HYPERBOLICITY OF UNITARY INVOLUTIONS

NIKITA A. KARPENKO

Abstract. We prove the so-called Unitary Hyperbolicity Theorem, a result on hyperbolicity of unitary involutions. The analogous previously known results for the orthogonal and symplectic involutions are formal consequences of the unitary one. While the original proofs in the orthogonal and symplectic cases were based on the incompressibility of generalized Severi-Brauer varieties, the proof in the unitary case is based on the incompressibility of their Weil transfers.

1. Inroduction

We refer to [14] for terminology and basic facts concerning central simple algebras with involutions. We fix the following notation: F is a field, K/F a separable quadratic field extension, A a central simple K-algebra, σ an F -linear unitary involution on A.

Precisely as in the orthogonal and in the symplectic case, a right ideal I of the algebra A is isotropic (with respect to the involution σ), if σ(I) · I = 0. The involution σ is hyperbolic, if there exists an isotropic ideal of reduced dimension (deg A)/2. Note that the reduced dimension of a right ideal (or, more generally, of a right A-module) is defined as its dimension over K (not over F) divided by the degree deg A := \sqrt{dim_{K} A} of A.

In this note we prove (in Section 4)

Theorem 1.1 (Unitary Hyperbolicity Theorem). Assume that char F ≠ 2. If σ is not hyperbolic, then there exists a field extension F′/F such that K′ := K ⊗_{F} F′ is a field, the central simple K′-algebra A′ := A ⊗_{F} F′ is split, and the F′-linear unitary involution σ′ := σ_{F′} on A′ is still not hyperbolic.

Theorem 1.1 is the unitary analogue of the following result concerning orthogonal involutions:

Theorem 1.2 (Orthogonal Hyperbolicity Theorem [12, Theorem 1.1]). Assume that char F ≠ 2. Let B be a central simple F -algebra with an orthogonal involution τ. If τ is not hyperbolic, then there exists a field extension F′/F such that the central simple F′-algebra B′ := B ⊗_{F} F′ is split and the orthogonal involution τ_{F′} on B′ is still not hyperbolic.

In the case when the exponent of A is 2, Theorem 1.1 has been deduced from Theorem 1.2 in [18]. In our setting the exponent of A is arbitrary; our proof is a unitary adaptation of the proof of Theorem 1.2.

Date: July, 2010.

Key words and phrases. Algebraic groups, involutions, projective homogeneous varieties, Chow groups and motives, Steenrod operations. Mathematical Subject Classification (2010): 14L17; 14C25.

In fact, as we show in Section 5, Theorem 1.2 can be deduced from (the exponent 2 case of) Theorem 1.1. Since the symplectic analogue [18, Theorem 1] of these hyperbolicity results is a consequence of Theorem 1.2, Theorem 1.1 turns out to be the principle hyperbolicity result. However, as the orthogonal and symplectic hyperbolicity results are consequences of the corresponding isotropy results [6, Theorem 1] and [7, Theorem 1], the most powerful result in this whole story would be the Unitary Isotropy Theorem (which imply all the results mentioned above). But so far we are not able to prove the Unitary Isotropy Theorem; the attempt to adopt to the unitary case the proof of the orthogonal case (which succeeds for the hyperbolicity business in this paper) fails for the isotropy business. So, we can only state

Conjecture 1.3 (“Unitary Isotropy Theorem”). If σ is anisotropic over any finite odd degree field extension of F, then there exists a field extension F′/F such that K′ := K ⊗_{F} F′ is a field, the central simple K′-algebra A′ := A ⊗_{F} F′ is split, and the F′-linear unitary involution σ′ := σ_{F′} on A′ is still anisotropic.

Theorem 1.1 (as well as Conjecture 1.3) can easily be reduced to the case where the index of A is a power of 2. Indeed, first of all, the index of an arbitrary A becomes a power of 2 over an appropriate finite odd degree field extension L/F, for instance, over the field extension of F corresponding to a Sylow 2-subgroup of the Galois groups of the normal closure of E/F, where E is a separable finite odd degree field extension of K such that ind(A ⊗_{K} E) is a power of 2. According to [14, Corollary 6.16], if [L : F] is odd, the involution σ_{L} is still non-hyperbolic (and K ⊗_{F} L is a field). So, if Theorem 1.1 is proved for A ⊗_{F} L, we get it also for the original A.

