ISOTROPY OF UNITARY INVOLUTIONS

NIKITA KARPENKO AND MAKSIM ZHYKHOVICH

Abstract. We prove the so-called Unitary Isotropy Theorem, a result on isotropy of a unitary involution. The analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal, symplectic, and unitary involutions are formal consequences of this theorem. A component of the proof is a detailed study of the quasi-split unitary grassmannians.

Contents

1. Introduction 1 2. Operations sq and st 3 3. Chow ring of quasi-split unitary grassmannians 6 4. Steenrod operations for split unitary grassmannians 12 5. Some ranks of some motives 14 6. Unitary isotropy theorem 15 References 19

1. Introduction

Let K be an arbitrary field of characteristic different from 2, A a central simple Kalgebra, τ an involution on A (i.e., a self-inverse ring anti-automorphism), F ⊂ K the subfield of τ-invariant elements of K. In this paper we finish the proof of Isotropy Theorem saying that if τ becomes isotropic over any field extension of F splitting A, then τ becomes isotropic over some finite odd degree field extension of F (note that by the example of [13], τ over F does not need to be isotropic).

We refer to [12] for generalities on central simple algebras with involutions. The involutions τ is isotropic, if the algebra A contains a non-zero right ideal I satisfying τ(I)·I = 0. The algebra A is split, if it represents 0 ∈ Br(K) in the Brauer group of K, that is, if it is isomorphic to a full matrix algebra over K.

Isotropy Theorem has been proved in [5] for algebras A of exponent 2 (for the symplectic case see also [10]). More precisely, it has been reduced to the case of orthogonal τ by J.-P. Tignol and proved in the orthogonal case by the first named author. In the remaining case, considered in the present paper (see Theorem 6.1), the involution τ is of unitary type (and, in particular, K is of degree 2 over F).

Date: March 2011.

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

The proof in the unitary case, made in Section 6, goes along the line of the proof of the orthogonal case, but there are (at least) two important differences. First of all, the information on orthogonal grassmannians needed in the orthogonal case was already available: partially from topology (in the split case), partially from more recent works of A. Vishik [16], [14] (partially remaking in algebraic terms the available topological material). In contrast with this, almost no information (even in the quasi-split case) on unitary grassmannians was available. It seems to be, almost completely, a terra incognito in the literature. Sections 3 and 4 cover this gap. (More study of unitary grassmannian is undertaken in [2], a successor paper).

To explain the second difference, we have to sketch the proof. It is easily reduced to the case of A of even 2-primary index. Let Y be the F -variety of isotropic right ideals in A of reduced dimension ind A. Let X be the F -variety of all right ideals in D of reduced dimension (ind A)/2, where D is a central division K-algebra Brauer-equivalent to A. Considering Chow motives with coefficients in 𝔽_{2} := ℤ/2ℤ of smooth projective F -varieties, we manage to show, that certain indecomposable direct summand of the motive of X (the so-called upper motive of X introduced in [6]) is isomorphic to a direct summand of the motive of Y. This produces a cycle class π in the modulo 2 Chow group Ch_{dim Y} (Y × Y) (the corresponding projector namely). With some more effort, we come to the case where π is symmetric (i.e., invariant under the factor exchange automorphism of the Chow group). We finish by applying to π a certain operation which transforms it to a 0-cycle class in Ch_{0}(Y × Y) of degree 1 (modulo 2) and therefore terminates the proof.

The shortest way to explain where the operation comes from is as follows. By [17], the projector π can be lifted to the algebraic cobordism (modulo 2). Then, applying an appropriate (modulo 2) symmetric operation of [15] and projecting back from cobordism to the Chow group, we get the required 0-cycle class.

Fortunately, the symmetric operations and algebraic cobordism theory are not really needed here (and thus we are not restricted to the characteristic 0). Actually, we succeed to compute the above symmetric operation because it can be described (on symmetric projectors) in terms of the Steenrod operations on the modulo 2 Chow groups. (The need of Steenrod operations explains our characteristic assumption.) This is done in Section 2.

The needed operation is related with the difference of two other operations: sq, given by the squaring, and st, given by a Steenrod operation. The proof succeeds if the value of one operation turns out to be trivial and the value of the other one – non-trivial (see Lemma 2.10). The value sq(π) is computed due to its relation with the rank of the motive (see Lemmas 2.5 and 2.2); the needed ranks are calculated in Section 5. To compute the value st(π) we use the information on the Steenrod operations on quasi-split unitary grassmannians obtained in Section 4.

The second difference between the orthogonal and the unitary cases is as follows: st(π) is the trivial value in the orthogonal case while sq(π) is the trivial value in the unitary case. In particular, we have to check the non-triviality of the more sophisticated st(π) here, which is certainly more difficult than to show its triviality (in the orthogonal case).

To finish the introduction, let us mention Hyperbolicity Theorem – a corollary of Isotropy Theorem: if τ becomes hyperbolic over any field extension of F splitting A, then τ is hyperbolic (over F). This corollary has been already proved (in a different way) in [9] (the exponent 2 case) and in [3] (the unitary case). We also note that the orthogonal and symplectic cases of Isotropy Theorem are formal consequences of its unitary case – Theorem 6.1 (cf. [3, §5] and [10]).

Acknowledgements. The authors thank Alexander Merkurjev for permission to include Lemma 2.1 and Burt Totaro for information about the state of study of unitary grassmannians in topology.

2. Operations sq and st

Let F be a field of characteristic ≠ 2. Let X be a connected smooth projective variety over F.

We write CH for the integral Chow group and we write Ch for CH /2 CH (the Chow group with coefficients in 𝔽_{2}).

We are going to use the following statement due to A. Merkurjev:

Lemma 2.1. Let δ : X → X × X be the diagonal morphism. For any 𝔞, 𝔟 ∈ CH(X × X) one has deg(𝔟^{t} · 𝔞) = deg(δ^{∗}(𝔟 ∘ 𝔞)), where · stands for the product in the Chow ring, ∘ stands for the composition of correspondences, and ^{t} stands for the transposition of correspondences.

Proof. The commutative diagram

X × X ⟶^{e} X × X × X ⟶^{s} X × X × X × X
pr^{1}↓  ^{pr}^{13}↓
X ⟶^{δ} X × X

where e(x, y) = (x, y, x) and s(x, y, z) = (x, y, y, z), yields the commutative diagram

CH(X × X) ⟵^{e^{∗}} CH(X × X × X) ⟵^{s^{∗}} CH(X × X × X × X)
pr^{1}^{∗}↓  ^{pr}^{13}^{∗}↓
CH(X) ⟵^{δ^{∗}} CH(X × X).

We have pr_{1}_{∗}(𝔟^{t} · 𝔞) = pr_{1}_{∗}e^{∗}s^{∗}(𝔞 × 𝔟) = δ^{∗}pr_{13}_{∗}s^{∗}(𝔞 × 𝔟) = δ^{∗}(β ∘ α), hence deg(𝔟^{t} · 𝔞) = deg pr_{1}_{∗}(𝔟^{t} · 𝔞) = deg δ^{∗}(𝔟 ∘ 𝔞). □

Our first basic operation is a map sq : Ch(X × X) → ℤ/4ℤ defined as follows. For any α ∈ Ch(X × X) we take its integral representative 𝔞 ∈ CH(X × X) and set

sq(α) := deg(𝔞^{2}) mod 4,

where by 𝔞^{2} we mean the product of cycles 𝔞·𝔞 and not the composition of correspondences 𝔞 ∘ 𝔞. Since any other integral representative of the same α is of the form 𝔞 + 2𝔟 with some 𝔟 ∈ CH(X × X) and deg((𝔞 + 2𝔟)^{2}) ≡ deg(𝔞^{2}) (mod 4), the definition is correct.

We also define an auxiliary operation sq′ : Ch(X × X) → ℤ/4ℤ as follows. For any α ∈ Ch(X ×X) we take its integral representative 𝔞 ∈ CH(X ×X) and set sq′(α) := deg(𝔞^{t} ·𝔞) mod 4. Since any other integral representative of the same α is of the form 𝔞 + 2𝔟 with some 𝔟 ∈ CH(X × X) and deg((𝔞 + 2𝔟)^{t} · (𝔞 + 2𝔟)) ≡ deg(𝔞^{2}) (mod 4), because

deg(𝔟^{t} · 𝔞) = deg ((𝔟^{t} · 𝔞)^{t}) = deg(𝔞^{t} · 𝔟),

the definition is correct.

