HYPERBOLICITY OF HERMITIAN FORMS OVER BIQUATERNION ALGEBRAS

NIKITA A. KARPENKO

Abstract. We show that a non-hyperbolic hermitian form over a biquaternion algebra over a field of characteristic ≠ 2 remains non-hyperbolic over a generic splitting field of the algebra.

Contents

1. Introduction 1 2. Notation 2 3. Krull-Schmidt principle 3 4. Splitting off a motivic summand 5 5. Rost correspondences 7 6. Motivic decompositions of some isotropic varieties 12 7. Proof of Main Theorem 14 References 16

1. Introduction

Throughout this note (besides of §3 and §4) F is a field of characteristic ≠ 2. The basic reference for the staff 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.

Conjecture 1.1. Let A be a central simple F -algebra endowed with an orthogonal involution σ. If σ becomes hyperbolic over the function field of the Severi-Brauer variety of A, then σ is hyperbolic (over F).

In a stronger version of Conjecture 1.1, each of two words “hyperbolic” is replaced by “isotropic”, cf. [10, Conjecture 5.2].

Here is the complete list of indices ind A and coindices coind A = deg A/ ind A of A for which Conjecture 1.1 is known (over an arbitrary field of characteristic ≠ 2), given in the chronological order:

• ind A = 1 — trivial; • coind A = 1 (the stronger version) — [10, Theorem 5.3];

Date: January 2008.

Key words and phrases. Algebraic groups, projective homogeneous varieties, Chow groups, algebraic K-theory 2000 Mathematical Subject Classifications: 14L17; 14C25.

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

• coind A odd — [6, appendix by Zainoulline] and independently [7, Theorem 3.3];

• ind A = 4 and coind A = 2 — [18, Proposition 3]. Let us note that Conjecture 1.1 for any given (A, σ) with coind A = 2 implies the

stronger version version of Conjecture 1.1 for this (A, σ): indeed, by [7, 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. In particular, the stronger version of Conjecture 1.1 is known for A with ind A = 4 and coind A = 2 (the last case in the above list, also discussed below).

The main result of this note is as follows:

Theorem 1.2 (Main Theorem). Conjecture 1.1 holds for any A of index 4.

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 a quadratic form of dimension 8 and of trivial discriminant. Assume that the index of the Clifford algebra C of φ is 4. If φ becomes isotropic over the function field F (X_{1}) of the Severi-Brauer variety X_{1} of C, then φ is isotropic (over F).

The following alternate proof of Theorem 1.3, given by Vishik, is a prototype of our proof of Main Theorem (Theorem 1.2). Let Y be the projective quadric of φ and let X_{2} be the Albert quadric of the 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 [19, 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]).

In §4 we produce a replacement of [19, 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 an index 4 central simple algebra with an isotropic orthogonal involution. Finally, in §7 we prove Main Theorem (Theorem 1.2).

2. Notation

Unlike [5], 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 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 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 form h, we write X(i; (V, h)) for the variety of the totally isotropic D-submodules in V of reduced dimension i and we write

X(i_{1} ⊂ ⋯ ⊂ i_{r}; (V, h))

for the variety of flags of the totally isotropic D-submodules in V of reduced dimensions i_{1}, …, i_{r}.

Let us observe that all the varieties introduced in this section are projective homogeneous.

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 will 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, for any its element x we have 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 motive M(X) ∈ CM(F, Λ) 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 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. 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.

Proof. Let F̄/F be a field extension such that the motive M(X)F̄ is a finite sum of Tate motives. Let f be an endomorphism of M(X). Since our coefficient ring Λ is finite, it follows that the ring End(M(X)F̄) is finite. Therefore a power of fF̄ is idempotent by Lemma 3.1, and we may assume that fF̄ is idempotent. Since X satisfies the nilpotence principle, the element ε := f^{2} − f is nilpotent. Let n be a positive integer such that ε^{n} = 0 = nε. Then (f + ε)^{n^{n}} = f^{n^{n}} because the binomial coefficients \binom{n^{n}}{i} for i < n are divisible by n. Therefore f^{n^{n}} is a projector. □

Lemma 3.3 (cf. [4, Theorem 28]). Assume that the coefficient ring Λ is finite. Let X be a geometrically split variety satisfying the nilpotence principle and let p ∈ End(M(X)) be a projector. Then the motive (X, p) decomposes into a finite sum of indecomposable motives.

Proof. If (X, p) does not decompose this way, we get an infinite sequence

p_{0} =p, p_{1}, p_{2}, … ∈ End(M(X))

of pairwise distinct projectors such that p_{i} ∘ p_{j} = p_{j} = p_{j} ∘ p_{i} for any i < j.

