Motivic decomposition of a generalized Severi-Brauer variety

K.Zainoulline

February 5, 2006

Abstract

Let A and B be two central simple algebras of a prime degree n over a field F generating the same subgroup in the Brauer group Br(F). We show that the Chow motive of a Severi-Brauer variety SB(A) is a direct summand of the motive of a generalized Severi-Brauer variety SB_{d}(B) if and only if [A] = ±d[B] in Br(F). The proof uses methods of Schubert calculus and combinatorial properties of Young tableaux, e.g., the Robinson-Schensted correspondence. Keywords: Severi-Brauer variety, Chow motive, Robinson-Schensted correspondence, Grassmannian

1 Introduction

Let X be a twisted flag G-variety for a linear algebraic group G. The main result of paper [CPSZ] says that under certain restrictions the Chow motive of X can be expressed in terms of motives of “minimal” flag varieties, i.e., those which correspond to maximal parabolic subgroups of G. A natural question arises: is it possible to decompose the motive of such a “minimal” flag?

A particular case of such a decomposition was already provided in [CPSZ]. More precisely, a “minimal flag” X was a generalized Severi-Brauer variety of ideals of reduced dimension 2 in a division algebra B of degree 5 and the decomposition was (see [CPSZ, Theorem 2.5])

M(SB_{2}(B)) ≃ M(SB(A)) ⊕ M(SB(A))(2),

where [B] = 2[A] in the Brauer group.

In the present paper we provide an affirmative answer on this question for any adjoint group G of inner type A_{n} of a prime rank n. Namely, we show that the motive of a generalized Severi-Brauer variety always contains (as a direct summand) the motive of a Severi-Brauer variety.

1.1 Theorem. Let A and B be two central simple algebras of a prime degree n over a field F generating the same subgroup in the Brauer group Br(F). Then the motive of a Severi-Brauer variety SB(A) is a direct summand of the motive of a generalized Severi-Brauer variety SB_{d}(B) if and only if

[A] = ±d [B] in Br(F).  (1)

1.2 Remark. Theorem 1.1 may also be considered as a generalization of the result by N. Karpenko (see [Ka00, Criterion 7.1]) which says that the motives of Severi-Brauer varieties SB(A) and SB(B) of central simple algebras A and B are isomorphic if and only if [A] = ±[B] in Br(F).

1.3 Remark. We expect that Theorem 1.1 holds in the case when d and n are coprime (see Section 7). Observe that if d and n are not coprime, then the theorem fails. It can be already seen on the level of generating functions. Namely, consider the Grassmann variety 𝔾_{2}(4) of 2-planes in a 4-dimensional affine space (n = 4 and d = 2). Then for the generating functions we have

P (𝔾_{2}(4), t) = (t^{2} + 1)(t^{2} + t + 1) and P (ℙ^{3}, t) = t^{3} + t^{2} + t + 1.

Obviously P (ℙ^{3}, t) doesn’t divide P (𝔾_{2}(4), t). Indeed, it can be shown that P (ℙ^{n−1}, t) divides P (𝔾_{d}(n), t) if and only if d and n are coprime.

1.4 Remark. For motives with ℤ/nℤ-coefficients there is the following decomposition (see [CPSZ, Proposition 2.4])

M(SB_{d}(B)) ≃ ⨁_{i} M(SB(B))(i)^{⊕a_{i}},  (2)

where the integers a_{i} are coefficients of the quotient of Poincaré polynomials \frac{P (𝔾_{d}(n),t)}{P (ℙ^{n−1},t)} = ∑_{i} a_{i}t^{i} (see 2.8). Note that in this case the motives of Severi-Brauer varieties corresponding to different classes of algebras generating the same subgroup in the Brauer group are isomorphic (see [Ka00, Section 7]), i.e., M(SB(A)) ≃ M(SB(B)).

The proofs are based on Rost Nilpotence Theorem for projective homogeneous varieties proved by V. Chernousov, S. Gille and A. Merkurjev in [CGM05]. Briefly speaking, this result reduces the problem of decomposing the motive of a variety X over F into the question about algebraic cycles in the Chow ring CH(X_{s} × X_{s}) over the separable closure F_{s}. Namely, the motive of SB(A) is a direct summand of the motive of SB_{d}(B) if there exist two cycles f and g in CH(ℙ^{n−1} × 𝔾_{d}(n)) such that both cycles belong to the image of the restriction map CH(SB(A) × SB_{d}(B)) → CH(ℙ^{n−1} × 𝔾_{d}(n)), and the correspondence product g^{t} ∘ f is the identity.

We define f and g to be the Schur functions of total Chern classes of certain bundles on ℙ^{n−1} ×𝔾_{d}(n). Using the language and properties of Schur functions we show that the identity g^{t} ∘ f = id is a direct consequence of the Robinson-Schensted correspondence, one of classical combinatorial facts about Young tableaux.

The paper is organized as follows. In section 2 we remind several definitions and notation used in the proofs. These include Chow motives, rational cycles, and generalized Severi-Brauer varieties. In section 3 we describe the subgroup of rational cycles of the Chow group of the product of two generalized Severi-Brauer varieties. Indeed, we provide an explicit set of generators for this subgroup modulo n in terms of Schur functions. In section 4 we use this description for proving some known results on motives of Severi-Brauer varieties. Section 5 is devoted to the proof of the main theorem. In section 6 we prove the crucial congruence used in the proof of the main theorem. In the last section we discuss the case of 𝔾_{2}(n), where n is an odd integer (not necessarily prime).

