Chow motives of twisted flag varieties

B.Calmès, V.Petrov, N.Semenov, K.Zainoulline

May 11, 2005

Abstract

Let G be an adjoint simple algebraic group of inner type. We express the Chow motive (with integral coefficients) of some anisotropic projective G-homogeneous varieties in terms of motives of simpler G-homogeneous varieties, namely, those that correspond to maximal parabolic subgroups of G. We decompose the motive of a generalized Severi-Brauer variety SB_{2}(A), where A is a division algebra of degree 5, into a direct sum of two indecomposable motives. As an application we provide another counter-example to the uniqueness of a direct sum decomposition in the category of motives with integral coefficients.

Keywords: anisotropic homogeneous variety, twisted flag variety, Chow motive, generalized Severi-Brauer variety

1 Introduction

Let G be an adjoint simple algebraic group of inner type over a field k. Let X be a twisted flag variety, i.e., a projective G-homogeneous variety over k. The main purpose of the paper is to express the Chow motive of X in terms of motives of “minimal” flags, i.e., those G-homogeneous varieties that correspond to maximal parabolic subgroups of G.

Observe that the motive of an isotropic G-homogeneous variety can be decomposed in terms of motives of simpler G-homogeneous varieties using the techniques developed by Chernousov, Gille, Merkurjev [CGM] and Karpenko [Ka01]. For G-varieties, when G is isotropic, one obtains a similar decomposition following the arguments of Brosnan [Br03]. In the case of G-varieties, where G is anisotropic, no general decomposition methods are known except several particular cases of quadrics (see for example Rost [Ro98]) Severi-Brauer varieties (see Karpenko [Ka95]) and exceptional varieties of type F_{4} (see [NSZ]).

In the present paper we provide methods that allow to decompose the motives of some anisotropic twisted flag G-varieties, where the root system of G is of types A_{n}, B_{n}, C_{n}, G_{2} and F_{4}, i.e., has a Dynkin diagram which does not branch.

As an application, we provide another counter-example to the uniqueness of a direct sum decomposition in the category of Chow motives with integral coefficients (see 2.6). Observe that such a counter-example was already constructed by Chernousov and Merkurjev (see [CM04, Example 9.4]) and is given by a G-homogeneous variety, where G is a product of two simple groups. Our example is given by a G-variety, where G is a simple group. Apart from this, our example involves an (indecomposable) object in the category of integral Chow motives which is isomorphic over ℤ[\frac{1}{2}] and ℤ[\frac{1}{3}] to the motive of the Severi-Brauer variety of a central simple division algebra of degree 5, but not integrally.

The paper is organized as follows. In section 2 we state the main results. We then provide some technical facts that are extensively used in the proofs (section 3). In the other sections we give proofs of the results for varieties of type A_{n} (section 4), of types B_{n} and C_{n} (section 6), and exceptional varieties of types G_{2} and F_{4} (section 7). Section 5 is devoted to the motivic decomposition of generalized Severi-Brauer varieties.

Notation and Conventions By G we denote an adjoint simple algebraic group of inner type over k and by n its rank. All varieties that appear in the paper are projective G-homogeneous varieties over k. They can be considered as twisted forms of the varieties G/P, where P is a parabolic subgroup of G. The Chow motive of a variety X is denoted by M(X). By A we denote a central simple algebra over k of index ind(A) and by SB(A) the corresponding Severi-Brauer variety. I is always a right ideal of A and rdim I stands for its reduced dimension. V is a vector space over k.

By λ = (λ_{1}, λ_{2}, …, λ_{l}) we denote a partition λ_{1} ≥ λ_{2} ≥ … ≥ λ_{l} ≥ 0 with |λ| = λ_{1} + λ_{2} + … + λ_{l}. Integers d_{1}, d_{2}, …, d_{k} always satisfy the condition 1 ≤ d_{1} < d_{2} < … < d_{k} ≤ n and are the dimensions of some flag. For each i = 0, …, k we define δ_{i} to be the difference d_{i+1} − d_{i} (assuming here d_{0} = 0 and d_{k+1} = n + 1).

2 Statements of Results

We follow [MPW96, Appendix] and [CG05] for the description of projective G-homogeneous varieties that appear below. According to the type of the group G, we obtain the following results.

A_{n}: In this case G = PGL_{1}(A), where A is a central simple algebra of degree n + 1, n > 0, and the set of points of a projective G-homogeneous variety X can be identified with the set of flags of (right) ideals

X(d_{1}, …, d_{k}) = {I_{1} ⊂ I_{2} ⊂ … ⊂ I_{k} ⊂ A}

of fixed reduced dimensions 1 ≤ d_{1} < d_{2} < … < d_{k} ≤ n. Observe that this variety is a twisted form of G/P, where P is the standard parabolic subgroup corresponding to the simple roots on the Dynkin diagram, numbered by d_{i}.

∘_{1}  ∘_{2}  ∘_{3}  ∘_{n−2} ∘_{n−1} ∘_{n}

The following result reduces the computation of the motive of X to the motives of “smaller” flags.

2.1 Theorem. Suppose that gcd(ind(A), d_{1}, …, d̂_{m}, …, d_{k}) = 1, then

M(X(d_{1}, …, d_{k})) ≃ ⨁_{λ} M(X(d_{1}, …, d̂_{m}, …, d_{k}))(δ_{m}δ_{m−1} − |λ|),

where the sum is taken over all partitions λ = (λ_{1}, …, λ_{δ_{m−1}}) such that δ_{m} ≥ λ_{1} ≥ … ≥ λ_{δ_{m−1}} ≥ 0.

Proof. See 4.8.

As a consequence, for the variety of complete flags we obtain

2.2 Corollary. The motive of the variety X = X(1, …, n) of complete flags is isomorphic to

M(X) ≃ ⨁_{i=0}^{n(n−1)/2} M(SB(A))(i)^{⊕a_{i}},

where the a_{i} are the coefficients of the polynomial φ_{n}(z) = ∑_{i} a_{i}z^{i} = ∏_{k=2}^{n} \frac{z^{k}−1}{z−1}.

Proof. Apply Theorem 2.1 recursively to the sequence of varieties X(1, …, n), X(1, …, n − 1), …, X(1, 2) and X(1) = SB(A).

Another interesting example is the “incidence” variety X(1, n):

2.3 Corollary. The motive of X(1, n) is isomorphic to

M(X(1, n)) ≃ ⨁_{i=0}^{n−1} M(SB(A))(i).

In order to complete the picture we need to know how to decompose the motive of a “minimal” flag, i.e., a generalized Severi-Brauer variety.

Note that for some rings of coefficients (fields, discrete valuation rings) one easily obtains the desired decomposition by using Krull-Schmidt Theorem (the uniqueness of a direct sum decomposition). More precisely, consider the subcategory M(G, R) of the category of motives with coefficients in a ring R that is the pseudo-abelian completion of the category of motives of projective G-homogeneous varieties (see [CM04, section 8]). Then we have the following

2.4 Proposition. Let X(d) = SB_{d}(A), 1 < d < n, be a generalized Severi-Brauer variety for a central simple algebra A of degree n + 1 such that gcd(ind(A), d) = 1. Let R be a ring such that Krull-Schmidt Theorem holds in the category M(G, R). Then the motive of SB_{d}(A) with coefficients in R is isomorphic to

M(SB_{d}(A)) ≃ ⨁_{i∈I} M(SB(A))(i)^{⊕a_{i}},

where the integers a_{i} are the coefficients of the polynomial \frac{φ_{n}(z)}{φ_{d}(z)φ_{n+1−d}(z)} at terms z^{i} and the set of indices I = {i | a_{i} ≠ 0}.

