Motivic decomposability of generalized Severi-Brauer varieties

Maksim Zhykhovich

26 octobre 2009

Abstract

Let F be an arbitrary field. Let p be a positive prime number and D a central division F -algebra of degree p^{n}, with n ≥ 1. We write SB(p^{m}, D) for the generalized Severi-Brauer variety of right ideals in D of reduced dimension p^{m} for m = 0, 1, …, n − 1. We note by M(SB(p^{m}, D)) the Chow motive with coefficients in 𝔽_{p} of the variety SB(p^{m}, D). It was proven by Nikita Karpenko that this motive is indecomposable for any prime p and m = 0 and for p = 2, m = 1 (cf. [7]). We prove decomposability of M(SB(p^{m}, D)) in all the other cases (p = 2, 1 < m < n and p > 2, 0 < m < n).

Résumé

Soient F un corps arbitraire, p un nombre premier positif et D une F -algèbre de division de degré p^{n}. On écrit SB(p^{m}, D) pour la variété de Severi-Brauer généralisée des idéaux à droite de dimension réduite p^{m}, m = 0, 1, …, n−1. On note par M(SB(p^{m}, D)) le motif de Chow à coefficients dans 𝔽_{p} de la variété SB(p^{m}, D). Il a été demontré par Nikita Karpenko que ce motif est indecomposable pour le nombre premier p arbitraire et m = 0 et pour p = 2, m = 1 (cf. [7]). Nous prouvons la décomposabilité de M(SB(p^{m}, D)) dans tous les autres cas (p = 2, 1 < m < n and p > 2, 0 < m < n).

1 Chow motives with finite coefficients

Our basic reference for Chow groups and Chow motives (including notations) is [4]. We fix an associative unital commutative ring Λ (most frequently Λ will be the finite field 𝔽_{p} of p elements, where p is prime) and for a variety (i.e., a separated scheme of finite type over a field) X we write Ch(X) for its Chow group with coefficients in Λ (while we write CH(X) for its integral Chow group). 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 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.

Let M be a geometrically split motive. Over an extension of F the 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 the splitting field extension: they are the ranks of the free Λ-modules colim_{L/F} Ch_{∗}(M_{L}) and colim_{L/F} Ch_{i}(M_{L}) respectively, where the colimit is taken over all field extensions L/F.

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 1.1. ([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.

Lemma 1.2. 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.

2 Motivic decomposability of generalized Severi-Brauer varieties

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 SB(p^{m}, D) for the generalized Severi-Brauer variety of right ideals in D of reduced dimension p^{m} for m = 0, 1, …, n − 1. For the main Theorem 2.6 we will need the following definition.

Definition 2.1. Let G_{r}(𝔸^{n}) be the Grassmann variety of r-planes in 𝔸^{n}. Let c_{1} = c_{1}(T av) ∈ Ch^{1}(G_{r}(𝔸^{n})), where T av is a tautological r-dimensional vector bundle on G_{r}(𝔸^{n}). We define t_{p}(r, n) as a maximal integer k, such that c_{1}^{k} ≠ 0.

Remark 2.2. By [5, Example 14.6.6], the integer t_{p}(r, n) does not depend on the base field and we have an inequality max{r, n − r} ≤ t_{p}(r, n) ≤ r(n − r) = dim G_{r}(𝔸^{n}) .

Lemma 2.3. Let E/F be a splitting field extension for X = SB(1, D). Then the subgroup of F -rational cycles in Ch_{dimX}(X_{E} × X_{E}) is generated by a diagonal class.

Proof. We write C̄h(X) for the image of the homomorphism Ch(X) → Ch(X_{E}). By [8, 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 nonzero, it generates all the group.

Corollary 2.4. (cf. [8, Theorem 2.2.1]). The motive with coefficients in 𝔽_{p} of the Severi-Brauer variety X = SB(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 Theorem 2.3, 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 1.2 gives us π = 1 in the second case.

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

Theorem 2.5. (cf. [7, 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 SB(2, D) is indecomposable.

Taking into account the Corollary 2.4, Theorem 2.5 and the fact that any generalized Severi-Brauer variety SB(p^{m}, D) is p-incompressible [7, Theorem 4.3] (this condition is weaker than motivic indecomposability), one can expect that the Chow motive with coefficients in 𝔽_{p} of any generalized Severi-Brauer variety SB(p^{m}, D) is indecomposable. But the following theorem gives us the opposite answer.

Theorem 2.6. Let D be a central division F -algebra of degree p^{n} with n ≥ 1. Then the motive with coefficients in 𝔽_{p} of the variety SB(p^{m}, D) is decomposable for p = 2, 1 < m < n and for p > 2, 0 < m < n. In these cases M(SB(1, D))(k) is a summand of M(SB(p^{m}, D)) for 2 ≤ k ≤ t_{p}(p^{m}, p^{n}).