2 Preliminaries

In the present section we remind definition of the category of Chow motives over a field F following [Ma68] and [Ka01]. We recall the notion of a rational cycle and state the Rost Nilpotence Theorem for idempotents following [CGM05]. We recall several auxiliary facts concerning generalized Severi-Brauer varieties following [KMRT98].

2.1 (Chow motives). Let F be a field and Var_{F} be the category of smooth projective varieties over F. We define the category Cor_{F} of correspondences over F. Its objects are non-singular projective varieties over F. For morphisms, called correspondences, we set Mor(X, Y) := CH^{dim X}(X × Y). For 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 product of 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. 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 diagonal cycle Δ_{X}.

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

By construction M_{F} 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 contravariant Chow functor CH : Var_{F} → ℤ-Ab (to the category of ℤgraded abelian groups) factors through M_{F}, i.e., one has the commutative diagram of functors

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

where Γ : f ↦ Γ_{f} is the (contravariant) graph functor and R : M_{F} → ℤ-Ab is the (covariant) realization functor given by R : (X, p) ↦ im(p^{∗}), where p^{∗} is the composition

p^{∗} : CH(X) ⟶^{pr_{1}^{∗}} CH(X × X) ⟶^{·p} CH(X × X) ⟶^{pr_{2}_{∗}} CH(X).

Consider the morphism (id, e) : ℙ^{1} × {pt} → ℙ^{1} × ℙ^{1}. The image 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 Lefschetz motive. For a motive M and a 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.

2.2 (Product of cellular varieties). Let G be a split linear algebraic group over a field F. 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

(f_{β} × g_{β}) ∘ (f_{α} × g_{α}) = deg(g_{α} · f_{β})(f_{α} × g_{β}),  (3)

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

2.3 (Rational cycles). Let X be a projective variety of dimension n over F. Let F_{s} denote the separable closure of F. Consider the scalar extension X_{s} = X ×_{F} F_{s}. We say a cycle J ∈ CH(X_{s}) is rational if it lies in the image of the pull-back 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.

Let E be a vector bundle over X. Then the total Chern class c(E_{s}) of the pull-back induced by the scalar extension F_{s}/F is rational. Let L/F be a finite separable field extension which splits X, i.e., there is an induced isomorphism CH(X_{L}) ⟶^{≃} CH(X_{s}). Then the cycle deg(L/F)·J, J ∈ CH(X_{s}), is rational. Observe that all linear combinations, intersections and correspondence products of rational cycles are rational.

2.4 (Rost nilpotence). We will use the following fact (see [CGM05, Cor. 8.3]) that follows from the Rost Nilpotence Theorem. Let X be a twisted flag Gvariety for a semisimple group G of inner type over F. 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 ×_{F} F_{s} = p_{s}. Hence, 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)  (4)

2.5 (Tautological and quotient bundles). Let A be a central simple algebra of degree n over F. Consider a generalized Severi-Brauer variety SB_{d}(A) of ideals of reduced dimension d of A. Over the separable closure it becomes isomorphic to the Grassmannian 𝔾_{d}(n) of d-dimensional planes in a n-dimensional affine space.

There is a tautological vector bundle over SB_{d}(A) of rank dn denoted by τ_{d}^{A} and given by the ideals of A of reduced dimension d considered as vector spaces over F. Over the separable closure F_{s} this bundle becomes isomorphic to Hom(E_{n}, τ_{d}) = τ_{d}^{⊕n}, where E_{n} denotes the trivial bundle of rank n and τ_{d} the tautological bundle over the Grassmannian 𝔾_{d}(n).

The universal quotient bundle over SB_{d}(A), denoted by κ_{d}^{A}, is the quotient E_{A}/τ_{d}^{A} of a trivial bundle E_{A} of rank n^{2} modulo the tautological bundle τ_{d}^{A}. Clearly, κ_{d}^{A} is of rank (n − d)n. Over F_{s} this bundle becomes isomorphic to Hom(E_{n}, κ_{d}) = κ_{d}^{⊕n}, where κ_{d} is the universal quotient bundle over 𝔾_{d}(n).

We will extensively use the following fact

2.6 Lemma. Let A be a central simple algebra over F, r a positive integer and B a division algebra which represents the class of the r-th tensor power of A in the Brauer group. Let X = SB(A) × SB_{d}(B^{op}). Then the bundle T_{s} = pr_{1}^{∗}(τ_{1}^{⊗r}) ⊗ pr_{2}^{∗}(τ_{d}) over X_{s} is a pull-back of some bundle T over X. As a consequence, any Chern class of T_{s} is a rational cycle.

Proof. According to [Pa94, 10.2] the image of the restriction map on K_{0}

K_{0}(SB(A) × SB_{d}(B^{op})) → K_{0}(ℙ^{n−1} × 𝔾_{d}(n))

is generated by classes of bundles ind(A^{i} ⊗ B^{−j}) · [pr_{1}^{∗}(τ_{1}^{⊗i}) ⊗ pr_{2}^{∗}(τ_{d}^{⊗j})].

2.7 Remark. Observe that the similar fact holds if one replaces the tensor power by an exterior (lambda) power. This is due to the fact that [Λ^{r}A] = [A^{⊗r}] in Br(F) (see [KMRT98, 10.A.]).