Proof. See 4.10.

It turns out that the motives of some generalized Severi-Brauer varieties with integral coefficients can still be decomposed but in a slightly unexpected way.

2.5 Theorem. Let SB_{2}(A) be a generalized Severi-Brauer variety for a division algebra of degree 5. Then there is an isomorphism

M(SB_{2}(A)) ≃ H ⊕ H(2),

where the motive H i) taken with R-coefficients, where 2 or 3 is invertible in a ring R, becomes isomorphic to the motive of SB(A); ii) is indecomposable over ℤ; iii) is not isomorphic to the motive of SB(A) over ℤ.

Proof. See 5.11.

As an immediate consequence of Theorems 2.1 and 2.5 we obtain

2.6 Corollary. The Krull-Schmidt Theorem fails in the category of motives M(PGL_{1}(A), ℤ) where A is a division algebra of degree 5.

Proof. Apply Theorem 2.1 recursively to the sequences of varieties X(1, 2), X(1) and X(1, 2), X(2), where X(1, 2) is the twisted flag G-variety for G = PGL_{1}(A). We obtain two decompositions of the motive of X(1, 2)

⨁_{i=0}^{3} M(SB(A))(i) ≃ M(X(1, 2)) ≃ M(SB_{2}(A)) ⊕ M(SB_{2}(A))(1).

Apply now Theorem 2.5 to the components of the second decomposition. We obtain two different decompositions of the motive M(X(1, 2)) into indecomposable objects

⨁_{i=0}^{3} M(SB(A))(i) ≃ M(X(1, 2)) ≃ ⨁_{i=0}^{3} H(i).

This finishes the proof of the corollary.

2.7 Remark. Observe that the counter-example provided by Chernousov and Merkurjev (see [CM04, Ex.9.4]) is the product of two Severi-Brauer varieties X = SB(A)×SB(B) which is a G-homogeneous variety for the semisimple group G = PGL_{1}(A) × PGL_{1}(B), where A and B are two division algebras satisfying some conditions. The example that we provide, i.e., the flag X(1, 2), is a G-homogeneous variety for the simple group G = PGL_{1}(A).

B_{n}: We assume that the characteristic of the base field k is different from 2. It is known that G = O^{+}(V, q), where (V, q) is a regular quadratic space of dimension 2n + 1, n > 0, and projective G-homogeneous varieties can be described as flags of totally q-isotropic subspaces

X(d_{1}, …, d_{k}) = {V_{1} ⊂ … ⊂ V_{k} ⊂ V }.

of fixed dimensions 1 ≤ d_{1} < … < d_{k} ≤ n. Observe that this variety is a twisted form of G/P, where P is the standard parabolic subgroup corresponding to the simple roots on the Dynkin diagram, numbered by d_{i}.

∘_{1}  ∘_{2}  ∘_{3}  ∘_{n−2} ∘_{n−1} ^{>} ∘_{n}

The following result shows that some motives of flag varieties can be decomposed into a direct sum of twisted motives of “smaller” flags.

2.8 Theorem. Suppose that m < k, then

M(X(d_{1}, …, d_{k})) ≃ ⨁_{λ} M(X(d_{1}, …, d̂_{m}, …, d_{k}))(δ_{m}δ_{m−1} − |λ|),

where the sum is taken over all partitions λ = (λ_{1}, …, λ_{δ_{m−1}}) such that δ_{m} ≥ λ_{1} ≥ … ≥ λ_{δ_{m−1}} ≥ 0.

Proof. See 6.5.

In particular, for the variety of complete flags we obtain a formula similar to the one of Corollary 2.2.

2.9 Corollary. The motive of the variety of complete flags X = X(1, 2, …, n) is isomorphic to

M(X) ≃ ⨁_{i=0}^{n(n−1)/2} M(X(n))(i)^{⊕a_{i}},

where the a_{i} are the coefficients of the polynomial φ_{n}(z) = ∑_{i} a_{i}z^{i} = ∏_{k=2}^{n} \frac{z^{k}−1}{z−1}. and X(n) is the twisted form of the maximal orthogonal Grassmannian.

C_{n}: We assume that the characteristic of the base field k is different from 2. In this case G = Aut(A, σ), where A is a central simple algebra of degree 2n, n ≥ 2, with an involution σ of symplectic type on A, and a projective G-homogeneous variety can be described as the set of flags of (right) ideals

X(d_{1}, …, d_{k}) = {I_{1} ⊂ … ⊂ I_{k} ⊂ A | I_{i} ⊆ I_{i}^{⊥}}

of fixed reduced dimensions 1 ≤ d_{1} < … < d_{k} ≤ n, where I^{⊥} = {x ∈ A | σ(x)I = 0} is the right ideal of reduced dimension 2n − rdim I. Observe that this variety is a twisted form of G/P, where P is the standard parabolic subgroup corresponding to the simple roots on the Dynkin diagram, numbered by d_{i}.

∘_{1}  ∘_{2}  ∘_{3}  ∘_{n−2} ∘_{n−1} ^{<} ∘_{n}

Again, the motives of some flag varieties can be decomposed into a direct sum of twisted motives of “smaller” flags.

2.10 Theorem. Suppose that d_{i} is odd for some i < k and d_{k} − d_{k−1} = 1. Then

M(X(d_{1}, …, d_{k})) ≃ ⨁_{i=0}^{2n−2d_{k−1}−1} M(X(d_{1}, …, d_{k−1}))(i).

In particular, for the variety of complete flags we obtain

2.11 Corollary. The motive of the variety of complete flags X = X(1, 2, …, n) is isomorphic to

M(X(1, …, n)) ≃ ⨁_{i=0}^{n(n−1)} M(SB(A))(i)^{⊕a_{i}},

where a_{i} are the coefficients of the polynomial ψ_{n}(z) = ∏_{k=1}^{n−1} \frac{z^{2k}−1}{z−1}.

G_{2}: We suppose that the characteristic of k is not 2. It is known that G = Aut(C), where C is a Cayley algebra over k. By an i-space, where i = 1, 2, we mean an i-dimensional subspace V_{i} of C such that uv = 0 for every u, v ∈ V_{i}. The only flag variety corresponding to a non-maximal parabolic is the variety of complete flags X(1, 2) which is described as follows

X(1, 2) = {V_{1} ⊂ V_{2} | V_{i} is a i-subspace of C}.

We enumerate the simple roots on the Dynkin diagram as follows:

∘_{1} ^{<} ∘_{2}

In this case we obtain 2.12 Theorem. The motive of the variety of complete flags X = X(1, 2) is isomorphic to

M(X) ≃ M(X(2)) ⊕ M(X(2))(1).

Proof. See 7.5

Observe that by the result of Bonnet [Bo03] the motives of X(1) and X(2) are isomorphic (here X(1) is a 5-dimensional quadric).

F_{4}: We suppose that the characteristic of k is not 2 and 3. It is known that G = Aut(J), where J is an exceptional Jordan algebra of dimension 27 over k. Set I = {1, 2, 3, 6}. By an i-space, i ∈ I, we mean an i-dimensional subspace V of J such that every u, v ∈ V satisfy the following condition:

tr(u) = 0, u × v = 0, and if i < 6 then u(va) = v(ua) for all a ∈ J.

A projective G-homogeneous variety can be described as the set of flags of subspaces

X(d_{1}, …, d_{k}) = {V_{1} ⊂ … ⊂ V_{k} | V_{i} is a d_{i}-subspace of J}.

where the integers d_{1} < … < d_{k} are taken from the set I. Observe that this variety is a twisted form of G/P, where P is the standard parabolic subgroup corresponding to the simple roots on the Dynkin diagram, numbered by d_{i}.

