved the first statement of Lemma 4.4. As to the second statement, let

p′ ∈ CH_{dim X}(X × X; Λ) and q′ ∈ CH_{dim Y} (Y × Y; Λ)

be projectors such that M_{X} = (X, p′) and M_{Y} = (Y, q′). Replacing α and β by q′ ∘ α ∘ p′ and p′ ∘ β ∘ q′, we get isomorphic upper motives (X, p) and (Y, q) which are summands of M_{X} and M_{Y}. □

Remark 4.5. Assume that the coefficient ring Λ is finite. Let X be a geometrically split irreducible smooth complete variety satisfying the nilpotence principle. Then the complete motivic decomposition of X contains precisely one upper summand and it follows by Corollary 3.3 (or by Lemma 4.4) that an upper indecomposable summands of M(X) is unique up to an isomorphism. (Of course, the same is true for the lower summands.)

Here comes the needed replacement of [20, Theorem 4.15]:

Proposition 4.6. Assume that the coefficient ring Λ is finite. Let X be a geometrically split, geometrically irreducible variety satisfying the nilpotence principle and let M be a motive. Assume that there exists a field extension E/F such that

(1) the field extension E(X)/F (X) is purely transcendental;

(2) the upper indecomposable summand of M(X)_{E} is also lower and is a summand of

M_{E}. Then the upper indecomposable summand of M(X) is a summand of M.

Proof. We may assume that M = (Y, p, n) for some irreducible smooth complete F -variety Y, a projector p ∈ CH_{dim Y} (Y × Y; Λ), and an integer n.

By the assumption (2), we have morphisms of motives f : M(X)_{E} → M_{E} and g : M_{E} → M(X)_{E} with mult(g ∘ f) = 1. By [9, Lemma 2.14], in order to prove Proposition 4.6, it suffices to construct morphisms f′ : M(X) → M and g′ : M → M(X) (over F) with mult(g′ ∘ f′) = 1.

Let ξ : Spec F (X) → X be the generic point of the (irreducible) variety X. For any F - scheme Z, we write ξ_{Z} for the morphism ξ_{Z} = (ξ × id_{Z}) : Z_{F (X)} = Spec_{F (X)} ×Z → X × Z. Note that for any α ∈ CH(X ×Z), the image ξ_{Z}^{∗}(α) ∈ CH(Z_{F (X)}) of α under the pull-back homomorphism ξ_{Z}^{∗} : CH(X × Z, Λ) → CH(Z_{F (X)}, Λ) coincides with the composition of correspondences α ∘ [ξ], [5, Proposition 62.4(2)], where [ξ] ∈ CH_{0}(X_{F (X)}, Λ) is the class of the point ξ:

(∗)  ξ_{Z}^{∗}(α) = α ∘ [ξ].

In the commutative square

CH(X_{E} × Y_{E}; Λ) ⟶^{ξ_{Y_{E}}^{∗}} CH(Y_{E(X)}; Λ)
res_{E/F} ↑  ^{res_{E(X)/F (X)}}↑
CH(X × Y; Λ) ⟶^{ξ_{Y}^{∗}} CH(Y_{F (X)}; Λ)

the change of field homomorphism res_{E(X)/F (X)} is surjective^{1} because of the assumption (1) by the homotopy invariance of Chow groups [5, Theorem 57.13] and by the localization property of Chow groups [5, Proposition 57.11]. Moreover, the pull-back homomorphism ξ_{Y}^{∗} is surjective by [5, Proposition 57.11]. It follows that there exists an element f′ ∈ CH(X × Y; Λ) such that ξ_{Y_{E}}^{∗}(f′_{E}) = ξ_{Y_{E}}^{∗}(f).

Recall that mult(g ∘ f) = 1. On the other hand, mult(g ∘ f′_{E}) = mult(g ∘ f). Indeed, mult(g ∘ f) = deg ξ_{X_{E}}^{∗}(g ∘ f) by [5, Lemma 75.1], where deg : CH(X_{E(X)}) → Λ is the degree homomorphism. Furthermore, ξ_{X_{E}}^{∗}(g ∘ f) = (g ∘ f) ∘ [ξ_{E}] by (∗). Finally, (g ∘ f) ∘ [ξ_{E}] = g ∘ (f ∘ [ξ_{E}]) and f ∘ [ξ_{E}] = ξ_{Y_{E}}^{∗}(f) = ξ_{Y_{E}}^{∗}(f′_{E}) by the construction of f′.