2.8 (Poincaré polynomial). By [Fu97] the Poincaré polynomial of a Chow group of a Grassmannian 𝔾_{d}(n) is given by the Gaussian polynomial

P (𝔾_{d}(n), t) = \binom{n}{d}(t) = \frac{(1 − t^{n})(1 − t^{n−1}) … (1 − t^{n−d+1})}{(1 − t)(1 − t^{2}) … (1 − t^{d})}.

Observe that for a projective space, i.e., for d = 1, the respective polynomial takes the most simple form

P (ℙ^{n−1}, t) = \frac{1 − t^{n}}{1 − t} .

Observe also that if n is a prime integer, then the polynomial P (ℙ^{n−1}, t) always divides P (𝔾_{d}(n), t).

3 The subgroup of rational cycles

The goal of the present section is to provide an explicit set of generators for the image of the restriction map CH(X) → CH(X_{s}) modulo n, where X is a product of a Severi-Brauer variety by a generalized Severi-Brauer variety corresponding to division algebras of a prime degree n.

3.1 (Grassmann bundle structure). Let A be a division algebra of degree n over F, r a positive integer and B a division algebra which represents the class of the r-th tensor power of A in Br(F). According to [IK00, Proposition 4.3] the product SB(A)×SB_{d}((A^{⊗r})^{op}) can be identified with the Grassmann bundle 𝔾_{d}(V) over SB(A), where V = (τ_{1}^{A})^{⊗r} is a locally free sheaf of (right) A^{⊗r}-modules. By Morita equivalence we may replace A^{⊗r} by B and, hence, obtain that the product X = SB(A) × SB_{d}(B^{op}) is the Grassmann bundle 𝔾_{d}(W) over SB(A) for a locally free sheaf of (right) B-modules W. The tautological bundle T over 𝔾_{d}(W) is the bundle

T = pr_{1}^{∗}(W) ⊗_{B} pr_{2}^{∗}(τ_{B}^{op}).

Let Q = pr_{1}^{∗}(W)/T denote the universal quotient bundle over 𝔾_{d}(W). Over the separable closure it can be identified with

Q_{s} = pr_{1}^{∗}(τ_{1}^{⊗r}) ⊗ pr_{2}^{∗}(κ_{d})

where τ_{1} and κ_{d} are the tautological and quotient bundles over ℙ^{n−1} and 𝔾_{d}(n) respectively. We shall write this bundle simply as τ_{1}^{⊗r} ⊗ κ_{d} meaning the respective pull-backs.

3.2 (Grassmann bundle theorem). According to Grassmann bundle theorem the Chow ring CH(X) is a free CH(SB(A))-module with the basis Δ_{λ}(c(Q)), where λ runs through the set of all partitions λ = (λ_{1}, …, λ_{d}) with n − d ≥ λ_{1} ≥ … ≥ λ_{d} ≥ 0, c(Q) is the total Chern class of the quotient bundle Q and Δ_{λ} is the Schur function. In other words, for any k we have the decomposition

CH^{k}(X) ≅ ⨁_{λ} Δ_{λ}(c(Q)) · pr_{1}^{∗}(CH^{k−|λ|}(SB(A)))  (5)

3.3. Observe that the decomposition (5) is compatible with the scalar extension F_{s}/F. Hence, any rational cycle α ∈ CH^{k}(X_{s}) can be represented uniquely as the sum of cycles

α = ∑_{λ} Δ_{λ}(c(Q_{s})) · α_{λ},  (6)

where α_{λ} ∈ CH^{k−|λ|}(ℙ^{n−1}) is rational. If n is a prime integer, then all rational cycles of positive codimensions in CH(ℙ^{n−1}) are divisible by n (see [Ka95, Corollary 4]). Hence, considering (6) modulo n we obtain

3.4 Lemma. If n is a prime integer, the cycles Δ_{λ}(c(Q_{s})), where λ runs through the set of all partitions, generate the subgroup of rational cycles of CH(X_{s}) modulo n. In particular case d = 1 the basis of the subgroup of rational cycles of CH^{k}(ℙ^{n−1}×ℙ^{n−1}) modulo n consist of Chern classes c_{k}(τ_{1}^{⊗r}⊗κ_{1}), where k = 0 … n − 1.

4 Applications to Severi-Brauer varieties

The following two lemmas were proven by N. Karpenko (see [Ka96, Theorem 2.2.1] and [Ka00, Criterion 7.1]). In the present section we provide short proofs of these results restricting to algebras of prime degrees.

4.1 Lemma. The motive of a Severi-Brauer variety of a division algebra A of a prime degree n is indecomposable.

Proof. Consider the product X = SB(A) × SB(A). It can be identified with the product SB(A) × SB(B^{op}), where [B] = [A^{n−1}]. Apply Lemma 3.4 to the case d = 1 and r = n − 1. We obtain that in codimension k = n − 1 there is only one basis element of the subgroup of rational cycles modulo n

Δ_{n−1} = c_{n−1}(τ_{1}^{⊗n−1} ⊗ κ_{1}) = ∑_{i=0}^{n−1}(1 − n)^{i} · H^{i} × H^{n−1−i} ∈ CH^{n−1}(X_{s})

which is congruent modulo n to the diagonal cycle Δ = ∑_{i=0}^{n−1} H^{i} × H^{n−1−i}.

If the motive of SB(A) splits, then there must exist a rational projector p ∈ CH^{n−1}(X_{s}) such that Δ ± p and p are non-trivial. By composition rule (3) if p is a projector, then p = ∑_{i} ±H^{i} × H^{n−1−i}, where the index i runs through a subset of {0 … n − 1}. From the other hand, since p is rational, it must be a multiple of Δ modulo n. So the only possibility for p is to coincide either with ±Δ or with 0, contradiction.