Because of the above reduction, we always assume below that the index of A is a power of 2.

We will prove that Theorem 1.1 holds for F′ being the function field of the Weil transfer R_{K/F} (X) of the Severi-Brauer K-variety X of A. Clearly, K′ is a field and A′ is split for such F′, so that we only need to check the non-hyperbolicity of σ′.

We start our work by writing down in Section 2 some consequences of the general motivic decompositions of [1] applied to the case of some varieties related to unitary involutions.

We start getting new results in Section 3 by establishing the unitary analogue of the results of [11] (which are about orthogonal involutions). The involution σ′ is adjoint to certain (uniquely determined by σ′ up to an isomorphism and a non-zero factor from F′) K′/F′-hermitian form on a vector K′-space; the Witt index ind σ′ of σ′ is the Witt index of this hermitian form. The characteristic assumption char F ≠ 2 is dropped in the following result (proved in Section 3):

Theorem 1.4. We are assuming that the (Schur) index of A is a power of 2. Let F′ be the function field of the Weil transfer R_{K/F} (X) of the Severi-Brauer variety X of A. Then the Witt index ind σ′ of σ′ is divisible by the Schur index ind A of A.

2. Motivic decompositions of some isotropic varieties of unitary type

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

By a variety we mean a separated scheme of finite type over a field. A variety is called anisotropic, if the degree of any its closed point is even.

Example 2.1. Any K-variety T, considered as an F -variety via the composition T → Spec K → Spec F, is anisotropic because the residue field of any point on T contains K.

Example 2.2. Let V be a finite-dimensional vector space over K with a K/F -hermitian form h. If h is anisotropic, then the F -variety of 1-dimensional totally isotropic subspaces in V is also anisotropic (that is, h remains anisotropic over any finite odd degree field extension of F). This classical fact is a consequence of the Springer theorem [3, Corollary 18.5] for quadratic forms (applied to the quadratic form v ↦ h(v, v) ∈ F on V considered as a vector space over F).

We write Ch for the Chow group with coefficients in 𝔽_{2} (the finite field of 2 elements). Motives below are the Chow motives with coefficients in 𝔽_{2}, [3, §64]. We write M(Y) for the motive of a smooth projective variety Y. The motive of the spectrum of the base field is denoted by 𝔽_{2}. For any integer i, the motive 𝔽_{2}(i) (the ith shift of the motive 𝔽_{2}) is called a Tate motive.

Note that the Krull-Schmidt principle [13, Corollary 2.2] holds for the motives of quasihomogeneous varieties, [13, Definition of §2].

Let D be a central division K-algebra of a index a power of 2 (possibly of index 1 = 2^{0}) with a fixed F -linear unitary involution τ. Let V′ be a finite-dimensional right Dmodule with a hermitian (with respect to the involution τ) form. Let ℍ be the hermitian hyperbolic D-plane. By definition, ℍ is the right D-module D ⊕ D equipped with the

hermitian form of the matrix (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}). Let V be the orthogonal sum of ℍ and V′. Let

Y be the F -variety of the totally isotropic submodules in V of reduced dimension deg D (that is, of D-dimension 1). The variety Y is a closed subvariety of the Weil transfer with respect to the field extension K/F of the K-variety of the submodules in V of the reduced dimension deg D. Similarly, we define Y′ as the F -variety of the totally isotropic submodules in V′ of reduced dimension deg D.

Lemma 2.3. The motive of Y is the sum of two Tate motives, a shift of the motive of Y′, and shifts of the motives of some anisotropic F -varieties.

Proof. According to [9, Theorem 15.8], the variety Y is a relative cellular space (as defined in [3, §66]) over the (non-connected) variety Z of triples (I, J, N), where I and J are right ideals in D and where N is a submodule in V′ such that the submodule I ⊕ J ⊕ N ⊂ V is a point of Y (that is, τ(I) · J = 0, N is totally isotropic, and the reduced dimension of the D-module I ⊕ J ⊕ N is equal to deg D). Therefore, by [3, Corollary 66.4], the motive of Y is the sum of shifts of the motives of the components of Z.