Since mult(g ∘ f′_{E}) = 1 and the indecomposable upper summand of M(X)_{E} is lower, we have mult((f′_{E})^{t} ∘ g^{t}) = 1. Therefore we may apply the above procedure to the dual morphisms

g^{t} : M(X)_{E} → (Y, p, n + dim X − dim Y)_{E}

and (f′_{E})^{t} : (Y, p, n + dim X − dim Y)_{E} → M(X)_{E}.

This way we get a morphism g′ : M → M(X) such that mult((f′)^{t} ∘ (g′)^{t}) = 1. It follows that mult(g′ ∘ f′) = 1. □

Remark 4.7. Replacing CM(F, Λ) by \overline{CM}(F, Λ) in Proposition 4.6, we get a weaker version of Proposition 4.6 which is still sufficient for our application. The nilpotence principle is no more needed in the proof of the weaker version. Because of that, there is no more need to assume that X satisfies the nilpotence principle.

5. Rost correspondences

In this section, X stands for a smooth complete geometrically irreducible variety of a positive dimension d.

The coefficient ring Λ of the motivic category is 𝔽_{2} in this section. We write Ch(−) for the Chow group CH(−; 𝔽_{2}) with coefficients in 𝔽_{2}. We write deg_{X/F} for the degree homomorphism Ch_{0}(X) → 𝔽_{2}.

1In fact, res_{E(X)/F (X)} is even an isomorphism, but we do not need its injectivity (which can be obtained with a help of a specialization).

Definition 5.1. An element ρ ∈ Ch_{d}(X × X) is called a Rost correspondence (on X), if ρ_{F (X)} = χ_{1} × [X_{F (X)}] + [X_{F (X)}] × χ_{2} for some 0-cycle classes χ_{1}, χ_{2} ∈ Ch_{0}(X_{F (X)}) of degree 1. A Rost projector is a Rost correspondence which is a projector.

Remark 5.2. Our definition of a Rost correspondence differs from the definition of a special correspondence in [16]. Our definition is weaker in the sense that a special correspondence on X (which is an element of the integral Chow group CH_{d}(X ×X)) considered modulo 2 is a Rost correspondence but not any Rost correspondence is obtained this way. This difference gives a reason to reproduce below some results of [16]. Actually, some of the results below are formally more general than the corresponding results of [16]; their proofs, however, are essentially the same.

Remark 5.3. Clearly, the set of all Rost correspondences on X is stable under transposition and composition. In particular, if ρ is a Rost correspondence, then its both symmetrizations ρ^{t} ∘ ρ and ρ ∘ ρ^{t} are (symmetric) Rost correspondences. Writing ρ_{F (X)} as in Definition 5.1, we have (ρ^{t} ∘ ρ)_{F (X)} = χ_{1} × [X_{F (X)}] + [X_{F (X)}] × χ_{1} (and (ρ ∘ ρ^{t})_{F (X)} = χ_{2} × [X_{F (X)}] + [X_{F (X)}] × χ_{2}).

Lemma 5.4. Assume that the variety X is projective homogeneous. Let ρ ∈ Ch_{d}(X × X) be a projector. If there exists a field extension E/F such that ρ_{E} = χ_{1} × [X_{E}] + [X_{E}] × χ_{2} for some 0-cycle classes χ_{1}, χ_{2} ∈ Ch_{0}(X_{E}) of degree 1, then ρ is a Rost projector.

Proof. According to [3, Theorem 7.5], there exist some integer n ≥ 0 and for i = 1, …, n some integers r_{i} > 0 and some projective homogeneous varieties X_{i} satisfying dim X_{i}+r_{i} < d such that for M = ⨁_{i=1}^{n} M(X_{i})(r_{i}) the motive M(X)_{F (X)} decomposes as 𝔽_{2}⊕M ⊕𝔽_{2}(d). Since there is no non-zero morphism between different summands of this three terms decomposition, the ring End M(X) decomposes in the product of rings

End 𝔽_{2} × End M × End 𝔽_{2}(d) = 𝔽_{2} × End M × 𝔽_{2}.

Let χ ∈ Ch_{0}(X_{F (X)}) be a 0-cycle class of degree 1. We set

ρ′ = χ × [X_{F (X)}] + [X_{F (X)}] × χ ∈ 𝔽_{2} × 𝔽_{2} ⊂ 𝔽_{2} × End M × 𝔽_{2} = End M(X)_{F (X)} = Ch_{d}(X_{F (X)} × X_{F (X)})

and we show that ρ_{F (X)} = ρ′. The difference ε = ρ_{F (X)}−ρ′ vanishes over E(X). Therefore ε is a nilpotent element of End M. Choosing a positive integer m with ε^{m} = 0, we get

ρ_{F (X)} = ρ_{F (X)}^{m} = (ρ′ + ε)^{m} = (ρ′)^{m} + ε^{m} = (ρ′)^{m} = ρ′.  □

Lemma 5.5. Let ρ ∈ Ch_{d}(X × X) be a projector. The motive (X, ρ) is isomorphic to 𝔽_{2} ⊕ 𝔽_{2}(d) iff ρ = χ_{1} × [X] + [X] × χ_{2} for some some 0-cycle classes χ_{1}, χ_{2} ∈ Ch_{0}(X) of degree 1.