Lemma 2.2. One has sq′(α) = sq(α) for any symmetric projector α ∈ Ch(X × X).

Proof. Indeed, such α has a symmetric integral representative: if 𝔞 is any integral representative, then 𝔞^{t} ∘ 𝔞 is a symmetric integral representative of α. Computing sq(α) and sq′(α) with the help of a symmetric integral representative of α, we get the same. □

The operations sq and sq′ are natural at least in the following sense:

Lemma 2.3. Each of the operations sq and sq′ commutes with the change of field homomorphism with respect to any field extension of F. □

The operation sq′ also enjoys the additivity with respect to sums of orthogonal correspondences:

Lemma 2.4. For any orthogonal correspondences α, β ∈ Ch(X × X) one has

sq′(α + β) = sq′(α) + sq′(β).

Proof. Let 𝔞, 𝔟 ∈ CH(X × X) be integral; representatives of α, β. It suffices to show that deg(𝔟^{t} · 𝔞) ≡ 0 (mod 2). By Lemma 2.1, deg(𝔟^{t} · 𝔞) = deg(δ^{∗}(𝔟 ∘ 𝔞)). Since the correspondences β and α are orthogonal, 𝔟 ∘ 𝔞 ∈ 2 CH(X × X). □

We are working with the Chow motives over F with coefficients in 𝔽_{2}, [1, Chapter XII]. A motive is split, if it is isomorphic to a (finite) direct sum of Tate motives. A motive is geometrically split, if it splits over a field extension of F. The rank rk M of a geometrically split motive M is the number of Tate summands in the decomposition of M_{E} for a field extension E/F such that M_{E} is split (this number does not depend on the choice of E). If α is a projector on a smooth projective variety X such that the motive (X, α) is geometrically split, we set rk α := rk(X, α).

Lemma 2.5. Let α ∈ Ch(X × X) be a projector and assume that the motive (X, α) is geometrically split (so that the rank rk(α) ∈ ℤ of α is defined). Then sq′(α) = rk(α) mod 4.

Proof. By the naturality of sq′ (Lemma 2.3), we may assume that the motive (X, α) is split, that is, that (X, α) is isomorphic to a finite sum of Tate motives. The number of the summands is the rank. By additivity of sq′ (Lemma 2.4), we may assume that the rank is 1. In this case α has an integral representative of the form a × b with some homogeneous a, b ∈ CH(X) having odd deg(a · b). It follows that sq′(α) = 1 mod 4. □

Now we are going to define our second basic operation. We write S^{•} for the modulo 2 total cohomological Steenrod operation, [1, Chapter XI]. Let pr : X × X → X be the projection onto the first factor.

Lemma 2.6. For any α, β ∈ Ch(X × X) one has

deg S^{•}(β^{t} ∘ α) = deg ( pr_{∗}(S(α)) · pr_{∗}(S(β)) · c_{•}(−T_{X})),

where T_{X} is the tangent bundle of X, and c_{•} is the total (modulo 2) Chern class.

Note that deg in Lemma 2.6 refers to the degree homomorphism on the modulo 2 Chow group taking its values in ℤ/2ℤ.

Here is the definition of the operation st : Ch(X ×X) → ℤ/4ℤ. For any α ∈ Ch(X ×X) we choose an integral representative 𝔞 ∈ CH(X × X) of S^{•}(α) ∈ Ch(X × X) and set

st(α) := deg ( pr_{∗}(𝔞)^{2} · c_{•}(−T_{X})) (mod 4),

where c_{•} refers now to the integral total Chern class. Clearly, the definition is correct because the choice of 𝔞 does not affect the value of the operation.

The operation st is a sort of refinement modulo 4 of a modulo 2 operation related with the Steenrod operation:

Lemma 2.7. We have st(α) mod 2 = deg S^{•}(α^{t} ∘ α). In particular,

st(α) mod 2 = deg S^{•}(α)

if the correspondence α is a symmetric projector.

Proof. This is the particular case β = α of Lemma 2.6. □

The operation st is natural:

Lemma 2.8. The operation st commutes with the change of field homomorphism with respect to any field extension of F. □

The operation st also enjoys a sort of additivity:

Lemma 2.9. If deg S(β^{t} ∘ α) = 0, then

st(α + β) = st(α) + st(β).

In particular, the additivity holds for orthogonal symmetric correspondences α, β.

Proof. Let 𝔞, 𝔟 ∈ CH(X × X) be integral; representatives of S^{•}(α), S^{•}(β). Then 𝔞 + 𝔟 is an integral representative of S^{•}(α + β) and it suffices to show that

deg ( pr_{∗}(𝔞) · pr_{∗}(𝔟) · c_{•}(−T_{X})) ≡ 0 (mod 2).

This is indeed so by Lemma 2.6 and the condition on α, β. □

The two operations sq and st are related as follows:

Lemma 2.10. Let d = dim X. For any symmetric projector α ∈ Ch^{d}(X × X) one has sq(α) ≡ st(α) (mod 2). If moreover X has no closed points of odd degree, then sq(α) = st(α).

Proof. The value sq(α) is given by the degree of certain integral representative 𝔞 of α^{2}. By Lemma 2.7, the value st(α) is given by the degree of certain integral representative 𝔟 of S^{d}α. Since S^{d}α = α^{2}, it follows that 𝔞 − 𝔟 ∈ 2 CH(X × X). Since deg CH(X × X) = deg CH(X), we get that deg 𝔞 − deg 𝔟 ∈ 2 deg CH(X). In particular, deg 𝔞 − deg 𝔟 ∈ 4ℤ + 2 deg CH(X) as claimed. □

Lemma 2.10 shows that the difference st − sq is a sort of weak replacement for a certain symmetric operation. This replacement suffices for our further purposes. One advantage of this replacement is that it is available over any F with char F ≠ 2, not only over F with char F = 0 like the symmetric operation itself.

3. Chow ring of quasi-split unitary grassmannians

Let V be a finite-dimensional vector space over a field K (of arbitrary characteristic). We set n := dim V. (Although in relation with our main purpose we are only interested in the case of even n, we treat the case of odd n because it differs from the even one only in a few places.) For any subset I ⊂ {1, 2 …, [n/2]} (where [n/2] = n/2 for even n and [n/2] = (n − 1)/2 for odd n), we write G_{I}(V) for the variety of flags of subspaces in V of dimensions given by I. In particular, for any integer k ∈ [1, [n/2]], the variety G_{{k}}(V) (which we simply denote as G_{k}(V)) is the grassmannian of k-planes.

Let us consider the closed subvariety H_{I} = H_{I}(V) of the product G_{I}(V) × G_{I}(V^{#}), where V^{#} is the dual vector space of V, defined by the orthogonality condition: H_{I} is the variety of pairs of flags such that each space of the first flag is orthogonal with respect to the corresponding space of the second flag (or, equivalently, the biggest space of the first flag is orthogonal with the biggest space of the second flag).

Example 3.1. The variety H_{k} is the variety of pairs of k-planes U ⊂ V, U′ ⊂ V^{#} such that U · U′ = 0. It is canonically isomorphic to the variety of flags in V consisting of a k-plane contained in a (n − k)-plane.

Any isomorphism of the vector spaces V and V^{#} provides a (non-canonical) isomorphism of the varieties G_{I}(V) and G_{I}(V^{#}). Since the automorphism group of V acts trivially on CH(G_{I}(V)), the ring CH(G_{I}(V)) is canonically identified with CH(G_{I}(V^{#})). In particular, we obtain a canonical involution (i.e., a self-inverse automorphism) of the ring

CH(G_{I}(V) × G_{I}(V^{#})) = CH(G_{I}(V)) ⊗ CH(G_{I}(V^{#})) = CH(G_{I}(V)) ⊗ CH(G_{I}(V))

given by the exchange of the factors in the last tensor product.

Lemma 3.2. There exists one and only one involution of CH(H_{I}) such that the following square of the involutions and a pull-back commutes:

CH(G_{I}(V) × G_{I}(V^{#})) ⟶ CH(G_{I}(V) × G_{I}(V^{#}))
↓  ↓
CH(H_{I})  ⟶  CH(H_{I}).