The rational points (0, D, 0) and (D, 0, 0) of Z are components of Z which produce the two promised Tate summands. Since D is a 2-primary division algebra, any other odd degree rational point of Z lies on the component of the triples (0, 0, N). This component is naturally identified with Y′. □

Remark 2.4. Since we are trying to avoid unnecessary precision, we do not determine the shifting numbers of the summands in the decomposition of Lemma 2.3. A more detailed analysis of the situation shows that the shifting numbers of the two Tate motives are 0 and dim Y; the shifting number of the motive of Y′ is (dim Y − dim Y′)/2.

3. Characterization of Witt index of hermitian forms

In this section we still allow to the base field F to be of arbitrary characteristic (including 2). We recall that the index of the K-algebra A is a 2-power.

For any integer i ∈ {0, 1, …, (deg A)/2} we write Y_{i} for the F -variety of the isotropic reduced dimension i ideals in A. This is a closed subvariety of the Weil transfer R_{K/F} X_{i} of the generalized Severi-Brauer K-variety X_{i} of the reduced dimension i right ideals in A. The K-variety (Y_{i})_{K} is identified with the variety of the flags I ⊂ J of right ideals I, J in A of reduced dimensions i, deg A − i (see [14, Proposition 2.15] and [9, Lemma 15.5]).

Now we assume that the algebra A is split and we study the variety Y_{1}. More precisely, we assume that A = End_{K}(V) for some finite-dimensional vector K-space V (of dimension deg A). In this case, the involution σ is adjoint to some K/F -hermitian form h on V. By the Morita equivalence, the variety Y_{1} is the variety of 1-dimensional totally isotropic with respect to h subspaces in V. So, the variety Y_{1} is the unitary analogue of a projective quadric (which we have in the orthogonal case). And it turns out that the Chow group of Y_{1} contains the information about the precise value of the Witt index of h exactly as in the orthogonal case the Chow group of the projective quadric contains the information about the precise value of the Witt index of the quadratic form [3, Corollary 72.6].

The K-variety (Y_{1})_{K} is a hypersurface in ℙ(V) × ℙ^{#}(V), where ℙ(V) is the projective space of V, i.e., the variety of 1-dimensional subspaces in V, and ℙ^{#}(V) is the dual projective space of V, i.e., the variety of 1-codimensional subspaces in V, which can be identified with ℙ(V^{#}), the projective space of the vector space V^{#} dual to V. More precisely, (Y_{1})_{K} ⊂ ℙ(V) × ℙ^{#}(V) is the flag hypersurface, the hypersurface of the pairs of subspaces (U, W) of the vector space V satisfying the condition U ⊂ W.

We write C for the image of the composition

Ch(Y_{1}) → Ch((Y_{1})_{K}) → Ch(ℙ(V) × ℙ^{#}(V)).

For any variety ℙ isomorphic to a projective space and any integer i ∈ {0, 1, …, dim ℙ}, we write l_{i} for the class in Ch_{i}(ℙ) of an i-dimensional linear subspace in ℙ.

The following statement is the unitary analogue of [3, Corollary 72.6]:

Lemma 3.1. Assume that A is split. For any i ∈ {0, 1, …, (deg A)/2 − 1} the following statements are equivalent:

(1) the Witt index of σ is > i;

(2) l_{i} × l_{i} ∈ C;

(3) the motive of Y_{1} contains the Tate summand 𝔽_{2}(2i).

Proof. (1) ⇒ (2). If the Witt index of σ is > i, there exists an isotropic ideal I ⊂ A of reduced dimension i + 1. The variety of the reduced dimension 1 right ideals contained in I is then a closed subvariety of Y_{1} whose class in C is equal to l_{i} × l_{i}.

(2) ⇒ (3). If l_{i} × l_{i} ∈ C, we write α for an element of Ch_{2i}(Y_{1}) whose class in C is equal to l_{i} × l_{i}. The variety Y_{1} is a hypersurface in R_{K/F} (ℙ(V)). For j := dim ℙ(V) − i, we consider the cycle class R_{K/F} (l_{j}) ∈ Ch_{2j}(R_{K/F} (ℙ(V))) (R_{K/F} in the expression R_{K/F} (l_{j}) here is the Weil transfer on the algebraic cycle classes, see [10, §3]) and let β ∈ Ch_{2j−1}(Y_{1}) be the pull-back of R_{K/F} (l_{j}) with respect to the imbedding in : Y_{1} ↪ ℙ(V). By the projection formula, the product α · β is a 0-cycle class on Y_{1} of degree 1. Indeed,

deg(α · β) = deg in_{∗}(α · β) = deg ( in_{∗}(α) · R_{K/F} (l_{j})) =

deg ( in_{∗}(α)_{K} · R_{K/F} (l_{j})_{K}) = deg ((l_{i} × l_{i}) · (l_{j} × l_{j})) = deg(l_{0} × l_{0}) = 1.