4.2 Lemma. Let A and B be two division algebras of a prime degree n generating the same subgroup in the Brauer group. Then the motives of SB(A) and SB(B^{op}) are isomorphic iff A = B or B^{op}.

Proof. Take r such that [B] = [A^{⊗r}] in Br(F). and apply Lemma 3.4 for X = SB(A) × SB(B^{op}). We obtain that the subgroup of rational cycles of codimension k = n − 1 is generated by the cycle

Δ_{n−1} = c_{n−1}(τ_{1}^{⊗r} ⊗ κ_{1}) = ∑_{i=0}^{n−1}(−r)^{i} · H^{i} × H^{n−1−i} ∈ CH^{n−1}(X_{s}).

According to composition rule (3) and Rost nilpotence theorem (see 2.4) any motivic isomorphisms between SB(A) and SB(B^{op}) is given by a lifting of a rational cycle of the kind ∑_{i=0}^{n−1} ±H^{i} × H^{n−1−i} ∈ CH^{n−1}(X_{s}) and vice versa. To finish the proof observe that a rational cycle of this kind is a multiple of Δ_{n−1} modulo n iff r ≡ ±1 mod n.

5 Generalized Severi-Brauer varieties

In the present section we prove the main theorem of the paper which is formulated as follows

5.1 Theorem. Let A and B be two division algebras of a prime degree n generating the same subgroup in the Brauer group. Take an integer r such that [B] = [A^{⊗r}]. Then the motive of a Severi-Brauer variety SB(A) is a direct summand of the motive of a generalized Severi-Brauer variety SB_{d}(B) if and only if

d · r ≡ ±1 mod n.

The cases d = 1 and d = n − 1 were considered in Lemma 4.2. From now on we assume 1 < d < n − 1. Moreover, by duality we may assume d ≤ [\frac{n}{2}].

Proof (⇒) Consider the product X = SB(A) × SB_{d}(B^{op}). According to Theorem 3.4 the subgroup of rational cycles of CH^{k}(X_{s}) is generated (modulo prime n) by the cycles

Δ_{λ} = Δ_{λ}(c(τ_{1}^{⊗r} ⊗ κ_{d})),

where λ runs through all partitions with |λ| = k. By [Fu98, Example A.9.1] we have

Δ_{λ} = ∑_{μ⊂λ} d_{λ̃,μ̃} · c_{1}(τ_{1}^{⊗r})^{k−|μ|} · Δ_{μ}(c(κ_{d})) =

= ∑_{i=0}^{k}(−r)^{k−i} · H^{k−i} × (∑_{{μ⊂λ \atop |μ|=i}} d_{λ̃,μ̃}ω_{μ}) ,  (7)

where λ̃ denotes the conjugate partition for λ, i.e., obtained by interchanging rows and columns in the respective Young diagram, H is the class of a hyperplane section of ℙ^{n−1}, ω_{μ} denotes the additive generator of CH^{|μ|}(𝔾_{d}(n)) corresponding to a partition μ and the coefficients d_{λ̃,μ̃} are the binomial determinants

d_{λ̃,μ̃} = |\binom{λ̃_{i} + n − d − i}{μ̃_{j} + n − d − j}|_{1≤i,j≤n−d}

Let k = N, where N = d(n − d) is the dimension of 𝔾_{d}(n). In this codimension there is only one partition λ with |λ| = N, namely, the maximal one λ = (n − d, …, n − d). Let g denote the cycle (7) corresponding to this maximal partition, i.e.,

g = ∑_{m=0}^{n−1}(−r)^{m} · H^{m} × ( ∑_{|μ|=N−m} d_{λ̃,μ̃}ω_{μ}) ,  (8)

where λ̃ = (d, d, …, d) and the coefficients d_{λ̃,μ̃} are given by

d_{λ̃,μ̃} = |\binom{n − i}{μ̃_{j} + n − d − j}|_{1≤i,j≤n−d}

From now on we denote the coefficient d_{λ̃,μ̃} by d_{μ′}, where μ′ is the dual partition (n − d − μ_{d}, …, n − d − μ_{1}). Observe that |μ′| = N − |μ|.

For an integer m denote by g^{(m)} the summand of (8) for the chosen index m and by d_{μ′}^{(m)} the respective coefficients, i.e.,

g^{(m)} = (−r)^{m} · H^{m} × ( ∑_{|μ′|=m} d_{μ′}^{(m)}ω_{μ})

Consider the summands g^{(0)} and g^{(1)}. Since the Chow group CH(𝔾_{d}(n)) has only one additive generator in the last two codimensions N and N − 1 denoted by ω_{N} and ω_{N−1} respectively, we obtain that

g^{(0)} = 1 × ω_{N} and g^{(1)} = −rd · H × ω_{N−1}  (9)

Now we are ready to finish the (⇒) part of the theorem.

Assume that the motive of SB_{d}(B^{op}) contains the motive of SB(A) as a direct summand. Then there must exist two rational cycles

α ∈ CH^{n−1}(ℙ^{n−1} × 𝔾_{d}(n)) and β ∈ CH^{N}(𝔾_{d}(n) × ℙ^{n−1})

such that β ∘ α = id. According to the composition rule (3) this can happen only if the coefficients before the monomials ω_{N} ×1 and ω_{N−1}×H of the cycle β are equal to ±1. But all rational cycles in codimension N are generated (modulo n) by the transposed cycle g^{t} which has coefficients 1 and −rd before the respective monomials (see (9)). This can only be possible if rd ≡ ±1 mod n.

