HERMITIAN FORMS OVER QUATERNION ALGEBRAS

NIKITA A. KARPENKO AND ALEXANDER S. MERKURJEV

Abstract. We study a hermitian form h over a quaternion division algebra Q over a field (h is supposed to be alternating if the characteristic of the field is 2). For generic h and Q, for any integer i ∈ [1, n/2], where n := dim_{Q} h, we show that the variety of idimensional (over Q) totally isotropic right subspaces of h is 2-incompressible. The proof is based on a computation of the Chow ring for the classifying space of a certain parabolic subgroup in a split simple adjoint affine algebraic group of type C_{n}. As an application, we determine the smallest value of the J-invariant of a non-degenerate quadratic form divisible by a 2-fold Pfister form; we also determine the biggest values of the canonical dimensions of the orthogonal Grassmannians associated to such quadratic forms.

Contents

1. Introduction 2 2. The norm subgroup of a quaternion algebra 2 3. Projective Q-spaces 3 4. Smooth Q-fibrations 5 5. Chow rings of some classifying spaces 6 5a. Chow rings of classifying spaces 6 5b. Semidirect products 7 5c. The parabolic subgroup P 8 5d. Quaternion Azumaya algebra associated to a P -torsor 9 5e. Chow ring of BP 10 6. Generic flag variety 12 7. Projective Q-bundles for constant Q 13 8. Generic maximal Grassmannian 15 9. Essential motives of Q-Grassmannians 17 10. Generic Grassmannians 18 11. Connection with quadratic forms 19 References 21

Date: May 12, 2013. Key words and phrases. Algebraic groups, classifying spaces, hermitian and quadratic forms, projective

homogeneous varieties, Chow groups and motives. Mathematical Subject Classification (2010): 14L17; 14C25.

The first author acknowledges the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-12-BL01-0005.

The work of the second author has been supported by the NSF grant DMS #1160206.

1. Introduction

We study a hermitian form h over a quaternion division algebra Q over a field. If the characteristic of the field is 2, we additionally assume that h is alternating (as defined in [12, §4.A]). For generic h and Q (defined as in the beginning of Section 6), for any integer i ∈ [1, n/2], where n := dim_{Q} h, we show (Theorem 10.1) that the variety of i-dimensional (over Q) totally isotropic right subspaces of h is 2-incompressible (see [6] for definition). The proof is based on a computation of the Chow ring for the classifying space of a certain parabolic subgroup in a split simple adjoint affine algebraic group of type C_{n} made in Section 5e. As an application, we determine the smallest value of the Jinvariant of a non-degenerate quadratic form divisible by a 2-fold Pfister form (Corollary 11.3); we also determine the biggest values of the canonical dimensions of the orthogonal Grassmannians associated to such quadratic forms (see Corollary 11.2).

The general outline of the paper follow the pattern of [10], where hermitian forms over quadratic extension fields (in place of quaternion algebras) and quadratic forms divisible by 1-fold Pfister forms have been treated.

We write Ch for the Chow group with coefficient in the field 𝔽_{2} := ℤ/2ℤ. So, Ch = CH /2 CH, where CH is the Chow group with integer coefficients.

Variety here is just a separated scheme of finite type over a field.

We are using the Grothendieck Ch-motive and write M(X) for the motive of a smooth complete variety X. Acknowledgements. We thank Burt Totaro for useful advices. The first author gratefully acknowledges hospitality of the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario), the Department of Mathematics of University of Toronto, and Thematic Program on Torsors, Nonassociative Algebras and Cohomological Invariants (January–June 2013).

2. The norm subgroup of a quaternion algebra

A quaternion algebra Q over a variety X is a rank 4 Azumaya algebra over X. Let s : C → X be the corresponding conic over X, i.e., the Severi-Brauer scheme of Q (the scheme of right ideals in Q of reduced rank 1).

The norm subgroup N_{Q} of Q is the image of the push-forward homomorphism

s_{∗} : Ch(C) → Ch(X).

By the projection formula, N_{Q} is an ideal in Ch(X) if X is smooth. The fiber C_{x} of s over a point x ∈ X is the conic curve over the residue field F (x) corresponding to the quaternion F (x)-algebra Q_{x}. If Q_{x} is split, C_{x} has a rational point (in fact C_{x} ≃ ℙ^{1}), hence [x] ∈ N_{Q}. If Q_{x} is not split, the subgroup of CH(X) generated by s_{∗}([y]) for all y in C_{x} is equal to the subgroup generated by 2[x] = 0. It follows that N_{Q} is generated by the classes [x] of the points x ∈ X such that Q_{x} is split.

Lemma 2.1. Let Q and Q′ be two quaternion algebras over a variety X. If [Q] = [Q′] in Br(X), then N_{Q} = N_{Q′}.

Proof. For any x ∈ X, we have [Q_{x}] = [Q′_{x}] in Br F (x) and hence Q_{x} ≃ Q′_{x}. It follows from the description of N_{Q} and N_{Q′} before the lemma that N_{Q} = N_{Q′}. □

Note that if X is irreducible and smooth, [Q] = [Q′] in Br(X) if and only if the classes of the generic fibers of Q and Q′ are equal in Br F (X), see [14, Corollary IV.2.6]).

We will need the following functorial property of N_{Q}:

Lemma 2.2. Let g : X′ → X be a morphism of smooth schemes and let Q be a quaternion algebra over a scheme X. Let Q′ be the pull-back of Q with respect to g. Then the inverse image homomorphism g^{∗} : Ch(X) → Ch(X′) takes N_{Q} to N_{Q′}. In particular, g^{∗} yields a homomorphism

Ch(X)/N_{Q} ⟶ Ch(X′)/N_{Q′}.

Proof. Let s : C → X be the conic associated to Q. Then the conic curve s′ : C′ → X′ associated with Q′ is the pull-back of s and we have a commutative diagram (see [16, Proposition 12.5])

\xymatrix{Ch(C) \ar[r]^{h^{∗}} \ar[d]_{s′_{∗}} & Ch(C′) \ar[d]^{s_{∗}} \\ Ch(X) \ar[r]^{g^{∗}} & Ch(X′),}

where h : C′ → C is the induced morphism. □

3. Projective Q-spaces

Let Q be a quaternion division algebra over a field F. For any finite-dimensional right Qvector space V, the projective Q-space Qℙ(V) is defined as the F -variety of 1-dimensional subspaces in V. The dimension of subspaces here is taken over Q; the dimension over F is 4; the reduced dimension (defined as the dimension over F divided by deg Q = 2) is therefore 2. For any n ≥ 0, we define Qℙ^{n} as Qℙ(Q^{n+1}). In particular, Qℙ^{0} is the point pt = Spec F. For any n ≥ 0, Qℙ^{n} is a smooth projective F -variety of dimension 4n.

Of course, the variety Qℙ(V) (and therefore Qℙ^{n}) can be defined for an arbitrary (nonsplit or split) quaternion algebra Q as the variety of Q-submodules in V (resp., in Q^{n+1}) of reduced dimension 2 (where V is a right Q-module of even reduced dimension). Then Qℙ(V)_{L} = Q_{L}ℙ(V_{L}) and Qℙ_{L}^{n} = Q_{L}ℙ^{n} for any extension field L/F. Once Q is split, an identification of Q with the matrix algebra M_{2}(F) identifies Qℙ^{n} with the Grassmannian of 2-planes in the 2(n + 1)-dimensional F -vector space F^{2(n+1)}. Our main case of interest, however, is the case where Q is a division algebra.

The following statement is true for Q non-split or split but only needs to be proved in the split case:

Lemma 3.1. Let a ∈ Ch^{4}(Qℙ^{n}) be the 4-th Chern class of the (rank 4) tautological vector bundle on Qℙ^{n}. Then deg(a^{n}) = 1.

Proof. Replacing F by a field extension splitting Q, we may assume that Q itself is split. In this case there is an isomorphism of Qℙ^{n} with the Grassmannian X of 2-planes in F^{2(n+1)} such that the tautological bundle on Qℙ^{n} corresponds to a vector bundle on X isomorphic to T ⊕ T , where T is the tautological bundle on X. It follows that the element a corresponds to c_{2}^{2}(T ). By [4, Proposition 14.6.5], c_{2}^{2n}(T ) is a generator of the group Ch_{0}(X). Therefore deg(a^{n}) = deg(c_{2}^{2n}(T )) = 1. □

Lemma 3.2. The Ch-motive M(Qℙ^{n}) decomposes in a direct sum of three summands: M(pt), ⨁_{i=1}^{2n} M(C)(i), and M(Qℙ^{n−1})(4).

Proof. This is a particular case of [5, Theorem 10.9 and Corollary 10.19], where the characteristic of the base field is assumed to be different from 2. We refer to [1] for a characteristic-free treatment. □

Corollary 3.3. The motive of Qℙ^{n} decomposes in a direct sum of two summands: the first one is ⨁_{i=0}^{n} M(pt)(4i) and the second one is a direct sum of shifts of M(C). □

Since the motive of the product C × C is isomorphic to the sum M(C) ⊕ M(C)(1), we get

Corollary 3.4. For any integers n_{1}, …, n_{r} ≥ 0, the motive of the product

P := Qℙ^{n_{1}} × ⋯ × Qℙ^{n_{r}}

decomposes in a direct sum of (n_{1} + 1) … (n_{r} + 1) shifts of M(pt) and several shifts of M(C).