∘_{1}  ∘_{2} ^{<} ∘_{3}  ∘_{6}

In this case we obtain

2.13 Theorem. Suppose that m < k and either d_{m+1} < 6 or d_{m} = 1, then

M(X(d_{1}, …, d_{k})) ≃ ⨁_{λ} M(X(d_{1}, …, d̂_{m}, …, d_{k}))(δ_{m}δ_{m−1} − |λ|),

where the sum is taken over all partitions λ = (λ_{1}, …, λ_{δ_{m−1}}) such that δ_{m} ≥ λ_{1} ≥ … ≥ λ_{δ_{m−1}} ≥ 0.

Proof. See 7.10.

3 Preliminaries

In the present section, we introduce the category of Chow motives following [Ma68]. We formulate and prove the Grassmann Bundle Theorem. At the end we recall the notion of functor of points following [Ka01, section 8] and provide some examples.

3.1 (Chow motives). Let k be a field and Var_{k} be the category of smooth projective varieties over k. We define the category Cor_{k} of correspondences over k. Its objects are smooth projective varieties over k. As morphisms, called correspondences, we set Mor(X, Y) := CH^{dim X}(X × Y). For any two correspondences α ∈ CH(X × Y) and β ∈ CH(Y × Z) we define the composition β ∘ α ∈ CH(X × Z)

β ∘ α = pr_{13}_{∗}(pr_{12}^{∗}(α) · pr_{23}^{∗}(β)),

where pr_{ij} denotes the projection on the i-th and j-th factors of X × Y × Z respectively and pr_{ij}_{∗}, pr_{ij}^{∗} denote the induced push-forwards and pull-backs for Chow groups. Observe that the composition ∘ induces a ring structure on the abelian group CH^{dim X}(X × X). The unit element of this ring is the class of the diagonal Δ_{X}.

The pseudo-abelian completion of Cor_{k} is called the category of Chow motives and is denoted by M_{k}. The objects of M_{k} are pairs (X, p), where X is a smooth projective variety and p is a projector, that is, p ∘ p = p. The motive (X, Δ_{X}) will be denoted by M(X).

3.2. By the construction M_{k} is a self-dual tensor additive category, where the duality is given by the transposition of cycles α ↦ α^{t} and the tensor product is given by the usual fiber product (X, p) ⊗ (Y, q) = (X × Y, p × q). Moreover, the Chow functor CH : Var_{k} → ℤ-Ab (to the category of ℤ-graded abelian groups) factors through M_{k}, i.e., one has the commutative diagram of functors

\xymatrix{Var_{k} \ar[rr]^{CH} \ar[dr]_{Γ} &  & ℤ-Ab \\  & M_{k} \ar[ur]_{R}}

where Γ : f ↦ Γ_{f} is the graph and R : (X, p) ↦ im(p^{∗}) is the realization.

3.3. Consider the morphism (id, e) : ℙ^{1} × {pt} → ℙ^{1} × ℙ^{1}. Its image by means of the induced push-forward (id, e)_{∗} doesn’t depend on the choice of a point e : {pt} → ℙ^{1} and defines the projector in CH^{1}(ℙ^{1} × ℙ^{1}) denoted by p_{1}. The motive L = (ℙ^{1}, p_{1}) is called the Lefschetz motive. For a motive M and an nonnegative integer i we denote by M(i) = M ⊗ L^{⊗i} its twist. Observe that

Mor((X, p)(i), (Y, q)(j)) = q ∘ CH^{dim X+i−j}(X, Y) ∘ p.

We will extensively use the following fact that easily follows from Manin’s Identity Principle and Grassmann Bundle Theorem for Chow groups [Ful].

3.4 Proposition (Grassmann Bundle Theorem). Let X be a variety over k and E be a vector bundle over X of rank n. Then the motive of the Grassmann bundle Gr(d, E) over X is isomorphic to

M(Gr(d, E)) ≃ ⨁_{λ} M(X)(d(n − d) − |λ|),

where the sum is taken over all partitions λ = (λ_{1}, …, λ_{d}) such that n − d ≥ λ_{1} ≥ … ≥ λ_{d} ≥ 0.

Proof. We follow notations of [Ma68]. Denote Gr(d, E) by Y and the canonical projection of Y to X by ψ. It is known (see [Ful, Prop. 14.6.5]) that CH^{∗}(Y) as CH^{∗}(X)-module (via ψ^{∗}) has a basis consisting of elements Δ_{λ} ∈ CH^{|λ|}(Y) parametrized by partitions λ. For any partition λ denote by λ^{op} the partition defined by λ_{i}^{op} = n − d − λ_{d+1−i}. Then for every two partitions λ and μ with |λ| + |μ| ≤ d(n − d) and any element a ∈ CH^{∗}(X) the following duality formula holds (see [Ful, Prop. 14.6.3]):

ψ_{∗}(Δ_{λ}Δ_{μ}ψ^{∗}(a)) = \{ \begin{matrix}a, μ = λ^{op}, \\ 0, otherwise. \end{matrix}

For every partition λ, we define an element f_{λ} of End(M(Y)) inductively. For the unique partition λ_{max} with |λ_{max}| = d(n − d) set f_{λ_{max}} = c(ψ) ∘ c(ψ)^{t}. Now, by decreasing induction on |λ|, set

f_{λ} = f_{λ_{max}} ∘ c_{Δ_{λ^{op}}} ∘ (Δ_{Y} − ∑_{|μ|>|λ|} c_{Δ_{μ}} ∘ f_{μ}).

Finally, set p_{λ} = c_{Δ_{λ}} ∘ f_{λ}.

Now, the duality formula implies that

(p_{λ})_{e}(∑_{μ} Δ_{μ}ψ^{∗}(a_{μ})) = Δ_{λ}ψ^{∗}(a_{λ}).

Therefore, by Manin’s identity principle, the p_{λ} form a complete orthogonal system of projectors. Identification of their images with twisted motives of X can be done as in [Ma68, §7] and we omit it.

3.5 (Functors of Points). In sections 4, 6 and 7 we use the functorial language, that is consider k-schemes as functors from the category of kalgebras to the category of sets. Fix a scheme X. By an X-algebra we mean a pair (R, x), where R is a k-algebra and x is an element of X(R). X-algebras form a category with obvious morphisms. Morphisms φ: Y → X can be considered as functors from the category of X-algebras to the category of sets, by sending a pair (R, x) to its preimage in Y (R).

3.6. Let X be a variety over k. To any vector bundle F over X we can associate the Grassmann bundle Y = Gr(d, F). Fix an X-algebra (R, x). The value of the functor corresponding to Gr(d, F) at (R, x) is the set of direct summands of rank d of the projective R-module F_{x} ⊗_{k} R (cf. [Ka01, section 9]), where F_{x} = F(R, x).

4 Groups of type A_{n}

The goal of the present section is to prove Theorem 2.1 and Proposition 2.4. We use the notation of section 2.

4.1. Let G be an adjoint group of inner type A_{n} defined over a field k. It is well known that G = PGL_{1}(A), where A is a central simple algebra of degree n + 1 and points of projective G-homogeneous varieties are flags of (right) ideals of A

X(d_{1}, …, d_{k}) = {I_{1} ⊂ … ⊂ I_{k} ⊂ A | rdim I_{i} = d_{i}}.

For convenience we set d_{0} = 0, d_{k+1} = n + 1, I_{0} = 0, I_{k+1} = A.

4.2. The value of the functor of points corresponding to the variety X(d_{1}, …, d_{k}) at a k-algebra R (see 3.5) equals the set of all flags I_{1} ⊂ … ⊂ I_{k} of right ideals of A_{R} = A ⊗_{k} R having the following properties (see [IK00, section 4])

• the injection of A_{R}-modules I_{i} ↪ A_{R} splits;

• rdim I_{i} = d_{i}.