Proof (⇐) Assume that the congruence rd ≡ ±1 mod n holds. We want to produce two rational cycles

α ∈ CH^{n−1}(ℙ^{n−1} × 𝔾_{d}(n)) and β ∈ CH^{N}(𝔾_{d}(n) × ℙ^{n−1})

such that β ∘α ∈ CH^{n−1}(ℙ^{n−1}×ℙ^{n−1}) is the identity morphism, i.e., the diagonal cycle ∑_{m=0}^{n−1} H^{n−1−m} × H^{m}. This will show that M(SB_{d}(B^{op})) contains M(SB(A)) as a direct summand.

Assume rd ≡ 1 mod n. Consider the bundle κ_{1} ⊗ Λ^{d}(τ_{d}) of rank n − 1 on the product ℙ^{n−1} × 𝔾_{d}(n) and define the cycle f as

f = c_{n−1}(κ_{1} ⊗ Λ^{d}(τ_{d})) = ∑_{m=0}^{n−1} c_{n−1−m}(κ_{1})c_{1}(τ_{d})^{m} =

= ∑_{m=0}^{n−1}(−1)^{m} · H^{n−1−m} × ω_{1}^{m} = ∑_{m=0}^{n−1}(−1)^{m} · H^{n−1−m} × ( ∑_{|ρ|=m} c_{ρ}^{(m)}ω_{ρ}) .

Observe that f is a rational cycle, since [A] = [B^{⊗d}]. If rd ≡ −1 mod n, then we take the bundle Λ^{n−d}(κ_{d}) instead of Λ^{d}(τ_{d}) and obtain the same formulae but without the coefficient (−1)^{m}.

The coefficients c_{ρ}^{(m)} appearing in the presentation of ω_{1}^{m} in terms of additive generators ω_{ρ} of CH^{m}(𝔾_{d}(n)) have the following nice property 5.2 Lemma. For any partition ρ with m = |ρ| < n the coefficient c_{ρ}^{(m)} is coprime with n.

Proof. According to [Fu98, Example 14.7.11.(ii)] we have

c_{ρ}^{(m)} = deg(a_{0}, …, a_{d−1}) = \frac{m!}{a_{0}! a_{1}! … a_{d−1}!} ∏_{i>j}(a_{i} − a_{j}),  (10)

where the set a = (a_{0}, a_{1}, …, a_{d−1}) is defined by a_{i} = ρ_{d−i}+i, i = 0, …, d−1. Observe that the set a has the property 0 ≤ a_{0} < … < a_{d−1} ≤ n − 1 and corresponds to a Schubert variety of dimension m which is dual to the Schubert variety of codimension m corresponding to the partition ρ (see [Fu98, 14.7]).

5.3 Remark. The constructed cycle f ∈ CH^{n−1}(ℙ^{n−1} × 𝔾_{d}(n)) has the following geometric interpretation. Consider the Plücker embedding SB_{d}(B) → SB(Λ^{d}(B)). Its graph defines a correspondence Γ ∈ CH(SB(Λ^{d}(B))×SB_{d}(B)). It is known that the motive of SB(Λ^{d}(B)) splits as a direct sum of twisted motives of SB(A) (see [Ka96, Corollary 1.3.2]). Let i : M(SB(A)) → M(SB(Λ^{d}(B))) be the respective splitting. Then over the separable closure the composition Γ ∘ i coincides with the cycle f corresponding to the case rd ≡ −1 mod n. After replacing B by B^{op} the respective composition Γ ∘ i will give the cycle f corresponding to the case rd ≡ 1 mod n.

Consider the transposed cycle g^{t} introduced in the first part of the proof

g^{t} = ∑_{m=0}^{n−1}(−1)^{m} · ( ∑_{|μ′|=m}(r^{m}d_{μ′}) · ω_{μ}) × H^{m}

Consider the m-th summand of the composition g^{t} ∘ f

(g^{t} ∘ f)^{(m)} = (r^{m} ∑_{|μ′|=m} c_{μ′}^{(m)} · d_{μ′}^{(m)}) · H^{n−1−m} × H^{m}

Note that if rd ≡ −1 mod n, then the coefficient (−1)^{m} will appear on the right hand side.

Assume that the following formulae holds

∑_{|μ′|=m} c_{μ′}^{(m)} · d_{μ′}^{(m)} ≡ d^{m} mod n.  (11)

Then the coefficient of (g^{t} ∘ f)^{(m)} is congruent to 1 modulo n. We claim that it is possible to modify the cycles g^{t} and f by adding cycles divisible by n in such a way that the coefficient of (g^{t} ∘ f)^{(m)} becomes equal to 1 for each m.

First, we modify the cycle f. For each m, 0 ≤ m ≤ n − 1, we do the following procedure. Consider the m-th summand

f^{(m)} = (−1)^{m} ∑_{|μ′|=m} c_{μ′} · H^{n−1−m} × ω_{μ′}

and its coefficients c_{μ′}^{(m)}. For m = 0 and 1 there is only one additive generator of CH^{m}(𝔾_{d}(n)) (ω_{0} and ω_{1}) and the respective coefficients are c^{(0)} = c^{(1)} = 1. So we set α^{(0)} = f^{(0)} and α^{(1)} = f^{(1)}. For 1 < m ≤ n − 1 the number of generators of CH^{m}(𝔾_{d}(n)) is greater than 1 and all the coefficients c_{μ′}^{(m)} are coprime with n, in particular, they are all non-zero. In this case we can modify each c_{μ′}^{(m)} modulo n by adding cycles of the kind a · nω_{μ′}, a ∈ ℤ, to the cycle f^{(m)} in such a way that the greatest common divisor of resulting coefficients, denoted by α_{μ′}^{(m)}, becomes equal to 1. As a result, we obtain a new cycle α^{(m)} having the coefficients α_{μ′} instead of c_{μ′}.