For an arbitrary smooth F -variety X, let us consider the (constant) Azumaya algebra Q_{X} over X given by the pull-back of Q. The norm subgroup N_{Q_{X}} is defined in Section 2 as the image of the push-forward homomorphism Ch(C_{X}) → Ch(X) with respect to the projection C_{X} := X × C → X. (Note that C_{X} is the conic over X associated with the quaternion algebra Q_{X}.)

If Q is split, the quotient Ch(X)/N_{Q_{X}} is 0. For a non-split Q we, for instance, have Ch(pt)/N_{Q} = 𝔽_{2} := ℤ/2ℤ and Ch(C)/N_{Q_{C}} = 0.

Proposition 3.5. Assume that Q is division algebra. For P as in Corollary 3.4, let

a_{1}, …, a_{r} ∈ Ch(P)/N_{Q_{P}}

be the elements given by the 4-th Chern classes of the tautological vector bundles of the factors of P. The 𝔽_{2}-algebra Ch(P)/N_{Q_{P}} is generated by these elements subject to the relations a_{1}^{n_{1}+1} = 0, …, a_{r}^{n_{r}+1} = 0.

Proof. Since the degree of any closed point on C is divisible by 2, the degree of any closed point on P × C is also divisible by 2 so that the degree homomorphism deg : Ch(P) → 𝔽_{2} is defined on the quotient Ch(P)/N_{Q_{P}}.

The monomials a_{1}^{i_{1}} … a_{r}^{i_{r}} with 0 ≤ i_{1} ≤ n_{1}, …, 0 ≤ i_{r} ≤ n_{r}, are linearly independent. Indeed, if a linear combination α of the monomials with coefficients α_{i_{1}...i_{r}} ∈ 𝔽_{2} is 0, then

0 = deg(α · a_{1}^{n_{1}−i_{1}} … a_{r}^{n_{r}−i_{r}}) = α_{i_{1}...i_{r}}

for every i_{1}, …, i_{r}.

It follows that the dimension (over 𝔽_{2}) of the 𝔽_{2}-subalgebra in Ch(P)/N_{Q_{P}} generated by a_{1}, …, a_{r} is at least the product (n_{1} + 1) … (n_{r} + 1). Since this product is equal to dim_{𝔽_{2}} Ch(P)/N_{Q_{P}} (see Corollary 3.4), the 𝔽_{2}-algebra Ch(P)/N_{Q_{P}} is generated by a_{1}, …, a_{r}.

The elements a_{1}, …, a_{r} satisfy the indicated relations simply by dimension reason: a_{i} is given by the pull-back of an element in Ch^{4}(Qℙ^{n_{i}}) and dim Qℙ^{n_{i}} = 4n_{i}. The resulting 𝔽_{2}-algebra epimorphism

𝔽_{2}[t_{1}, …, t_{r}]/(t_{1}^{n_{1}+1}, …, t_{r}^{n_{r}+1}), t_{i} ↦ a_{i}

is an isomorphism by dimension reason once again. □

4. Smooth Q-fibrations

The following lemma is a distant descendant of [21, Statement 2.13]:

Lemma 4.1. Let f : X′ → X be a smooth morphism of smooth F -varieties, Q a quaternion algebra on X, Q′ = f^{∗}Q and r an integer. Let B ⊂ Ch(X′) be a homogeneous Ch(X)-submodule containing N_{Q′}. Suppose that for any integer i and any point x ∈ X of codimension i such that the restriction of Q on x is not split, the composition

(4.2)  B^{r−i} ↪ Ch^{r−i}(X′) → Ch^{r−i}(X′_{x}) → Ch^{r−i}(X′_{x})/N_{Q′_{x}}^{r−i},

where Q′_{x} is the restriction of Q′ on the fiber X′_{x} of f over x, is surjective. Then B^{r} = Ch^{r}(X′).

Proof. First of all we notice that Ch(X′_{x})/N_{Q′_{x}} = 0 if the restriction of Q on x is split. Therefore the composition (4.2) is surjective for any point x ∈ X of codimension i.

For any integer i, we write F^{i} Ch^{r}(X′) for the subgroup in Ch^{r}(X′) generated by the classes of cycles on X′ whose images in X have codimension ≥ i; these are terms of a descending ring filtration on Ch^{r}(X′). For any point x ∈ X of codimension i, we have a homomorphism Ch^{r−i}(X′_{x}) → F^{i} Ch^{r}(X′)/F^{i+1} Ch^{r}(X′) mapping the class of a point x′ ∈ X′_{x} to the the class modulo F^{i+1} Ch^{r}(X′) of the class of x′ considered as a point of X′. The composition

Ch^{r−i}(X′) → Ch^{r−i}(X′_{x}) → F^{i} Ch^{r}(X′)/F^{i+1} Ch^{r}(X′)

is the multiplication by [x] ∈ Ch^{i}(X). The sum ⨁_{x} Ch^{r−i}(X′_{x}) over all points x ∈ X of codimension i surjects onto the quotient F^{i} Ch^{r}(X′)/F^{i+1} Ch^{r}(X′).

Let α ∈ Ch^{r}(X′). Inducting on i, we will show that α ∈ B^{r} + F^{i} Ch^{r}(X′) for any i. With a sufficiently large i this will give the required statement.

The case of i ≤ 0 being trivial, we assume that α ∈ B^{r} + F^{i} Ch^{r}(X′) for some i ≥ 0, and we show that α ∈ B^{r} + F^{i+1} Ch^{r}(X′). We write α as a sum of an element of B^{r} and some β ∈ F^{i} Ch^{r}(X′)

The class of β modulo F^{i+1} Ch^{r}(X′) decomposes into a sum of elements of two kinds. An element of the first kind is in the image of the composition

B^{r−i} ↪ Ch^{r−i}(X′) → Ch^{r−i}(X′_{x}) → F^{i} Ch^{r}(X′)/F^{i+1} Ch^{r}(X′)

for some x, and therefore is represented by an element of B^{r−i} · [x] ⊂ B^{r}.

An element of the second kind is in the image of the composition

N_{Q′_{x}}^{r−i} ↪ Ch^{r−i}(X′_{x}) → F^{i} Ch^{r}(X′)/F^{i+1} Ch^{r}(X′).

We claim that any element of this image is represented by an element of N_{Q′}^{r}. Since N_{Q′}^{r} ⊂ B^{r}, Lemma 4.1 is proved with this claim.

To prove the claim, we recall that N_{Q′_{x}}^{r−i} is generated by [x′] with x′ ∈ X′_{x} of codimension r − i such that the restriction of Q′_{x} on x′ is split. The image of such a generator in F^{i} Ch^{r}(X′)/F^{i+1} Ch^{r}(X′) is represented by the class [x′] ∈ Ch(X′) of x′ viewed as a point of X′. Since the restriction of Q′_{x} on x′ ∈ X′_{x} coincides with the restriction of Q′ on x′ ∈ X′, the class [x′] ∈ Ch^{r}(X′) is in N_{Q′}^{r} ⊂ Ch^{r}(X′). □

Example 4.3. Let V be a right Q-module of even reduced rank 2(n + 1). The corresponding projective Q-bundle Qℙ(V) is then defined as the X-scheme of Q-submodules in V of reduced rank 2. The structure morphism Qℙ(V) → X is smooth and proper; its fiber over a point x ∈ X is the projective Q_{x}-space Q_{x}ℙ(V_{x}) defined in the previous section. It follows by Propositions 3.5 and Lemma 4.1 that the Ch(X)/N_{Q}-algebra Ch(Qℙ(V))/N_{Q} is generated by the 4-th Chern class a of the tautological vector bundle on Qℙ(V). More precisely, the Ch(X)/N_{Q}-module Ch(Qℙ(V))/N_{Q} is generated by the powers a^{i} with i = 0, …, n.

5. Chow rings of some classifying spaces

5a. Chow rings of classifying spaces. Let G be an algebraic group over a field F. Write BG for the classifying space of G viewed as the stack of G-torsors over the category of F -varieties (see [22]).

In [19], Totaro defined the Chow ring CH(BG) as follows. Let V be a generically free linear representation of G over F and U ⊂ V a G-invariant open subscheme admitting a G-torsor U → U/G for a variety U/G. For any integer i ≥ 0, the Chow group CH^{i}(U/G) does not depend (up to canonical isomorphism) on the choice of V and U provided that codim_{V} (V \ U) > i. We write CH^{i}(BG) for CH^{i}(U/G) and refer to U/G as to an i-th approximation of the classifying space BG of G.

By [19, Theorem 1.3], the group CH^{i}(BG) is naturally identified with the set of functorial assignments α to every smooth variety X over F with a G-torsor E over X of an element α(E) ∈ CH^{i}(X).