Therefore, α and β, considered as morphisms of motives

α : 𝔽_{2}(2i) → M(Y_{1}) and β : M(Y_{1}) → 𝔽_{2}(2i),

identify 𝔽_{2}(2i) with a summand of M(Y_{1}).

Note that the same α and β are morphisms β : 𝔽_{2}(dim Y_{1} − 2i) → M(Y_{1}) and α : M(Y_{1}) → 𝔽_{2}(dim Y_{1} −2i) showing that the Tate motive 𝔽_{2}(dim Y_{1} −2i) is also a summand of M(Y_{1}) (this summand is dual to the previous one, cf. [3, §65]).

(3) ⇒ (1). Let j be the Witt index of σ. Note that j ≤ (deg A)/2. According to Lemma 2.3 (applied j times), there exists a motivic decomposition of Y_{1} containing 2j Tate summands and such that each of the remaining summands is a shift of the motive of an anisotropic variety (take Example 2.2 into account). According to [12, Lemma 6.3], the complete motivic decomposition of an anisotropic variety does not contain Tate summands. It follows that the complete motivic decomposition of M(Y_{1}) contains precisely 2j Tate summands. But during the previous step, we have already constructed 2j Tate summands of M(Y_{1}) (with pairwisely different shifting numbers):

𝔽_{2}, 𝔽_{2}(2), …, 𝔽_{2}(2j − 2) and 𝔽_{2}(d), 𝔽_{2}(d − 2), …, 𝔽_{2}(d − 2j + 2),

where d = dim Y_{1} = 2 deg A − 3. Note that the shifting number of every Tate summand in the first group is ≤ deg A − 2 while the shifting number of every Tate summand in the second group is ≥ deg A − 1. Therefore, M(Y_{1}) contains 𝔽_{2}(2i) with some i < (deg A)/2 only if j > i. □

Proof of Theorem 1.4. We may assume that ind σ′ > 0. Let r be the biggest multiple of ind A with r < ind σ′. We will show that ind σ′ = r + ind A.

Let T be the Severi-Brauer variety of a central division K-algebra Brauer-equivalent to A. We write R instead of R_{K/F} and we write Y instead of Y_{1} for short. By Lemma 3.1, there exists α′ ∈ Ch_{2r}(Y_{F (R(T))}) with the image l_{r} × l_{r} in Ch_{2r}(Y_{K(R(T))}). Let

α ∈ Ch_{2r+2 dim T} (R(T) × Y )

be a preimage of α′ with respect to the epimorphism

Ch_{2r+2 dim T} (R(T) × Y ) →→ Ch_{2r}(Y_{F (R(T))})

given by the pull-back with respect to the morphism Y_{F (R(T))} → R(T) × Y induced by the generic point of the (integral) variety R(T).

According to [11, Lemma 3.1] and since r is a multiple of ind A lying on the interval [0, dim X], there exists an element

β′ ∈ Ch_{dim X−r}(X × T) = Ch^{r+dim T} (X × T)

with the image under the composition of the push-forward Ch(X × T) → Ch(X) with respect to the projection X × T → X followed by the change of field homomorphism Ch(X) → Ch(X_{K(X)}) being equal to l_{dim X−r} ∈ Ch(X_{K(X)}). Let

β ∈ Ch^{2r+2 dim T} (Y × R(T))

be the pull-back of R(β′) ∈ Ch (R(X) × R(T)) to Ch (Y × R(T)) with respect to the closed imbedding Y ×R(T) ↪ R(X)×R(T) induced by the closed imbedding Y ↪ R(X).

