Decompositions of Motives

of Generalized Severi-Brauer Varieties

Maksim Zhykhovich

Received: October 13, 2011 Revised: November 11, 2011

Communicated by Alexander Merkurjev

Abstract. Let p be a positive prime number and X be a Severi-Brauer variety of a central division algebra D of degree p^{n}, with n ≥ 1. We describe all shifts of the motive of X in the complete motivic decomposition of a variety Y, which splits over the function field of X and satisfies the nilpotence principle. In particular, we prove the motivic decomposability of generalized Severi-Brauer varieties X(p^{m}, D) of right ideals in D of reduced dimension p^{m}, m = 0, 1, …, n − 1, except the cases p = 2, m = 1 and m = 0 (for any prime p), where motivic indecomposability was proven by Nikita Karpenko.

2010 Mathematics Subject Classification: 14L17; 14C25 Keywords and Phrases: Central simple algebras, generalized Severi-Brauer varieties, Chow groups and motives.

Contents

1. Introduction 152 2. Chow motives with finite coefficients 153 3. Main results 154 4. Complete motivic decompositions 159 References 164

1. Introduction

Let F be an arbitrary field and p be a prime numbre. For any integer l, we write v_{p}(l) for the exponent of the highest power of p dividing l. Let D be a central division F -algebra of degree p^{n}, with n ≥ 1. We write X(p^{m}, D) for the generalized Severi-Brauer variety of right ideals in D of reduced dimension p^{m} for m = 0, 1, …, n. In particular, X(p^{n}, D) = Spec F and X(1, D) is the usual Severi-Brauer variety of D. The generalized Severi-Brauer varieties are twisted forms of grassmannians (see [11, §I.1.C]). For each integer m = 0, ..., n we define an upper motive M_{m,D} in the category of Chow motives with coefficients in 𝔽_{p}. This is the summand of the complete motivic decomposition of the variety X(p^{m}, D) such that the 0-codimensional Chow group of M_{m,D} is non-zero. Let A be a central simple F -algebra, such that the p-primary component of A is Brauer equivalent to D. Let 𝔛_{A} be the class of finite direct products of projective (Aut A)-homogeneous F -varieties (the class 𝔛_{A} includes the generalized Severi-Brauer varieties of the algebra A). Nikita Karpenko proved the following theorem [9, Theorem 3.8]. Any variety X from 𝔛_{A} decomposes into a sum of shifts of the motives M_{m,D} with m ≤ v_{p}(ind A_{F (X)}). This theorem shows that the motivic indecomposable summands M_{m,D} of the generalized Severi-Brauer varieties X(p^{m}, D) are some kind of “basic material” to constuct the motives of more general class of varieties. This gives us a motivation to understand the structure of the upper motives M_{m,D} themselves. It was known that in the cases m = 0 (Severi-Brauer case, see Corollary 3.2) and m = 1, p = 2 ( [9, Theorem 4.2]) the motive M_{m,D} coincides with the whole motive of the variety X(p^{m}, D) (that is, the motive of this variety is indecomposable). Taking into account these cases and the fact that any generalized Severi-Brauer variety X(p^{m}, D) is p-incompressible [9, Theorem 4.3] (this condition is weaker than motivic indecomposability), one probably expected that the Chow motive with coefficients in 𝔽_{p} of any variety X(p^{m}, D) is indecomposable. But, except the two already mentioned cases, the motivic decomposability of generalized Severi-Brauer variety X(p^{m}, D) was proven in [14]. This article is an extended version of [14]. To show that the motive of the variety X(p^{m}, D) is decomposable, we prove in [14] that some shifts of M_{0,D} are the motivic summands of X(p^{m}, D). Let Y be a F -variety satisfying the nilpotence principle and such that it splits over the function field of X(1, D). For example, one can take for Y any generalized Severi-Brauer variety X(p^{m}, D) and, more generally, any variety from 𝔛_{A}. The main result of the present article (Theorem 3.4) find all shifts of M_{0,D} in the complete motivic decomposition of the variety Y in terms of some subgroups of rational cycles. These subgroups can be described in the case of generalized Severi-Brauer variety X(p^{m}, D) (see Proposition 3.7). As consequence, we prove the motivic decomposability of these varieties in Corollary 3.8. With Theorem 3.4 in hand, we find in §4 more examples (comparing to [14]) of complete motivic decompositions of generalized Severi-Brauer varieties X(p^{m}, D) and therefore we describe the upper motives M_{m,D} in that cases. Theorem 3.4 also permits to prove differently (see Corollary 3.12, [3, Corollary 5]) a particular case of the following conjecture.

Conjecture 1.1. Let D be a central division F -algebra. Let K/F be a field extension such that D_{K} is still division. Then (M_{m,D})_{K} is still indecomposable.

Acknowledgements. I would like to express particular gratitude to Nikita Karpenko, my Ph.D. thesis adviser, for introducing me to the subject, raising the question studied here, and guiding me during this work. I am also very grateful to Olivier Haution, Sergey Tikhonov and Skip Garibaldi for very useful discussions.