4.3. On the scheme X = X(d_{1}, …, d_{k}) there are “tautological” vector bundles J_{i}, i = 0, …, k + 1, of ranks (n + 1)d_{i}. The value of J_{i} on an X-algebra (R, x), where x = (I_{1}, …, I_{k}), is the ideal I_{i} considered as a projective Rmodule. The bundle J_{i} also has a structure of right A_{X}-module, where A_{X} is the constant sheaf of algebras on X determined by A.

For every m ∈ {1, …, k} there exists an obvious morphism

X(d_{1}, …, d_{k}) → X(d_{1}, …, d̂_{m}, …, d_{k})
(I_{1}, …, I_{k}) ↦ (I_{1}, …, Î_{m}, …, I_{k})

that turns X(d_{1}, …, d_{k}) into a X(d_{1}, …, d̂_{m}, …, d_{k})-scheme.

4.4 Lemma. Denote X(d_{1}, …, d_{k}) by Y and X(d_{1}, …, d̂_{m}, …, d_{k}) by X. Assume there exists a vector bundle E over X such that A_{X} ≃ End_{O_{X}}(E). Consider the vector bundle

F = J_{m+1}E/J_{m−1}E = J_{m+1}/J_{m−1} ⊗_{A_{X}} E

of rank d_{m+1} − d_{m−1}. Then Y as a scheme over X can be identified with the Grassmann bundle Z = Gr(d_{m} − d_{m−1}, F) over X.

Proof. We use essentially the same method as in [IK00, Proposition 4.3].

Fix an X-algebra (R, x) where x = (I_{1}, …, Î_{m}, …, I_{k}). The fiber of Y over x, i.e., the value at (R, x), can be identified with the set of all ideals I_{m} satisfying the conditions 4.2 such that I_{m−1} ⊂ I_{m} ⊂ I_{m+1}. The fiber of Z over x is the set of all R-submodules N of F_{y} = F(R, y) such that the injection N ↪ F_{y} splits and rk_{R} N = d_{m} − d_{m−1}.

We define a natural bijection between the fibers of Y and Z over x as follows.

Consider the following mutually inverse bijections between the set of all right ideals of reduced dimension r in A_{R} (satisfying 4.2) and the set of all direct summands of rank r of the R-module E_{x}

Φ : I ↦ IE_{x}
Ψ : N ↦ Hom_{R}(E_{x}, N) ⊂ End_{R}(E_{x}) ≃ A_{R}

Observe that these bijections preserve the respective inclusions of ideals and modules. So ideals of reduced dimension d_{m} between I_{m−1} and I_{m} correspond to submodules of rank d_{m} between I_{m−1}E_{x} and I_{m+1}E_{x}, and, therefore, to submodules of rank d_{m+1} − d_{m−1} in I_{m+1}E_{x}/I_{m−1}E_{x} = F_{x}. This gives the desired natural bijection on the fibers.

4.5 Lemma. Suppose that gcd(ind(A), d_{1}, …, d_{k}) = 1. Then there exists a vector bundle E over X = X(d_{1}, …, d_{k}) of rank n + 1 such that A_{X} ≃ End_{O_{X}}(E).

Proof. We have to prove that the class [A_{X}] in Br(X) is trivial. Since X is a regular Noetherian scheme the canonical map

Br(X) → Br(K)

where K = k(X) is the function field of X, is injective by [Gr, 1.10] and [AG60, Theorem 7.2]. So it is enough to prove that A ⊗_{k} K splits. But the generic point of X defines a flag of ideals of A ⊗_{k} K of reduced dimensions d_{1}, …, d_{k}. Since the index ind(A ⊗_{k} K) divides d_{1}, …, d_{k} and ind A, by the assumption of the lemma it must be equal to 1. So A ⊗_{k} K is split and this finishes the proof of the lemma.

4.6 Remark. In the case d_{1} = 1 one can take E = J_{1}^{∨}.

4.7 Remark. It can be shown using the Index Reduction Formula (see [MPW96]) that the condition on the gcd is necessary and sufficient for the central simple algebra A_{k(X)} to be split.

We are now ready to finish the proof of 2.1.

4.8 (Proof of Theorem 2.1). By Lemma 4.5 there exists a vector bundle E over X = X(d_{1}, …, d̂_{m}, …, d_{k}) of rank n + 1 such that A_{X} ≃ End_{O_{X}}(E). By Lemma 4.4 we conclude that Y = X(d_{1}, …, d_{k}) is a Grassmann bundle over X. Now by Proposition 3.4 we obtain the isomorphism of 2.1.

4.9 Remark. Note that the assumption of Theorem 2.1 on the reduced dimensions d_{1}, …, d_{k} is essential. Indeed, suppose the Theorem holds for any twisted flag variety. Consider the flag X = X(1, d) with gcd(ind(A), d) > 1. Then we have an isomorphism of motives

M(X) ≃ ⨁_{i=0}^{d−1} M(SB_{d}(A))(i)

which appears after applying Theorem to the flags X(1, d) and X(d). Consider the group CH_{0}(X) = Mor_{M_{k}}(M(pt), M(X)). The isomorphism above induces the isomorphism of groups

Coker(CH_{0}(X) ⟶^{res} CH_{0}(X_{k_{s}})) ≅ Coker(CH_{0}(SB_{d}(A)) ⟶^{res} CH_{0}(Gr(d, n + 1)))
≅ ℤ/(\frac{ind(A)}{gcd(ind(A),d)})ℤ,

where res is the pull-back induced by the scalar extension k_{s}/k (here k_{s} denotes the separable closure of k) and the last isomorphism follows by [Bl91, Theorem 3]. On the other hand, applying Theorem 2.1 to the flags X(1, d) and X(1) we obtain the isomorphism

M(X) ≃ ⨁_{λ} M(SB(A))((n + 1 − d)(d − 1) − |λ|)

which induces the isomorphism of groups

Coker(CH_{0}(X) ⟶^{res} CH_{0}(X_{k_{s}})) ≅ Coker(CH_{0}(SB(A)) ⟶^{res} CH_{0}(ℙ^{n}))
≅ ℤ/ ind(A)ℤ,

that leads to a contradiction.

We now prove Proposition 2.4.

4.10 (Proof of Proposition 2.4). Let G = PGL_{1}(A) and let M(G, R) be the symmetric tensor category of motives of G-homogeneous varieties with coefficients in a ring R for which the Krull-Schmidt theorem holds. It is the case, e.g., when R is a field or, more general, a discrete valuation ring (see [CM04, Theorem 9.6]).

Consider the G-homogeneous variety X(1, d), 1 < d < n. Apply Theorem 2.1 to the sequences of flags X(1, d), X(d) and X(1, d), X(1). We obtain two isomorphisms in M(G, R)

⨁_{i=0}^{d−1} M(SB_{d}(A))(i) ≃ M(X) ≃ ⨁_{λ} M(SB(A))((n + 1 − d)(d − 1) − |λ|), (*)

where the sum on the right hand side is taken over all partitions λ = (λ_{1}, …, λ_{d−1}) such that n+1−d ≥ λ_{1} ≥ … ≥ λ_{d−1} ≥ 0. Since Krull-Schmidt Theorem holds in M(G, R), the motive SB(A) has a unique decomposition into the direct sum of indecomposable objects H_{i}, i ∈ I, and their twists

M(SB(A)) ≃ ⨁_{i∈I}(⊕_{j∈J_{i}}H_{i}(j)).