Let E → pt be a G-torsor and^{E}G := Aut_{G}(E) the twist of G by E. The correspondence I ↦ Iso_{G}(I, E) gives rise to an equivalence between BG and B(^{E}G). In particular, there is a natural ring isomorphism between CH(BG) and CH(B^{E}G).

Example 5.1. The ring CH(B GL_{n}) is the polynomial ring on the Chern classes

c_{1}, c_{2}, …, c_{n}

of the tautological vector bundle on B GL_{n} (see [19]). A representation ρ : G → GL_{n} yields the pull-back homomorphism ρ^{∗} : CH(B GL_{n}) → CH(BG). The elements ρ^{∗}(c_{i}) are the Chern classes of the pull-back of the tautological vector bundle under

Bρ : BG → B GL_{n} .

Example 5.2. Let A be a central simple algebra of degree n over F and G = GL_{1}(A). For every N > 0, the group G acts on the open subvariety U_{N} of nondegenerate elements in the free A-module A^{N}. Then BG is approximated by the Grassmannian varieties Aℙ^{N−1} := U_{N}/G of A-submodules in A^{N} of reduced dimension n. We write BG = Aℙ^{∞} (an infinite “projective space” over A). The left multiplication action of A on itself yields a representation GL_{1}(A) → GL(A) = GL_{n^{2}}. The pull-back of the tautological vector bundle on B GL_{n^{2}} is the tautological vector bundle on BG = Aℙ^{∞}.

Example 5.3. The projective linear group PGL_{2} is the automorphism group of the matrix algebra M_{2}. The twisted forms of M_{2} are the quaternion algebras. It follows that every PGL_{2}-torsor over a scheme X is isomorphic to the torsor of isomorphisms between a unique (up to canonical isomorphism) quaternion algebra Q over X and the matrix algebra M_{2}. The algebra Q carries a canonical (symplectic) involution τ. The kernel Q^{0} of the endomorphism 1 + τ on Q is a sub-bundle of Q of rank 3. The natural homomorphism

PGL_{2} = Aut(M_{2}) → Aut((M_{2})^{0}) = O_{3}^{+}

is an isomorphism between the group PGL_{2} of Dynkin type A_{1} and the split special orthogonal group O_{3}^{+} of type C_{1} (see [12, §15]). It is proved in [19, §16] (see also [15]) that if char(F) ≠ 2, then the ring CH(B PGL_{2}) = CH(B O_{3}^{+}) is generated by the Chern classes c_{2}(Q^{0}) and c_{3}(Q^{0}) with the relation 2c_{3}(Q^{0}) = 0. If char(F) = 2, the result (and the proof) still holds. Moreover, in this case, Q^{0} contains the trivial line sub-bundle generated by 1, hence c_{3}(Q^{0}) = 0.

Let V be a generically free representation of G and let U/G be an approximation of BG as above. The morphism U/G → BG induced by the versal G-torsor U → U/G yields a ring homomorphism CH(BG) → CH(U/G).

Lemma 5.4. The ring homomorphism CH(BG) → CH(U/G) is surjective.

Proof. Suppose that an algebraic group G acts on a variety X over F. The G-equivariant Chow group CH^{G}(X) was defined in [2]. In particular, CH^{G}(pt) = CH(BG). The homomorphism CH(BG) → CH(U/G) coincides with the composition

CH(BG) = CH^{G}(pt) ⟶^{α} CH^{G}(V) ⟶^{β} CH^{G}(U) = CH(U/G),

where the pull-back homomorphism α is an isomorphism by the homotopy invariance property and the restriction β is surjective by localization. □

5b. Semidirect products. Let an algebraic group K over F act on another algebraic group H by group automorphisms (so that we can form a semidirect product H ⋊ K). For a K-torsor E over pt we can twist H by E. The resulting group is denoted by ^{E}H.

Proposition 5.5. Let S = H ⋊ K be a semidirect product of algebraic groups over F, f : S → K the corresponding split surjective homomorphism. Let U/S and W/K be i-th approximations of BS and BK respectively, and let E → Spec F be the fiber of W → W/K over a point x ∈ (W/K)(F). Then (U × W)/S is an i-th approximation of BS and the fiber of the natural morphism (U × W)/S → W/K over x is an i-th approximation of B(^{E}H), where ^{E}H is the twist of H by E.

Proof. Write E_{sep} = K_{sep}w for w in W_{sep}. Then the isomorphism

U_{sep}/H_{sep} ⟶ p^{−1}(x)_{sep}, H_{sep}v ↦ K_{sep}(v, w)

over F_{sep} descents to an isomorphism between ^{E}U/^{E}H and p^{−1}(x) over F. Note that the variety ^{E}U/^{E}H is an i-th approximation of B(^{E}H). □

5c. The parabolic subgroup P. Let G̃ = Sp_{2n} be the symplectic group of the alternating form on the 2n-dimensional vector space V with a symplectic basis {v_{i}, w_{i}}, i = 1, 2, …, n. Write T for the maximal torus T = (G_{m})^{n} acting on the basis vector by tv_{i} = t_{i}v_{i} and tw_{i} = t_{i}^{−1}w_{i}. Write {e_{i}} for the standard basis of the character group T^{∗} = ℤ^{n}. The simple roots of G̃ are α_{1} = e_{1} − e_{2}, α_{2} = e_{2} − e_{3}, …, α_{n−1} = e_{n−1} − e_{n}, α_{n} = 2e_{n} (see [12, §24]).

Consider the subset Λ = {α_{1}, α_{3}, … } of all odd simple roots. Let S̃ be a (reductive) subgroup of G̃ generated by the torus T and the root subgroups U_{α} with α in the root system ±Λ ≃ A_{1} + ⋯ + A_{1}. The lattice T^{∗} splits accordingly into a direct sum of rank 2 lattices ℤe_{2i−1} ⊕ ℤe_{2i} for i = 1, 2, …, m := [n/2] (and ℤe_{n} if n is odd). It follows that

(5.6)  S̃ ≃ \{ \begin{matrix}(GL_{2})^{m}, & if n is even (n = 2m); \\ (GL_{2})^{m} × SL_{2}, & if n is odd (n = 2m + 1). \end{matrix}

The group S̃ is the Levi subgroup of the parabolic subgroup P̃ of G̃ corresponding to the set of simple roots Λ. The group P̃ is the stabilizer of the flag

{0} = W_{0} ⊂ W_{1} ⊂ W_{2} ⊂ ⋯ ⊂ W_{m} ⊂ V

of totally isotropic subspaces W_{i} := span(v_{1}, v_{2}, …, v_{2i}). The projection of S̃ onto the i-th component GL_{2} in (5.6), i = 1, …, m, is given by the action on the factor space W_{i}/W_{i−1}, i.e., coincides with S̃ → GL(W_{i}/W_{i−1}) = GL_{2}.

The projective symplectic group G = PGSp_{2n} is the factor group of G̃ by μ_{2}. Write P for the parabolic subgroup P̃/μ_{2} in G. The Levi subgroup S of P is S̃/μ_{2} with the μ_{2} embedded diagonally into the product of GL_{2} and SL_{2} with respect to the decomposition (5.6). By the above, we have

S = \{ \begin{matrix}(GL_{2})^{m}/μ_{2}, & if n is even; \\ ((GL_{2})^{m} × SL_{2})/μ_{2}, & if n is odd. \end{matrix}

Let

H := \{ \begin{matrix}(GL_{2})^{m−1} × G_{m}, & if n is even; \\ (GL_{2})^{m}, & if n is odd. \end{matrix}

We view H as a subgroup of S via the map

(s_{1}, s_{2}, …, s_{m−1}, λ^{2}) ↦ (s_{1}λ, s_{2}λ, …, s_{m−1}λ, λ)μ_{2}

if n is even and

(s_{1}, s_{2}, …, s_{m}) ↦ (s_{1}, s_{2}, …, s_{m}, 1)μ_{2}

if n is odd.

We also view the group K := PGL_{2} as a subgroup of P embedded diagonally. Note that S coincides with the the semidirect product H ⋊ K and K acts on H by component-wise conjugation.

Consider the representations

ρ_{i} : S → GL_{4}

defined by

ρ_{i}((a_{1}, a_{2}, … a_{m})μ_{2}) = a_{i} ⊗ a_{m}, i = 1, 2, …, m − 1,

if n is even and by

ρ_{i}((a_{1}, a_{2}, … a_{m+1})μ_{2}) = a_{i} ⊗ a_{m+1}, i = 1, 2, …, m,

is n is odd.

A quaternion algebra Q over F can be viewed as a K-torsor. Twisting by Q the composition of the embedding of the i-th component GL_{2} ↪ H and ρ_{i} restricted to H, we get a natural representation

GL_{1}(Q) → GL_{4} .

If n is even, write τ : S → G_{m} for the homomorphism

τ((a_{1}, a_{2}, … a_{m})μ_{2}) = det(a_{m}).

5d. Quaternion Azumaya algebra associated to a P -torsor. Let G = PGSp_{2n} be a split adjoint group G of type C_{n}. There is a natural embedding of G into PGL_{2n}. Therefore, for every G-torsor h : E → X we have associated a PGL_{2n}-torsor, i.e., an Azumaya algebra A(h) over X of degree 2n.