5.4 Definition. Define a cycle α as α = ∑_{m=0}^{n−1} α^{(m)}. By construction of α^{(m)} we have

• α^{(0)} = α^{(1)} = 1;

• α is rational (congruent modulo n to f);

• all the coefficients α_{μ′} are coprime with n;

• for each m the g.c.d of coefficients α_{μ′}^{(m)} is 1;

• for each m the coefficient of (g^{t} ∘ α)^{(m)} is congruent to 1 modulo n.

Next we modify the second cycle g^{t}. For each m we apply the following obvious observation

5.5 Lemma. Let a_{1}, …, a_{l} be a finite set of integers with g.c.d. = 1. Assume ∑_{i} a_{i}b_{i} ≡ 1 mod n for some integers b_{i}. Then there exist integers b′_{i} ≡ b_{i} mod n such that ∑_{i} a_{i}b′_{i} = 1. to the congruence

∑_{|μ′|=m} c_{μ′}^{(m)}(r^{m}d_{μ′}^{(m)}) ≡ 1 mod n,

where a_{i} = α_{μ′}^{(m)} are coefficients of the cycle α^{(m)} and b_{i} = r^{m}d_{μ′}^{(m)} are coefficients of the cycle (g^{t})^{(m)}. As a result, we obtain a new cycle β^{(m)} having coefficients b′_{i} instead of b_{i}.

5.6 Definition. Define a cycle β as β = ∑_{m=0}^{n−1} β^{(m)}. By construction of β^{(m)} we have

• β is rational (congruent modulo n to g^{t})

• for each m the coefficient of (β ∘ α)^{(m)} is equal to 1.

To finish the proof observe that the constructed cycles α and β are rational and the composition β ∘ α is the diagonal cycle. This implies that the composition α∘β is a rational projector which gives rise to the decomposition of motives with integral coefficients (see (4))

M(SB_{d}(B^{op})) ≃ M(SB(A)) ⊕ H

for some motive H.

6 The proof of (11)

In the present section we prove the following

6.1 Lemma.

∑_{|μ′|=m} c_{μ′}^{(m)} · d_{μ′}^{(m)} ≡ d^{m} mod n,

Proof. First, we express the coefficients c_{μ′} and d_{μ′} in terms of binomial determinants. According to [Fu98, Example 14.7.11] we have

c_{μ′} = deg(ω_{1}^{m} · ω_{μ}) = \frac{m! · b_{0}! b_{1}! … b_{d−1}!}{a_{0}! a_{1}! … a_{d−1}!} · |\binom{a_{i}}{b_{j}}|_{0≤i,j≤d−1}  (12)

where a_{i} = n−d+i−μ_{i+1} and b_{j} = j for i, j = 0 … d−1. Observe that the sets a = (a_{0}, …, a_{d−1}) and b = (b_{0}, …, b_{d−1}) with 0 ≤ a_{0} < … < a_{d−1} ≤ n−1 and 0 ≤ b_{0} < … < b_{d−1} ≤ n − 1 correspond to the classes of Schubert varieties ω_{μ} and ω_{N} = {pt} of dimensions ∑_{i=0}^{d−1}(a_{i} − i) = m and ∑_{j=0}^{d−1}(b_{j} − j) = 0 respectively.

From another hand side, we have

d_{μ′} = |\binom{n − i}{μ̃_{j} + n − d − j}|_{1≤i.j≤n−d} = |(ã_{i} ; b̃_{j})|_{0≤i,j,≤n−d−1}

where ã_{i} = d + i and b̃_{j} = μ̃_{n−d−j} + j for i, j = 0 … n − d − 1. Here the sets ã = (ã_{0}, …, ã_{n−d−1}) and b̃ = (b̃_{0}, …, b̃_{n−d−1}) with 0 ≤ ã_{0} < … ã_{n−d−1} ≤ n−1 and 0 ≤ b̃_{0} < … < b̃_{n−d−1} ≤ n − 1 correspond to the classes of Schubert varieties on the dual Grassmannian 𝔾_{n−d}(n) of dimensions N and N − m respectively. Observe that the Schubert variety corresponding to the set b̃ is dual to the Schubert variety corresponding to the partition μ̃. Hence, we have

ω_{1}^{m} = \frac{m! · b̃_{0}! b̃_{1}! … b̃_{n−d−1}!}{ã_{0}! ã_{1}! … ã_{n−d−1}!}d_{μ′} · ωb̃

And by duality we obtain

\frac{m! · b̃_{0}! b̃_{1}! … b̃_{n−d−1}!}{ã_{0}! ã_{1}! … ã_{n−d−1}!}d_{μ′} = \frac{m! · b_{0}! b_{1}! … b_{d−1}!}{a_{0}! a_{1}! … a_{d−1}!} · |\binom{a_{i}}{b_{j}}|_{0≤i,j≤d−1}