Proof. The uniqueness is a consequence of the surjectivity of the pull-back which we have because the ring CH(H_{I}) is generated by the Chern classes of the tautological bundles on H_{I} which are pull-backs of the tautological bundles on G_{I}(V) × G_{I}(V^{#}).

To show the existence, we fix a non-degenerate symmetric bilinear form on V (giving a self-dual isomorphism V ≃ V^{#}). This provides the variety G_{I}(V) × G_{I}(V^{#}) with the switch involution (inducing our involution on the Chow group), and the subvariety H_{I} is stable under it. □

The canonical involution on CH(H_{I}) just constructed will be denoted by σ.

Example 3.3. Assume that the field K is separable quadratic over some subfield F ⊂ K and let h be a K/F -hermitian form on V. Let Y_{I} be the flag variety of totally isotropic subspaces in V. The K-variety (Y_{I})_{K} is canonically isomorphic to H_{I}. The non-trivial automorphism of K/F induces an automorphism of CH(Y_{I})_{K} identified with σ. The image of the change of field homomorphism CH(Y_{I}) → CH(H_{I}) is contained in the subring CH(H_{I})^{σ} ⊂ CH(H_{I}) of the σ-invariant elements. Moreover, if h is hyperbolic, the change of field homomorphism CH(Y_{I}) → CH(H_{I}) is injective and its image coincides with CH(H_{I})^{σ} so that we have a canonical identification CH(Y_{I}) = CH(H_{I})^{σ}; the ideal (1 + σ) CH(H_{I}) ⊂ CH(H_{I})^{σ} coincides with the image of the norm homomorphism CH(Y_{I})_{K} → CH(Y_{I}).

We are going to study the subring CH(H_{I})^{σ} ⊂ CH(H_{I}) of the σ-invariant elements. More precisely, we will study the quotient of this subring by its “elementary part” – the norm ideal (1 + σ) CH(H_{I}). We are basically interested in the case of #I = 1.

We start with the case I = {1}. Let h be the hyperplane class in CH^{1}(G_{1}(V)) or in CH^{1}(G_{1}(V^{#})) and let us define the elements a, b ∈ CH^{1}(H_{1}) as the pull-backs of h × 1 and 1 × h. For any i ≥ 0, one has a^{i} = c_{i}(−A) and b^{i} = c_{i}(−B), where A and B are the corresponding tautological vector bundles on H_{1}.

The ring CH(H_{1}) is generated by the two elements a, b subject to the relations a^{n} = 0 and a^{n−1} + a^{n−2}(−b) + ⋯ + (−b)^{n−1} = 0 (implying b^{n} = 0). The involution σ exchanges the generators a and b.

Let T be the vector bundle on H_{1} whose fiber over a point (U, U′) is U ⊕ U′ (i.e., T = A ⊕ B). (Note that the isomorphism (Y_{1})_{K} = H_{1} of Example 3.3 transforms T to the tautological vector bundle on (Y_{1})_{K} (defined over F).) We write c_{i} for c_{i}(−T ). We have c_{i} = a^{i} + a^{i−1}b + ⋯ + b^{i} ∈ CH(H_{1})^{σ}.

The elements c_{n−1}, c_{n}, … are divisible by 2. Indeed,

c_{n−1}/2 = a^{n−1} + a^{n−3}b^{2} + ⋯ + ab^{n−2} = a^{n−2}b + ⋯ + b^{n−1}

and c_{n−1+i}/2 = (c_{n−1}/2) · a^{i} = (c_{n−1}/2) · b^{i} for any i ≥ 0.

Lemma 3.4. The ring CH(H_{1})^{σ}/(1 + σ) CH(H_{1}) is additively generated by the classes of the following elements:

c_{0}, c_{1}, …, c_{n−2} and c_{n−1}/2, c_{n}/2, ….

Moreover, for any odd i ≤ n − 2 the class of c_{i} is 0, for any even i ≥ n − 1 the class of c_{i}/2 is 0, and for any i > 2n − 3 the class of c_{i}/2 is 0.

Proof. The group CH^{<n−1}(H_{1}) is freely generated by a^{i}b^{j} with i + j < n − 1. Therefore the quotient CH(H_{1})^{σ}/(1 + σ) CH(H_{1}) in codimensions < n − 1 is (additively) generated by the classes of a^{i}b^{i} (2i < n − 1) which are also represented by c_{2i}.

For any i = n − 1, n, …, 2n − 3, the group CH^{i}(H_{1}) is generated by the elements

a^{n−1}b^{i−(n−1)}, a^{n−2}b^{i−(n−2)}, …, a^{i−(n−1)}b^{n−1}

whose alternating sum is 0, and this is the only relation on the generators. The quotient of the subgroup of σ-invariant elements by the norms is therefore trivial for even i and generated by the class of c_{i}/2 for odd i.

Finally, for i > 2n − 3 = dim H_{1}, the group CH^{i}(H_{1}) is trivial. □

Remark 3.5. Here is a complete analysis of the graded ring

R := CH(H_{1})^{σ}/(1 + σ) CH(H_{1}),

which is now easily done. Similarities as well as differences with the Chow ring of a split projective quadric are striking.

In the case of even n, the ring R is generated (as a ring) by two elements: (the classes of) ab ∈ R^{2} and c := c_{n−1}/2 ∈ R^{n−1} (R^{2} and R^{n−1} are the graded components of R). The relations are: (ab)^{n/2} = 0 and c^{2} = 0. The non-zero homogeneous elements of R are as follows:

(ab)^{i} = c_{2i}, c(ab)^{i} = c_{n−1+2i}/2, with i = 0, 1, …, (n − 2)/2.

If n is odd, the ring R is generated by two elements: (the classes of) ab ∈ R^{2} and c := c_{n}/2 ∈ R^{n}. The relations are: (ab)^{(n−1)/2} = 0 and c^{2} = 0. The non-zero homogeneous elements of R are as follows:

(ab)^{i} = c_{2i}, c(ab)^{i} = c_{n+2i}/2, with i = 0, 1, …, (n − 3)/2.

The geometric description of the generators (for arbitrary parity of n) is as follows. The element (ab)^{i} is the pullback of h^{i} × h^{i} ∈ CH^{2i} (G_{1}(V) × G_{1}(V^{#})). To describe c(ab)^{i}, we take some orthogonal subspaces U ⊂ V and U′ ⊂ V^{#} of dimension [n/2] − i. Then c(ab)^{i} is the class of the (closed) subvariety L_{i} ⊂ H_{1} of pairs of lines: one line in U, the other in U′.

Now we start to study the case of I = {k} where k satisfies 1 ≤ k ≤ [n/2]. We write T_{k} for the vector bundle on H_{k} whose fiber over a point (U, U′) is U ⊕ U′ (in particular, T_{1} = T ). Note that the isomorphism (Y_{k})_{K} = H_{k} of Example 3.3 transforms T_{k} to the tautological vector bundle on (Y_{k})_{K} (defined over F).

We consider the natural projections π_{1} : H_{{1,k}} → H_{1} and π_{k} : H_{{1,k}} → H_{k}.

Lemma 3.6 (cf. [16, Proposition 2.1]). For any integer i one has

c_{i}(−T_{k}) = (π_{k})_{∗}π_{1}^{∗}c_{i+2(k−1)}(−T_{1}).

Proof. For any smooth scheme X with a rank k vector bundle E one has

c_{i}(−E) = π_{∗}c_{i+k−1}(−O(−1)),

where π is the morphism ℙ(E) → X and O(−1) is the tautological (line) bundle on ℙ(E). If now E_{1}, E_{2} are two rank k vector bundles on X and π is the morphism ℙ(E_{1})×_{X} ℙ(E_{2}) → X, we get that

c_{i}(−(E_{1} ⊕ E_{2})) = π_{∗}c_{i+2(k−1)}( − (O_{1}(−1) ⊕ O_{2}(−1))).

In particular, taking π = π_{k}, we see that

c_{i}(−T_{k}) = (π_{k})_{∗}c_{i+2(k−1)}( − (O_{1}(−1) ⊕ O_{2}(−1))).