2. Chow motives with finite coefficients

A variety is a separated scheme of finite type over a field. Our basic reference for Chow groups and Chow motives (including notations) is [4]. We fix an associative unital commutative ring Λ. Given a variety X over a field F, we write Ch(X) and CH(X) respectively for its Chow group with coefficients in Λ and for its integral Chow group. For a field extension L/F we denote by X_{L} the respective extension of scalars. An element of Ch(X_{L}) is called F -rational, if it lies in the image of the homomorphism Ch(X) → Ch(X_{L}). Our category of motives is the category CM(F, Λ) of graded Chow motives with coefficients in Λ, [4, definition of § 64]. By a sum of motives we always mean the direct sum. We also write Λ for the motive M(SpecF) ∈ CM(F, Λ). A Tate motive is the motive of the form Λ(i) with i an integer. Let X be a smooth complete variety over F and let M be a motive. We call M split if it is a finite sum of Tate motives. We call X split, if its integral motive M(X) ∈ CM(F, ℤ) (and therefore the motive of X with an arbitrary coefficient ring Λ) is split. We call M or X geometrically split, if it splits over a field extension of F. For a geometrically split variety X over F, we denote by X̄ the scalar extension of X to a splitting field of its motive and we write C̄h(X) for the subring of F -rational cycles in Ch(X̄). Note that the rings Ch(X̄) and C̄h(X) are independent on the choice of a splitting field. Over an extension of F the geometrically split motive M becomes isomorphic to a finite sum of Tate motives. We write rk M and rk_{i} M for respectively the number of all summands and the number of summands Λ(i) in this decomposition, where i is an integer. Note that these two numbers do not depend on the choice of a splitting field extension. We say that X satisfies the nilpotence principle, if for any field extension E/F and any coefficient ring Λ, the kernel of the change of field homomorphism End(M(X)) → End(M(X)_{E}) consists of nilpotents. Any projective homogeneous (under an action of a semisimple affine algebraic group) variety is geometrically split and satisfies the nilpotence principle, [4, Theorem 92.4 with Remark 92.3].

A complete decomposition of an object in an additive category is a finite direct sum decomposition with indecomposable summands. We say that the Krull-Schmidt principle holds for a given object of a given additive category, if every direct sum decomposition of the object can be refined to a complete one (in particular, a complete decomposition exists) and there is only one (up to a permutation of the summands) complete decomposition of the object. We have the following theorem:

Theorem 2.1. ([2, Theorem 3.6 of Chapter I]). Assume that the coefficient ring Λ is finite. The Krull-Schmidt principle holds for any shift of any summand of the motive of any geometrically split F -variety satisfying the nilpotence principle.

We will use the following two statements in the next section.

Lemma 2.2. Assume that the coefficient ring Λ is a field. Let X be a split variety. Then the bilinear form 𝔟 : Ch(X) × Ch(X) → Λ, 𝔟(x, y) = deg(x · y) is non-degenerate.

Proof. Since the motive of X decomposes into a finite sum of Tate motives, we have the following decomposition for the diagonal class Δ ∈ Ch_{dim X}(X × X):

Δ = a_{1} × b_{1} + ... + a_{n} × b_{n} ,

where a_{1}, ..., a_{n} and b_{1}, ..., b_{n} are the homogeneous elements in Ch(X), such that for any i, j = 1, ..., n the degree deg(a_{i} · b_{j}) ∈ Λ is 0 for i ≠ j and 1 for i = j. Note that dim_{Λ} Ch(X) = rk M(X) = n < ∞. Therefore, to prove the lemma it suffices to show that rad 𝔟 = {0}. Suppose that x ∈ rad 𝔟 (this means 𝔟(x, y) = 0 for any y ∈ Ch(X)). Then we have

x = Δ_{∗}(x) = ∑_{i=1}^{n} deg(x · a_{i})b_{i} = ∑_{i=1}^{n} 𝔟(x, a_{i})b_{i} = 0 .

□

Lemma 2.3. Assume that the coefficient ring Λ is finite. Let X be a variety satisfying the nilpotence principle. Let f ∈ End(M(X)) and 1_{E} = f_{E} ∈ End(M(X)_{E}) for some field extension E/F. Then f^{n} = 1 for some positive integer n.

Proof. Since X satisfies the nilpotence principle, we have f = 1 + ε, where ε is nilpotent. Let n be a positive integer such that ε^{n} = 0 = nε. Then f^{n^{n}} = (1 + ε)^{n^{n}} = 1 because the binomial coefficients \binom{n^{n}}{i} for i < n are divisible by n. □

3. Main results

Let p be a positive prime integer. The coefficient ring Λ is 𝔽_{p} in this section. Let F be a field. Let D be a central division F -algebra of degree p^{n}. We write X(p^{m}, D) for the generalized Severi-Brauer variety of right ideals in D of reduced dimension p^{m} for m = 0, 1, …, n.

Lemma 3.1. Let E/F be a splitting field extension for X = X(1, D). Then the subgroup of F -rational cycles in Ch_{dimX}(X_{E} ×X_{E}) is generated by the diagonal class.

Proof. By [7, Proposition 2.1.1], we have C̄h^{i}(X) = 0 for i > 0. Since the (say, first) projection X^{2} → X is a projective bundle, we have a (natural with respect to the base field change) isomorphism Ch_{dim X}(X^{2}) ≃ Ch(X). Passing to C̄h, we get an isomorphism C̄h_{dim X}(X^{2}) ≃ C̄h(X) = C̄h^{0}(X) showing that dim_{𝔽_{p}} C̄h_{dim X}(X^{2}) = 1. Since the diagonal class in C̄h_{dim X}(X^{2}) is non-zero, it generates all the group. □

Corollary 3.2. (cf. [7, Theorem 2.2.1]). The motive with coefficients in 𝔽_{p} of the Severi-Brauer variety X = X(1, D) is indecomposable.

Proof. To prove that our motive is indecomposable it is enough to show that End(M(X))= Ch_{dimX}(X × X) does not contain nontrivial projectors. Let π ∈ Ch_{dimX}(X × X) be a projector. By Lemma 3.1, π_{E} is zero or equal to 1_{E}. Since X satisfies the nilpotence principle, π is nilpotent in the first case, but also idempotent, therefore π is zero. Lemma 2.3 gives us π = 1 in the second case. □

Nikita Karpenko proved the motivic indecomposability of generalized Severi-Brauer varieties also in the case p = 2, m = 1.