Let P be the parabolic subgroup of G introduced in Section 5c. We have the composition P → S → K = PGL_{2}. Therefore, for every P -torsor f : I → X we have associated a PGL_{2}-torsor, i.e., a quaternion Azumaya algebra Q = Q(f) over X.

Every P -torsor f : I → X yields a G-torsor res_{G/P} (f) := G ×_{P} I → X.

Lemma 5.7. For a P -torsor f : I → X, we have [A(res_{G/P} (f))] = [Q(f)] in Br(X).

Proof. Consider the group G′ = GSp_{2n} of symplectic similitudes (see [12, §12]). The group G is the factor group of G′ by the center G_{m} of scalar matrices. Let P′ the inverse image of P in G′. By Section 5c, the diagram

[figure: G_{m}; G_{m}; G_{m}; G_{m}; _{GL_{2}}; P′; α; G′; _{GL_{2n}}; _{PGL_{2}}; P; β; G; PGL_{2n},]

where α is the composition P′ → S′ → GL(W_{1}) = GL_{2} and β is the composition P → S = H ⋊ K → K = PGL_{2}, is commutative. It follows that the diagram

[figure: H_{ét}^{1}(X, P); H_{ét}^{1}(X, PGL_{2}); Br(X); H_{ét}^{1}(X, G); H_{ét}^{1}(X, PGL_{2n}); Br(X)]

is also commutative, whence the result. □

5e. Chow ring of BP.

Lemma 5.8. Let P be a parabolic subgroup in a semisimple group G defined over F, R the unipotent radical of P and S a Levi subgroup of P. Then

(1) the map H^{1}(F, S) → H^{1}(F, P) is a bijection;

(2) the map A^{p}(Spec F, K_{q}) → A^{p}(R, K_{q}) of K-cohomology groups is an isomorphism for all p and q.

Proof. (1) We have P = R ⋊ S, hence the map s : H^{1}(F, P) → H^{1}(F, S) is split surjective. By [12, Proposition 28.11], for any cocycle ξ ∈ Z^{1}(F, S) there is a surjection from H^{1}(F,^{ξ}R) to the fiber of s over ξ. The group ^{ξ}R is the unipotent radical of the parabolic subgroup ^{ξ}P of the semisimple group ^{ξ}G. Hence by [18, Proposition. 16.1.1], ^{ξ}R is split over F, therefore, H^{1}(F,^{ξ}R) = 1. It follows that s is a bijection.

(2) As R is split, there is a sequence of normal subgroups 1 = R_{0} ⊂ R_{1} ⊂ ⋯ ⊂ R_{n} = U such that R_{i+1}/R_{i} is isomorphism to the additive group G_{a} for all i. Hence every fiber of the natural morphism R/R_{i} → R/R_{i+1} is isomorphic to an affine line. By homotopy invariance, the map A^{p}(R/R_{i+1}, K_{q}) → A^{p}(R/R_{i}, K_{q}) is an isomorphism for all p and q. □

Proposition 5.9. Let P be a parabolic subgroup in a semisimple group G defined over F and S a Levi subgroup of P. Then the natural map CH(BS) → CH(BP) is an isomorphism.

Proof. By Lemma 5.8(1), every fiber of (an approximation of) the natural morphism BP → BS over a point ξ has a rational point and hence is isomorphic to ^{ξ}(P/S) ≃ ^{ξ}R, where R is the unipotent radical of P. The result follows from Lemma 5.8(2) applied to the parabolic subgroup ^{ξ}P of the semisimple group ^{ξ}G and [3, Proposition 52.10]. □

Let P be the parabolic subgroup in the split adjoint group of type C_{n} introduced in Section 5c, S the Levi subgroup of P, K and H as in Section 5c.

A point x of BK over a field L is a K-torsor over L, i.e., a quaternion algebra Q_{x} over L. Let H_{x} be the twist of H by x, i.e.,

H_{x} = \{ \begin{matrix}GL_{1}(Q_{x})^{m−1} × G_{m}, & if n is even; \\ GL_{1}(Q_{x})^{m}, & if n is odd, \end{matrix}

Determine the space BH_{x}. By Example 5.2, B GL_{1}(Q_{x}) = Q_{x}ℙ^{∞}. In follows that

(5.10)  BH_{x} = \{ \begin{matrix}B GL_{1}(Q_{x})^{m−1} × B G_{m} = ℙ^{∞} × (Q_{x}ℙ^{∞})^{m−1}, & if n is even; \\ B GL_{1}(Q_{x})^{m} = (Q_{x}ℙ^{∞})^{m}, & if n is odd. \end{matrix}

Let Q′ be the pull-back to BS of the tautological quaternion algebra over BK. Let J be the following subset of Ch(BS) (see Section 5c):

J = \{ \begin{matrix}\{ρ_{1}^{∗}(c_{4}), ρ_{2}^{∗}(c_{4}), …, ρ_{m−1}^{∗}(c_{4}), τ^{∗}(c_{1})\}, & if n is even; \\ \{ρ_{1}^{∗}(c_{4}), ρ_{2}^{∗}(c_{4}), …, ρ_{m}^{∗}(c_{4})\}, & if n is odd. \end{matrix}

Theorem 5.11. The Ch(BK)-algebra Ch(BS) is generated by J ∪ N_{Q′}.

Proof. Let r > 0 be an integer and choose the r-th approximations

f : Y := (U × W)/S → W/K =: X

of the morphism BS → BK as in Proposition 5.5.

We apply Lemma 4.1 to the Ch(X)-submodule B of Ch(Y) generated by J ∪ N_{Q′}. Let x be a point in X of codimension i such that the restriction Q_{x} of Q on x is not split. By Proposition 5.5, the fiber of f over x is an r-th approximation of BH_{x} as in (5.10).

By Proposition 3.5, the ring Ch(BH_{x})/N_{Q_{x}} is generated by the first Chern class of the (line) tautological bundle on ℙ^{∞} and by the 4-th Chern classes of the (rank 4) tautological bundles on the m − 1 factors Qℙ^{∞} if n is even. If n is odd, the ring Ch(BH_{x})/N_{Q_{x}} is generated by the 4-th Chern classes of the tautological bundles on the m factors Qℙ^{∞}. By Example 5.2 and the end of Section 5c, these Chern classes are the restrictions of the Chern classes in J. □

By Example 5.3, the ring CH(BK) is generated by two elements: one in codimension 2 and the other one is 2-torsion in codimension 3.

Let Q be the quaternion algebra over BP associated to a P -torsor U → U/P approximating BP. By definition, Q is the pull-back of Q′ under BP → BS. By Proposition 5.9, the natural map CH(BS) → CH(BP) is an isomorphism. We have proved

Corollary 5.12. The ring Ch(BP)/N_{Q} is generated by a set of elements of degree at most 4 (with all elements of degree 3 in the set being represented by 2-torsion elements in CH^{3}(BP)). □

Let U → U/P be a P -torsor approximating BP with the associated quaternion algebra Q(U). By Lemma 5.4, the natural ring homomorphism CH(BP) → CH(U/P) is surjective. The induced surjection Ch(BP) → Ch(U/P) takes N_{Q} to N_{Q(U)} (see Section 2). We have proved

Corollary 5.13. Let U → U/P be a P -torsor with U/P approximating BP with the associated quaternion algebra Q(U). Then the ring Ch(U/P)/N_{Q(U)} is generated by a set of elements of degree at most 4 (with all elements of degree 3 in the set being represented by 2-torsion elements in CH^{3}(U/P)). □

Consider a G-torsor U → U/G with U/G approximating BG. The generic fiber f : E → Spec(K), where K = F (U/G) is the associated generic G-torsor. Let L/K be a field extension. The fiber of g : E → E/P =: X over an L-point of X is a P -torsor h : I → Spec(L) over L. Clearly, res_{G/P} (h) = f_{L}. Therefore, [Q(h)] = [A(f)_{L}] in Br(L) by Lemma 5.7.

Now take L = F (X). We have the following two quaternion Azumaya algebras over X_{L}. One is Q(g_{L}) for the P -torsor g_{L} : E_{L} → X_{L}. Another one is the constant algebra coming from the L-algebra Q(h).

We claim that the pull-backs to L(X) = L(X_{L}) of both algebras are Brauer equivalent (and hence isomorphic). We see that the two pull-backs are obtained from each other by the automorphism of the field L(X) induced by the exchange automorphism of X × X. Since the Brauer class of Q(h) comes from the field F, the exchange automorphisms acts identically on Q(h) proving the claim.

As a consequence of the claim, the norm subgroups N in Ch(X_{L}) for both quaternion algebras are the same by Lemma 2.1.

Corollary 5.14. Let f : E → Spec K be a generic G-torsor and let L be the function field of X := E/P. Then the image of the natural ring homomorphism Ch(X) → Ch(X_{L}) is generated by elements of codimensions 1, 2, 4 and the norm subgroup N_{Q} ⊂ Ch(X_{L}) of the constant quaternion algebra Q on X_{L} Brauer equivalent to A(f) lifted to X_{L}.