Since O_{1}(−1) ≃ π_{1}^{∗}A and O_{2}(−1) ≃ π_{1}^{∗}B, we are done. □

Corollary 3.7. The (σ-invariant) elements

c_{n−2k+1}(−T_{k}), c_{n−2k+2}(−T_{k}), …, c_{2n−2k−1}(−T_{k}) ∈ CH(H_{k})^{σ}

are divisible by 2. □

We consider the projections π_{k} : H_{{k,k+1}} → H_{k} and π_{k+1} : H_{{k,k+1}} → H_{k+1}. The vector bundle π_{k}^{∗}T_{k} is a subbundle of the vector bundle π_{k+1}^{∗}T_{k+1} (the quotient is a direct sum of two line bundles), and we write α ∈ CH^{2}(H_{{k,k+1}}) for c_{2}(π_{k+1}^{∗}T_{k+1}/π_{k}^{∗}T_{k}). Lemma 3.8 (cf. [16, Lemma 2.5]). For i ∈ {0, 1, …, n − 2k} one has

π_{k}^{∗}c_{i}(−T_{k}) ≡ π_{k+1}^{∗}c_{i}(−T_{k+1}) + α · c_{i−2}(−T_{k+1}) (mod 1 + σ).

For i ≥ n − 2k + 1 one has

π_{k}^{∗}c_{i}(−T_{k})/2 ≡ π_{k+1}^{∗}c_{i}(−T_{k+1})/2 + α · c_{i−2}(−T_{k+1})/2 (mod 1 + σ).

Proof. We play with the following commutative diagram

[figure: H_{{1,k}}; H_{1}; H_{{1,k,k+1}}; H_{k}; H_{{1,k+1}}; H; H_{{k,k+1}}; H_{k+1}; 1; 3; 2; 5; 6; 7; 11; 4; 9; 10; 12; 8]

where H is defined as the fiber product of H_{{1,k+1}} and H_{{k,k+1}} over H_{k+1}. The variety H_{{1,k,k+1}} is naturally a closed subvariety (of codimension 2) in H and 6 is the closed imbedding. Note that π_{k} = 8 and π_{k+1} = 12. By Lemma 3.6, the elements c_{i}(−T_{k}) and c_{i}(−T_{k})/2 are 3_{∗}1^{∗}(x) for certain x ∈ CH(H_{1})^{σ}. Let us compute y := 8^{∗}3_{∗}1^{∗}(x) for an arbitrary x ∈ CH(H_{1})^{σ}.

The square 3-8-7-2 is transversal cartesian. Therefore 8^{∗}3_{∗} = 7_{∗}2^{∗}. By commutativity of the square 1-2-5-4, 2^{∗}1^{∗} = 5^{∗}4^{∗} so that y = 7_{∗}5^{∗}4^{∗}(x). By commutativity of the triangles 5-6-9 and 6-7-10, y = 10_{∗}6_{∗}6^{∗}9^{∗}4^{∗}(x) = 10_{∗}[H_{{1,k,k+1}}] · 9^{∗}4^{∗}(x). The class [H_{{1,k,k+1}}] ∈ CH^{2}(H) is computed modulo 1 + σ as 9^{∗}4^{∗}(ab) + 10^{∗}(α). It follows that y ≡ 10_{∗}9^{∗}4^{∗}(abx) + α · 10_{∗}9^{∗}4^{∗}(x). Since the square 9-10-12-11 is transversal cartesian, 10_{∗}9^{∗} = 12^{∗}11_{∗}, so that we finally get y ≡ 12^{∗}11_{∗}4^{∗}(abx) + α · 12^{∗}11_{∗}4^{∗}(x) (mod 1 + σ).

We get the first (resp. second) desired congruence taking x = c_{i+2(k−1)}(−T_{1}) (resp. x = c_{i+2(k−1)}(−T_{1})/2) by Lemma 3.6, because abc_{i+2(k−1)}(−T_{1}) ≡ c_{i+2(k+1)}(−T_{1}) (mod 1 + σ) (resp. abc_{i+2(k−1)}(−T_{1})/2 ≡ c_{i+2(k+1)}(−T_{1})/2 (mod 1 + σ)) for the corresponding values of i (cf. Remark 3.5). □ Proposition 3.9 (cf. [16, Proposition 2.9]). The ring CH(H_{k})^{σ} is generated (as a ring) modulo the ideal (1 + σ) CH(H_{k}) by the elements c_{i}(−T_{k}) with even i satisfying 0 ≤ i ≤ n − 2k and the elements c_{i}(−T_{k})/2 with odd i satisfying n − 2k + 1 ≤ i ≤ 2n − 2k − 1. Proof. For each integer l with 1 ≤ l ≤ k, we consider the projection π_{l} : H_{{1,...,k}} → H_{l} and the elements

(∗) π_{l}^{∗}c_{i}(−T_{l}) with even i satisfying 0 ≤ i ≤ n − 2l and π_{l}^{∗}c_{i}(−T_{l})/2 with odd i satisfying n − 2l + 1 ≤ i ≤ 2n − 2l − 1.

Lemma 3.10 (cf. [16, Lemma 2.10]). The ring CH(H_{{1,...,k}})^{σ} is generated modulo the ideal (1 + σ) CH(H_{k}) by the elements (∗) (with l running over 1, …, k).

Proof. Since the statement does not depend on the base field K, we may assume that K is quadratic separable over some subfield F. Then we fix a hyperbolic K/F -hermitian form on V and replace CH(H_{k})^{σ} by CH(Y_{k}) (see Example 3.3).

We do induction on k. The case k = 1 is Lemma 3.4. To pass from k − 1 to k, we apply a variant of [16, Statement 2.11] taking as Y → X the projection Y_{{1,2,...,k}} → Y_{{1,2,...,k−1}} and taking as B the subgroup generated by the norms and the elements (∗) with l = k. Each fiber of this projection is a hermitian quadric given by a hyperbolic hermitian space of dimension n − 2(k − 1). Let us check that condition (a) of [16, Statement 2.11] is verified. The restriction of π_{k}^{∗}T_{k} to the generic fiber of the projection is isomorphic to the direct sum of T_{1} and a trivial vector bundle (of rank 2(k − 1)). Therefore the pull-backs of the elements (∗) to the generic fiber give the elements

c_{i}(−T_{k}) with even i satisfying 0 ≤ i ≤ n − 2k and c_{i}(−T_{k})/2 with odd i satisfying n − 2k + 1 ≤ i ≤ 2n − 2k − 1.

which generate the group CH(Y) modulo the norms by Lemma 3.4 (note that 2(n−2(k − 1)) − 3 ≤ 2n − 2k − 1).

However condition (b) of [16, Statement 2.11] is not satisfied: the specialization homomorphism from the Chow group of the generic fiber to the Chow group of the fiber over a point x is not surjective in general. It is surjective if the residue field of x does not contain a subfield isomorphic to K. We finish the proof by showing that in the opposite case the image of CH(Y_{x}) in the associated graded group of the filtration on CH(Y) is in the image of 1 + σ. We are speaking about the filtration on CH(Y) whose term F^{r} CH(Y) is the subgroup generated by the classes of cycles on Y whose image in X has codimension ≥ r.

Let T be the closure of x in X. Let Y_{T} = Y ×_{X} T ↪ Y be the preimage of T under Y → X. The image of the homomorphism CH(Y_{x}) → F^{r} CH(Y)/F^{r+1} CH(Y), where r = codim_{X} x, is in the image of the push-forward CH(Y_{T}) → F^{r} CH(Y)/F^{r+1} CH(Y). Since x is the generic point of T and F (x) = F (T) ⊃ K, a non-empty open subset U ⊂ T possesses a morphism to Spec K. Its preimage Y_{U} ⊂ Y_{T} is open and also possesses a morphism to Spec K. Therefore (Y_{U})_{K} ≃ Y_{U} ∐ Y_{U} and, in particular, the push-forward CH((Y_{U})_{K}) → CH(Y_{U}) is surjective.