Theorem 3.3. (cf. [9, Theorem 4.2]). Let D be a central division F -algebra of degree 2^{n} with n ≥ 1. Then the motive with coefficients in 𝔽_{2} of the variety X(2, D) is indecomposable.

Corollary 3.8 of the following main theorem will show that Corollary 3.2 and Theorem 3.3 give us the only cases when the motive of generalized Severi-Brauer variety is indecomposable.

Theorem 3.4. Let D be a central division F -algebra of degree p^{n} with n ≥ 1. Let X be the Severi-Brauer variety X(1, D) and Y be a variety satisfying nilpotence principle, such that Y is split over the function field of X. Then for any integer k the number of copies M(X)(k) in the complete motivic decomposition of Y is equal to dim_{𝔽_{p}} f_{∗} C̄h_{dim Y −k}(X × Y), where f is a projection onto the second factor.

Proof. We fix an integer k and we note the motive M(X)(k) simply by M. Let r be the number of copies of M in the complete motivic decomposition of Y. We note V := f_{∗} C̄h_{dim Y −k}(X × Y) and r′ := dim_{𝔽_{p}} V. We want to show that r = r′. Let A_{1}, ..., A_{m} and B_{1}, ..., B_{n} be the motives. We recall that a morphism between the motives ⨁_{i=1}^{m} A_{i} and ⨁_{j=1}^{n} B_{j} is given by an n × m-matrix of morphisms A_{i} → B_{j}. The composition of morphisms is the matrix multiplication.

The motive M^{⊕r} is a summand of the motive M(Y). Therefore there exist two morphisms α = (α_{1}, ..., α_{r})^{t} ∈ Hom(M^{⊕r}, M(Y)) and β = (β_{1}, ..., β_{r}) ∈ Hom(M(Y), M^{⊕r}), such that

β ∘ α = (β_{j} ∘ α_{i})_{1≤i,j≤r} = (δ_{i,j})_{1≤i,j≤r} ,

where (δ_{i,j})_{1≤i,j≤r} is the identity morphism in Hom(M^{⊕r}, M^{⊕r}) (that is δ_{i,j} is zero if i ≠ j and δ_{i,j} is the diagonal class Δ in Corr_{0}(X, X) if i = j). Let E = F (X), then E/F is a splitting field extension for the varieties X and Y (here we use the condition of the theorem) and X_{E} ≃ ℙ^{d}, where d = p^{n} −1. We know that Δ_{E} = ∑_{i=0}^{d} h^{i} × h^{d−i}, where h is the hyperplane class in Ch^{1}(X_{E}). For any 1 ≤ i ≤ r we have

(β_{i})_{E} ∘ (α_{i})_{E} = (δ_{i,i})_{E} = Δ_{E} =

= h^{0} × h^{d} + ∑_{i=1}^{d} h^{i} × h^{d−i} = [X_{E}] × [pt] + ∑_{i=1}^{d} h^{i} × h^{d−i} ,

where [pt] is the class of a rational point in Ch(X_{E}). Therefore the correspondences β_{i} ∈ Ch_{dim Y −k}(Y_{E} × X_{E}) and α_{i} ∈ Ch_{d+k}(X_{E} × Y_{E}) have to be of the following form:

(3.5)  (β_{i})_{E} = b_{i} × [pt] + …,

where b_{i} ∈ Ch^{k}(Y_{E}) is non-zero and where “… ” stands for a linear combination of only those terms whose first factor has codimension > k,

(3.6)  (α_{i})_{E} = [X_{E}] × b_{i}^{∗} + …,

where b_{i}^{∗} ∈ Ch_{k}(Y_{E}) is such that deg(b_{i} · b_{i}^{∗}) = 1 and where “… ” stands for a linear combination of only those terms whose second factor has dimension > k. For any i ≠ j we have (β_{j})_{E} ∘ (α_{i})_{E} = 0, this implies that deg(b_{j} · b_{i}^{∗}) = 0. Therefore the system of vectors {b_{1}^{∗}, ..., b_{r}^{∗}} from the vector space Ch(Y_{E}) is dual to the system of vectors {b_{1}, ..., b_{r}} with respect to the bilinear form 𝔟 : Ch(Y_{E}) × Ch(Y_{E}) → 𝔽_{p}, 𝔟(x_{1}, x_{2}) = deg(x_{1} · x_{2}). It follows that the vectors b_{1}, ..., b_{r} are linearly independent. Since b_{i} = f_{∗}((β_{i}^{t})_{E}), then b_{i} ∈ V for any 1 ≤ i ≤ r. Therefore r ≤ r′. Let now b_{1}, ..., b_{r′} be a basis of V. We want to show that M^{⊕r′} is a motivic summand of Y. By the definition of V, there exist correspondences β_{1}, ..., β_{r′} ∈ Ch_{dim Y −k}(Y ×X) of the form (3.5), such that b_{i} = f_{∗}((β_{i}^{t})_{E}). Since the variety Y_{E} is split, then by Lemma 2.2 the bilinear form 𝔟 is non-degenerate. It follows that there exists a system of vectors {b_{1}^{∗}, ..., b_{r′}^{∗}} from the vector space Ch(Y_{E}), which is dual to the system of vectors {b_{1}, ..., b_{r′}}. For any 1 ≤ i ≤ r′ we construct the correspondence α_{i} ∈ Ch_{d+k}(X × Y), such that (α_{i})_{E} is of the form (3.6), by the following way. The pull-back homomorphism