Let F̄/F be a splitting field of X. Since the ring End(M(X)F̄) is finite, we have (p_{i})F̄ = (p_{j})F̄ for some i < j. The difference p_{i} −p_{j} is nilpotent and idempotent, therefore p_{i} = p_{j}. □

A (non necessarily commutative) ring is called local, if the sum of any two non-invertible elements differs from 1 in the ring. Since the sum of two nilpotents is never 1, we have

Lemma 3.4. A ring, where each non-invertible element is nilpotent, is local. In particular, by Corollary 3.2, so is the ring End(M(X)), if Λ is finite and X is a geometrically split variety satisfying the nilpotence principle and such that the motive M(X) is indecomposable. □

Theorem 3.5 ([1, Theorem 3.6 of Chapter I]). If the endomorphism ring of any indecomposable object of a pseudo-abelian category is local, then any object of this category has at most one decomposition in a finite direct sum of indecomposable objects.

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 which is an immediate consequence of Lemmas 3.3 and 3.4 and the above theorem:

Corollary 3.6 (cf. [4, Corollary 35]). 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 cellular F -varieties satisfying the nilpotence principle. □

Remark 3.7. 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 cellular 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 4.5), let us do some warm up.

Definition 4.1. A motive M ∈ CM(F, Λ) is called outer, if CH^{0}(M; Λ) ≠ 0. A summand M of the motive of a smooth complete variety is called outer, if this summand is an outer motive.

For instance, the motive of any smooth complete non-empty variety is outer and an outer summand of itself.

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. [10, Corollary 1.7]). In particular, multiplicity of a projector is idempotent and therefore ∈ {0, 1} because the coefficient ring Λ is connected.

Lemma 4.2. Let X be a smooth complete irreducible variety. The motive (X, p) given by a projector p ∈ CH_{dim X}(X × X; Λ) is outer iff mult p = 1.

Proof. The group CH^{0}((X, p); Λ) (defined as Hom ((X, p), Λ)) is the image of the endomorphism of CH^{0}(X; Λ) = Λ · [X] given by the multiplication by mult p. □

Lemma 4.3. Assume that a motive M decomposes into a sum of Tate motives. Then M is outer iff the Tate motive Λ is present in the decomposition.

Proof. For any i ∈ ℤ we have: CH^{0}(Λ(i); Λ) ≠ 0 iff i = 0. □

The following statement generalizes (the finite coefficient version of) [19, 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 outer summand of M(X) isomorphic to an outer summand of M(Y). Moreover, for any outer summand M_{X} of M(X) and any outer summand M_{Y} of M(Y), there is an outer summand of M_{X} isomorphic to an outer 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 outer 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 outer motives (X, p) and (Y, q) which are summands of M_{X} and M_{Y}. □

Here comes the needed replacement of [19, Theorem 4.15]:

Proposition 4.5. 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 E-motive M(X)_{E} ∈ CM(E, Λ) of the E-variety X_{E} is indecomposable (this implies that the F -motive M(X) ∈ CM(F, Λ) is indecomposable);

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

(3) the motive M(X)_{E} is a summand of the motive M_{E}. Then the motive M(X) is a summand of the motive 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 (3), we have morphisms of motives f : M(X)_{E} → M_{E} and g : M_{E} → M(X)_{E} with g ∘ f = id_{M(X)_{E}}.

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 (2) 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).

The multiplicity mult(g ∘ f) of the correspondence given by the composition g ∘ f = id_{M(X)_{E}} is equal to 1. The multiplicity mult(g ∘ f′_{E}) coincides with 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 construction of f′.

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 specialization).

Since the multiplicity is multiplicative, the multiplicity of (g ∘ f′_{E})^{∘n} is still 1 and therefore (g ∘ f′_{E})^{∘n} ≠ 0 for any n ≥ 1. On the other hand, by Corollary 3.2, (g ∘ f′_{E})^{∘n} is a projector for some n ≥ 1. By the assumption (1) (indecomposability of M(X)_{E}), it follows that (g ∘ f′_{E})^{∘n} = id_{M(X)_{E}} for this n. We set g′ = (g ∘ f′_{E})^{∘(n−1)} ∘ g. Then for the new morphisms f′ : M(X) → M and g′ : M_{E} → M(X_{E}) we still have g′ ∘ f′_{E} = id_{M(X)_{E}}.