We play with the following commutative diagram:

[figure: Y_{K}; Y; X; (Y_{T})_{K}; Y_{T}; T; (Y_{U})_{K}; Y_{U}; U]

It follows that the image of the push-forward CH((Y_{T})_{K}) → CH(Y_{T}) generates CH(Y_{T}) modulo the image of CH(Y_{T} \ Y_{U}). Since the image of CH(Y_{T} \ Y_{U}) → CH(Y) is in F^{r+1} CH(Y), it follows that the image of CH(Y_{T}) in the quotient of the filtration on CH(Y) is contained in the image of F^{r} CH(Y_{K}), that is, in the image of 1 + σ. □

Let I = [1, k] = {1, 2, …, k}. For every i ∈ I, let A_{i} and B_{i} the the tautological vector bundles on H_{i} (so that T_{i} = A_{i} ⊕ B_{i}). We define a_{i}, b_{i} ∈ CH^{1}(H_{I}) as the first Chern classes of the linear bundles (H_{I} → H_{i})^{∗}A_{i}/(H_{I} → H_{i−1})^{∗}A_{i−1} and (H_{I} → H_{i})^{∗}B_{i}/(H_{I} → H_{i−1})^{∗}B_{i−1}.

For any l ∈ I, we identify CH(H_{[l ,k]}) with a subring in CH(H_{I}) via the pull-back. Note that a_{i}, b_{i} ∈ CH(H_{[l ,k]}) for i ∈ [l + 1 , k].

By induction on l ∈ I, we prove the following statement: the ring CH(H_{[l, k]})^{σ} is generated modulo (1 + σ) by the elements of Proposition 3.9 and the elements {a_{i}b_{i}}_{i∈[l+1, k]}. Note that this statement for l = k is the statement of Proposition 3.9.

The induction base l = 1 follows from Lemma 3.10 and Lemma 3.8 (the latter showing that the missing generators of Lemma 3.10 are expressible in terms of the kept generators and the added generators). Let us do the passage from l − 1 to l.

The projection H_{[l−1, k]} → H_{[l, k]} is (canonically isomorphic to) a product of two rank l − 1 projective bundles (given by the dual of the rank l tautological vector bundles A_{l} and B_{l} on H_{[l, k]}). The CH(H_{[l, k]})-algebra CH(H_{[l−1, k]}) is therefore generated by the two elements a_{l}, b_{l} subject to the two relations

∑_{i=0}^{l} c_{i}(A_{l})a_{l}^{l−i} = 0, ∑_{i=0}^{l} c_{i}(B_{l})b_{l}^{l−i} = 0.

In particular, the CH(H_{[l, k]})-module CH(H_{[l−1, k]}) is free, a basis is given by the products a_{l}^{i}b_{l}^{j} with i, j ∈ [0, l − 1].

The involution σ exchanges a and b. Therefore the module CH(H_{[l−1, k]})^{σ}/(1 + σ) over the ring CH(H_{[l, k]})^{σ}/(1+σ) is free of rank l, a basis is given by the (classes of the) products a_{l}^{i}b_{l}^{i} with i ∈ [0, l−1]. In particular, the CH(H_{[l, k]})^{σ}/(1+σ)-algebra CH(H_{[l−1, k]})^{σ}/(1+σ) is generated by a_{l}b_{l}. This generator satisfy the relation

∑_{i=0}^{l} c_{2i}(T_{l})(a_{l}b_{l})^{l−2i} = 0

(this is an equality in the quotient CH(H_{[l−1, k]})^{σ}/(1 + σ) !). This is the only relation on the generator because its powers up to l − 1 form a basis.

Now let C ⊂ CH(H_{[l, k]})^{σ}/(1 + σ) be the subring generated by the elements of Proposition 3.9 and the elements {a_{i}b_{i}}_{i∈[l+1, k]}. Note that the coefficients of the above relation are in C: they are expressible in terms of c_{i}(−T_{l}) (which are non-zero modulo 1 + σ only for i = 0, 2, …, n − 2l by Lemmas 3.6 and 3.4. Therefore the subring of CH(H_{[l−1, k]})^{σ}/(1 + σ) generated by C and a_{l}b_{l} is also a free C-module of rank l. On the other hand, this subring coincides with the total ring by the induction hypothesis and it follows that C = CH(H_{[l, k]})^{σ}/(1 + σ). □

Remark 3.11 (Geometric description of the generators). Proposition 3.9 provides us with generators of the ring CH(Y_{k}) modulo the K/F −norms via the identification CH(Y_{k}) = CH(H_{k})^{σ} of Example 3.3. These generators have precisely the same geometric description as the standard generators of the Chow ring of an orthogonal grassmannian. Namely, they are obtained as via the composition (Y_{{1,k}} → Y_{k})_{∗} ∘ (Y_{{1,k}} → Y_{1})^{∗} out of the additive generators of CH(Y_{1}) modulo the norms. Moreover, for any odd i satisfying n − 2k + 1 ≤ i ≤ 2n − 2k − 1, the generator c_{i}(−T_{k})/2 is the class of the Schubert subvariety of the subspaces intersecting non-trivially a fixed totally isotropic subspace in V of certain K-dimension. This is a consequence of Remark 3.5 and Lemma 3.6.

4. Steenrod operations for split unitary grassmannians

In this section, dimension n of the K-vector space V is supposed to be even.

Let H = H_{k}. One more tool for study of CH(H) is given by the morphism in : H → X, where X is the variety of totally isotropic 2k-planes of the hyperbolic quadratic form ℍ(V) = V ⊕ V^{#}. The morphism associates to a point (U, U′) of H the point U ⊕ U′ of X. This is a closed imbedding by [7, Corollary 10.4].

Note that the image of the pull-back in^{∗} : CH(X) → CH(H) is contained in CH(H)^{σ}. Indeed, fixing a non-degenerated symmetric bilinear form on V giving an identification of V with V^{#}, we get the exchange involution on H (inducing σ on CH(H)) and an involution on X given by the automorphism V ⊕ V^{#} = V^{#} ⊕ V. The imbedding H ↪ X commute with these involutions, and the involution induced on CH(X) is the identity because V is of even dimension.

The power of this tool of studying CH(H) is explained by the fact that CH(X) (in contrast to CH(H)) is very well studied. One more advantage of the variety X is that (in contrast to H) it has twisted forms with “high” degrees of the closed points.

The meaning of the imbedding H ↪ X is as follows. Assume that V is endowed with a K/F -hermitian form h. We consider the variety Y_{k}. Let X_{2k} be the variety of 2k-planes in the vector F -space V totally isotropic with respect to the quadratic form on V given by h. We have a natural closed imbedding in : Y_{k} ↪ X_{2k} which becomes the above imbedding over K. Choosing a hyperbolic h, we get another proof of the fact that the image of CH(X) → CH(H) is in CH(H)^{σ}: this is so because CH(X) = CH(X_{2k}) and CH(Y_{k}) = CH(H)^{σ}.

Recall (see [16]) that the ring CH(X) is generated by certain elements w_{i} ∈ CH^{i}(X), i = 0, 1, …, n − 2k and z_{i} ∈ CH^{i}(X), i = n − 2k, n − 2k + 1, …, 2n − 2k − 1. They satisfy w_{i} = c_{i}(−T_{X}) for all i and z_{i} = c_{i}(−T_{X})/2 for i ≠ n − 2k, where T_{X} is the tautological vector bundle on X.

Lemma 4.1. The pull-back CH(X) → CH(H)^{σ}/(1 + σ) is surjective. The image of each z_{i} with even i ≠ n − 2k is 0.

Remark 4.2. One may show (see [2]) that the pull-back in^{∗} : CH(X_{2k}) → CH(Y_{k})/(1 + σ) is surjective (for any h). Moreover, the push-forward in_{∗} induces an injection in_{∗} : CH(Y_{k})/(1 + σ) → Ch(X_{2k}), where Ch := CH /2. It follows that the ring CH(Y_{k})/(1 + σ) is naturally identified with Ch(X_{2k}) modulo the kernel of the multiplication by [Y_{k}] ∈ Ch(X_{2k}). In the case of k = n/2 and hyperbolic h, the computation of the class [Y_{k}] given below together with the computation of Ch(X_{2k}) given [1], provides the following presentation of CH(Y_{n/2})/(1 + σ) by generators and relations: generators are e_{i} ∈ CH^{i}, i = 1, 3, …, n − 1; relations are e_{i}^{2} = 0 for each i.