Proof. Recall that E is the generic fiber of U → U/G for an appropriate U, hence X is a localization of U/P. By the localization property for Chow groups, the pull-back ring homomorphism Ch(U/P) → Ch(X) is surjective. By Lemma 2.2 and the discussion before the theorem, it takes the standard norm group in Ch(U/P) to the norm group N_{Q}. The result follows from Corollary 5.13 and from the fact that the (integral) Chow group CH(X_{L}) is torsion free as the (integral) CH-motive of the projective homogeneous variety X_{L} is a sum of shifts of several copies of the motives of the point and of a conic (see [5] and [1]). □

6. Generic flag variety

Let k be a field, n an integer ≥ 2, F the field of rational functions in n + 2 variables t, t′, t_{1}, …, t_{n}, Q the quaternion (division) F -algebra given by the elements t, t′, and h the hermitian form ⟨t_{1}, …, t_{n}⟩ on the right Q-module Q^{n}. Note that in the characteristic 2 case, the hermitian form h is alternating (as defined in [12, §4.A]). By [12, Theorem 4.2], this means that the adjoint to h involution on the matrix algebra M_{n}(Q) is symplectic (in any characteristic).

The pair Q, h is generic in the following sense: there exist a smooth connected k-variety X (namely, the affine space of dimension n + 2), a quaternion algebra Q̃ over X and a hermitian form h̃ on Q̃^{n} such that:

(1) the pair Q, h is Q̃, h̃ restricted to the generic point of X and

(2) for any quaternion algebra Q′ over an extension field F′/k with a non-degenerate hermitian form h′ on Q′ (alternating in characteristic 2), there exists an F′-point of X such that Q′, h′ is isomorphic to the restriction of Q̃, h̃ to the point.

Indeed, h′ can be diagonalized and the diagonal entries are elements of Q which are symmetric (alternating in characteristic 2) with respect to the canonical symplectic involution on Q. It follows by [12, Proposition 2.6] that the diagonal entries are elements of F′.

A different construction of such a generic pair occurs in the proof of the following

Proposition 6.1. Let Y be the F -variety of flags of totally isotropic subspaces in Q^{n} of Q-dimensions 1, 2, …, [n/2] (i.e., of reduced dimensions 2, 4, …, 2[n/2]). Then the ring \overline{Ch}(Y)/\overline{N}_{Q_{Y}} is generated by codimension 1, 2, and 4.

Proof. Let P be the parabolic subgroup of the split simple adjoint affine algebraic group G of type C_{n} over the field k considered (P and G) in Section 5c. Let E be a generic G-torsor over an extension field K/k, and let us consider the projective homogeneous K-variety Y′ := E/P. The torsor E is given by a central simple K-algebra A of degree 2n with a symplectic involution σ. The variety Y′ is isomorphic to the variety of flags of σ-isotropic right ideals in A of reduced dimensions 2, 4, 6, …, 2[n/2]. There is a quaternion algebra Q′ over Y′ Brauer-equivalent to A_{Y′}.

Let X be the generalized Severi-Brauer variety SB_{2}(A). By Example 4.3, the Ch(Y′)- algebra Ch(Y′ ×_{K} X)/N_{Q′_{Y′×X}} is generated by an element of codimension 4. It follows by Corollary 5.14 that the ring \overline{Ch}(Y′_{K(X)})/\overline{N}_{Q′_{Y{′ \atop K(X)}}} is generated by codimensions 1, 2, 4.

By a specialization argument similar to that used in [10, Proof of Corollary 4.8], the ring \overline{Ch}(Y′_{K(X)})/\overline{N}_{Q′_{Y{′ \atop K(X)}}} is isomorphic to the ring \overline{Ch}(Y)/\overline{N}_{Q_{Y}}. □

7. Projective Q-bundles for constant Q

In this section, Q is a quaternion algebra over F, X is a smooth F -variety, V is a right Q_{X}-module of reduced dimension 2(n + 1).

The following statement generalizes Lemma 3.1:

Lemma 7.1. As usual, let a be the 4-th Chern class of the tautological vector bundle on Qℙ(V). Let π be the structure morphism Qℙ(V) → X. Then π_{X∗}(a^{i}) = 0 for any i = 0, …, n − 1 and π_{X∗}(a^{n}) = [X].

Proof. For i < n, we have π_{X∗}(a^{i}) = 0 by dimension reason. The remaining formula π_{X∗}(a^{n}) = [X] may be checked over an extension of F so that we may assume Q is split. In this case Qℙ(V) is identified with the Grassmannian of 2-planes in a rank 2(n+1) vector bundle over X the way that a = c_{2}^{2}, where c_{2} is the 2-nd Chern class of the tautological vector bundle on the Grassmannian. The desired formula becomes a particular case of Duality Theorem [4, 14.6.3]. □

Proposition 7.2. For a Q-module V of even reduced rank 2(n + 1), the Ch(X)/N_{Q_{X}}- module Ch(Qℙ(V))/N_{Q_{Qℙ(V )}} is free with the basis {a^{i}}_{i=0}^{n}.

Proof. We know already by Example 4.3 that the system {a^{i}}_{i=0}^{n} generates the module. It remains to check that it is free.

Assuming that α := ∑_{i=0}^{n} α_{i}a^{i} ∈ N_{Q_{Qℙ(V )}} for some α_{0}, …, α_{n} ∈ Ch(X), we show that all α_{i} are in N_{Q_{X}} using a descending induction on i. Let i be the biggest index for which we didn’t prove α_{i} ∈ N_{Q_{X}} yet. Calculating π_{X∗}(α · a^{n−i}) ∈ N_{Q_{X}} using Lemma 7.1, we get that α_{i} ∈ N_{Q_{X}}. □

Now we are going to look at the reduced Chow ring \overline{Ch}(X) defined as the quotient of Ch(X) by the ideal of the elements vanishing over an extension field of F. Note that for any extension field L/F, the change of field homomorphism \overline{Ch}(X) → \overline{Ch}(X_{L}) is injective. We write \overline{N}_{Q_{X}} for the image of N_{Q_{X}} in \overline{Ch}(X).

Proposition 7.3. The element a (more precisely, its class) in the \overline{Ch}(X)/\overline{N}_{Q_{X}}-algebra \overline{Ch}(Qℙ(V))/\overline{N}_{Q_{Qℙ(V )}} satisfies the relation

∑_{i=0}^{n+1} c_{4i}a^{n+1−i} = 0,

where c_{i} := c_{i}(V) is the i-th Chern class of the vector bundle V.

Proof. The tautological vector bundle T on Y := Qℙ(V) is a subbundle of the vector bundle V_{Y}. The exact sequence of vector bundles

0 ⟶ T ⟶ V_{Y} ⟶ V_{Y} /T ⟶ 0

gives the relation c(T )c(V_{Y} /T ) = c(V) of the total Chern classes. In particular,

(7.4)  a_{4} + a_{3}b_{1} + a_{2}b_{2} + a_{1}b_{3} + b_{4} = c_{4}(V) ; a_{4}b_{4} + a_{3}b_{5} + a_{2}b_{6} + a_{1}b_{7} + b_{8} = c_{8}(V) ; ⋮ ; a_{4}b_{4n−4} + a_{3}b_{4n−3} + a_{2}b_{4n−2} + a_{1}b_{4n−1} + b_{4n} = c_{4n}(V) ; a_{4}b_{4n} = c_{4n+4}(V),

where a_{i} := c_{i}(T ) and b_{i} := c_{i}(V_{Y} /T ).

We claim that the classes of a_{1}, a_{2}, a_{3} in \overline{Ch}(Y)/\overline{N}_{Q_{Y}} are zero. Indeed, let L/F be an extension field splitting Q. Choosing a simple left module M over the split quaternion L-algebra Q_{L} and defining T′ as the tensor product of T_{L} and M over Q_{L}, we get an isomorphism T_{L} ≃ T′ ⊕ T′ of vector bundles over Y_{L}. It follows that the images of a_{1} and a_{3} in Ch(Y_{L}) are 0. Therefore a_{1} and a_{3} are zero already in \overline{Ch}(Y) – before factorization by \overline{N}_{Q_{Y}}.

It remains to deal with the image of a_{2} in Ch(Y_{L}) which is equal to c_{1}^{2}(T′). It suffices to prove the following

Lemma 7.5. The element c_{1}^{2}(T′) is in the image of the composition Ch(Y × C) → Ch(Y) → Ch(Y_{L}).

Proof. The following square commutes:

Ch(Y × C) ⟶ Ch(Y × C)_{L}
↓  ↓
Ch(Y) ⟶ Ch(Y_{L}).

Let C be the tautological vector bundle on C. This is a right Q-module, but we may view it as a left Q-module via the canonical involution on Q in order to take the tensor product F of T_{Y ×C} and C_{Y ×C} over Q_{Y ×C}. Pulling-back the (Y × C)-vector bundle F to (Y × C)_{L} = Y_{L} × C_{L}, we get a vector bundle isomorphic to the tensor product of T′ and C′, where C′ is the (rank 1) vector bundle on C_{L} defined as the tensor product over Q_{L} of C and M.