g : Ch(X × Y) → Ch(Y_{F (X)}) = Ch(Y_{E})

with respect to the morphism Y_{F (X)} = (Spec F (X)) × Y → X × Y given by the generic point of X is surjective by [4, Corollary 57.11]. We define α_{i} ∈ Ch(X × Y) as a cycle whose image in Ch(Y_{E}) under the surjection g is b_{i}^{∗}. We have (α_{i})_{E} = [X_{E}] × b_{i}^{∗} + …, so (α_{i})_{E} is of the form (3.6). The r′-tuples (α_{1}, ..., α_{r′})^{t} and (β_{1}, ..., β_{r′}) give us respectively two morphisms α ∈ Hom(M^{⊕r′}, M(Y)) and β ∈ Hom(M(Y), M^{⊕r′}). By the construction of α and β, the matrix (mult((β_{j})_{E} ∘ (α_{i})_{E}))_{1≤i,j≤r} is an identity matrix. Then, by Lemma 3.1, β_{E} ∘ α_{E} = ((β_{j})_{E} ∘ (α_{i})_{E})_{1≤i,j≤r} = 1_{E}, where we note simply by 1 the identity morphism ((δ_{i,j}))_{1≤i,j≤r} in Hom(M^{⊕r}, M^{⊕r}). Let 𝔛 be a disjoint union of r′ copies of X, then Hom(M(𝔛), M(𝔛)) = Hom(M^{⊕r′}, M^{⊕r′}). According to [4, Theorem 92.4] the variety 𝔛 satisfies the nilpotence principle. By Lemma 2.3, there exist a positive integer n, such that (β ∘ α)^{n} = 1 (we apply Lemma 2.3 to the variety 𝔛 and to the morphism β ∘ α ∈ Hom(M(𝔛), M(𝔛))). The morphisms α and (β ∘ α)^{n−1} ∘ β give the isomorphism between the motive M^{⊕r′} and a direct summand of M(Y). Therefore r′ ≥ r and then finally r′ = r. □

Proposition 3.7. Let D be a central division F -algebra of degree p^{n} with n ≥ 1. Let X and Y be respectively the varieties X(1, D) and X(p^{m}, D), 0 ≤ m < n. Let E/F be a splitting field extension for the variety X, let T_{1} and T_{p^{m}} be the tautological bundles of rank 1 and p^{m} on X_{E} and Y_{E} respectively. Then the subring of F -rational cycles in Ch(X_{E} ×Y_{E}) is generated by the Chern classes of the vector bundle T_{1} ⊠ (−T_{p^{m}})^{∨} (we lift the bundles T_{1} and T_{p^{m}} on X_{E} × Y_{E} and then take a product).

Proof. Let T av be the tautological vector bundle on X. The product X × Y considered over X (via the first projection) is isomorphic (as a scheme over X) to the Grassmann bundle G_{r}(T av) of r-dimensional subspaces in T av (cf. [6, Proposition 4.3]), where r = p^{n} − p^{m}. Let T be the tautological r-dimensional vector bundle on G_{r}(T av). By [5, Example 14.6.6], the Chow ring Ch(G_{r}(T av)) as an algebra over Ch(X) is generated by Chern classes c_{0}(T), c_{1}(T), ..., c_{r}(T). By [7, Proposition 2.1.1], we have C̄h(X) = C̄h^{0}(X) = ℤ · [X_{E}]. Therefore the Chow ring C̄h(X ×Y) ≃ C̄h(G_{r}(T av)) is generated (as a ring) by Chern classes c_{0}(T_{E}), ..., c_{r}(T_{E}). Since there exists an isomorphism (cf. [6, Proposition 4.3]): T_{E} ≃ T_{1} ⊠ (−T_{p^{m}})^{∨}, we are done.

□

Corollary 3.8. The motive with coefficients in 𝔽_{p} of the variety X(p^{m}, D) is decomposable for p = 2, 1 < m < n and for p > 2, 0 < m < n. In these cases M(X(1, D))(k) is a summand of M(X(p^{m}, D)) for 2 ≤ k ≤ p^{n} − p^{m}.

Proof. We use the notations: X = X(1, D), Y = X(p^{m}, D), d = dim(X(1, D)) = p^{n} − 1, r = p^{n} − p^{m}. Let E = F (X), then E/F is a splitting field extension for the variety X (and also for Y). Over the field E the algebra D becomes isomorphic to End_{E}(V) for some E-vector space V of dimension d + 1 = p^{n}. We have X_{E} ≃ ℙ^{d}(V) and Y_{E} ≃ G_{p^{m}}(V). Let T_{1} and T_{p^{m}} be the tautological bundles of rank 1 and p^{m} on X_{E} and Y_{E} respectively. We note by T the r-dimensional vector bundle T_{1} ⊠ (−T_{p^{m}})^{∨} on X_{E} × Y_{E}. By Proposition 3.7, the ring C̄h(X × Y) is generated by Chern classes of the vector bundle T. Let h = c_{1}(T_{1}) ∈ Ch^{1}(X_{E}) (then −h is the hyperplane class in Ch^{1}(X_{E})) and c_{i} = c_{i}((−T_{p^{m}})^{∨}) ∈ Ch^{i}(Y_{E}), 0 ≤ i ≤ r. Then by [5, Remark 3.2.3(b)]

(3.9)  c_{t}(T) = c_{t}(T_{1} ⊠ (−T_{p^{m}})^{∨}) = ∑_{i=0}^{r}(1 + (h × 1)t)^{r−i}(1 × c_{i})t^{i} .