Consider the subcategory M(G, R)_{I} additively generated by the motives H_{i}, i ∈ I, and their twists. The abelian group of isomorphism classes of objects of this category can be equipped with a structure of a free module over the polynomial ring R[z]. Namely, multiplication by z is given by the twist. Clearly, the classes [H_{i}], i ∈ I, form the basis of this R[z]-module.

By (*) we have M(SB_{d}(A)) ∈ M(G, R)_{I} and the isomorphisms (*) can be rewritten as

\frac{z^{d} − 1}{z − 1}[SB_{d}(A)] = \frac{φ_{n}(z)}{φ_{d−1}(z)φ_{n+1−d}(z)}[SB(A)]

= \frac{z^{d} − 1}{z − 1} \frac{φ_{n}(z)}{φ_{d}(z)φ_{n+1−d}(z)}[SB(A)]

where φ_{n}(z) = ∏_{k=2}^{n} \frac{z^{k}−1}{z−1}. This immediately implies the equality

[SB_{d}(A)] = \frac{φ_{n}(z)}{φ_{d}(z)φ_{n+1−d}(z)}[SB(A)],

i.e., the isomorphism in M(G, R)_{I} between M(SB_{d}(A)) and the respective sum of twists of M(SB(A)). This finishes the proof of the proposition.

5 Motivic decomposition of SB_{2}(A)

This section is devoted to the proof of Theorem 2.5.

First, we need to recall some properties of rational cycles on projective homogeneous varieties.

5.1. Let G be a split linear algebraic group over k. Let X be a projective G-homogeneous variety, i.e., X = G/P, where P is a parabolic subgroup of G. The abelian group structure of CH(X), as well as its ring structure, is well-known. Namely, X has a cellular filtration and the generators of Chow groups of the bases of this filtration correspond to the free additive generators of CH(X) (see [Ka01]). Note that the product of two projective homogeneous varieties X × Y has a cellular filtration as well, and CH^{∗}(X × Y) ≅ CH^{∗}(X) ⊗ CH^{∗}(Y) as graded rings. The correspondence product of two cycles α = f_{α} × g_{α} ∈ CH(X × Y) and β = f_{β} × g_{β} ∈ CH(Y × X) is given by (cf. [Bo03, Lem. 5])

(f_{β} × g_{β}) ∘ (f_{α} × g_{α}) = deg(g_{α} · f_{β})(f_{α} × g_{β}),

where deg : CH(Y) → CH({pt}) = ℤ is the degree map.

5.2. Let X be a projective variety of dimension n over a field k. Let k_{s} be the separable closure of the field k. Consider the scalar extension X_{s} = X ×_{k} k_{s}. We say a cycle J ∈ CH(X_{s}) is rational if it lies in the image of the pullback homomorphism CH(X) → CH(X_{s}). For instance, there is an obvious rational cycle Δ_{X_{s}} on CH^{n}(X_{s} × X_{s}) that is given by the diagonal class. Clearly, linear combinations, intersections and correspondence products of rational cycles are rational.

5.3. We will use the following fact (see [CGM, Cor. 8.3]) that follows from the Rost Nilpotence Theorem. Let p_{s} be a non-trivial rational projector in CH^{n}(X_{s} × X_{s}), i.e., p_{s} ∘ p_{s} = p_{s}. Then there exists a non-trivial projector p on CH^{n}(X × X) such that p ×_{k} k_{s} = p_{s}. Hence, the existence of a non-trivial rational projector p_{s} on CH^{n}(X_{s} × X_{s}) gives rise to the decomposition of the Chow motive of X

M(X) ≅ (X, p) ⊕ (X, Δ_{X} − p)

Our goal is to find such a projector in the case X = SB_{d}(A).

5.4. An isomorphism between (X, p)(m) and (Y, q)(l) can be given by correspondences j_{1} ∈ CH^{dim X−l+m}(X × Y) and j_{2} ∈ CH^{dim Y −m+l}(Y × X) such that q ∘ j_{1} = j_{1} ∘ p, p ∘ j_{2} = j_{2} ∘ q, j_{1} ∘ j_{2} = q, j_{2} ∘ j_{1} = p. If X and Y lie in the category M(G, ℤ) then by the Rost nilpotence theorem (see [CM04, Theorem 8.2] and [CGM, Corollary 8.4]) it suffices to give a rational j_{1} and some j_{2} satisfying these conditions over separable closure (note that j_{2} will automatically be rational).

We now recall some properties of Grassmann varieties and describe their Chow rings.

5.5. Consider the Grassmann variety Gr(d, n + 1), 1 ≤ d ≤ n, of d-planes in the (n + 1)-dimensional affine space. It has the dimension d(n + 1 − d). The twisted form of it is a generalized Severi-Brauer variety SB_{d}(A), where A is a central simple algebra of degree n + 1. For any two integers d and d′, 1 ≤ d, d′ ≤ n, there is the fiber product diagram

\xymatrix{Gr(d, n + 1) × Gr(d′, n + 1) \ar[r]^{Seg} \ar[d] & Gr(dd′, (n + 1)^{2}) \ar[d]^{≃} \\ SB_{d}(A) × SB_{d′}(A^{op}) \ar[r]^{Seg} & SB_{dd′}(A ⊗ A^{op})}  (1)

where the horizontal arrows are Segre embeddings given by the tensor product of ideals (resp. linear subspaces) and the vertical arrows are canonical maps induced by the scalar extension k_{s}/k (here k_{s} is the separable closure of k). Observe that the right vertical arrow is an isomorphism since A ⊗ A^{op} splits.

5.6. The diagram (1) induces the commutative diagram of rings

\xymatrix{CH(Gr(d, n + 1) × Gr(d′, n + 1)) & CH(Gr(dd′, (n + 1)^{2})) \ar[l]_{^{Seg^{∗}}} \\ CH(SB_{d}(A) × SB_{d′}(A^{op})) \ar[u]_{res} & CH(SB_{dd′}(A ⊗ A^{op})) \ar[l]_{_{Seg_{∗}}} \ar[u]^{≃}_{res}}  (2)

where all maps are the induced pull-backs. Consider a vector bundle E over Gr(dd′, (n + 1)^{2}). It is easy to see that the pull-back of the total Chern class Seg^{∗}(c(E)) is a rational cycle on CH(Gr(d, n + 1) × Gr(d′, n + 1)) = CH(Gr(d, n + 1)) ⊗ CH(Gr(d′, n + 1)). In particular, if E = τ_{dd′} is the tautological bundle of Gr(dd′, (n + 1)^{2}) we obtain the following

5.7 Lemma. The total Chern class c(pr_{1}^{∗}τ_{d} ⊗ pr_{2}^{∗}τ_{d′}) of the tensor product of the pull-backs (induced by the projection maps) of the tautological bundles τ_{d} and τ_{d′} of Gr(d, n + 1) and Gr(d′, n + 1) respectively is rational.

From now on we restrict ourselves to the case n = 4, d = 2 and d′ = 1, i.e., to the Grassmannian Gr(2, 5) and the projective space ℙ^{4} = Gr(1, 5).