The composition of correspondences

β ∘ α ∈ Ch_{2 dim T} (R(T) × R(T))

is a degree 0 correspondence on R(T). The multiplicity of β ∘ α is 1. Indeed, the multiplicity of β ∘ α coincides with the degree of the 0-cycle class (β ∘ α)_{∗}(l_{0} × l_{0}), where l_{0} × l_{0} ∈ Ch_{0}(R(T)_{K(T)}) is the 0-cycle class of degree 1 on R(T)_{K(T)}. Since α_{∗}(l_{0} × l_{0}) = l_{r} × l_{r} ∈ Ch_{2r}(Y_{K(T)}) and β_{∗}(l_{r} × l_{r}) = l_{0} × l_{0} by the construction of α and β, the 0-cycle class (β ∘ α)_{∗}(l_{0} × l_{0}) is equal to l_{0} × l_{0} and has the degree 1.

We have constructed morphisms of motives

α : M(R(T)) → M(Y)(−2r) and β : M(Y)(−2r) → M(R(T))

such that the multiplicity of the composition β ∘α is 1. It follows by [8, Lemma 2.14] that a non-zero summand of the motive of R(T) is isomorphic to a summand of M(Y)(−2r). On the other hand, according to [5, Theorem 1.2], the motive of the variety R(T) is indecomposable. Therefore, the whole motive of R(T) is isomorphic to a summand of M(Y)(−2r). Since over the function field of R(T), the Tate motive 𝔽_{2}( dim R(T)) is a summand of M(R(T))_{F (R(T))}, the motive of Y_{F (R(T))} contains a Tate summand with the shifting number 2r + dim R(T) = 2(r + dim T) = 2(r + ind A − 1). It follows by Lemma 3.1 that ind σ′ > r + ind A − 1, that is, ind σ′ ≥ r + ind A. Now, the definition of r ensures that ind σ′ = r + ind A. □

4. Proof of Theorem 1.1

We follow the lines of [12, §7] doing the necessary modifications. First of all we refix our setting. Our base field F is now an arbitrary field of characteristic ≠ 2, K/F is a quadratic field extension, A is a non-split central simple K-algebra whose index is a power of 2, σ is an F -linear unitary involution on A. We assume that σ becomes hyperbolic over the function field of the Weil restriction R(X) = R_{K/F} (X) of the Severi-Brauer variety X of A. Equivalently, σ becomes hyperbolic over the function field of the Weil restriction R(T) = R_{K/F} (T) of the Severi-Brauer variety T of a central division K-algebra D Brauer-equivalent to A.

According to Theorem 1.4, coind A := (deg A)/(ind A) is 2n for some integer n ≥ 1. We assume that Theorem 1.1 is already proved for all algebras (over all fields) of index < ind A as well as for all algebras of index = ind A and coindex < 2n.^{1}

Let us fix an arbitrary F -linear unitary involution τ on D and an isomorphism of F -algebras A ≃ End_{D}(D^{2n}). Let h be a hermitian (with respect to τ) form on the right D-module D^{2n} such that σ is adjoint to h. Then h_{F (R(T))} is hyperbolic. Since the anisotropic kernel of h also becomes hyperbolic over F (R(T)), our induction hypotheses ensure that h is anisotropic. Moreover, h_{L} is hyperbolic for any field extension L/F such that K ⊗_{F} L is field and h_{L} is isotropic (we avoid the case where K ⊗_{F} L is not a field because we did not give a definition of hyperbolic in this case). It follows by [14, Corollary 6.16] that h_{L} is anisotropic for any finite odd degree field extension L/F.

1We are inducting on the index and on the coindex of A. The index induction base is the trivial case of the index 1. The coindex induction base is the case of the coindex 1 proved by Theorem 1.4 (see also [4, Example 4.5] for a simpler proof).

Let Y be the variety of totally isotropic submodules in D^{2n} of reduced dimension n deg D (that is, of D-dimension n). The variety Y is a closed subvariety of the Weil transfer with respect to the field extension K/F of the K-variety of the submodules in D^{2n} of the reduced dimension n deg D. Note that by [15], the F (Y)-algebra D_{F (Y)} is still a division algebra. Also note that Y (K) ≠ ∅ because coind A is even.

We are going to apply to the motive of Y the following slightly modified version of [12, Proposition 4.6]. Here and below, for any irreducible smooth projective variety S we write U(S) for the upper motive of S defined as the indecomposable upper motivic summand of S, see [8].