It follows from the conditions of the corollary that the binomial coefficients \binom{p^{n}−p^{m}}{2}, \binom{p^{n}−p^{m}}{p^{m}−1} are divisible by p and \binom{p^{n}−p^{m}−1}{p^{m}−2} ≡ (−1)^{p^{m}−2} mod p. Therefore

c_{1}(T) = (p^{n} − p^{m})h × 1 + 1 × c_{1} = 1 × c_{1} ,

c_{2}(T) = \binom{p^{n} − p^{m}}{2}h^{2} × 1 + (p^{n} − p^{m} − 1)h × c_{1} + 1 × c_{2} = −h × c_{1} + 1 × c_{2} ,

c_{p^{m}−1}(T) = \binom{p^{n} − p^{m}}{p^{m} − 1}h^{p^{m}−1} × 1 + \binom{p^{n} − p^{m} − 1}{p^{m} − 2}h^{p^{m}−2} × c_{1} + … =

= (−1)^{p^{m}−2}h^{p^{m}−2} × c_{1} + …,

where “… ” stands for a linear combination of only those terms whose second factor has codimension > 1. For the top Chern class we have:

c_{r}(T) = ∑_{i=0}^{r} h^{r−i} × c_{i} .

For any integer k ≥ 2 we define β_{k} = c_{r}(T)c_{p^{m}−1}(T)c_{2}(T)c_{1}(T)^{k−2} = (−h)^{d} × c_{1}^{k} + … = [pt] × c_{1}^{k} + …, where “… ” stands for a linear combination of only those terms whose second factor has codimension > k and where [pt] is the class of a rational point in Ch(X_{E}). Let f : X × Y → X be a projection onto the first factor. The cycle β_{k} is F -rational and f_{∗}(β_{k}) = c_{1}^{k}. By [5, Example 14.6.6], the cycle c_{1}^{k} is non-zero for 2 ≤ k ≤ p^{n} − p^{m}. Therefore dim_{𝔽_{p}} f_{∗} C̄h_{dim Y −k}(X × Y) ≥ 1 for 2 ≤ k ≤ p^{n} − p^{m}. The statement follows from Theorem 3.4. □

Remark 3.10. The Corollary 3.8 also gives us some information about the integral motive of the variety X(p^{m}, D). Indeed, according to [12, Corollary 2.7] the decomposition of M(X(p^{m}, D)) with coefficients in 𝔽_{p} lifts (and in a unique way) to the coefficients ℤ/p^{N}ℤ for any N ≥ 2. Then by [12, Theorem 2.16] it lifts to ℤ (uniquely for p = 2 and p = 3 and non-uniquely for p > 3). See also Remark 4.14.

Remark 3.11. Let l be an integer such that 0 < l < p^{n} and gcd(l, p) = 1. The complete decomposition of the motive M(X(l, D)) with coefficients in 𝔽_{p} is described in [1, Proposition 2.4].

Corollary 3.12. Let D be a central division F -algebra of p-primary index. Let K/F be a field extension, such that D_{K} is still division. Then the motive (M_{1,D})_{K} is still indecomposable.

Proof. We note by X and Y respectively the varieties X(1, D) and X(p, D). We note by M the motive M(X). By [9, Theorem 3.8] the complete motivic decomposition of the variety Y consists of the motive M_{1,D} and of the sum of motives M (we neglect the shifts in this proof). Suppose that the motive (M_{1,D})_{K} is decomposable, then by the same theorem, M_{K} is a summand of (M_{1,D})_{K}. Therefore, the number of motives M_{K} in the complete motivic decomposition of Y_{K} is greater then the number of motives M in the complete motivic decomposition of Y. Let E/K be a splitting field extension for the algebra D. By Proposition 3.7, the subspace of K-rational cycles in Ch(X_{E} ×Y_{E}) coincides with the subspace of F -rational cycles in Ch(X_{E} × Y_{E}). Therefore the Theorem 3.4 gives a contradiction. □

4. Complete motivic decompositions

In the Corollary 3.8 we proved that the motive of the variety X(p^{m}, D) is decomposable for p = 2, 1 < m < n and for p > 2, 0 < m < n. Moreover, in these cases the Corollary 3.8 gives us a list of some motivic summands of the variety X(p^{m}, D). By duality, we can extend this list. It happens, that in two small-dimensional cases p = 3, m = 1, n = 2 and p = 2, m = 2, n = 3 this is already a complete list of indecomposable motivic summands of the variety X(p^{m}, D). Note that in general it is not true (see Example 4.8).

Example 4.1. In this example we describe the complete motivic decomposition of Y := X(3, D) for a division F -algebra D of degree 9. We note by X the variety X(1, D) and by M the motive M(X). Note that dim X = 8 and dim Y = 18. By [9, Theorem 3.8], any indecomposable motivic summand of Y, besides the upper motive M_{1,D}, is some shift of M. By Corollary 3.8, the motives M(2), M(3), M(4), M(5), M(6) and by duality M(8), M(7) are direct summands of M(Y). Suppose that there is at least one more motive M(t) (for some integer t ≥ 0) in the complete motivic decomposition of Y. Since by [9, Theorem 4.3] the variety Y is 3-incompressible, we have

rk_{0} M(Y) = rk_{0} M_{1,D} = rk_{dim Y} M_{1,D} = rk_{dim Y} M(Y) = 1 .

It follows that rk_{0} M(t) = rk_{dim Y} M(t) = 0. We have

1 ≤ t ≤ dim Y − dim X − 1 = 9 .