Since the integers a_{0}, …, a_{d−1} form the complement of b̃_{0}, …, b̃_{n−d−1} in the set of integers from 0 to n − 1 (see [Fu98, Example 14.7.5]), we obtain

d_{μ′} = |\binom{a_{i}}{b_{j}}|_{0≤i,j≤d−1}  (13)

According to [Fu98, Example 14.7.11] we express the binomial determinant appearing in (12) and (13) in terms of Vandermonde determinant D_{a} = ∏_{i<j}(a_{j} − a_{i}) and get

c_{μ′} = \frac{m! · D_{a}}{a_{0}! a_{1}! … a_{d−1}!},  d_{μ′} = \frac{D_{a}}{0! 1! … (d − 1)!}.  (14)

Then the formulae (11) we want to prove turns into

\frac{m!}{0! 1! … (d − 1)!} · ∑_{a} \frac{D_{a}^{2}}{a_{0}!a_{1}! … a_{d−1}!} ≡ d^{m} mod n,  (15)

where the sum is taken over all sets of integers a = (a_{0}, …, a_{d−1}) such that 0 ≤ a_{0} < … < a_{d−1} ≤ n − 1 and ∑_{i=0}^{d−1} a_{i} = m + \frac{d(d−1)}{2}.

The case d = 2. In this case (15) follows from the following elementary fact

\frac{m!}{2} · ∑_{{0≤x_{1},x_{2} \atop x_{1}+x_{2}=m+1}} \frac{(x_{1} − x_{2})^{2}}{x_{1}!x_{2}!} = 2^{m}  (16)

To prove it consider the following chain of identities

\frac{m! · (x_{1} − x_{2})^{2}}{2 · x_{1}!x_{2}!} = \frac{(m + 1)!}{x_{1}!(m + 1 − x_{1})!} · (\frac{m + 1}{2} − \frac{2x_{1}(m + 1 − x_{1})}{m + 1}) =

= \frac{m + 1}{2}\binom{m + 1}{x_{1}} − 2m\binom{m − 1}{x_{1} − 1}

Then taking the sum we obtain the desired equality

∑_{x_{1}=0}^{m+1} \frac{m + 1}{2}\binom{m + 1}{x_{1}} − 2m\binom{m − 1}{x_{1} − 1} = \frac{m + 1}{2} · 2^{m+1} − 2m · 2^{m−1} = 2^{m}

Observe that for m < n − 1 the left hand side of (15) coincides with the left hand side of (16), hence, we obtain the equality in (15) (not just a congruence modulo n). For m = n − 1 the left hand side of (15) is equal to 2^{n−1} − n.

The general case. It turns out that the identity (16) is a particular case of the following combinatorial identity known as Robinson-Schensted correspondence (see [Fu97, 4.3.(5)])

∑_{|ξ|=m} d_{ξ}(d) · f^{ξ} = d^{m},  (17)

where the sum is taken over all partitions ξ = (ξ_{1} ≥ ξ_{2} ≥ … ≥ ξ_{d} ≥ 0) with |ξ| = m, d_{ξ}(d) denote the number of Young tableaux on the shape ξ whose entries are taken from the set (1, …, d) and f^{ξ} denote the number of standard tableaux on the shape ξ.

By using Hook length formulae (see [Fu97, 4.3, Exercise 9]) we obtain

f^{ξ} = \frac{m! · D_{l}}{l_{0}! … l_{d−1}!},  (18)

where l = (l_{0}, …, l_{d−1}) is a strictly increasing set of non-negative integers defined from the partition ξ = (ξ_{1}, …, ξ_{d}) by l_{d−i} = ξ_{i} + d − i and D_{l} is the Vandermonde determinant for l. By definition we have ∑_{i=0}^{d−1}(l_{i} − i) = m.

By [GV85, Corollary 13] we have

d_{ξ}(d) = |\binom{l_{i}}{j}|_{0≤i,j,≤d−1} = \frac{D_{l}}{0! 1! … (d − 1)!}  (19)

Observe that if the set l is bounded by n − 1, i.e., l_{d−1} ≤ n − 1, the expressions (18) and (19) coincide with the expressions (14) defining the coefficients c_{μ′} and d_{μ′} respectively (take ξ = μ′ and a = l).

Assume that l_{d−1} ≥ n. Then l_{d−2} ≤ n − 1, i.e., l_{d−1} is the only element of l which exceeds n − 1. Indeed, if this is not the case then we have a sequence of inequalities

n − 1 + \frac{d(d−1)}{2} ≥ m + \frac{d(d−1)}{2} = 0 + 1 + … + l_{d−3} + l_{d−2} + l_{d−1} ≥
≥ 0 + 1 + … + (d − 3) + n + (n + 1) = \frac{(d−3)(d−2)}{2} + 2n + 1

which can be rewritten as

d ≥ \frac{n + 5}{2}.

But we have assumed from the beginning that d ≤ [\frac{n}{2}], contradiction. Moreover, by the similar arguments one can check that l_{d−1} < 2n. Now consider the product x = d_{ξ}(d) · f^{ξ}, when ξ is a partition for which

the respective l_{d−1} ≥ n. Since n is prime, the denominator of x is divisible by n but not by n^{2}. Since x is an integer, the numerator of x must be divisible by n as well. From this we conclude that D_{l} is divisible by n. But the numerator of x is the product of the square of D_{l} by something, hence, it is divisible by n^{2}. So we obtain that x must be divisible by n. This means that modulo n the left hand side of (17) is congruent to the left hand side of (15).