5.8. We describe the generators and relations of the Chow ring CH(Gr(2, 5)) following [Ful, section 14.7]. Set σ_{m} = c_{m}(Q), m = 1, 2, 3, where Q = O^{5}/τ_{2} is the universal quotient bundle of rank 3 over Gr(2, 5). It is known that the elements σ_{m} generate the Chow ring CH(Gr(2, 5)). More precisely, as an abelian group this ring is generated by the Schubert cycles Δ_{λ}(σ) that are parametrized by all partitions λ = (λ_{1}, λ_{2}) such that 3 ≥ λ_{1} ≥ λ_{2} ≥ 0. In particular, σ_{m} = Δ_{(m,0)}, m = 1, 2, 3. For other generators we set the following notation g_{2} = Δ_{(1,1)}, g_{3} = Δ_{(2,1)}, h_{4} = Δ_{(3,1)}, g_{4} = Δ_{(2,2)}, g_{5} = Δ_{(3,2)}, pt = Δ_{(3,3)}. These generators corresponds to the vertices of the Hasse diagram of Gr(2, 5)

[figure: σ_{3}; h_{4}; σ_{2}; pt; g_{5}; g_{3}; σ_{1}; 1; g_{4}; g_{2}]

The multiplication rules can be determined using Pieri formulae

Δ_{λ} · σ_{m} = ∑_{μ} Δ_{μ},

where the sum is taken over all partitions μ = (μ_{1}, μ_{2}) such that 3 ≥ μ_{1} ≥ λ_{1} ≥ μ_{2} ≥ λ_{2} ≥ 0.

5.9. Consider the tautological bundle τ_{2} of the Grassmannian Gr(2, 5). Its total Chern class is

c(τ_{2}) = c(Q)^{−1} = \frac{1}{1 + σ_{1} + σ_{2} + σ_{3}} = 1 − σ_{1} − σ_{2} + σ_{1}^{2} + …

where the rest consists of the summands of degree greater than 2. Hence, we obtain c_{1}(τ_{2}) = −σ_{1} and c_{2}(τ_{2}) = −σ_{2} + σ_{1}^{2} = g_{2}.

5.10. The Chow ring of the projective space ℙ^{4} can be identified with the factor ring ℤ[H]/(H^{5}), where H = c_{1}(O(1)) is the class of a hyperplane section. Thus, the first Chern class of the tautological bundle of ℙ^{4} equals to c_{1}(τ_{1}) = c_{1}(O(−1)) = −H.

We are now ready to prove Theorem 2.5.

5.11 (Proof of Theorem 2.5). By Lemma 5.7 we obtain the following rational cycles in CH^{∗}(Gr(2, 5) × ℙ^{4})

r = c_{1}(pr_{1}^{∗}(τ_{2}) ⊗ pr_{2}^{∗}(τ_{1})) = c_{1}(pr_{1}^{∗}(τ_{2}) + 2c_{1}(pr_{2}^{∗}(τ_{1})) = −σ_{1} × 1 − 2(1 × H),
ρ = c_{2}(pr_{1}^{∗}(τ_{2}) ⊗ pr_{2}^{∗}(τ_{1})) = c_{2}(pr_{1}^{∗}(τ_{2})) + c_{1}(pr_{1}^{∗}(τ_{2}))c_{1}(pr_{2}^{∗}(τ_{1})) + c_{1}(pr_{2}^{∗}(τ_{1}))^{2}

= g_{2} × 1 + σ_{1} × H + 1 × H^{2}

For two cycles x and y we shall write x =_{5} y if there exists a cycle z such that x − y = 5z. Note that =_{5} is an equivalence relation that preserves rationality of cycles. Then the following cycles are rational

ρ^{2} =_{5} 1 × H^{4} + 2σ_{1} × H^{3} + (σ_{2} + 3g_{2}) × H^{2} + 2g_{3} × H + g_{4} × 1,
ρ^{3} =_{5} (3σ_{2} + g_{2}) × H^{4} + (σ_{3} + 3g_{3}) × H^{3} + (g_{4} + 3h_{4}) × H^{2} + 3g_{5} × H + pt × 1.

and a direct computation shows that ρ^{3} ∘ (ρ^{2})^{t} =_{5} Δ_{ℙ^{4}} is the class of the diagonal in CH^{4}(ℙ^{4} × ℙ^{4}).

Consider the composition

(ρ^{2})^{t} ∘ ρ^{3} =_{5}(3σ_{2} + g_{2}) × g_{4} + (2σ_{3} + g_{3}) × g_{3}
+ (g_{4} + 3h_{4}) × (σ_{2} − 2g_{2}) + g_{5} × σ_{1} + pt × 1.

Note that the right-hand side is a rational projector (over ℤ) and, therefore, by Rost Nilpotence Theorem (see [CGM, Corollary 8.3]) has a form p ×_{k} k_{s} where p is a projector in End(M(SB_{2}(A))). The latter determines an object (SB_{2}(A), p) in the category of motives (actually in M(G, ℤ)) which we denoted by H.

Set q = Δ_{SB_{2}(A)} − p. We then show that

(M(SB_{2}(A)), q) ≃ (M(SB_{2}(A)), p^{t}) ≃ H(2),

which gives the claimed decomposition M(SB_{2}(A)) ≃ H ⊕ H(2).

Observe that an isomorphism (SB_{2}(A), q) ≃ (SB_{2}(A), p^{t}) is given by the two mutually inverse motivic isomorphisms p_{s}^{t}∘q_{s} and q_{s}∘p_{s}^{t} over k_{s} which are rational. An isomorphism H(2) ≃ (M(SB_{2}(A)), p^{t}) is given by the following two cycles

j_{1} = (3σ_{2} + g_{2}) × pt − (2σ_{3} + g_{3}) × g_{5}
+ (g_{4} + 3h_{4}) × (g_{4} + 3h_{4}) − g_{5} × (2σ_{3} + g_{3}) + pt × (3σ_{2} + g_{2}),
j_{2} = 1 × g_{4} − σ_{1} × g_{3} + (σ_{2} − 2g_{2}) × (σ_{2} − 2g_{2}) − g_{3} × σ_{1} + g_{4} × 1.

Note that j_{1} is rational, since j_{1} =_{5} (1 × (3σ_{2} + g_{2}))p, and 1 × (3σ_{2} + g_{2}) =_{5} 3(ρ + r^{2})^{t} ∘ ρ^{2} is rational.

We now check the properties of the motive H.

i) To prove that H is isomorphic to M(SB(A)) over R it is enough to provide rational cycles α ∈ CH^{4}(ℙ^{4} × Gr(2, 5)) and β ∈ CH^{6}(Gr(2, 5) × ℙ^{4}) such that α ∘ β = p_{s} and β ∘ α = Δ_{ℙ^{4}}. If 2 is invertible in R, set

α = (ρ^{2})^{t}, β = ρ^{3} − \frac{5}{2}(g_{5} × H + g_{3} × H^{3}).

If 3 is invertible, set

α = (ρ^{2})^{t} − \frac{5}{3}(H × g_{3} + H^{3} × σ_{1}) − 5H^{2} × g_{2}, β = ρ^{3}.

ii) We show that H is indecomposable over ℤ. It is enough to prove that it is indecomposable modulo 5. But over ℤ/5ℤ H is isomorphic to M(SB(A)) by item i). Let μ be a non-trivial projector in CH(SB(A) × SB(A))/5. Then μ_{s} = ∑_{i} ±H^{i}×H^{5−i}. Chose any inverse image ν of μ in CH(SB(A)×SB(A)). Then ν_{s} =_{5} ∑_{i} ±H^{i} × H^{5−i} is rational, and so we have a rational non-trivial projector over ℤ, that contradicts [Ka95, theorem 2.2.1].

iii) We claim that H ≄ M(SB(A)) over ℤ. Indeed, assume there is a motivic isomorphism φ: M(SB(A)) ⟶^{≃} H. Since H is a direct summand of M(SB_{2}(A)), there exist maps α′, β′ such that α′ ∘ β′ = p, β′ ∘ α′ = Δ_{H} (see the diagram).