Proof. A morphism 𝔽_{2} ⊕ 𝔽_{2}(d) → (X, ρ) is given by some

f ∈ Hom (𝔽_{2}, M(X)) = Ch_{0}(X) and f′ ∈ Hom (𝔽_{2}(d), M(X)) = Ch_{d}(X).

A morphism in the inverse direction is given by some

g ∈ Hom(M(X), 𝔽_{2}) = Ch^{0}(X) and g′ ∈ Hom(M(X), 𝔽_{2}(d)) = Ch^{d}(X).

The two morphisms 𝔽_{2} ⊕ 𝔽_{2}(d) ↔ (X, ρ) are mutually inverse isomorphisms iff ρ = f × g + f′ × g′ and deg_{X/F} (fg) = 1 = deg_{X/F} (f′g′). The degree condition means that f′ = [X] = g and deg_{X/F} (f) = 1 = deg_{X/F} (g′). □

Corollary 5.6. If X is projective homogeneous and ρ is a projector on X such that

(X, ρ)_{E} ≃ 𝔽_{2} ⊕ 𝔽_{2}(d)

for some field extension E/F, then ρ is a Rost projector. □

A smooth complete variety is called anisotropic, if the degree of its any closed point is even.

Lemma 5.7 ([16, Lemma 9.2], cf. [17, proof of Lemma 6.2]). Assume that X is anisotropic and possesses a Rost correspondence ρ. Then for any integer i ≠ d and any elements α ∈ Ch_{i}(X) and β ∈ Ch^{i}(X_{F (X)}), the image of the product α_{F (X)} · β ∈ Ch_{0}(X_{F (X)}) under the degree homomorphism deg_{X_{F (X)}/F (X)} : Ch_{0}(X_{F (X)}) → 𝔽_{2} is 0.

Proof. Let γ ∈ Ch^{i}(X × X) be a preimage of β under the surjection

ξ_{X}^{∗} : Ch^{i}(X × X) → Ch^{i}(Spec F (X) × X)

(where ξ_{X}^{∗} is as defined in the proof of Proposition 4.6). We consider the 0-cycle class

δ = ρ · ([X] × α) · γ ∈ Ch_{0}(X × X).

Since X is anisotropic, so is X × X, and it follows that deg_{(X×X)/F} δ = 0. Therefore it suffices to show that deg_{(X×X)/F} δ = deg_{X_{F (X)}/F (X)}(α_{F (X)} · β).

We have deg_{(X×X)/F} δ = deg_{(X×X)_{F (X)}/F (X)}(δ_{F (X)}) and

δ_{F (X)} = (χ_{1} × [X_{F (X)}] + [X_{F (X)}] × χ_{2}) · ([X_{F (X)}] × α_{F (X)}) · γ_{F (X)} = (χ_{1} × [X_{F (X)}]) · ([X_{F (X)}] × α_{F (X)}) · γ_{F (X)}

(because i ≠ d) where χ_{1}, χ_{2} ∈ Ch_{0}(X_{F (X)}) are as in Definition 5.1. For the first projection pr_{1} : X_{F (X)} × X_{F (X)} → X_{F (X)} we have

deg_{(X×X)_{F (X)}/F (X)} δ_{F (X)} = deg_{X_{F (X)}/F (X)}(pr_{1})_{∗}(δ_{F (X)})

and by the projection formula

(pr_{1})_{∗}(δ_{F (X)}) = χ_{1} · (pr_{1})_{∗}(([X_{F (X)}] × α_{F (X)}) · γ_{F (X)}).

Finally,

(pr_{1})_{∗}(([X_{F (X)}] × α_{F (X)}) · γ_{F (X)}) = mult (([X_{F (X)}] × α_{F (X)}) · γ_{F (X)}) · [X_{F (X)}]

and

mult (([X_{F (X)}] × α_{F (X)}) · γ_{F (X)}) = mult (([X] × α) · γ).