7 Grassmannian 𝔾(2, n)

The goal of the present is to extend Theorem 1.1 to the case of algebras of an arbitrary odd degree n ≥ 5 and d = 2 7.1 Theorem. Let A and B be two central simple algebras of an odd degree n over a field F generating the same subgroup in the Brauer group Br(F). Then the motive of a Severi-Brauer variety SB(A) is a direct summand of the motive of a generalized Severi-Brauer variety SB_{2}(B) if and only if

[A] = ±2 [B] in Br(F).

Proof Consider the cycles g and f defined in Section 5. Clearly, g and f are rational, since Lemma 2.6 holds for any algebras.

(⇒) Repeating the arguments of 3.3 for an odd integer n and d = 2 one obtains (in view of [Ka95, Corollary 4]) that the subgroup of rational cycles of CH(X_{s}) modulo n is generated in codimension N = dim 𝔾_{d}(n) by the cycles

\frac{n}{(n,N−|λ|)} · H^{N−|λ|} × 1 · Δ_{λ}(c(Q_{s})),

for all partitions λ with N − (n − 1) ≤ |λ| ≤ N. Observe that in the case |λ| = N one obtains precisely the cycle g.

Since the coefficient \frac{n}{(n,N−|λ|)} is divisible by n when |λ| = N − 1, we may exclude the cycles with |λ| = N − 1 from the set of generators. The latter implies that the last two non-trivial monomials (see (9)) of any rational cycle in CH^{N}(X_{s}) come only from the cycle g. And we finish the proof as in Section 5.

(⇐) We claim that it is still possible to modify cycles f and g^{t} modulo n in such a way that the obtained rational cycles α and β will satisfy the property β ∘ α = id, hence, providing the desired motivic decomposition for M(SB_{2}(B)).

The following lemma allows us to take α = f (see Definition 5.4).

7.2 Lemma. Fix a codimension m, 0 ≤ m ≤ n − 1. For any partition ρ with |ρ| = m let c_{ρ}^{(m)} be the coefficient appearing in Lemma 5.2. Then the greatest common divisor of all the coefficients c_{ρ}^{(m)} is 1.

Proof. Since d = 2, for any codimension m which is less than n − 1, there is a partition ρ = (m, 0) with c_{ρ}^{(m)} = 1. In the last codimension m = n − 1 consider two partitions ρ = (1, n − 2) and ρ′ = (2, n − 3). By (10) the respective coefficients c_{ρ}^{(n−1)} and c_{ρ′}^{(n−1)} are equal to deg(1, n − 1) = n − 2 and deg(2, n−2) = \frac{(n−1)(n−4)}{2}. Since n is odd, n−2 and \frac{(n−1)(n−4)}{2} are coprime.

Choose β as in the Section 5. To finish the proof we have to prove the congruence (11). But this was done already in Section 6, where we treated the case d = 2. Namely, we proved that for m < n − 1 there is an exact equality (not just a congruence modulo n) and for m = n−1 the left hand side of (11) is, indeed, equal to 2^{n−1} − n. This finishes the proof of Theorem 7.1.

Acknowledgements I would like to thank B. Calmés, N. Karpenko, I. Panin, M. Rost and N. Semenov for stimulating discussions on the subject of the present paper. I am grateful to Max-Planck-Institut für Mathematik in Bonn and University of Bielefeld for hospitality and support. This work was partially supported by RTN-Network HPRN-CT-2002-00287.

References

[CPSZ] B. Calmès, V. Petrov, N. Semenov, K. Zainoulline, Chow motives of twisted flag varieties, Compositio Math. (to appear).

[CGM05] V. Chernousov, S. Gille and A. Merkurjev, Motivic decomposition of isotropic projective homogeneous varieties, Duke Math. J. 126 (2005), no. 1, 137–159.

[Fu98] W. Fulton, Intersection Theory, Second edition, Springer-Verlag, Berlin-Heidelberg, 1998.

[Fu97] W. Fulton, Young tableaux. With applications to representation theory and geometry. London Mathematical Society Student Texts, 35. Cambridge University Press, Cambridge, 1997. x+260 pp.

[GV85] I. Gessel, G. Viennot, Binomial determinants, paths, and hook length formulae. Adv. in Math. 58 (1985), no. 3, 300–321.

[IK00] O. Izhboldin and N. Karpenko, Some new examples in the theory of quadratic forms, Math. Z. 234 (2000), 647–695.

[Ka95] N. Karpenko, On topological filtration for Severi-Brauer varieties., Proc. Symp. Pure Math. 58.2 (1995), 275–277.

[Ka96] N. Karpenko, Grothendieck Chow-motives of Severi-Brauer varieties, St.-Petersburg Math. J. 7 (1996), no. 4, 649–661.

[Ka00] N. Karpenko, Criteria of motivic equivalence for quadratic forms and central simple algebras, Math. Ann. 317 (2000), 585–611.

[Ka01] N. Karpenko, Cohomology of relative cellular spaces and isotropic flag varieties, St. Petersburg Math. J. 12 (2001), no. 1, 1–50.

[KMRT98] M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol. The book of involutions. AMS Colloquium Publications, vol. 44, 1998.

[Ma68] Y. Manin, Correspondences, motives and monoidal transformations, Matematicheskij Sbornik 77 (119) (1968), no. 4, 475–507 (in Russian). Engl. transl.: Math. USSR Sb. 6 (1968), 439–470.

[Pa94] I. Panin, On the Algebraic K-Theory of Twisted Flag Varieties. K-Theory J. 8 (1994), 541–585.