Since the decomposition of any of eight motives M(2), M(3), ..., M(8), M(t) into the sum of Tate motives over the splitting field contains a Tate motive 𝔽_{3}(9), then rk_{9} M_{1,D} ≤ rk_{9} M(Y) − 8. According to [13, §2.5], we have rk_{9} M(Y) = 8, therefore rk_{9} M_{1,D} = 0. By [8, Corollary 10.19], we have the following motivic decomposition of Y over the function field L = F (Y):

(4.2)  M(Y)_{L} = ⨁_{i+j+k=3} M(X(i, C) × X(j, C) × X(k, C)) ,

where C is a central division L-algebra (of degree 3) Brauer-equivalent to D_{L}. Note that the triples (3, 0, 0), (0, 3, 0), (0, 0, 3) correspond to three Tate motives 𝔽_{3}, 𝔽_{3}(9) and 𝔽_{3}(18). Let M̃ = M(X(1, C)), then by [8, Example 10.20], M_{L} = M̃ ⊕ M̃(3) ⊕ M̃(6). It follows that the complete decomposition of M_{L} does not contain 𝔽_{3}(9). Therefore 𝔽_{3}(9) is a direct motivic summand of (M_{1,D})_{L} and we have a contradiction with rk_{9} M_{1,D} = 0. The complete motivic decomposition of the variety X(3, D) with coefficients in 𝔽_{3} is the following one: (4.3) M(X(3, D)) = M_{1,D} ⊕ M(2) ⊕ M(3) ⊕ M(4) ⊕ M(5) ⊕ M(6) ⊕ M(7) ⊕ M(8) .

Example 4.4. Similarly, as in the previous example, we can find the complete motivic decomposition of Y := X(4, D) for a division F -algebra D of degree 8. We note by M the motive M(X(1, D)). By Corollary 3.8, the motives M(2), M(3), M(4) and by duality M(7), M(6), M(5) are direct summands of M(Y). We have

M(X(4, D)) = M(2) ⊕ ... ⊕ M(7) ⊕ N

for some motive N. Assume that N is decomposable. Then by [9, Theorem 3.8], and Theorems 3.2, 3.3, the motive N has an indecomposable summand which is some shift of either M_{0,D} = M or M_{1,D} = M(X(2, D)). But the second case is impossible because

70 = \binom{8}{4} = rk M(Y) < 6 rkM + rk M(X(2, D)) = 6 · 8 + \binom{8}{2} = 76

(see [9, Example 2.18] for the computations of ranks). Therefore M(t) is a summand of N for some integer t. According to [8, Corollary 10.19], we can write the complete decomposition of N over the function field L = F (Y):

N_{L} = 𝔽_{2} ⊕ M̃(1) ⊕ M(X(2, C))(4) ⊕ M(X(2, C))(8) ⊕ M̃(12) ⊕ 𝔽_{2}(16) ,

where C is a central division L-algebra (of degree 4) Brauer-equivalent to D_{L} and where M̃ = M(X(1, C)). It follows from this decomposition that the motive M(t)_{L} = M̃(t) ⊕ M̃(t + 4) can not be a direct summand of N_{L}. We have a contradiction. Therefore the motive N is indecomposable and N ≃ M_{2,D}. Now we can write the complete motivic decomposition of X(4, D) with coefficients in 𝔽_{2}:

(4.5) M(X(4, D)) = M_{2,D} ⊕ M(2) ⊕ M(3) ⊕ M(4) ⊕ M(5) ⊕ M(6) ⊕ M(7) .

Let us consider the following class of generalized Severi-Brauer varieties.

Definition 4.6. We say that the generalized Severi-Brauer variety X(p^{m}, D) is of type 0, if the complete decomposition of M(X(p^{m}, D)) consists only of the upper motive M_{m,D} and some (possibly zero) shifts of the motive M_{0,D} = M(X(1, D)).

For example, by [9, Theorem 3.8], the variety X(p, D) is of this type. Let Y be a generalized Severi-Brauer X(p^{m}, D) variety of type 0. By Theorem 3.4, the subspace of F -rational cycles in Ch(X_{E} × Y_{E}) describes the complete motivic decomposition of Y, where X = X(1, D), E = F (X). Note that the structure of the ring Ch(X_{E} × Y_{E}) = Ch(X_{E}) × Ch(Y_{E}) is well-known (cf. [5, § 14]) and by Proposition 3.7 we can compute the subring C̄h(X × Y) ⊂ Ch(X_{E}×Y_{E}). Therefore we can say that the complete motivic decomposition of any generalized Severi-Brauer variety X(p^{m}, D) of type 0 can be “theoretically” found in a finite time using computer.

Remark 4.7. We do not possess a single example of a variety X(p^{m}, D), which is not of type 0. Therefore, it may happen that the generalized Severi-Brauer variety X(p^{m}, D) is always of type 0 (for any division F -algebra D of degree p^{n} and for any integer m, 0 ≤ m ≤ n). Note that if this is true, then Conjecture 1.1 holds (one can follow the lines of the proof of Corollary 3.12).