Since mult χ = deg_{X_{F (X)}/F (X)} ξ_{X}^{∗}(χ) for any element χ ∈ Ch_{d}(X × X) by [5, Lemma 75.1], it follows that

mult (([X] × α) · γ) = deg(α_{F (X)} · β).  □

For anisotropic X, we consider the homomorphism deg/2 : Ch_{0}(X) → 𝔽_{2} induced by the homomorphism CH_{0}(X) → ℤ, α ↦ deg(α)/2.

Corollary 5.8. Assume that X is anisotropic and possesses a Rost correspondence. Then for any integer i ≠ d and any elements α ∈ Ch_{i}(X) and β ∈ Ch^{i}(X) with β_{F (X)} = 0 one has (deg/2)(α · β) = 0.

Proof. Let β′ ∈ CH^{i}(X) be an integral representative of β. Since β_{F (X)} = 0, we have β′_{F (X)} = 2β′′ for some β′′ ∈ CH^{i}(X_{F (X)}). Therefore

(deg/2)(α · β) = deg_{X_{F (X)}/F (X)} (α_{F (X)} · (β′′ mod 2)) = 0

by Lemma 5.7. □

Corollary 5.9. Assume that X is anisotropic and possesses a Rost correspondence ρ. For any integer i ∉ {0, d} and any α ∈ Ch_{i}(X) and β ∈ Ch^{i}(X) one has

(deg/2)((α × β) · ρ) = 0.

Proof. Let α′ ∈ CH_{i}(X) and β′ ∈ CH^{i}(X) be integral representatives of α and β. Let ρ′ ∈ CH_{d}(X × X) be an integral representative of ρ. It suffices to show that the degree of the 0-cycle class (α′ × β′) · ρ′ ∈ CH_{0}(X × X) is divisible by 4.

Let χ_{1} and χ_{2} be as in Definition 5.1. Let χ′_{1}, χ′_{2} ∈ CH_{0}(X_{F (X)}) be integral representatives of χ_{1} and χ_{2}. Then ρ′_{F (X)} = χ′_{1} × [X_{F (X)}] + [X_{F (X)}] × χ′_{2} + 2γ for some γ ∈ CH_{d}(X_{F (X)} × X_{F (X)}). Therefore (since i ∉ {0, d})

(α′_{F (X)} × β′_{F (X)}) · ρ′_{F (X)} = 2(α′_{F (X)} × β′_{F (X)}) · γ.

Applying the projection pr_{1} onto the first factor and the projection formula, we get twice the element α′_{F (X)} · (pr_{1})_{∗}(([X_{F (X)}] × β′_{F (X)}) · γ) whose degree is even by Lemma 5.7 (here we use once again the condition that i ≠ d). □

Lemma 5.10. Assume that X is anisotropic and possesses a Rost correspondence ρ. Then (deg/2)(ρ^{2}) = 1.

Proof. Let χ_{1} and χ_{2} be as in Definition 5.1. Let χ′_{1}, χ′_{2} ∈ CH_{0}(X_{E}) be integral representatives of χ_{1} and χ_{2}. The degrees of χ′_{1} and χ′_{2} are odd. Therefore, the degree of the cycle class

(χ′_{1} × [X_{F (X)}] + [X_{F (X)}] × χ′_{2})^{2} = 2(χ′_{1} × χ′_{2}) ∈ CH_{0}(X_{F (X)} × X_{F (X)})

is not divisible by 4.

Let ρ′ ∈ CH_{d}(X × X) be an integral representative of ρ. Since ρ′_{F (X)} is χ′_{1} × [X_{F (X)}] + [X_{F (X)}] × χ′_{2} modulo 2, (ρ′_{F (X)})^{2} is (χ′_{1} × [X_{F (X)}] + [X_{F (X)}] × χ′_{2})^{2} modulo 4. Therefore (deg/2)(ρ^{2}) = 1. □

Theorem 5.11 ([16, Theorem 9.1], see also [17, proof of Lemma 6.2]). Let X be an anisotropic smooth complete geometrically irreducible variety of a positive dimension d over a field F of characteristic ≠ 2 possessing a Rost correspondence. Then the degree of the highest Chern class c_{d}(−T_{X}), where T_{X} is the tangent bundle on X, is not divisible by 4.

Proof. In this proof, we write c_{•}(−T_{X}) for the total Chern class ∈ Ch(X) in the Chow group with coefficient in 𝔽_{2}. It suffices to show that (deg/2)(c_{d}(−T_{X})) = 1.