\xymatrix{M(SB_{2}(A)) \ar[r]_{β′} & H \ar[l]_{α′} \ar[r]_{φ^{−1}} & M(SB(A)). \ar[l]_{φ}}

Consider the cycles α = α′ ∘ φ and β = φ^{−1} ∘ β′. Clearly α ∘ β = p. Consider this equality over the separable closure k_{s}. We have two rational cycles

α_{s} = ∑_{i=0}^{4} H^{i} × a_{4−i} ∈ CH^{4}(ℙ^{4} × Gr(2, 5)) and

β_{s} = ∑_{j=0}^{6} b_{6−j} × H^{j} ∈ CH^{6}(Gr(2, 5) × ℙ^{4})

for some elements a_{i} ∈ CH^{i}(Gr(2, 5)) and b_{j} ∈ CH^{j}(Gr(2, 5)). Their composition can be written as

α_{s} ∘ β_{s} = ∑_{i=0}^{4} b_{6−i} × a_{i} = p_{s}.

Therefore,

α_{s} = ε_{1}1 × g_{4} + ε_{2}H × g_{3} + ε_{3}H^{2} × (σ_{2} − 2g_{2}) + ε_{4}H^{3} × σ_{1} + H^{4} × 1 and

β_{s} = pt×1+ε_{4}g_{5}×H+ε_{3}(g_{4}+3h_{4})×H^{2}+ε_{2}(2σ_{3}+g_{3})×H^{3}+ε_{1}(3σ_{2}+g_{2})×H^{4},

where ε_{i} = ±1. Now let c denotes the rational cycle (ρ^{2})^{t} − α_{s} and b the rational cycle ρ^{3} − β_{s}. Then the cycle

3(b ∘ c) =_{5} H × H^{3} + H^{3} × H + 3(1 − ε_{3})^{2}H^{2} × H^{2} + 3(1 − ε_{1})^{2}1 × H^{4} = d is rational in CH^{4}(ℙ^{4} × ℙ^{4}). Now d^{∘4} gives a non-trivial rational projector on ℙ^{4}, i.e., the motive M(SB(A)) is decomposable. This leads to a contradiction with indecomposability of M(SB(A)) over ℤ, and finishes the proof of the theorem.

6 Groups of types B_{n} and C_{n}

The goal of the present section is to prove Theorems 2.8 and 2.10.

6.1. Let G be an adjoint group of type B_{n}. From now on we suppose that the characteristic of k is not 2. It is known that G = O^{+}(V, q), where (V, q) is a regular quadratic space of dimension 2n+1 and projective G-homogeneous varieties can be described as flags of q-totally isotropic subspaces

X(d_{1}, …, d_{k}) = {V_{1} ⊂ … ⊂ V_{k} ⊂ V | dim V_{i} = d_{i}}.

6.2. The value of the functor corresponding to the variety X(d_{1}, …, d_{k}) at a k-algebra R equals the set of all flags V_{1} ⊂ … ⊂ V_{k}, where V_{i} is a q_{R}-totally isotropic direct summand of V_{R} of rank d_{i}.

For convenience we set d_{0} = 0, V_{0} = 0.

6.3. On the scheme X = X(d_{1}, …, d_{k}) there are “tautological” vector bundles V_{i} of ranks d_{i}. The value of V_{i} on an X-algebra (R, x) is V_{i}, where x = (V_{1}, …, V_{k}). For every m there exists an obvious morphism

X(d_{1}, …, d_{k}) → X(d_{1}, …, d̂_{m}, …, d_{k})
(V_{1}, …, V_{k}) ↦ (V_{1}, …, V̂_{m}, …, V_{k})

which makes X(d_{1}, …, d_{k}) into a X(d_{1}, …, d̂_{m}, …, d_{k})-scheme.

6.4 Lemma. Denote X(d_{1}, …, d_{k}) by Y and X(d_{1}, …, d̂_{m}, …, d_{k})) by X. Suppose that m < k. Then Y as a scheme over X can be identified with the Grassmann bundle Z = Gr(d_{m} − d_{m−1}, V_{m+1}/V_{m−1}) over X.

Proof. Fix an X-algebra (R, x), where x = (V_{1}, …, V̂_{m}, …, V_{k}). We define a natural bijection between the fibers over the point x of Y and Z as follows. The fiber of Y over x can be identified with the set of all direct summands V_{m} of V_{R} of rank d_{m} such that V_{m−1} ⊂ V_{m} ⊂ V_{m+1} (note that V_{m} is automatically q_{R}-isotropic since V_{m+1} is so). This fiber is clearly isomorphic to the fiber of Z over x which is the set of all direct summands of (V_{m+1}/V_{m−1})_{x} = V_{m+1}/V_{m−1} of rank d_{m}.

6.5 (Proof of Theorem 2.8). Apply Lemma 6.4 to the varieties Y = X(d_{1}, …, d_{k}) and X = X(d_{1}, …, d̂_{m}, …, d_{k}). We obtain that Y is a Grassmann bundle over X. To finish the proof apply Proposition 3.4.

6.6. Let G be an adjoint group of type C_{n} over k. It is known that G = Aut(A, σ), where A is a central simple algebra of degree 2n with an involution σ of symplectic type on A, and projective G-homogeneous varieties can be described as flags of (right) ideals of A

X(d_{1}, …, d_{k}) = {I_{1} ⊂ … ⊂ I_{k} ⊂ A | I_{i} ⊆ I_{i}^{⊥}, rdim I_{i} = d_{i}}.

Here I^{⊥} = {x ∈ A | σ(x)I = 0} is a right ideal of reduced dimension 2n − rdim I.

6.7. The value of the functor corresponding to the variety X(d_{1}, …, d_{k}) at a k-algebra R equals to the set of all flags I_{1} ⊂ … ⊂ I_{k} of right ideals of A_{R} = A ⊗_{k} R having the following properties

• the injection of A_{R}-modules I_{i} ↪ A_{R} splits;

• I_{i} ⊆ I_{i}^{⊥};

• rdim I_{i} = d_{i}.

For convenience we set I_{0} = 0.

6.8. On the scheme X = X(d_{1}, …, d_{k}) there are “tautological” vector bundles J_{i} of ranks 2nd_{i} and their “orthogonal complements” J_{i}^{⊥} of rank 2n(2n− d_{i}). The value of J_{i} (resp. J_{i}^{⊥}) on an X-algebra (R, x), where x = (I_{1}, …, I_{k}), is I_{i} (resp. I_{i}^{⊥}) considered as a projective R-module. The bundles J_{i} and J_{i}^{⊥} also have structures of right A_{X}-modules, where A_{X} is a constant sheaf of algebras on X determined by A. There exists an obvious morphism

X(d_{1}, …, d_{k}) → X(d_{1}, …, d_{k−1})
(I_{1}, …, I_{k}) ↦ (I_{1}, …, I_{k−1}),

which makes X(d_{1}, …, d_{k}) into a X(d_{1}, …, d_{k−1})-scheme.

6.9 Lemma. Denote X(d_{1}, …, d_{k}) by Y and X(d_{1}, …, d_{k−1}) by X. Suppose that d_{k} = d_{k−1} + 1 and there exists a vector bundle E over X such that A_{X} ≃ End_{O_{X}}(E). Consider the vector bundle

F = J_{k−1}^{⊥}E/J_{k−1}E = J_{k−1}^{⊥}/J_{k−1} ⊗_{A_{X}} E

of rank 2(n − d_{k−1}). Then Y as a scheme over X can be identified with the projective bundle Z = ℙ(F) = Gr(1, F) over X.

Proof. Fix an X-algebra (R, x), where x = (I_{1}, …, I_{k−1}). We define a natural bijection between the fibers over the point x of Y and Z. The fiber of Y can be identified with the set of all ideals I_{k} containing I_{k−1} and satisfying the conditions 6.7. The fiber of Z is the set of all direct summands of F_{x} = F(R, x) of rank 1.