Proposition 4.1. Let S be a geometrically split variety satisfying the nilpotence principle and let M be a motive. Assume that there exists a field extension E/F such that

(1) S_{E} is irreducible and the motive U(S_{E}) is lower [8, Definition 2.10];

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

(3) the upper motive U(S_{E}) of the variety S_{E} is a summand of M_{E}. Then the upper motive U(S) of the variety S is a summand of M.

Proof. The only difference with [12, Proposition 4.6] (besides of the particular current choice 𝔽_{2} for the coefficient ring in place of an arbitrary finite connected ring) is that now we do not require S to be geometrically irreducible and we only require instead that S is irreducible over E. This change does not affect the original proof. □

We will apply Proposition 4.1 in the case where the variety S is quasi-homogeneous in the sense of [13, §2]. Such S is geometrically split and satisfies the nilpotence principle by [13, Theorem 2.1]. In Example 4.2 below, we describe a sufficient amount of varieties for which Condition (1) of Proposition 4.1 is satisfied.

Example 4.2. Let i be a 2-power of the interval [1, deg D]. Let S be the generalized Severi-Brauer K-variety of the reduced dimension i right ideals in D. According to [5], the upper motive U(R_{K/F} (S)) is lower (this fact is equivalent to the 2-incompressibility of the variety R_{K/F} (S) established in [5, Theorem 1.1], c.f. [4, Lemma 2.7 or Theorem 5.1]). The K-motive U(S) is also lower by [8, Theorem 4.1]. Since the F -motive cor_{K/F} U(S) is indecomposable by [13, Proposition 3.1] (see [13, §3] for the definition of the motivic functor cor_{K/F}), we have U(cor_{K/F} S) ≃ cor_{K/F} U(S) and it follows that the motive U(cor_{K/F} S) is also lower. (The F -variety cor_{K/F} S is simply the K-variety S considered as an F -variety via the composition S → Spec K → Spec F.)

Corollary 4.3. The motive U(Y) has the following property: U(Y)_{F (Y)} is a sum of Tate motives.

Proof. According to [13], each summand of the complete motivic decomposition of Y_{F (Y)} is a shift of the upper motive of R(S)_{F (Y)} or of cor(S)_{F (Y)}, where S/K is a generalized Severi-Brauer variety of Example 4.2, R = R_{K/F}, and cor = cor_{K/F}.

Let us neglect the shifts. Proposition 4.1 with E = F (Y) allows to split off from the motive of Y (over F) the summand U(R(S)) for each motivic summand U(R(S)_{F (Y)}) of Y_{F (Y)} with S ≠ Spec K, and it allows to split off the summand U(cor(S)) for each motivic summand U(cor(S)_{F (Y)}) of Y_{F (Y)} with any S. Indeed, Condition (1) of Proposition 4.1 is satisfied by Example 4.2 (and because D_{E} is still a division algebra); Condition (2) is satisfied because the variety Y_{K} is rational and also for any S ≠ Spec K the variety Y_{R(S)} is rational.

The remaining part of M(Y) is an upper motivic summand of Y which, if considered over F (Y), becomes a sum of Tate motives. Therefore, the upper motive of Y has the same property. □

It will turn out below, that in fact the upper motive of Y is binary: U(Y)_{F (Y)} is the sum of two Tate motives.

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, v ≥ 2 (because D_{F (Y)} is a division algebra). Let Y′ be the variety of totally isotropic submodules in V of reduced dimension deg D (that is, of D-dimension 1). Writing F̃ for an odd degree field extension of F (Y) with isotropic VF̃, we have Y′(F̃) ≠ ∅ (because DF̃ is a division algebra). Therefore there exists a correspondence of odd multiplicity (that is, of multiplicity 1 ∈ 𝔽_{2}) α ∈ Ch_{dim Y} (Y × Y′).

The variety Y′ is projective homogeneous (in particular, irreducible) of dimension

dim Y′ = (deg D)^{2}(2v − 3)

which is not a power of 2 minus 1 (because 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). Therefore, Lemma 4.4 below contradicts [12, Corollary 5.14] thus proving Theorem 1.1.^{2}

Lemma 4.4. There is a Rost projector ([12, Definition 5.1]) on Y′.