Let Sq_{•}^{X} : Ch(X) → Ch(X) be the modulo 2 homological Steenrod operation, [5, §59]. We have a commutative diagram

[figure: Ch_{d}(X × X); Ch_{d}(X); Ch_{0}(X × X); Ch_{0}(X); Ch_{0}(X); 𝔽_{2}; (pr_{1})_{∗}; Sq_{d}^{X×X}; Sq_{d}^{X}; (pr_{1})_{∗}; deg/2; (pr_{2})_{∗}; deg/2; deg/2]

Since (pr_{1})_{∗}(ρ) = [X] and Sq_{d}^{X}([X]) = c_{d}(−T_{X}) [5, formula (60.1)], it suffices to show that

(deg/2)( Sq_{d}^{X×X}(ρ)) = 1.

We have Sq_{•}^{X×X} = c_{•}(−T_{X×X}) · Sq_{X×X}^{•}, where Sq^{•} is the cohomological Steenrod operation, [5, §61]. Therefore

Sq_{d}^{X×X}(ρ) = ∑_{i=0}^{d} c_{d−i}(−T_{X×X}) · Sq_{X×X}^{i}(ρ).

The summand with i = d is Sq_{X×X}^{d}(ρ) = ρ^{2} by [5, Theorem 61.13]. By Lemma 5.10, its image under deg/2 is 1.

Since c_{•}(−T_{X×X}) = c_{•}(−T_{X}) × c_{•}(−T_{X}) and Sq^{0} = id, the summand with i = 0 is

(∑_{j=0}^{d} c_{j}(−T_{X}) × c_{d−j}(−T_{X})) · ρ.

Its image under deg/2 is 0 because

(deg/2)((c_{0}(−T_{X}) × c_{d}(−T_{X})) · ρ) = (deg/2)(c_{d}(−T_{X})) =

(deg/2)((c_{d}(−T_{X}) × c_{0}(−T_{X})) · ρ)

while for j ∉ {0, d}, we have (deg/2)((c_{j}(−T_{X}) × c_{d−j}(−T_{X})) · ρ) = 0 by Corollary 5.9. Finally, for any i with 0 < i < d the ith summand is the sum

∑_{j=0}^{d−i} (c_{j}(−T_{X}) × c_{d−i−j}(−T_{X})) · Sq_{X×X}^{i}(ρ).

We shall show that for any j the image of the jth summand under deg/2 is 0. Note that the image under deg/2 coincides with the image under the composition (deg/2) ∘ (pr_{1})_{∗} and also under the composition (deg/2)∘(pr_{2})_{∗} (look at the above commutative diagram). By the projection formula we have

(pr_{1})_{∗}((c_{j}(−T_{X}) × c_{d−i−j}(−T_{X})) · Sq_{X×X}^{i}(ρ)) =

c_{j}(−T_{X}) · (pr_{1})_{∗}(([X] × c_{d−i−j}(−T_{X})) · Sq_{X×X}^{i}(ρ))

and the image under deg/2 is 0 for positive j by Corollary 5.8 applied to α = c_{j}(−T_{X}) and β = (pr_{1})_{∗}(([X]×c_{d−i−j}(−T_{X}))·Sq_{X×X}^{i}(ρ)). Corollary 5.8 can be indeed applied, because

since ρ_{F (X)} = χ_{1} × [X_{F (X)}] + [X_{F (X)}] × χ_{2} and i > 0, we have Sq_{(X×X)_{F (X)}}^{i}(ρ)_{F (X)} = 0 and therefore β_{F (X)} = 0.

For j = 0 we use the projection formula for pr_{2} and Corollary 5.8 with α = c_{d−i}(−T_{X}) and β = (pr_{2})_{∗}( Sq_{X×X}^{i}(ρ)). □

Remark 5.12. The reason of the characteristic exclusion in Theorem 5.11 is that its proof makes use of Steenrod operations on Chow groups with coefficients in 𝔽_{2} which are not available in characteristic 2.

We would like to mention

Lemma 5.13 ([16, Lemma 9.10]). Let X be an anisotropic smooth complete equidimensional variety over a field of arbitrary characteristic. If dim X + 1 is not a power of 2, then the degree of the integral 0-cycle class c_{dim X}(−T_{X}) ∈ CH_{0}(X) is divisible by 4.

6. Motivic decompositions of some isotropic varieties

The coefficient ring Λ is 𝔽_{2} in this section. Throughout this section, D is a central division F -algebra of degree 2^{r} with some positive integer r.

We say that motives M and N are quasi-isomorphic and write M ≈ N, if there exist decompositions M ≃ M_{1} ⊕ ⋯ ⊕ M_{m} and N ≃ N_{1} ⊕ ⋯ ⊕ N_{n} such that

M_{1}(i_{1}) ⊕ ⋯ ⊕ M_{m}(i_{m}) ≃ N_{1}(j_{1}) ⊕ ⋯ ⊕ N_{n}(j_{n})

for some (shift) integers i_{1}, …, i_{m} and j_{1}, …, j_{n}.