Now we apply the same 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 morphisms g′′ : M → M(X) and f′′ : M(X) → M such that

(f′′_{E})^{t} ∘ (g′′_{E})^{t} = id_{M(X)_{E}^{∗}}, that is, g′′_{E} ∘ f′′_{E} = id_{M(X)_{E}}.

It follows by the nilpotence principle that (g′′ ∘ f′′)^{∘n} = id_{M(X)} for some integer n ≥ 1. Therefore M(X) is a summand of M. □

Remark 4.6. Replacing CM(F, Λ) by \overline{CM}(F, Λ) in Proposition 4.5, we get a weaker version of Proposition 4.5 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 absolutely irreducible variety of dimension d > 0.

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}.

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].

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 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 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 4.5). 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 absolutely irreducible variety of dimension d > 0 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 arbitrary in the first half of this section and Λ = 𝔽_{2} in the second half (starting from Lemma 6.3). Throughout this section, D is a biquaternion division algebra, X_{1} is the Severi-Brauer variety X(1; D) of D, and X_{2} is the generalized Severi-Brauer variety X(2; D).

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 (twist) integers i_{1}, …, i_{m} and j_{1}, …, j_{n}. Lemma 6.1. Tensor product of any finite number of copies of M(X_{1}) by any finite number of copies of M(X_{2}) is quasi-isomorphic to a sum of finite number of copies of M(X_{1}) and M(X_{2}). (In other words, the Tate subcategory of CM(F, Λ) generated by M(X_{1}) and M(X_{2}) is a tensor subcategory, cf. [4, Theorem 24].) Proof. The product X_{1} × X_{1} is a projective bundle over X_{1}. Therefore M(X_{1}) ⊗ M(X_{1}) is quasi-isomorphic to a sum of several copies of M(X_{1}), [5, Theorem 63.10].

The product X_{2} × X_{1} is a grassmann bundle over X_{1}. Therefore M(X_{2}) ⊗ M(X_{1}) is also quasi-isomorphic to a sum of several copies of M(X_{1}).

Finally, by [4], the motive of the product X_{2} × X_{2} quasi-decomposes into two copies of M(X_{2}) and one copy of the motive of the flag variety X(1 ⊂ 2 ⊂ 3; D). The projections

X(1 ⊂ 2 ⊂ 3; D) → X(1 ⊂ 2; D) → X(1; D) = X_{1}

are grassmann bundles. Consequently, M(X(1 ⊂ 2 ⊂ 3; D)) is quasi-isomorphic to a sum of several copies of M(X_{1}). □

For the rest of this section, we fix an orthogonal involution on the biquaternion 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(4n; (D^{2n}, h)) (of the maximal totally isotropic submodules). Then the motive M(Y) is isomorphic to a finite sum of several twisted copies of the motives M(X_{1}), M(X_{2}), and Λ including one non-twisted copy of Λ. Proof. By [9, ��������� 15.9] (cf. [3]), the motive of the variety Y is quasi-isomorphic 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 [9, ��������� 10.10] (cf. [3]). We finish by Lemma 6.1. The non-twisted 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).

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 4 (= ind D). Then the complete motivic decomposition of M(Y′) ∈ CM(F, 𝔽_{2}) (cf. Corollary 3.6) contains one summand 𝔽_{2}, one summand 𝔽_{2}(dim Y′), and does not contain any other Tate motive.

Proof. According to [9, ��������� 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 ≤ 4). The choices i = j = 0 and i = j = 4 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 = 4 is anisotropic by the assumption involving the odd degree field extensions), and we are done by Lemma 6.3. □

Lemma 6.5. The motives M(X_{1}) and M(X_{2}) in the category CM(F, 𝔽_{2}) are indecomposable.

Proof. Since the varieties X_{1} and X_{2} satisfy the nilpotence principle, it suffices to show that the reduced Chow groups \overline{Ch}_{dim X_{1}}(X_{1} × X_{1}) and \overline{Ch}_{dim X_{2}}(X_{2} × X_{2}) do not contain non-trivial projectors. The statement on X_{1} is a consequence of [8, ����������� 2.1.1].