Example 4.8. Let D be a central division F -algebra of degree 27. In this example we find complete motivic decomposition of the variety Y = X(3, D), which is of type 0. We take the same notations as in the proof of Corollary 3.8: X = X(1, D), E = F (X), T = T_{1} ⊠ (−T_{3})^{∨}, where T_{1} and T_{3} are the tautological bundles of rank 1 and 3 on X_{E} and Y_{E} respectively (the vector bundle T is of the rank 24). We note also by V_{∗} the graded 𝔽_{3}-vector space f_{∗} C̄h_{dim Y −∗}(X × Y), where f is a projection onto the second factor. By Theorem 3.4, for any integer k the number of motives M(k) in the complete motivic decomposition of Y is equal to dim_{𝔽_{3}} V_{k}, where M = M(X). By duality, this number is also equal to the number of motives M(dim Y −dim X − k) = M(46 − k) in the same decomposition. Therefore the vector space V_{≤23} describes the complete motivic decomposition of Y. Let h = c_{1}(T_{1}) ∈ Ch^{1}(X_{E}) and c_{i} = c_{i}((−T_{3})^{∨}) ∈ Ch^{i}(Y_{E}), 0 ≤ i ≤ 24. Using the formula 3.9 we can compute the following Chern classes of the vector bundle T:

c_{1}(T) = 1 × c_{1} , c_{2}(T) = −h × c_{1} + 1 × c_{2} ,
c_{7}(T) = 1 × c_{7} , c_{8}(T) = −h × c_{7} + 1 × c_{8} ,
c_{24}(T) = h^{24} × 1 + ∑_{i=1}^{24} h^{24−i} × c_{i} .

We have:

\begin{matrix}c_{1}^{2} = f_{∗}(c_{24}(T)(c_{2}(T))^{2}) ∈ V_{2} , & c_{1}c_{7} = f_{∗}(c_{24}(T)c_{2}(T)c_{8}(T)) ∈ V_{8} , \\ c_{7}^{2} = f_{∗}(c_{24}(T)(c_{8}(T))^{2}) ∈ V_{14} . \end{matrix}

Also we have the following property:

(4.9)  x ∈ V_{∗} ⇒ (xc_{1} ∈ V_{∗+1} and xc_{7} ∈ V_{∗+7}) .

Indeed, if x ∈ V_{∗}, then x = f_{∗}(y) for some y ∈ C̄h_{dim Y −∗}(X_{E} × Y_{E}). Therefore xc_{1} = f_{∗}(y · c_{1}(T)) ∈ V_{∗+1} and xc_{7} = f_{∗}(y · c_{7}(T)) ∈ V_{∗+7}. This property gives us the following elements in V_{i}:

(4.10)

c_{1}^{i} if 2 ≤ i ≤ 7 , c_{1}^{i}, c_{1}^{i−7}c_{7}, c_{1}^{i−14}c_{7}^{2} if 14 ≤ i ≤ 20 ,
c_{1}^{i}, c_{1}^{i−7}c_{7} if 8 ≤ i ≤ 13 , c_{1}^{i}, c_{1}^{i−7}c_{7}, c_{1}^{i−14}c_{7}^{2}, c_{1}^{i−21}c_{7}^{3} if 21 ≤ i ≤ 23 .

We also define a sequence b_{i}, i ∈ ℤ:

b_{i} = \{ \begin{matrix}0 & if i < 2 \\ 1 & if 2 ≤ i ≤ 7 \\ 2 & if 8 ≤ i ≤ 13 \\ 3 & if 14 ≤ i ≤ 20 \\ 4 & if 21 ≤ i ≤ 23 \\ b_{46−i} & if i > 23. \end{matrix}

Note that for any i ≤ 23 the number of elements lying in V_{i} from the list 4.10 is equal to b_{i}. We are going to show that all elements from the list 4.10 are linearly independent (to apply then Theorem 3.4). The 𝔽_{3}-vector space V_{∗} is a subspace of Ch^{∗}(Y_{E}). We note c̃_{i} = c_{i}(T_{3}^{∨}), i = 1, 2, 3, where T_{3} is the tautological bundle of rank 3 on Y_{E}. According to [5, Example 14.6.6] the graded ring Ch^{∗}(Y_{E}) is generated by Chern classes c̃_{1}, c̃_{2}, c̃_{3}, c_{1}, ..., c_{24} modulo the homogeneous relations

(4.11)  c_{r} + c_{r−1}c̃_{1} + c_{r−2}c̃_{2} + c_{r−3}c̃_{3} = 0 for r = 1, ..., 27 ,

where c_{i} = 0 for i < 0 or i > 24. It follows that the graded ring Ch^{∗}(Y_{E}) is generated by only three elements c̃_{1}, c̃_{2}, c̃_{3} modulo some homogeneous relations of degree greater than 23. Therefore we have an isomorphism:

Ch^{∗ ≤23}(Y_{E}) ≃ 𝔽_{3}[c̃_{1}, c̃_{2}, c̃_{3}]_{≤23} .

Using relations 4.11 we can compute that c_{1} = −c̃_{1} and c_{7} = −c̃_{1}^{7} + c̃_{1}^{4}c̃_{3} − c̃_{1}^{3}c̃_{2}^{2} + c̃_{1}c̃_{2}^{3}. Now we consider the elements from the list 4.10 as polynomials in 𝔽_{3}[c̃_{1}, c̃_{2}, c̃_{3}]. To show that all of them are linearly independent, it suffices to check this for four elements c_{1}^{21}, c_{1}^{14}c_{7}, c_{1}^{7}c_{7}^{2}, c_{7}^{3} (they are in our list) from V_{21}. Since the polynomial ring 𝔽_{3}[c̃_{1}, c̃_{2}, c̃_{3}] is factorial and c_{7} is not divisible by c̃_{1}^{2}, then for any α, β, γ, δ ∈ 𝔽_{3} we have

αc_{1}^{21} + βc_{1}^{14}c_{7} + γc_{1}^{7}c_{7}^{2} + δc_{7}^{3} = 0 ⟹ α = β = γ = δ = 0 .