We shall use the following

Theorem 6.1 ([9, Theorems 3.8 and 4.1]). For any integer l = 0, 1, …, r, the upper indecomposable summand M_{l} of the motive of the generalized Severi-Brauer variety X(2^{l}; D) is lower. Besides of this, the motive of any finite direct product of any generalized Severi-Brauer varieties of D is quasi-isomorphic to a finite sum of M_{l} (with various l).

For the rest of this section, we fix an orthogonal involution on the algebra D.

Lemma 6.2. Let n be a positive integer. Let h be a hyperbolic hermitian form on the right D-module D^{2n} and let Y be the variety X(n deg D; (D^{2n}, h)) (of the maximal totally isotropic submodules). Then the motive M(Y) is isomorphic to a finite sum of several shifted copies of the motives M_{0}, M_{1}, …, M_{r} = Λ including one non-shifted copy of Λ.

Proof. By [10, ��������� 15.9] (cf. [3] or [2]), the motive of the variety Y is quasiisomorphic to the motive of the “total” variety

X(∗; D^{n}) = ∐_{i∈ℤ}X(i; D^{n}) = ∐_{i=0}^{4n} X(i; D^{n})

of D-submodules in D^{n}. Furthermore, M(X(∗; D^{n})) ≈ M(X(∗; D))^{⊗n} by [10, ��������� 10.10] (cf. [3] or [2]). We finish by Theorem 6.1. The non-shifted copy of Λ is obtained as Λ = M(X(0; D^{n})) = M(X(0; D))^{⊗n}. □

As before, we write Ch(−) for the Chow group CH(−; 𝔽_{2}) with coefficients in 𝔽_{2}. We recall that a smooth complete variety is called anisotropic, if the degree of its any closed point is even (the empty variety is anisotropic). The following statement is a particular

case of [9, Lemma 2.21].

Lemma 6.3. Let Z be an anisotropic F -variety with a projector p ∈ Ch_{dim Z}(Z × Z) such that the motive (Z, p)_{L} ∈ CM(L, 𝔽_{2}) for a field extension L/F is isomorphic to a finite sum of Tate motives. Then the number of the Tate summands is even. In particular, the motive in CM(F, 𝔽_{2}) of any anisotropic F -variety does not contain a Tate summand.

Proof. Mutually inverse isomorphisms between (Z, p)_{L} and a sum of, say, n Tate summands, are given by two sequences of homogeneous elements a_{1}, …, a_{n} and b_{1}, …, b_{n} in Ch(Z_{L}) with p_{L} = a_{1} × b_{1} + ⋯ + a_{n} × b_{n} and such that for any i, j = 1, …, n the degree deg(a_{i}b_{j}) is 0 for i ≠ j and 1 ∈ 𝔽_{2} for i = j. The pull-back of p via the diagonal morphism of Z is therefore a 0-cycle class on Z of degree n (modulo 2). □

Lemma 6.4. Let n be a positive integer. Let h′ be a hermitian form on the right D-module D^{n} such that h′_{L} is anisotropic for any finite odd degree field extension L/F. Let h be the hermitian form on the right D-module D^{n+2} which is the orthogonal sum of h′ and a hyperbolic D-plane. Let Y′ be the variety of totally isotropic submodules of D^{n+2} of reduced dimension 2^{r} (= ind D). Then the complete motivic decomposition of M(Y′) ∈ CM(F, 𝔽_{2}) (cf. Corollary 3.3) contains one summand 𝔽_{2}, one summand 𝔽_{2}(dim Y′), and does not contain any other Tate motive.

Proof. According to [10, ��������� 15.9], M(Y′) is quasi-isomorphic to the sum of the motives of the products

X(i ⊂ j; D) × X(j − i; (D^{n}, h′))

where i, j run over all integers (the product is non-empty iff 0 ≤ i ≤ j ≤ 2^{r}). The choices i = j = 0 and i = j = 2^{r} give the summands 𝔽_{2} and 𝔽_{2}(dim Y′). The variety obtained by any other choice of i, j is anisotropic (the variety with i = 0, j = 2^{r} is anisotropic by the assumption involving the odd degree field extensions), and we are done by Lemma 6.3. □

7. Proof of Main theorem

We fix a central simple algebra A of index > 1 with a non-hyperbolic orthogonal involution σ. Since the involution is an isomorphism of A with its dual, the exponent of A is 2; therefore, the index of A is a power of 2, say, ind A = 2^{r} for a positive integer r.