The variety X_{2} is isomorphic to the Albert quadric of D, [5, §16.A]. By [19, §7.3], the splitting pattern ([5, definition of §25]) of the Albert form of a biquaternion division algebra is (0, 1, 3). It follows by [5, Theorem 73.26] that in the case of decomposable M(X_{2}), one of the summands is given by a Rost projector on X_{2}. As already mentioned, there can’t be a Rost projector on X_{2} because X_{2} is anisotropic and dim X_{2} + 1 = 5 is not a power of 2. Here one may refer to [5, Corollary 80.11], or to a more general Theorem 5.11, or to a simpler argument for this specific X_{2} which uses [11, Lemma 6.5] and computation of K_{0}(X_{2}). □

7. Proof of Main Theorem

We fix a central simple algebra A of index 4 with a non-hyperbolic orthogonal involution σ. 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 [7, Theorem 3.3], coind A = 2n for some integer n ≥ 1. We assume that Main Theorem (Theorem 1.2) is already proven for all algebras (over all fields) of index 4 and of coindex < 2n.

Let D be a biquaternion division algebra Brauer-equivalent to A. Let X_{1} 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_{1})} is hyperbolic. Since the anisotropic kernel of h also becomes hyperbolic over F (X_{1}), 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 [2, 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 4n. (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 4n. Since σ is hyperbolic over F (X_{1}) and the field F is algebraically closed in F (X_{1}) (because the variety X_{1} 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_{1})} is hyperbolic, Y (F (X_{1})) ≠ ∅. Since the varieties Y_{+} and Y_{−} become isomorphic over F (X_{1}), each of them has an F (X_{1})-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_{1})} = [C_{−}]_{F (X_{1})} = 0 ∈ Br(F (X_{1})), 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_{+})} = 4, ind D_{F (Y_{−})} = 1. We replace Y by the component of Y whose function field does not reduce the index of D.

The coefficient ring Λ is 𝔽_{2} in this section. As before, we set X_{2} = X(2, D).

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(X_{1}) and M(X_{2}), 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, the motive M(Y)_{F (Y)} is isomorphic to a sum of several twisted copies of M(X_{1}), M(X_{2}), and 𝔽_{2} including a copy of 𝔽_{2}. If for some i = 1, 2 there is at least one copy of M(X_{i}) (with a twist j ∈ ℤ) in the decomposition, let us apply Proposition 4.5 taking as X the variety X_{i}, taking as M the motive M(Y)(−j), and taking as E the function field F (Y).

The condition (3) of Proposition 4.5 is fulfilled.

Since D_{E} is still a division algebra, the condition (1) is fulfilled by Lemma 6.5. Since ind D_{F (X)} < 4, the hermitian form h_{F (X)} is hyperbolic by [15]; therefore the variety Y_{F (X)} is rational and the condition (2) is fulfilled.

It follows that M(X_{i}) is a summand of M(Y)(−j). Let now M be the complement summand of M(Y)(−j). By Corollary 3.6, the complete decomposition of M_{F (Y)} is the complete decomposition of M(Y)(−j)_{F (Y)} with one copy of M(X_{i}) erased. If M_{F (Y)} contains one more copy of a twist of M(X_{i}) (for some i = 1, 2), we once again apply Proposition 4.5 to the variety X_{i} and an appropriate twist 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 4 (that is, of “D-dimension” 1). For n = 1 we set Y′ = Y.

The variety Y′ has dimension dim Y′ = 16v−26 which is even (and positive). 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). Since the summand R_{2} of M(Y) given by Lemma 7.1 is outer (cf. Definition 4.1 and Lemma 4.3), by Lemma 4.4 there is an outer 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 outer, 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 (see Corollary 3.6) has one copy of 𝔽_{2}, one copy of 𝔽_{2}(dim Y′), and no other Tate summands. By Corollary 3.6 and anisotropy of the variety Y′ (see Lemma 6.3), it follows that

(Y′, ρ)F̃ ≃ 𝔽_{2} ⊕ 𝔽_{2}(dim Y′).  □

Lemma 7.2 contradicts to the general results of §5 (namely, to Theorem 5.11 and Lemma 5.13) thus proving Main Theorem (Theorem 1.2). We can avoid the use of Lemma 5.13 by showing that deg c_{dim Y′}(−T_{Y′}) is divisible by 16 for our variety Y′. Indeed, let K be the field F (t_{1}, …, t_{4v}) of rational functions over F in 4v variables. Let us consider the (generic) diagonal quadratic form ⟨t_{1}, …, t_{4v}⟩ on the K-vector space K^{4v}. Let Y′′ be the variety of 4-dimensional totally isotropic subspaces in K^{4v}. The degree of any closed point on Y′′ is divisible by 2^{4}. In particular, the integer deg c_{dim Y′′}(−T_{Y′′}) is divisible by 2^{4}. Since over an algebraic closure K̄ of K the varieties Y′ and Y′′ become isomorphic, we have

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

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

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HYPERBOLICITY OF HERMITIAN FORMS 17

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