Since all elements from the list 4.10 are linearly independent, then dim_{𝔽_{3}} V_{i} ≥ b_{i} for i ≤ 23. Therefore for any integer i the motive M^{⊕b_{i}}(i) is a direct summand of M(Y). Indeed, the statement follows from Theorem 3.4 for i ≤ 23 and by duality it is also true for i > 23. We have

(4.12)  M(Y) = ⊕_{i∈ℤ}M^{⊕b_{i}}(i) ⊕ N

for some motive N over F. Now we want to understand the complete decomposition of N. Let L be a function field of the variety Y and C be a central division L-algebra (of degree 9) Brauer-equivalent to D_{L}. Using the motivic decomposition similar to the decomposition 4.2 from Example 4.1, we can show that the complete decomposition of M(Y)_{L} contains three indecomposable motives: M_{1,C}, M_{1,C}(27), M_{1,C}(54). Moreover any other summand in the complete motivic decomposition of M(Y)_{L} is a shift of the motive M̃:= M(X(1, C)). We know that M_{L} = M̃ ⊕ M̃(9) ⊕ M̃(18). It follows that N_{L} = M_{1,C} ⊕ M_{1,C}(27) ⊕ M_{1,C}(54) ⊕ N′ for some motive N′ over L and N′ is a sum of shifts of the motive M̃. Note that if M(k) is direct summand of N for some integer k, then M_{L}(k) is a direct summand of N′. Let S be a direct summand of the motive of a geometrically split variety. We write P (S, t) for the Poincaré polynomial of S:

P (S, t) = ∑_{i≥0} (rk_{i} S) · t^{i} .

Let us find the Poincaré polynomial of the motive N′. We have

P (N′, t) = P (M(Y), t) − (1 + t^{27} + t^{54})P (M_{1,C}, t) − ∑_{i≥0} b_{i}t^{i}P (M, t) .

Using the following formulas

P (M(Y), t) = \frac{(1 − t^{27})(1 − t^{26})(1 − t^{25})}{(1 − t)(1 − t^{2})(1 − t^{3})} , (according to [13, §2.5]),

P (M, t) = \frac{1 − t^{27}}{1 − t} = ∑_{i=0}^{26} t^{i} ,

P (M_{1,C}, t) = t^{6} + t^{12} + ∑_{i=0}^{26} t^{i} , (by Example 4.3) ,

we can compute P (N′, t). Since N′ is a sum of shifts of the motive M̃ then P (N′, t) is divisible by P (M̃, t) = (1 − t^{9})/(1 − t) = 1 + t + ... + t^{8}. Let Q(t) be a quotient of these two polynomials. After computations, we have

Q(t) = t^{7} +t^{13} +t^{16} +t^{18} +t^{19} +t^{20} +t^{22} +t^{24} +t^{26} +t^{28} +t^{29} +t^{30} +t^{34} +t^{35}+

t^{36} + t^{38} + t^{40} + t^{42} + t^{44} + t^{45} + t^{46} + t^{48} + t^{51} + t^{57} .

Now if M(k) is direct summand of N for some integer k, then M_{L}(k) = M̃(k) ⊕ M̃(k + 9) ⊕M̃(k + 18) is a direct summand of N′. Therefore in this case the decomposition of Q(t) contains t^{k} + t^{k+9} + t^{k+18} = P (M_{L}(k), t)/P (M̃, t). Only two values k = 20 and k = 26 satisfy this condition. Note that if complete decomposition of the motive N contains M(20) then by duality it contains also M(26). It follows that the question of the complete motivic decomposition of Y reduces to the question either dim_{𝔽_{3}} V_{20} = 3 or dim_{𝔽_{3}} V_{20} = 4? Let us show that we are in the second case.

Consider the following cycle e from V_{20}

e = f_{∗}(c_{2}^{11}(−T)c_{3}^{8}(−T)) = −c̃_{1}^{17}c̃_{3} + c̃_{1}^{16}c̃_{2}^{2} − c̃_{1}^{14}c̃_{2}^{3} − c̃_{1}^{14}c̃_{3}^{2} − c̃_{1}^{13}c̃_{2}^{2}c̃_{3} − c̃_{1}^{12}c̃_{2}^{4}+

c̃_{1}^{11}c̃_{2}^{3}c̃_{3} − c̃_{1}^{11}c̃_{3}^{3} − c̃_{1}^{10}c̃_{2}^{5} − c̃_{1}^{2}c̃_{2}^{9} ,

where c_{2}(−T) = −hc̃_{1} + c̃_{2}, c_{3}(−T) = h^{3} + h^{2}c̃_{1} + hc̃_{2} + c̃_{3}. The cycle e as a polynomial in 𝔽_{3}[c̃_{1}, c̃_{2}, c̃_{3}] is not divisible by c̃_{1}^{3}. It follows that the cycle e could not be a linear combination of three cycles c_{1}^{20}, c_{1}^{13}c_{7}, c_{1}^{6}c_{7}^{2} from the list 4.10. Therefore dim_{𝔽_{3}} V_{20} = 4. Consider a sequence (a_{i})_{i∈ℤ} defined by

a_{i} = \{ \begin{matrix}b_{i} + 1 & if i = 20 or i = 26 \\ b_{i} & else. \end{matrix}

The complete motivic decomposition of the variety Y is the following

(4.13)  M(Y) = ⊕_{i∈ℤ}M^{⊕a_{i}} ⊕ M_{1,D} .

Remark 4.14. We have the same decompositions as (4.3), (4.5) and (4.13) for the motives with the integral coefficients. To show this one can apply [12, Corollary 2.7] and then [12, Theorem 2.16].

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Maksim Zhykhovich UPMC Univ Paris 06 UMR 7586 Institut de Mathématiques de Jussieu (boite courrier 247) 4 place Jussieu F-75252 Paris CEDEX 05 France 166

Documenta Mathematica 17 (2012)