Since C′ is a line bundle and T′ is a vector bundle of rank 2, we have

c_{2}(T′ ⊗ C′) = c_{2}(T′) × [C_{L}] + c_{1}(T′) × c_{1}(C′) + [Y_{L}] × c_{1}^{2}(C′).

Note that the last summand is 0 by dimension reason and that deg(C′) = 1. Therefore the image of c_{2}(T′ ⊗ C′) under the push-forward to Ch(Y_{L}) is c_{1}(T′) showing that c_{1}(T′) is in the image of the composition Ch(Y × C) → Ch(Y) → Ch(Y_{L}). Therefore c_{1}^{2}(T′) is also in the image of the composition. □

Turning back to the proof of Proposition 7.3 and passing from Ch(Y) to \overline{Ch}(Y)/\overline{N}_{Q_{Y}} in the relations (7.4), we get the following simpler relations

a_{4} + b_{4} = c_{4}(V)
a_{4}b_{4} + b_{8} = c_{8}(V)
⋮
a_{4}b_{4n−4} + b_{4n} = c_{4n}(V)
a_{4}b_{4n} = c_{4n+4}(V).

Starting with the last relation and sequentially excluding b_{4i} for i = n, n − 1, … with the help of the previous relations, we get the relation desired. □

Corollary 7.6. The \overline{Ch}(X)/\overline{N}_{Q_{X}}-algebra \overline{Ch}(Qℙ(V))/\overline{N}_{Q_{Qℙ(V )}} is generated by the element a subject to one relation

∑_{i=0}^{n+1} c_{4i}a^{n+1−i} = 0,

where c_{i} := c_{i}(V) is the i-th Chern class of the vector bundle V.

Proof. Let A be an \overline{Ch}(X)/\overline{N}_{Q_{X}}-algebra generated by one element t subject to one relation

∑_{i=0}^{n+1} c_{4i}t^{n+1−i} = 0.

The \overline{Ch}(X)/\overline{N}_{Q_{X}}-algebra homomorphism A → \overline{Ch}(Qℙ(V))/\overline{N}_{Q_{Qℙ(V )}}, t ↦ a is well defined by Proposition 7.3 and surjective by Proposition 7.2. Since both algebras, considered as \overline{Ch}(X)/\overline{N}_{Q_{X}}-modules, are free of rank n + 1 (the right one is so by Proposition 7.2 once again), the epimorphism is an isomorphism. □

8. Generic maximal Grassmannian

We use the settings of the beginning of Section 6. Let m := [n/2]. We consider the F -variety X of m-dimensional totally isotropic subspaces in h.

Proposition 8.1. The components of positive codimension of the ring \overline{Ch}(X)/\overline{N}_{Q_{X}} are trivial.

Proof. Let I be the set of integers [1, m] := {1, 2, …, m}. For any subset J ⊂ I we consider the variety X_{J} of flags of totally isotropic subspaces in h of Q-dimensions given by J.

By induction on l ∈ I, we prove the following statement: the ring \overline{Ch}(X_{[l, m]})/\overline{N}_{Q_{X_{[l, m]}}} is generated by codimensions 1, 2, 4 and the Chern classes of the tautological rank 4l vector bundle T_{l} on X_{[l, m]} (which is the pull-back to X_{[l, m]} of the tautological vector bundle on X_{l}). This statement with l = m gives the required statement of Proposition 8.1 due to the following

Lemma 8.2. Let F be a field, Q a quaternion division F -algebra, h a hermitian form on Q^{n} which is hyperbolic for even n and almost hyperbolic for odd n, X the corresponding maximal Grassmannian. Then all the elements of codimensions 1, 2, 4 in the ring Ch(X)/N_{Q_{X}} as well as the elements given by the Chern classes of positive codimensions of the tautological vector bundle on X are 0.

Proof. It follows by [5, Theorem 15.8 and Corollary 15.14] and [1] that the motive of X decomposes in a direct sum with one summand M(pt), one summand M(pt)(3), and every of the remaining summands being either M(pt)(i) with i > 4 or a shift of M(C). Therefore all elements of codimensions 1, 2, 4 in the ring Ch(X)/N_{Q_{X}} are 0, and it remains to prove the statement about the Chern classes of the tautological bundle.

The ring Ch(X) imbeds into Ch(X̄) (see, e.g., Corollary 9.2). The variety X̄ is identified with the Grassmannian of 2m-planes in a 2n-dimensional vector space V which are totally isotropic with respect to a fixed non-degenerate alternating form on V. The tautological bundle on X gives rise to a vector bundle on X̄ isomorphic to T ⊕ T , where T is the tautological bundle on X̄.

Let us consider the case of even n first. We claim that in this case the Chern classes of the tautological bundle on X are trivial already in Ch(X̄). Indeed, there is an exact sequence

0 ⟶ T ⟶ VX̄ ⟶ T^{∗} ⟶ 0

relating the bundle T with its dual T^{∗} and the trivial vector bundle VX̄ (where the epimorphism VX̄ → T^{∗} is induced by the alternating form). Since c(T ) = c(T^{∗}) ∈ Ch(X̄), it follows that c(T ⊕ T ) = 1.

In the case of odd n, there is an exact sequence

0 ⟶ T^{⊥} ⟶ VX̄ ⟶ T^{∗} ⟶ 0,

where T^{⊥} is the orthogonal complement of T in VX̄. The bundle T^{⊥} contains T as a subbundle, the quotient T^{⊥}/T is of rank 2. We have

1 = c(T^{⊥})c(T^{∗}) = c(T )^{2}c(T^{⊥}/T ) ∈ Ch(X̄).

Multiplying by c(T^{⊥}/T ), we get that c(T^{⊥}/T ) ∈ Ch(X). Passing to the quotient by N_{Q_{X}}, we see that c(T^{⊥}/T ) = 1 ∈ Ch(X)/N_{Q_{X}} (because T^{⊥}/T is of rank 2 and the ring Ch(X)/N_{Q_{X}} has no non-zero elements in codimensions 1 and 2). Therefore c(T )^{2} = 1 ∈ Ch(X)/N_{Q_{X}}. □

We turn back to the inductive proof of the statement formulated in the beginning of the proof of Proposition 8.1. The induction base l = 1 follows from Proposition 6.1. Now, assuming that l ≥ 2, let us do the passage from l − 1 to l.

The projection X_{[l−1, m]} → X_{[l, m]} is the projective Q-bundle given by the dual of the vector Q-bundle T_{l}. Therefore, by Corollary 7.6, the \overline{Ch}(X_{[l, m]})/\overline{N}_{Q_{X_{[l, m]}}}-algebra

\overline{Ch}(X_{[l−1, m]})/\overline{N}_{Q_{X_{[l−1, m]}}} is generated by certain codimension 4 element a subject to one

relation ∑_{i=0}^{l} c_{4i}a^{l−i} = 0, where c_{i} := c_{i}(T_{l}). In particular, the \overline{Ch}(X_{[l, m]})/\overline{N}_{Q_{X_{[l, m]}}}-

module \overline{Ch}(X_{[l−1, m]})/\overline{N}_{Q_{X_{[l−1, m]}}} is free of rank l.

Now let C ⊂ \overline{Ch}(X_{[l, m]})/\overline{N}_{Q_{X_{[l, m]}}} be the subring generated by all c_{i} together with the elements of codimensions 1, 2, 4. The coefficients of the above relation are then in C. Therefore the subring of \overline{Ch}(X_{[l−1, m]})/\overline{N}_{Q_{X_{[l−1, m]}}} generated by C and a is also free (now as a C-module) of rank l. On the other hand, this subring coincides with the total ring by the induction hypothesis. Indeed, it contains all the elements of codimension 1, 2, 4 in \overline{Ch}(X_{[l−1, m]})/\overline{N}_{Q_{X_{[l−1, m]}}} because any such element is either equal to a or lies in the image

of \overline{Ch}(X_{[l, m]})/\overline{N}_{Q_{X_{[l, m]}}}. It also contains the Chern classes of the vector bundle T_{l−1} on X_{[l−1, m]} because these Chern classes are polynomials in c_{i} := c_{i}(T_{l}) and a := c_{4}(T_{l}/T_{l−1}) (we recall that the 1-st, 2-nd and 3-d Chern classes of the quotient T_{l}/T_{l−1} are trivial in \overline{Ch}(X_{[l−1, m]})/\overline{N}_{Q_{X_{[l−1, m]}}} as shown in the proof of Proposition 7.3).

It follows that C = \overline{Ch}(X_{[l, m]})/\overline{N}_{Q_{X_{[l, m]}}}. □

9. Essential motives of Q-Grassmannians

We fix the following notation. Let F be a field (of arbitrary characteristic). Let Q be a quaternion division F -algebra and C the corresponding conic. Let n be an integer ≥ 0. Let V be a right vector space over Q of dimension n. Let h be a non-degenerate hermitian (with respect to the canonical involution of Q) form on V. If charF = 2, we additionally assume that h is alternating. For any integer r, let X_{r} be the F -variety of totally isotropic subspaces in V of Q-dimension r (so that X_{0} = Spec F and X_{r} = ∅ for r outside of the interval [0, n/2]).