Proof of Lemma 4.1. The generators of the ring on the right-hand side given in Proposition 3.9 come from CH(X) because the pull-back of the tautological vector bundle T_{X} on X to H is T_{k} and the Chern classes c_{i}(−T_{X}) are divisible by 2 for i > n − 2k. This gives the surjectivity.

The image of z_{i} with even i ≠ n − 2k is 0 by Lemmas 3.6 and 3.4. □

Lemma 4.3. The element [H] ∈ CH(X) is a square.

Proof. Let x ∈ CH(X) be the class of the Schubert subvariety S ⊂ X of the subspaces U ⊂ V ⊕ V^{#} satisfying dim U ∩ V ≥ k. We claim that [H] = x^{2}. Indeed, x can be also represented by the Schubert subvariety S′ ⊂ X of the subspaces U ⊂ V ⊕ V^{#} satisfying dim U ∩V^{#} ≥ k. Since S ∩S′ = H and codim_{X} H = codim_{X} S +codim_{X} S′, [H] = [S]·[S′] by [1, Corollary 57.22]. □

We now pass to the modulo 2 Chow group Ch(X) = CH(X)/2 and we use the notion of level for elements of Ch(X) introduced in [5]. Namely, an element of Ch(X) is of level l if it can be written as a polynomial in the generators of the z-degree ≤ l (we use the same notation w_{i}, z_{i} for the classes of the integral generators). We recall that (see [5, proof of Proposition 12]) that by the formula of [16, Proposition 2.8] the cohomological Steenrod operation preserves the level. In particular, the squaring preserves the level.

We also recall that the generators satisfy the relation

z_{i}^{2} = z_{i}c_{i}(−T_{X}) − z_{i+1}c_{i−1}(−T_{X}) + z_{i+2}c_{i−2}(−T_{X}) − …

which shows that any element of Ch(X) can be written as a polynomial in the generators of z_{i}-degree ≤ 1 for each i. A polynomial satisfying this restriction is called standard below.

Corollary 4.4. The element [H] ∈ Ch(X) is of level k.

Proof. Since squaring does not affect the level, it suffices to show that the level of a homogeneous element x with x^{2} = [H] is k. The codimension of x is equal to

(dim X − dim H)/2 = (k(4n − 6k − 1) − k(2n − 3k))/2 = (k/2)(2n − 3k − 1) = (n − 2k) + (n − 2k + 1) + ⋯ + (n − k − 1),

and the minimal codimension of an element which is not of level k is this number plus n − k. □

Theorem 4.5 (cf. [5, Proposition 12]). Let F be a field of characteristic ≠ 2, K/F a quadratic field extension, V a vector space over K of even positive dimension n, h a K/F -hermitian form on V, k an integer satisfying 1 ≤ k ≤ n/2, Y the variety of totally isotropic k-planes in V. Then for any i > k(n − 2k), one has deg S Ch_{i}(Y_{k}) = 0, where S is the cohomological Steenrod operation and deg is the degree homomorphism on the modulo 2 Chow groups.

Proof. Assume that deg S Ch^{j}(Y) ≠ 0 for some j. Then deg S Ch^{j}(H)^{σ} ≠ 0. Since S commutes with σ, S is trivial on (1 + σ). Therefore deg S Ch^{j}(H)^{σ}/(1 + σ) ≠ 0. It follows by Lemma 4.1 that deg in^{∗} S Ch^{j}(X) ≠ 0, or, equivalently, deg S in^{∗} Ch^{j}(X) ≠ 0. Let y ∈ Ch^{j}(X) be a standard monomial in the generators with deg S in^{∗}(y) ≠ 0. Since in^{∗}(y) ≠ 0, the monomial y does not contain any z_{i} with even i ≠ n = 2k by the second half of Lemma 4.1. We may also assume that y does not contain z_{n−2k}. Indeed, in^{∗}(z_{n−2k}) is a polynomial in the generators of Ch(H)^{σ}/(1+σ) of codimension ≤ n−2k. In particular, in^{∗}(z_{n−2k}) is a polynomial in c_{i}(−T_{k}) with i ≤ n − 2k. Let P ∈ Ch^{n−2k}(X) be the same polynomial in c_{i}(−T_{X}) = w_{i}. Then in^{∗}(P) = in^{∗}(z_{n−2k}) and we may replace z_{n−2k} by P in y without changing in^{∗}(y).

We have

0 ≠ deg in^{∗} S(y) = deg in_{∗} in^{∗} S(y) = deg S([H] · y).

Since degree of any level 2k − 1 element is 0, [5, proof of Proposition 12], the element S([H]y) is not of level 2k−1. Since the Steenrod operation preserves the level, the product [H] · y is not of level 2k − 1. Since [H] is of level k by Corollary 4.4, y is not of level k − 1. The smallest possible codimension of a monomial of level not k − 1 without z-generators of even codimension is the sum of k summands

(n − 2k + 1) + (n − 2k + 3) + ⋯ + (n − 1) = k(n − 2k + 1) + k(k − 1) = k(n − k).

It follows that j < k(n − k). Since dim Y − k(n − k) = k(n − 2k), we are done. □

5. Some ranks of some motives

Let K/F be a separable quadratic field extension. Let M be a motive over F with coefficients in 𝔽_{2}. We assume that there exists a field extension F′/F such that F is algebraically closed in F′ and the motive M_{F′} decomposes in a sum of shifts of the motives of Spec F′ and Spec K′, where K′ is the field K ⊗_{F} F′. Note that the number of F′ and the number of K′ appearing in the decomposition do not depend on the choice of F′: if F′′ is another field like that, the Krull-Schmidt principle over the field F′ ⊗_{F} F′′ gives the equalities. The number of F′ in the decomposition is the F -rank rk_{F} of M, the number of K′ is the K-rank rk_{K} of M. The usual rank rk M is also defined for such M and is equal to rk_{F} M + 2 rk_{K} M.

Recall that there are functors

tr, cor : CM(K, 𝔽_{2}) → CM(F, 𝔽_{2}).

The first one (non-additive and not commuting with the shift, see [8]) is induced by the Weil transfer. The second one (additive and commuting with the shift, see [11]) is induced by the functor associating to a K-variety the same variety considered as a variety over F via the composition with Spec K → Spec F.

Here is an example of computation of ranks.

Lemma 5.1. Let M be a motive over K isomorphic to a sum of n shifts of the Tate motive. Then rk_{F} tr M = rk M = n, rk_{K} tr M = n(n − 1)/2, rk_{F} cor M = 0, and rk_{K} cor M = n.

Proof. Since cor M(Spec K) = M(Spec K), the formulas for cor follow. The formulas for tr follow from [4, Lemma 2.1]. □

Let D be a central division K algebra admitting a K/F -unitary involution, and assume that deg D = 2^{n} for some n ≥ 0. For an integer k ∈ [0, n − 1], let X_{k} be the Weil transfer with respect to K/F of the generalized Severi-Brauer variety X(2^{k}, D). The motive M(X_{k}) satisfy the above conditions (one may take as F′ the function field of the variety X_{0}) so that the ranks rk_{F} M and rk_{K} M are defined for any summand M of M(X_{k}). In particular, the ranks rk_{F} U(X_{k}) and rk_{K} U(X_{k}) are defined for the upper (indecomposable) motive U(X_{k}).

Proposition 5.2. v_{2}(rk_{F} U(X_{k})) = n − k, v_{2}(rk_{K} U(X_{k})) = n − k − 1.

Proof. We induct on k. Let us do the induction base k = 0. According to [4, Theorem 1.2], U(X_{0}) = M(X_{0}). Since rk M(X(1, D)) = 2^{n}, it follows from Lemma 5.1 that rk_{F} U(X_{0}) = 2^{n}, rk_{K} U(X_{0}) = 2^{n−1}(2^{n} − 1).

Now we assume that k > 0. Since rk M(X(2^{k}, D)) = b := \binom{2^{n}}{2^{k}}, it follows from Lemma 5.1 that rk_{F} M(X_{k}) = b and rk_{K} M(X_{k}) = b(b − 1)/2. In particular, v_{2}(rk_{F} M(X_{k})) = n − k > 0 and v_{2}(rk_{K} M(X_{k})) = n − k − 1. Therefore, it suffices to show that for each summand M of the complete motivic decomposition of X_{k} different from U(X_{k}) we have v_{2}(rk_{F} M) > n − k and v_{2}(rk_{K} M) > n − k − 1.