We assume that σ becomes hyperbolic over the function field of the Severi-Brauer variety of A and we are looking for a contradiction.

According to [8, Theorem 3.3], coind A = 2n for some integer n ≥ 1. We assume that Main theorem (Theorem 1.1) is already proven for all algebras (over all fields) of index < 2^{r} as well as for all algebras of index 2^{r} and coindex < 2n.

Let D be a central division algebra Brauer-equivalent to A. Let X_{0} be the Severi-Brauer variety of D. Let us fix an orthogonal involution on D and an isomorphism of F -algebras A ≃ End_{D}(D^{2n}). Let h be a hermitian form on the right D-module D^{2n} such that σ is adjoint to h. Then h_{F (X_{0})} is hyperbolic. Since the anisotropic kernel of h also becomes hyperbolic over F (X_{0}), our induction hypothesis ensures that h is anisotropic. Moreover, h_{L} is hyperbolic for any field extension L/F such that h_{L} is isotropic. It follows by [1, Proposition 1.2] that h_{L} is anisotropic for any finite odd degree field extension L/F.

Let Y be the variety of totally isotropic submodules in D^{2n} of reduced dimension n deg D. (The variety Y is a twisted form of the variety of maximal totally isotropic subspaces of a quadratic form studied in [5, Chapter XVI].) It is isomorphic to the variety of totally isotropic right ideals in A of reduced dimension (deg A)/2 (=n2^{r}). Since σ is hyperbolic over F (X_{0}) and the field F is algebraically closed in F (X_{0}) (because the variety X_{0} is geometrically integral), the discriminant of σ is trivial. Therefore the variety Y has two connected components Y_{+} and Y_{−} corresponding to the components C_{+} and C_{−} (cf. [5, Theorem 8.10]) of the Clifford algebra C(A, σ).

Since σ_{F (X_{0})} is hyperbolic, Y (F (X_{0})) ≠ ∅. Since the varieties Y_{+} and Y_{−} become isomorphic over F (X_{0}), each of them has an F (X_{0})-point.

The central simple algebras C_{+} and C_{−} are related with A by the formula [12, (9.14)]:

[C_{+}] + [C_{−}] = [A] ∈ Br(F).

Since [C_{+}]_{F (X_{0})} = [C_{−}]_{F (X_{0})} = 0 ∈ Br(F (X_{0})), we have [C_{+}], [C_{−}] ∈ {0, [A]} and it follows that [C_{+}] = [A], [C_{−}] = 0 up to exchange of the indices +, −.

By the index reduction formula for the varieties Y_{+} and Y_{−} of [14, page 594], we have (up to exchange of the indices +, −): ind D_{F (Y_{+})} = ind D, ind D_{F (Y_{−})} = 1. We replace Y by the component of Y whose function field does not reduce the index of D. Note that the variety Y is projective homogeneous.

The coefficient ring Λ is 𝔽_{2} in this section. We use the F -motives M_{0}, …, M_{r} introduced in Theorem 6.1. Note that for any field extension E/F such that D_{E} is still a division algebra, we also have the E-motives M_{0}, …, M_{r}.

Lemma 7.1. The motive of Y decomposes as R_{1} ⊕ R_{2}, where R_{1} is quasi-isomorphic to a finite sum of several copies of the motives M_{0}, …, M_{r−1}, and where (R_{2})_{F (Y)} is isomorphic to a finite sum of Tate motives including one exemplar of 𝔽_{2}.

Proof. According to Lemma 6.2 and since D_{F (Y)} is a division algebra, the motive M(Y)_{F (Y)} is isomorphic to a sum of several shifted copies of the F (Y)-motives M_{0}, …, M_{r} = 𝔽_{2} including a copy of 𝔽_{2}. If for some l = 0, …, r − 1 there is at least one copy of M_{l} (with a shift j ∈ ℤ) in the decomposition, let us apply Proposition 4.6 taking as X the variety X_{l} = X(2^{l}, D), taking as M the motive M(Y)(−j), and taking as E the function field F (Y).

Since, as mentioned already, D_{E} is a division algebra, condition (2) of Proposition 4.6 is fulfilled. Since ind D_{F (X)} < 2^{r}, the hermitian form h_{F (X)} is hyperbolic by the induction hypothesis; therefore the variety Y_{F (X)} is rational and condition (1) of Proposition 4.6 is fulfilled as well.