Proof. As explained above, 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). It follows that the upper motives of the varieties Y and Y′ are isomorphic. In particular, U(Y′)F̃, where F̃/F (Y) is a finite odd degree field extension with isotropic VF̃, is a sum of Tate motives.

On the other hand, (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 field F̃. Indeed, otherwise – if h′_{L} is isotropic for some such L, the module V_{L} contains a totally isotropic submodule W of D-dimension 2; any D-hyperplane V′ ⊂ V, considered over L, meets W non-trivially; it follows that V′_{L} is isotropic and this contradicts the minimality of V. (This is a very standard argument in the theory of quadratic forms over field which we applied now to a hermitian form over a division algebra.)

2We recall that [12, Corollary 5.14] is due to M. Rost [17]; its proof makes use of Steenrod operations on Ch which are available only over fields of characteristic ≠ 2. This explains the characteristic assumption of Theorem 1.1.

It follows by Lemma 2.3 that 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. The anisotropy of the variety Y′ implies now that

U(Y′)F̃ ≃ 𝔽_{2} ⊕ 𝔽_{2}(dim Y′),

that is, that the projector giving U(Y′) is a Rost projector on Y′. □

5. Relation with the orthogonal case

The aim of this Section is to explain the relationship between Theorem 1.1 and Theorem 1.2. It turns out that Theorem 1.2 is “contained” in Theorem 1.1 in the following sense: Theorem 1.2 is equivalent to Theorem 1.1 with exp A = 2.

It has been already shown in [18] that Theorem 1.2 implies Theorem 1.1 for algebras A of exponent 2. The inverse implication is based on the following construction (similar to the symplectic case construction of [18]).

Let F be a field of characteristic ≠ 2, B a central simple F -algebra with an orthogonal involution τ, x a variable, F̃/F (x) the quadratic extension F̃ = F (x)(\sqrt{x}) (the “generic quadratic extension over F ”). Let B̃ := B ⊗_{F} F̃ and let τ̃ be the F (x)-linear unitary involution on B̃ obtained as the tensor product of τ by the non-trivial automorphism of F̃/F (x).

Lemma 5.1 (cf. [18, Proposition 1]). If τ is anisotropic, then τ̃ is also anisotropic. If for some field extension L/F (x) such that L̃ := L ⊗_{F (x)} F̃ is a field, the involution τ_{L} is hyperbolic, then the involution τ̃_{L} is also hyperbolic.

Proof. If τ̃ is isotropic, there exists a non-zero element a ∈ B̃ with τ̃(a) · a = 0 which is a polynomial in t := \sqrt{x}, a = a_{n}t^{n} + ⋯ + a_{0} for some a_{n}, …, a_{0} ∈ B such that a_{n} ≠ 0. The coefficient of t^{2n} in the polynomial τ̃(a) · a is ±τ(a_{n}) · a_{n}. Therefore τ(a_{n}) · a_{n} = 0 and τ is isotropic.

If for some field extension L/F (x) such that L̃ := L ⊗_{F (x)} F̃ is a field, the involution τ_{L} is hyperbolic, the algebra B_{L} contains an isotropic ideal I of the reduced dimension (deg B)/2. The tensor product I⊗_{L}L̃ is then an isotropic ideal of the algebra B̃_{L} = B_{L}⊗_{L}L̃ of the same reduced dimension (deg B)/2 = (deg B̃)/2. Hence τ̃_{L} is also hyperbolic. □

Now Theorem 1.1 with A = B̃ implies Theorem 1.2 as follows. We may assume that τ is anisotropic. Then, by Lemma 5.1, the unitary involution τ̃ is also anisotropic. In particular, τ̃ is not hyperbolic and it follows by Theorem 1.1 that there exists a field extension L/F (x) such that L ⊗_{F (x)} F̃ is a field, the algebra B̃ ⊗_{F (x)} L is split, and τ̃_{L} is not hyperbolic. By Lemma 5.1, the involution τ_{L} is also not hyperbolic. Since the algebra B_{L} splits over the quadratic extension L̃/L, ind B_{L} ≤ 2. We finish by the orthogonal hyperbolicity theorem for index 2 algebras proved in [2] and independently in [16].

Acknowledgements. I am grateful to Maksim Zhykhovich for useful discussions.

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

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