By [11] and [4], M is a shift of the motive U(X_{l}) with some l ∈ [0, k − 1] or a shift of the motive cor_{K/F} U(X(2^{l}, D)) with some l ∈ [0, k]. In the first case we are done by the induction hypothesis. In the second case we have rk_{F} M = 0 and rk_{K} M = \binom{2^{n}}{2^{l}}. □

6. Unitary isotropy theorem

Let K be a field of characteristic ≠ 2, A a central simple K-algebra, τ a unitary involution on A, F the subfield of the elements of K fixed under τ. We say that τ is isotropic, if τ(I) · I = 0 for some non-zero right ideal I ⊂ A; otherwise we say that τ is anisotropic. Theorem 6.1 (Unitary Isotropy Theorem). If τ becomes isotropic over any field extension F′/F such that K′ := K ⊗_{F} F′ is a field and the central simple K′-algebra A′ := A⊗_{F} F′ is split, then τ becomes isotropic over some finite odd degree field extension of F. Proof. We can easily reduce this theorem to the case of 2-primary ind A. Indeed, it suffices to find a finite odd degree field extension L/F, such that A becomes 2-primary over L. For such L/F we can take 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 ⊗_{F} E) is 2-primary.

Because of the above redaction, we assume that the index of A is a power of 2.

We follow the lines of the proof of [5, Theorem 1]. We prove Theorem 6.1 over all fields simultaneously using an induction on ind A. The case of ind A = 1 is trivial. From now we are assuming that ind A = 2^{r} for some integer r ≥ 1, and we fix the following notations:

F is a field of characteristic different from 2;

K/F is a quadratic field extension;

A is a central simple K-algebra of the index 2^{r} (with r ≥ 1);

τ is an F -linear unitary involution on A;

D is a central division F -algebra (of degree 2^{r}) Brauer-equivalent to A;

V is a right D-module of D-dimension v with an isomorphism End_{D}(V) ≃ A (in particular, rdim V = deg A = 2^{r} · v, where rdim V := dim_{F} V/ deg D is the reduced dimension);

we fix an arbitrary F -linear unitary involution ε on D;

h is a hermitian (with respect to ε) form on V such that the involution τ is adjoint to h;

Y = X(2^{r}; (V, h)) ≃ X(2^{r}; (A, τ)) is the variety of totally isotropic submodules in V of reduced dimension rdim = 2^{r} which is isomorphic (via Morita equivalence) to the variety of right totally isotropic ideals in A of the same reduced dimension;

X is the Weil transfer (with respect to K/F) of the generalized Severi-Brauer K-variety X(2^{r−1}; D).

Let F′ be the function field of the Weil transfer of the Severi-Brauer variety X(1; D) of D. Clearly, K′ is a field and A′ is split for such F′. We assume that the involution τ′ (and therefore, the hermitian form h_{F′}) is isotropic and we want to show that h (and τ) becomes isotropic over a finite odd degree extension of F. According to [3, Theorem 1.4], the Witt index of h_{F′} is a multiple of 2^{r} = ind A. In particular, v ≥ 2. If the Witt index is greater than 2^{r}, we replace V by a submodule in V of D-codimension 1 and we replace h by its restriction on this new V. The Witt index of h_{F′} drops by 2^{r} or stays unchanged. We repeat the procedure until the Witt index becomes equal to 2^{r}. In particular, v is still ≥ 2.

The variety Y has an F′-point and the index of the central simple K ⊗_{F} F (X)-algebra A ⊗_{F} F (X) is equal to 2^{r−1} (note that K ⊗_{F} F (X) is a field). Consequently, by the induction hypothesis, the variety Y_{F (X)} has an odd degree closed point. We prove Theorem 6.1 by showing that the variety Y has an odd degree closed point.

We will use and we recall the following statement from [5].

Proposition 6.2. Let 𝔛 be a geometrically split, geometrically irreducible F -variety satisfying the nilpotence principle and let Y be a smooth complete F -variety. Assume that there exists a field extension E/F such that

(1) for some field extension \overline{E(𝔛)}/E(𝔛), the image of the change of field homomorphism Ch(Y_{E(𝔛)}) → Ch(Y\overline{E(𝔛)}) coincides with the image of the change of field homomorphism Ch(Y_{F (𝔛)}) → Ch(Y\overline{E(𝔛)});

(2) the E-variety 𝔛_{E} is p-incompressible;

(3) a shift of the upper indecomposable summand of M(𝔛)_{E} is a summand of M(Y)_{E}. Then the same shift of the upper indecomposable summand of M(𝔛) is a summand of

M(Y).

We are going to apply Proposition 6.2 (with p = 2) 𝔛 = X, Y = Y, and E = F (Y). We need to check that conditions (1) - (3) are satisfied for these X, Y, E. First of all, we need a motivic decomposition of Y over a field extension F̃/F, such that Y (F̃) ≠ ∅ and K̃ = K ⊗_{F} F̃ is a field. Over such F̃, the hermitian form h decomposes in the orthogonal sum of the hyperbolic D̃-plane and a hermitian form h′ on a right D̃-module V′ with rdim V′ = 2^{r}(v − 2), where D̃ is central simple K̃-algebra D ⊗_{F} F̃. Let L/F (X) be a finite odd degree extension such that Y (L) ≠ ∅. Recall that a smooth projective variety is anisotropic, if it has no odd degree closed points (by [9, lemma 6.3], the motive of an anisotropic variety does not contain a Tate summand).

Lemma 6.3. The shift of the motive of XF̃ and two Tate motives are the motivic summands of YF̃. In the case F̃ = L, any other motivic summand of Y_{L} is a shift of some anisotropic L-variety.

Proof. According to [7, Theorem 15.8], the variety YF̃ is a relative cellular space (as defined in [1, §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 YF̃ (that is, εF̃(I) · J = 0, N is totally isotropic, and the reduced dimension of the D̃-module I ⊕ J ⊕ N ⊂ V is equal to deg D̃). Therefore, by [1, Corollary 66.4], the motive of YF̃ is the sum of shifts of the motives of the components of Z.

The shift of the motive of XF̃ is given by the motive of the component of the triples {(I, J, 0)| rdim I = rdim J = (deg D̃)/2}. The rational points (0, D̃, 0) and (D̃, 0, 0) of Z are components of Z which produce the two promised Tate summands. In the case F̃ = L we have ind D̃ = (deg D̃)/2 = 2^{r−1}. Therefore to prove the second statement of this lemma, we only need to check that the component of Z of triples (0, 0, N) is anisotropic. It is true, because this component is naturally identified with anisotropic L-variety Y′ = X(2^{r}; (V′, h′)). □

Remark 6.4. Two Tate motives mentioned in Lemma 6.3 are clearly 𝔽_{2} and 𝔽_{2}(dim Y). In the case F̃ = L, by duality, the motivic summand M(X_{L}) of Y_{L} has as the shifting number the integer

n := (dim Y − dim X)/2 .

Since Y (F (Y)) ≠ ∅, the condition (3) of Proposition 6.2 is checked by Lemma 6.3. Let us check now the condition (2). By [4, Theorem 1.1], the variety X_{F (Y)} is 2-incompressible if (and only if) the K ⊗_{F} F (Y)-algebra D ⊗_{F} F (Y) is division. This is indeed the case:

Lemma 6.5. The algebra D ⊗_{F} F (Y) is division, that is, ind(D ⊗_{F} F (Y)) = ind D.

Proof. The proof is similar to the proof of [5, Lemma 6]. Assume that ind(D ⊗_{F} F (Y)) < ind D. Then we could prove as in [5, Lemma 6], that the upper indecomposable motivic summand of X is a motivic summand of Y. This implies (because the variety X is 2incompressible) that the complete motivic decomposition of the variety Y_{F (X)} contains the Tate summand 𝔽_{2}(dim X). By Lemma 6.3 and Remark 6.4 we get a contradiction. □

We have checked condition (2) of Proposition 6.2. To check the remaining condition (1), we will need the same property for the variety Y as in [5, Lemma 7]. We can prove it for more general class of varieties. Let Z be a projective homogeneous variety under an arbitrary absolutely simple adjoint affine algebraic group G of type A_{n} over a field k (we can replace “absolutely simple of type A_{n}” by the condition, that G is semisimple and becomes of inner type over some quadratic separable field extension of k). In other words, Z is a variety of flags of isotropic right ideals of a central simple algebra over a quadratic separable field extension of k endowed (the algebra) with a unitary k-linear involution.