The involution σ induces an isomorphism h: E_{x} ⊗ L → E_{x}^{∗} for some invertible R-module L (see [Knus, Lemma III.8.2.2]) such that

σ(f) ⊗ 1 = h^{−1}f^{∗}h for all f ∈ A
h^{∗}can ⊗ 1 = −h

where can : E_{x} → E_{x}^{∗∗} is the canonical isomorphism.

Let U_{1} and U_{2} be direct summands of E_{x}. We write U_{2} ⊆ U_{1}^{⊥} if h(u ⊗ l)(v) = 0 for all u ∈ U_{1}, v ∈ U_{2}, l ∈ L. We call a direct summand U of E_{x} totally isotropic if U ⊆ U^{⊥}. Note that any direct summand of rank 1 is totally isotropic (it can be proved easily using localization).

Define Φ and Ψ as in the proof of Theorem 4.4. Direct computations show that I_{1} ⊆ I_{2}^{⊥} if and only if Φ(I_{1}) ⊆ Φ(I_{2})^{⊥}.

So the fiber of Y over x is naturally isomorphic to the set of all totally isotropic direct summands U_{k} of E_{x} of rank d_{k} containing U_{k−1} = Φ(I_{k}). One can represent U_{k} as the direct sum U_{k−1} ⊕ U where U is a direct summand of rank 1 (since d_{k} = d_{k−1} + 1). This U is totally isotropic and, therefore, U_{k} is totally isotropic if and only if U_{k} ⊆ U_{k−1}^{⊥}. Hence the set of all U_{k−1} is naturally isomorphic to the set of all direct summands of Φ(I_{k−1}^{⊥}) of rank d_{k} containing Φ(I_{k−1}). The latter can be identified with ℙ(F_{x}). This finishes the proof of the lemma.

We are now ready to prove Theorem 2.10.

6.10 (Proof of Theorem 2.10). Consider the flag varieties Y = X(d_{1}, …, d_{k}) and X(d_{1}, …, d_{k−1}). Since ind A = 2^{r} for some r and there is an odd d_{i}, we have gcd(ind(A), d_{1}, …, d_{k−1}) = 1. By Lemma 4.5 there exists a bundle E over X such that A_{X} = End_{O_{X}}(E). Applying Lemma 6.9 to the varieties X, Y and the bundle E, we obtain that Y is a projective bundle over X. Now we use Proposition 3.4 to finish the proof.

7 Groups of types G_{2} and F_{4}

This section is devoted to proofs of Theorems 2.12 and 2.13.

7.1. Let G be a group of type G_{2}. We suppose that characteristic of k is not 2. It is known that G = Aut(C) where C is a Cayley algebra over k. By i-space where i = 1, 2 we mean an i-dimensional subspace V_{i} of C such that uv = 0 for every u, v ∈ V_{i}.

The only flag variety corresponding to a non-maximal parabolic is the full flag variety X(1, 2) which is described as follows (see [Bo03]):

X(1, 2) = {V_{1} ⊂ V_{2} | V_{i} is a i-subspace of C, dim V_{i} = i}.

Similarly one can describe the homogeneous flag variety corresponding to the maximal parabolic

X(2) = {V | V is a 2-subspace of C, dim V = 2}.

7.2. Let R be a k-algebra. By an i-submodule in C_{R} = C ⊗_{k} R we mean a direct summand V_{i} of C_{R} of rank i such that uv = 0 for every two elements u, v ∈ V_{i}. The value of the functor corresponding to the variety X(1, 2) (respectively X(2)) at a k-algebra R equals the set of all flags V_{1} ⊂ V_{2} (respectively submodules V_{2}) where V_{i} is an i-submodule of C_{R}.

7.3. On the scheme X = X(2) there is a “tautological” vector bundle V of rank 2. The value of V on an X-algebra (R, x) is V, where x = V.

There exists an obvious morphism

X(1, 2) → X(2)
(V_{1}, V_{2}) ↦ V_{2}

which makes X(1, 2) into a X(2)-scheme.

7.4 Lemma. X(1, 2) as a scheme over X(2) can be identified with the projective bundle ℙ(V) = Gr(1, V) over X(2).

Proof. The proof goes as in B_{n}-case (note that if V_{2} is a 2-submodule then each of its direct summands of rank 1 is a 1-submodule).

7.5 (Proof of Theorem 2.12). Apply Lemma 7.4 and Proposition 3.4.

7.6. Let G be a group of type F_{4}. We suppose that characteristic of k is not 2, 3. It is known that G = Aut(J) where J is an exceptional Jordan algebra of dimension 27 over k. Set I = {1, 2, 3, 6}. By i-space where i ∈ I we mean an i-dimensional subspace V_{i} of J such that every u, v ∈ V_{i} satisfy the following condition:

tr(u) = 0, u × v = 0, and if i < 6 then u(va) = v(ua) for all a ∈ J.

It is known that projective G-homogeneous varieties are parametrized by sequences of numbers d_{1} < … < d_{k} from I and can be described as follows:

X(d_{1}, …, d_{k}) = {V_{1} ⊂ … ⊂ V_{k} | V_{i} is a d_{i}-subspace of A, dim V_{i} = d_{i}}.

7.7. Let R be a k-algebra. By an i-submodule in J_{R} = J ⊗_{k} R we mean a direct summand V_{i} of J_{R} of rank i such that every two elements u, v ∈ V_{i} satisfy the conditions above. The value of the functor corresponding to the variety X(d_{1}, …, d_{k}) at a k-algebra R equals the set of all flags V_{1} ⊂ … ⊂ V_{k} where V_{i} is a d_{i}-submodule of J_{R}.

For convenience we set d_{0} = 0, V_{0} = 0.

7.8. On the scheme X = X(d_{1}, …, d_{k}) there are “tautological” vector bundles V_{i} of rank d_{i}. The value of V_{i} on an X-algebra (R, x) is V_{i}, where x = (V_{1}, …, V_{k}).

There exists an obvious morphism

X(d_{1}, …, d_{k}) → X(d_{1}, …, d̂_{m}, …, d_{k})
(V_{1}, …, V_{k}) ↦ (V_{1}, …, V̂_{m}, …, V_{k})

which makes X(d_{1}, …, d_{k}) into a X(d_{1}, …, d̂_{m}, …, d_{k})-scheme.

7.9 Lemma. Denote X(d_{1}, …, d_{k}) by Y and X(d_{1}, …, d̂_{m}, …, d_{k}) by X. Suppose that m < k and either d_{m+1} < 6 or d_{m} = 1. Then Y as a scheme over X can be identified with the Grassmann bundle Z = Gr(d_{m} − d_{m−1}, V_{m+1}/V_{m−1}) over X.

Proof. The proof goes as in B_{n}-case (note that under our restrictions if V_{m+1} is a d_{m+1}-submodule then each of its direct summands of rank d_{m} is a d_{m}- submodule).

7.10 (Proof of Theorem 2.13). Apply Lemma 7.9 to the varieties Y = X(d_{1}, …, d_{k}) and X = X(d_{1}, …, d̂_{m}, …, d_{k}). We obtain that Y is a Grassmann bundle over X. To finish the proof apply Proposition 3.4.

Acknowledgements We are sincerely grateful to N. Karpenko for the comments concerning rational cycles and Severi-Brauer varieties.

The first and the last authors would like to thank U. Rehmann and RTN-project for hospitality and support. The last author is also grateful to Alexander von Humboldt foundation for support.

The second and the third authors highly appreciate A. Bak’s hospitality and sympathy with their activity. The second author also would like to thank DAAD Jahresstipendium program and INTAS project 03-51-3251.

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