Lemma 9.1 ([5, Theorem 15.8 and Corollary 15.14] and [1]). Assume that the hermitian form h is isotropic: n is ≥ 2 and h ≃ ℍ⊥h′, where ℍ is the hyperbolic plane, h′ a hermitian form of dimension n − 2. For any integer r one has

M(X_{r}) ≃ M(X′_{r−1}) ⊕ M(X′_{r})(i) ⊕ M(X′_{r−1})(j) ⊕ M,

where X′_{r−1} and X′_{r} are the varieties of h′, i = (dim X_{r} − dim X′_{r})/2, j = dim X_{r} − dim X′_{r−1}, and M is a sum of shifts of the motive of C.

The following Corollary is also a consequence of a general result of [7] or of [1]:

Corollary 9.2. If h is split (meaning hyperbolic for even n or “almost hyperbolic” for odd n), then M(X_{r}) is a sum of shifts of M(pt) and of M(C). □

Corollary 9.3. There is a decomposition M(X_{r}) ≃ M_{r} ⊕ M such that the motive M is a sum of shifts of M(C) and for any field extension L/F with split h_{L} the motive M_{r} is split (meaning is a sum of Tate motives).

Proof. Apply [8, Proposition 4.1] inductively to E := F (X_{[n/2]}) and S := C. Note that the variety S_{E} is still irreducible and has indecomposable motive because the quaternion E-algebra Q ⊗_{F} E is non-split (see, e.g., [13]). Therefore Condition (1) of [8, Proposition 4.1] is satisfied.

Since X_{[n/2]}(F (C)) ≠ ∅, the field extension E(S)/F (S) is purely transcendental, that is, Condition (2) is satisfied as well.

Clearly, the hermitian form h_{E} is split so that the motive of X_{r} over E is a sum of shifts of M(pt) and of M(C) (Corollary 9.2). The inductive application of [8, Proposition 4.1] shows that the sum M of all copies of shifts of M(C) present in the complete decomposition of M(X_{r}) over E, can be extracted from M(X_{r}) over F. The complementary summand M_{r} of the motive of X_{r} has the desired property. □

Remark 9.4. The reduced Chow group (homological or cohomological one) of the motive M (as a subgroup of \overline{Ch}(X_{r})) is equal to \overline{N}_{Q_{Xr}} (it is evidently contained in \overline{N}_{Q_{Xr}} and coincides in fact with \overline{N}_{Q_{Xr}} because \overline{N}_{Q_{Xr}} intersects the reduced Chow group of M_{r} trivially). Therefore the reduced Chow group of M_{r} is identified with the quotient \overline{Ch}(X_{r})/\overline{N}_{Q_{Xr}}. In the case where h is hyperbolic or almost hyperbolic, the reduced Chow group in the above statements can be replaced by the usual Chow group. We refer to [3, §64] for the definition of homological and cohomological Chow group of a motive. The coincidence of descriptions of homological and cohomological Chow groups for the motives M and M_{r} is explained by their symmetry: M ≃ M^{∗}(dim X_{r}) (and the same for M_{r}), where M^{∗} is the dual motive, [3, §65].

Definition 9.5. The motive M_{r} (defined by X_{r} uniquely up to an isomorphism) will be called the essential motive of X_{r} (or the essential part of the motive of X_{r}).

It follows that the decomposition of the essential motive in the isotropic case has precisely the same shape as the decomposition of the motive of an isotropic orthogonal Grassmannian [9, Decomposition 2.6]:

Corollary 9.6. Under the hypotheses of Lemma 9.1, one has

M_{r} ≃ M′_{r−1} ⊕ M′_{r}(i) ⊕ M′_{r−1}(j),

where M′_{r−1} and M′_{r} are the essential motives of X′_{r−1} and X′_{r}, i = (dim X_{r} − dim X′_{r})/2, j = dim X_{r} − dim X′_{r−1}. □

According to the general result of [7], any summand of the complete motivic decomposition of the variety X_{r} is a shift of the upper motive U(X_{s}) for some s ≥ r or a shift of M(C). Therefore we get

Corollary 9.7. Any summand of the complete decomposition of the essential motive M_{r} is a shift of the upper motive U(X_{s}) for some s ≥ r. □

Remark 9.8. A motive is split if it is isomorphic to a finite direct sum of Tate motives. A motive is geometrically split if it becomes split over an extension of the base field. Dimension dim P of a geometrically split motive P is the maximum of the distance |i − j| between i and j running over the integers such that the Tate motives M(pt)(i) and M(pt)(j) are direct summands of P_{L}, where L/F is a field extension splitting N.

Since the quaternion F -algebra Q remains non-split over the function field L := F (X_{[n/2]}) and the motive of (X_{r})_{L} contains the Tate summands 𝔽_{2} and 𝔽_{2}(dim X_{r}), these Tate motives are summands of (M_{r})_{L}. It follows that dim M_{r} = dim X_{r}.

10. Generic Grassmannians

In the statements below we use the notion of the essential motive M_{r} of the variety X_{r}, introduced in the previous section. It turns out that in the generic case, this motive is indecomposable:

Theorem 10.1. For F, Q, n, and h as in the beginning of Section 6, for any r = 0, 1, …, [n/2], the essential motive M_{r} of the variety X_{r} is indecomposable, the variety X_{r} is 2-incompressible.

Proof. We induct on n in the proof of the first statement. The induction base is the trivial case of n < 2. Now we assume that n ≥ 2.

We do a descending induction on r. The case of the maximal r = [n/2] is an immediate consequence of Proposition 8.1. Indeed, one summand of the complete decomposition of the motive M_{r} for such r is the upper motive U(X_{r}) of X_{r}. The remaining summands (if any) are positive shifts U(X_{r})(i) (i > 0) of the upper motive, see Corollary 9.7. But if we have a summand U(X_{r})(i), then the reduced Chow group \overline{Ch}^{i}(M_{r}) is non-zero. However by Remark 9.4, \overline{Ch}(M_{r}) is isomorphic to \overline{Ch}(X_{r})/\overline{N}_{Q_{Xr}} which is 0 in positive codimensions by Proposition 8.1.

Now we assume that r < [n/2]. Since the case of r = 0 is trivial, we may assume that r ≥ 1 (and therefore n ≥ 4).

Let L := F (X_{1}). We have h_{L} ≃ ℍ⊥h′, where h′ is a hermitian form of dimension n − 2 and ℍ is the hyperbolic plane.

For any integer s, we write X′_{s} for the variety X_{s} of the hermitian form h′, and we write M′_{s} for the essential motive of the variety X′_{s}. By Corollary 9.6, the motive (M_{r})_{L} decomposes in a sum of three summands:

(10.2)  (M_{r})_{L} ≃ M′_{r−1} ⊕ M′_{r}(i) ⊕ M′_{r−1}(j),

where i := (dim X_{r} − dim X′_{r})/2 and j := dim X_{r} − dim X′_{r−1}. Let us check that each of three summands of decomposition (10.2) is indecomposable.

Let F′ be the function field of the variety of totally isotropic subspaces of Q-dimension 1 of the hermitian form ⟨t_{n−1}, t_{n}⟩. Then h_{F′} ≃ ℍ⊥ ⟨t_{1}, …, t_{n−2}⟩ so that we have a motivic decomposition similar to (10.2) where each of the three summands is indecomposable by the induction hypothesis. Since the field extension F′(X_{1})/F′ is purely transcendental, the complete decomposition of (M_{r})_{F′(X_{1})} has only three summands. Since L = F (X_{1}) ⊂ F′(X_{1}), the complete decomposition of (M_{r})_{L} has at most three summands so that the summands of decomposition (10.2) are indecomposable.

It follows by [9, Proposition 2.4]) that if the motive M_{r} is decomposable (over F), then it has a summand P with P_{L} ≃ M′_{r}(i) = U(X′_{r})(i). Note that U(X′_{r}) ≃ U((X_{r+1})_{L}). Again by [9, Proposition 2.4], P ≃ U(X_{r+1})(i), showing that U(X_{r+1})_{L} ≃ M′_{r}. By the induction hypothesis, the motive M_{r+1} is indecomposable, that is, U(X_{r+1}) = M_{r+1}. Therefore we have an isomorphism (M_{r+1})_{L} ≃ M′_{r} and, in particular, dim X_{r+1} = dim X′_{r} (see Remark 9.8). However dim X_{r+1} = (r + 1)(4n − 6(r + 1) + 1), dim X′_{r} = r(4(n − 2) − 6r + 1), and the difference is 4n − 4r − 5 > 2n − 5 > 0 (recall that n ≥ 4 now).