Lemma 6.6. Let k′/k be a finite odd degree field extension and let k̄ be an algebraic closure of k containing k′. Then Im(Ch(Z) → Ch(Zk̄)) = Im(Ch(Z_{k′}) → Ch(Zk̄)).

Proof. For any field extension E ⊂ k̄ of k, we write I_{E} for the image of Ch(Z_{E}) → Ch(Zk̄)). We only need to show that I_{k′} ⊂ I_{k} because, clearly, I_{k} ⊂ I_{k′}.

If G is of inner type, the variety Z is a variety of flags of right ideals of a central simple k-algebra. Therefore the group Aut(k̄/k) acts trivially on Ch(Zk̄). It follows that [k′ : k] · I_{k′} ⊂ I_{k} and therefore I_{k′} ⊂ I_{k}.

Now we assume that G is of outer type. Let K ⊂ k̄ be the separable quadratic field extension of k such that G_{K} is of inner type. Consider two subgroups Aut(k̄/K) and Aut(k̄/k′) of the group Aut(k̄/k). Acting on Ch(Zk̄), they act trivially on I_{k′}. The index of the first subgroup is 2 while the index of the second one is odd (a divisor of [k′ : k]). Indeed,

Aut(k̄/k) = Aut(k_{sep}/k), Aut(k̄/K) = Aut(k_{sep}/K),

where k_{sep} is the separable closure of k in k̄, so that Aut(k̄/k)/ Aut(k̄/K) = Aut(K/k); if k′′ is the separable closure of k in k′, then Aut(k̄/k′) = Aut(k̄/k′′), so that the index of Aut(k̄/k′) in Aut(k̄/k) is [k′′ : k].

It follows that Aut(k̄/k) acts trivially on I_{k′}. Therefore we still have the inclusion [k′ : k] · I_{k′} ⊂ I_{k} giving I_{k′} ⊂ I_{k}. □

Corollary 6.7. U(X)(n) is a motivic summand of Y.

Proof. As planned, we apply Proposition 6.2 to p = 2, 𝔛 = X, Y = Y, and E = F (Y). Since E(X) ⊂ L(Y), we have the commutative diagram

CH(Y_{E(X)}) ⟶ CH(Y_{L(Y)}) ⟶ CH(Y\overline{L(Y)})
↑  ↑
CH(Y_{F (X)}) ⟶ CH(Y_{L})

where the maps are the change of field homomorphisms and where \overline{L(Y)} is an algebraic closure of L(Y). We check condition (1) for \overline{E(𝔛)} = \overline{L(Y)}. For any field extension F ⊂ \overline{L(Y)} of F, we write I_{F} for the image of Ch(Y_{F}) → Ch(Y\overline{L(Y)}). We only need to show that I_{E(X)} ⊂ I_{F (X)}. We have I_{E(X)} ⊂ I_{L(Y)}. Since Y (L) ≠ ∅, the field extension L(Y)/L is purely transcendental. Therefore res_{L(Y)/L} is surjective and I_{L(Y)} = I_{L}. Finally, by Lemma 6.6, I_{L} = I_{F (X)}. We obtain the necessary inclusion I_{E(X)} ⊂ I_{L(Y)} = I_{L} = I_{F (X)}.

As already pointed out, condition (2) is satisfied by Lemma 6.5, and condition (3) is satisfied by Lemma 6.3. Therefore, by Proposition 6.2, a shift of U(X) is a motivic summand of Y. By Remark 6.4, it follows that the shifting number of this motivic summand U(X) is equal to n. □

As in [5] we need the following enhancement of Corollary 6.7.

Proposition 6.8. There exists a \underline{symmetric} projector π on Y such that the motive (Y, π) is isomorphic to U(X)(n).

Proof. We can follow the lines of the proof of [5, Proposition 9] if we know that the complete motivic decomposition of Y_{F (X)} could not contain two copies of 𝔽_{2}(n). This is true by Lemma 6.3 and Remark 6.4. □

The following proposition finishes the proof of Theorem 6.1.

Proposition 6.9. Let F be a field of characteristic ≠ 2. Let K/F be a separable quadratic field extension. Let D be a central division K algebra of degree 2^{r} with some r ≥ 1 admitting a K/F -unitary involution. Let X be the Weil transfer of the generalized Severi-Brauer variety X(2^{r−1}, D). Let A be a central simple K-algebra Brauer-equivalent to D endowed with a K/F -unitary involution. Let Y be the variety of isotropic rank 2^{r} right ideals in A. Assume that there is a symmetric projector π ∈ Ch_{dim Y} (Y × Y) such that the motive (Y, π) is isomorphic to a shift of the upper motive of X. Then Y has a closed point of odd degree.

Proof. By Lemma 2.10, it is enough to show that sq(π) ≠ st(π). Computing sq and st, we may go over any field extension of F. There exists a field extension F̃/F over which D is split, but K̃ := K ⊗_{F} F̃ is still a field. Since the motive of X over F̃ is a sum of shifts of the motives of Spec F̃ and Spec K̃, π decomposes in a sum of two orthogonal projectors α and β such that (YF̃, α) is a sum of shifts of the motive of Spec F̃ and (YF̃, β) is a sum of shifts of the motive of Spec K̃.

The projectors α and β are symmetric. Indeed, the subspace Ch_{∗}(YK̃, β) ⊂ Ch_{∗}(YK̃, π) coincides with

(1 + σ) Ch_{∗}(YK̃, π) = Im ( Ch_{∗}(YK̃, π) → Ch_{∗}(YF̃, π) → Ch_{∗}(YK̃, π)).

Since π is also a sum of the orthogonal projectors α^{t}, β^{t} and (XF̃, α^{t}) is still a sum of shifts of the motive of Spec F̃ and (YF̃, β^{t}) is still a sum of shifts of the motive of Spec K̃, we also have that the subspace Ch_{∗}(YK̃, β^{t}) ⊂ Ch_{∗}(YK̃, π) coincides with (1 + σ) Ch_{∗}(YK̃, π). This shows that β is symmetric. Therefore α = π − β also is symmetric.

Now we have (see Lemmas 2.2 and 2.5): sq(α) = rk_{F} (Y, π) (mod 4) = rk_{F} U(X) (mod 4) = 2 and sq(β) = 2 rk_{K}(Y, π) (mod 4) = 2 rk_{K} U(X) (mod 4) = 2 by Proposition 5.2. On the other hand, st(α) = 0. Indeed, α over F̃ is a sum of a×b with a, b ∈ Ch_{≥d}(YF̃), where d := (dim Y − dim X)/2 = (k(2n − 3k) − k^{2}/2)/2 with k := 2^{r} = ind A and n := deg A. Since d > k(n−2k), deg S(b) = 0 by Theorem 4.5. Therefore pr_{∗} S(a×b) = 0. It follows that pr_{∗}(𝔞) is divisible by 2 for an integral representative 𝔞 of S(a×b). Therefore pr_{∗}(𝔞) is divisible by 2 if now 𝔞 is an integral representative of S(α). It follows that pr_{∗}(𝔞)^{2} is divisible by 4 and consequently st(α) = 0.

Finally, let us check that st(β) = 2. The point is that β is in

(1 + σ) Ch(YK̃) = Im ( Ch(YK̃) → Ch(YF̃) → Ch(YK̃)).

Therefore β over K̃ is rational even if h is anisotropic (in which case the variety Y has no odd degree closed points). By Lemma 2.10, this shows that st(β) = sq(β).

We have calculated the values of the operations sq and st on α and β. We have by Lemmas 2.2, 2.4, and 2.9 that sq(π) = sq(α) + sq(β) = 0 and st(π) = st(α) + st(β) = 2. In particular, sq(π) ≠ st(π). □

Theorem 6.1 is proved. □

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UPMC Univ Paris 06, Institut de Mathématiques de Jussieu (boite courrier 247), 4 place Jussieu, F-75252 Paris CEDEX 05, FRANCE

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