It follows that the F -motive M_{l} is a summand of M(Y)(−j). Let now M be the complement summand of M(Y)(−j). By Corollary 3.3, the complete decomposition of M_{F (Y)} is the complete decomposition of M(Y)(−j)_{F (Y)} with one copy of M_{l} erased. If M_{F (Y)} contains one more copy of a shift of M_{l} (for some l = 0, …, r − 1), we once again apply Proposition 4.6 to the variety X_{l} and an appropriate shift of M. Doing this until we can, we get the desired decomposition in the end. □

Now let us consider a minimal right D-submodule V ⊂ D^{2n} such that V becomes isotropic over a finite odd degree field extension of F (Y). We set v = dim_{D} V. Clearly, 2 ≤ v ≤ n + 1. ^{2} For n > 1, let Y′ be the variety of totally isotropic submodules in V of reduced dimension 2^{r} (that is, of “D-dimension” 1). For n = 1 we set Y′ = Y. Note that the variety Y′ is projective homogeneous.

The variety Y′ is irreducible and has an even positive dimension. Moreover, the variety Y′ is anisotropic (because the hermitian form h is anisotropic and remains anisotropic over any finite odd degree field extension of the base field). Surprisingly, we can however prove the following

Lemma 7.2. There is a Rost projector (Definition 5.1) on Y′.

Proof. By the construction of Y′, there exists a correspondence of odd multiplicity (that is, of multiplicity 1 ∈ 𝔽_{2}) α ∈ Ch_{dim Y} (Y × Y′). On the other hand, since h_{F (Y′)} is isotropic, h_{F (Y′)} is hyperbolic and therefore there exist a rational map Y′ −−→ Y and a multiplicity 1 correspondence β ∈ Ch_{dim Y′}(Y′ × Y) (e.g., the class of the closure of the graph of the rational map). Since the summand R_{2} of M(Y) given by Lemma 7.1 is upper (cf. Definition 4.1 and Lemma 4.3), by Lemma 4.4 there is an upper summand of M(Y′) isomorphic to a summand of R_{2}.

Let ρ ∈ Ch_{dim Y′}(Y′ × Y′) be the projector giving this summand. We claim that ρ is a Rost projector. We prove the claim by showing that the motive (Y′, ρ)F̃ is isomorphic to 𝔽_{2} ⊕ 𝔽_{2}(dim Y′), cf. Corollary 5.6, where F̃/F (Y) is a finite odd degree field extension such that V becomes isotropic over F̃.

Since (R_{2})_{F (Y)} is a finite sum of Tate motives, the motive (Y′, ρ)F̃ is also a finite sum of Tate motives. Since (Y′, ρ)F̃ is upper, the Tate motive 𝔽_{2} is included (Lemma 4.3). Now, by the minimal choice of V, the hermitian form (h|_{V})F̃ satisfies the condition on h in Lemma 6.4: (h|_{V})F̃ is an orthogonal sum of a hyperbolic DF̃-plane and a hermitian form h′ such that h′_{L} is anisotropic for any finite odd degree field extension L/F̃ of the base field F̃ (otherwise any D-hyperplane V′ ⊂ V would become isotropic over some odd degree extension of F (Y)). Therefore the complete motivic decomposition of Y ′ ; F̃ has one copy of 𝔽_{2}, one copy of 𝔽_{2}(dim Y′), and no other Tate summands. By Corollary 3.3 and anisotropy of the variety Y′ (see Lemma 6.3), it follows that

(Y′, ρ)F̃ ≃ 𝔽_{2} ⊕ 𝔽_{2}(dim Y′).  □

2One probably always has v = n + 1 here, but we do not need to know the precise value of v.

Lemma 7.2 contradicts to the general results of §5 (namely, to Theorem 5.11 and Lemma 5.13) thus proving Main theorem (Theorem 1.1). We can avoid the use of Lemma 5.13 by showing that deg c_{dim Y′}(−T_{Y′}) is divisible by 2^{2^{r}} for our variety Y′. Indeed, let K be the field F (t_{1}, …, t_{v2^{r}}) of rational functions over F in v2^{r} variables. Let us consider the (generic) diagonal quadratic form ⟨t_{1}, …, t_{v2^{r}}⟩ on the K-vector space K^{v2^{r}}. Let Y′′ be the variety of 2^{r}-dimensional totally isotropic subspaces in K^{v2^{r}}. The degree of any closed point on Y′′ is divisible by 2^{2^{r}}. In particular, the integer deg c_{dim Y′′}(−T_{Y′′}) is divisible by 2^{2^{r}}. Since over an algebraic closure K̄ of K the varieties Y′ and Y′′ become isomorphic, we have

deg c_{dim Y′}(−T_{Y′}) = deg c_{dim Y′′}(−T_{Y′′}).

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UPMC Univ Paris 06, Institut de Mathématiques de Jussieu, F-75252 Paris, FRANCE

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