To show that X_{r} is 2-incompressible, we show that its canonical 2-dimension cd_{2} X_{r} equals dim X_{r}. By [6, Theorem 5.1], cd_{2} X_{r} = dim U(X_{r}). By the first part of Theorem 10.1, U(X_{r}) = M_{r}. Finally, dim M_{r} = dim X_{r} (see Remark 9.8). □

11. Connection with quadratic forms

Let F, Q, V, and h be as in the beginning of Section 9. For any v ∈ V the value h(v, v) is in F and the map q : V → F, v ↦ h(v, v) is a non-degenerate quadratic form on V considered this time as a vector space over F. Note that the dimension of q, that is, the dimension of V over F is the dimension n of V over Q multiplied by 4. Moreover, q is isomorphic to the tensor product of the 2-fold Pfister quadratic form Nrd_{Q} given by the reduced norm of Q by an n-dimensional non-degenerate symmetric bilinear form. Note that an arbitrary anisotropic 2-fold Pfister form over F is isomorphic to Nrd_{Q} for a unique up to an isomorphism quaternion division F -algebra Q ([3, Corollary 2.15]). Any non-degenerate quadratic form divisible by Nrd_{Q} arises the way described above from an appropriate hermitian form over Q (unique up to an isomorphism). For the case of characteristic not 2, we may refer to [17, §1 of Chapter 10].

The Witt indexes i(h) and i(q) of h and q are related as follows (and this relationship implies the above uniqueness statement, c.f. [10, Corollary 9.2], known as Jacobson Theorem):

Lemma 11.1. i(q) = 4i(h).

Proof. For any integer r ≥ 0, the inequality i(h) ≥ r implies i(q) ≥ 4r. Indeed, if i(h) ≥ r, V contains a totally h-isotropic Q-subspace W of dimension r. This W is also totally q-isotropic and has dimension 4r over F. Therefore i(q) ≥ 4r.

To finish, we prove by induction on r ≥ 0 that i(q) ≥ 4r − 3 implies i(h) ≥ r. This is trivial for r = 0. If r > 0 and i(q) ≥ 4r − 3, then q is isotropic. But any q-isotropic vector is also h isotropic, therefore the Q-vector space V decomposes in a direct sum of h-orthogonal subspaces V = U ⊕ V′ such that h|_{U} is a hyperbolic plane. The subspaces U and V′ are also q-orthogonal and q|_{U} is hyperbolic (of dimension 4). For h′ := h|_{V′} and q′ := q|_{V′} it follows that i(h′) = i(h) − 1 and i(q′) = i(q) − 4, and we are done by the induction hypothesis applied to h′ (of course, q′ is the quadratic form given by h′). □

For any integer r, let X_{r} be the variety of totally h-isotropic r-dimensional Q-subspaces in V and let Y_{r} be the variety of totally q-isotropic r-dimensional F -subspaces in V. The variety Y_{2n}, where n := dim_{Q} V, is not connected and has two isomorphic connected components; changing notation, we let Y_{2n} be one of its connected component in this case.

Corollary 11.2. For any r, the upper motives of the varieties X_{r}, Y_{4r}, Y_{4r−1}, Y_{4r−2}, and Y_{4r−3} are isomorphic. In particular, these varieties have the same canonical 2-dimension. This canonical dimension is maximal and equal to

dim X_{r} = r(4n − 6r + 1)

in the case of generic Q and h.

Proof. By Lemma 11.1, each of the three varieties possesses a rational map to each other. Therefore the upper motives are isomorphic by [11, Corollary 2.15]. For the first statement on canonical dimension see [6, Theorem 5.1]. The statement on the maximal canonical dimension follows from Theorem 10.1. □

Let us recall that according to the original definition [20, Definition 5.11(2)] due to A. Vishik of the J-invariant J(q) of a non-degenerate quadratic form q over F of dimension 4n, J(q) is a certain subset of the set of integers {0, 1, …, 2n − 1}. Note that in [3, §88], the name J-invariant and the notation J(q) stand for the complement of the above subset (with the “excuse” that this choice simplifies several formulas involving the J-invariant). In the present paper we are using the original definition and notation.

Theorem 10.1 with Corollary 11.2 make it possible to compute for any n the smallest (in the sense of inclusion) value of the J-invariant of a quadratic form given by tensor product an n-dimensional non-degenerate symmetric bilinear form by a 2-fold quadratic Pfister form. Only the case of even n is of interest, because the J-invariant is {2, 3, …, 2n − 1} (everything but 0 and 1) for odd n, if the 2-fold Pfister form is anisotropic.

Corollary 11.3. For any even n ≥ 2, the smallest value of J-invariant discussed right above is the set of the integers in the interval [0, 2n − 1] which are not congruent to 3 modulo 4.

Proof. If a quadratic form q over a field F is given by tensor product of an n-dimensional non-degenerate bilinear form by a 2-fold quadratic Pfister form, the Witt index of q over any extension field of F is divisible by 4. It follows by [3, Proposition 88.8] that the J-invariant of q contains the set indicated.

Let now Q and h be as in the beginning of Section 6. We calculate the J-invariant of the quadratic form q associated with h. Note that q is divisible by the 2-fold Pfister form Nrd_{Q}. We are using the above notation for the varieties associated to h and to q.

By [3, Theorem 90.3], the canonical 2-dimension of Y_{2n} is dim Y_{2n} = n(2n − 1) minus the sum of the elements of the J-invariant. On the other hand, by Corollary 11.2 and Theorem 10.1, the canonical 2-dimension of Y_{2n} is equal to the dimension of X_{n/2} which is

dim X_{n/2} = n(n + 1)/2 = n(2n − 1) − ∑_{j∈[0, 2n−1], j≢3 (mod 4)} j.

Therefore the J-invariant is equal to the set indicated. □

Lemma 11.1 and Corollaries 11.2 and 11.3 are analogues of [10, Lemma 9.1 and Corollaries 9.3 and 9.4]. The reader may discover on his own the analogues of the remaining statements of [10, §9].

References

[1] Chernousov, V., Gille, S., and Merkurjev, A. Motivic decomposition of isotropic projective homogeneous varieties. Duke Math. J. 126, 1 (2005), 137–159.

[2] Edidin, D., and Graham, W. Equivariant intersection theory. Invent. Math. 131, 3 (1998), 595– 634.

[3] Elman, R., Karpenko, N., and Merkurjev, A. The algebraic and geometric theory of quadratic forms, vol. 56 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2008.

[4] Fulton, W. Intersection theory, second ed., vol. 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 1998.

[5] Karpenko, N. A. Cohomology of relative cellular spaces and of isotropic flag varieties. Algebra i Analiz 12, 1 (2000), 3–69.

[6] Karpenko, N. A. Canonical dimension. In Proceedings of the International Congress of Mathematicians. Volume II (New Delhi, 2010), Hindustan Book Agency, pp. 146–161.

[7] Karpenko, N. A. Upper motives of outer algebraic groups. In Quadratic forms, linear algebraic groups, and cohomology, vol. 18 of Dev. Math. Springer, New York, 2010, pp. 249–258.

[8] Karpenko, N. A. Hyperbolicity of unitary involutions. Sci. China Math. 55, 5 (2012), 937–945.

[9] Karpenko, N. A. Sufficiently generic orthogonal Grassmannians. J. Algebra 372 (2012), 365–375.

[10] Karpenko, N. A. Unitary Grassmannians. J. Pure Appl. Algebra 216, 12 (2012), 2586–2600.

[11] Karpenko, N. A. Upper motives of algebraic groups and incompressibility of Severi-Brauer varieties. J. Reine Angew. Math. (Ahead of Print), doi: 10.1515/crelle.2012.011.

[12] Knus, M.-A., Merkurjev, A., Rost, M., and Tignol, J.-P. The book of involutions, vol. 44 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits.

[13] Merkurjev, A. S., Panin, I. A., and Wadsworth, A. R. Index reduction formulas for twisted flag varieties. I. K-Theory 10, 6 (1996), 517–596.

[14] Milne, J. S. Étale cohomology, vol. 33 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1980.

[15] Molina Rojas, L. A., and Vistoli, A. On the Chow rings of classifying spaces for classical groups. Rend. Sem. Mat. Univ. Padova 116 (2006), 271–298.

[16] Rost, M. Chow groups with coefficients. Doc. Math. 1 (1996), No. 16, 319–393 (electronic).

[17] Scharlau, W. Quadratic and Hermitian forms, vol. 270 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1985.

[18] Springer, T. A. Linear algebraic groups, second ed. Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston, MA, 2009.

[19] Totaro, B. The Chow ring of a classifying space. In Algebraic K-theory (Seattle, WA, 1997), vol. 67 of Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 1999, pp. 249–281.

[20] Vishik, A. On the Chow groups of quadratic Grassmannians. Doc. Math. 10 (2005), 111–130 (electronic).

[21] Vishik, A. Fields of u-invariant 2^{r} + 1. In Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, vol. 270 of Progr. Math. Birkhäuser Boston Inc., Boston, MA, 2009, pp. 661–685.

[22] Vistoli, A. Grothendieck topologies, fibered categories and descent theory. In Fundamental algebraic geometry, vol. 123 of Math. Surveys Monogr. Amer. Math. Soc., Providence, RI, 2005, pp. 1– 104.

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

E-mail address: karpenko at math.jussieu.fr

Department of Mathematics, University of California, Los Angeles, CA, USA

E-mail address: merkurev at math.ucla.edu