Proof. We use the notations: X = SB(1, D), Y = SB(p^{m}, D), d = dim(SB(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). We will show that M(X)(k) is a summand of M(Y). By Lemma 1.2 it suffices to construct two F -rational morphisms

α : M(X_{E})(k) → M(Y_{E}) and β : M(Y_{E}) → M(X_{E})(k)

satisfying β ∘α = 1 ∈ End(M(X_{E})(k)) = Ch_{d}(X_{E} ×X_{E}). By Theorem 2.3 we can replace condition β ∘ α = 1 by β ∘ α ≠ 0.

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]). Let T be the tautological r-dimensional vector bundle on G_{r}(T av). 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. Then we have an isomorphism (cf. [6, Proposition 4.3]): T_{E} ≃ T_{1} ⊠ (−T_{p^{m}})^{∨} (here we lift the bundles T_{1} and T_{p^{m}} on X_{E} × Y_{E}). Let h = c_{1}(T_{1}) ∈ Ch^{1}(X_{E}) (then −h is a hyperplane class in Ch^{1}(X_{E})) and c_{i} = c_{i}((−T_{p^{m}})^{∨}) ∈ Ch^{i}(Y_{E}). Then by [5, Remark 3.2.3(b)]

c_{t}(T_{E}) = 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 theorem 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_{E}) = (p^{n} − p^{m})h × 1 + 1 × c_{1} = 1 × c_{1} ,

c_{2}(T_{E}) = \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_{E}) = \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_{E}) = ∑_{i=0}^{r} h^{r−i} × c_{i} .

Let β_{1} = c_{r}(T_{E})c_{p^{m}−1}(T_{E})c_{2}(T_{E})c_{1}(T_{E})^{k−2} = (−h)^{d} × c_{1}^{k} + … = x × c_{1}^{k} + …, where “… ” stands for a linear combination of only those terms whose second factor has codimension > k and where x is the class of a rational point in Ch(X_{E}). We take β = β_{1}^{t}, where β_{1}^{t} is the transpose of β_{1}. Since the bundle T is defined over F, the morphism β ∈ Ch_{dimY −k}(Y_{E} × X_{E}) = Hom(M(Y_{E}), M(X_{E})(k)) is F -rational.

By Definition 2.1 the cycle c_{1}^{k} is non-zero. Let a ∈ Ch(Y_{E}) be the element dual to c_{1}^{k} with respect to the bilinear form Ch(Y_{E}) × Ch(Y_{E}) → 𝔽_{p}, (x_{1}, x_{2}) ↦ deg(x_{1} · x_{2}). The pull-back homomorphism f : 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 ]. Let α′ ∈ Ch(X ×Y) be a cycle whose image in Ch(Y_{E}) under the surjection f is a. We define α as α′_{E} and we have α = 1 × a + …, where “… ” stands for a linear combination of only those elements whose first factor is of positive codimension and where 1 = [X_{E}]. Then β ∘ α = 1 × x + …, where “… ” stands for a linear combination of only those terms whose first factor is of positive codimension. It follows that β ∘ α ≠ 0.

Remark 2.7. The Theorem 2.6 also gives us some information about the integral motive of the variety SB(p^{m}, D). Indeed, according to [10, Corollary 2.7] the decomposition of M(SB(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 [10, Theorem 2.16] it lifts to ℤ (uniquely for p = 2 and p = 3 and non-uniquely for p > 3). See also Remark 2.10.

Remark 2.8. Let l be an integer such that 0 < l < p^{n} and gcd(l, p) = 1. The complete decomposition of the motive M(SB(l, D)) with coefficients in 𝔽_{p} is described in [1, Proposition 2.4].

Example 2.9. As an application of Theorem 2.6 we describe the complete motivic decomposition of SB(4, D) for a division algebra D of degree 8. Let E/F be a splitting field extension for the algebra D. We note M = M(X). By Theorem 2.6 and Remark 2.2, the motives M(2), M(3), M(4) and by duality M(7), M(6), M(5) are direct summands of M(SB(4, D)). By [7, Theorem 4.1], we have an indecomposable direct summand M_{2,D} of M(SB(4, D)) with a property: 𝔽_{2}(0) and 𝔽_{2}(16) are presented in the decomposition of (M_{2,D})_{E}. By [7, Theorem 3.8], and Theorems 2.4, 2.5 any other indecomposable summand of M(SB(4, D)) is some shift of either M or M(SB(2, D)). But the second case is impossible because 70 = \binom{8}{4} = rk M(SB(4, D)) < 6 rkM + rk M(SB(2, D)) = 6 · 8 + \binom{8}{2} = 76.

We temporary note X = SB(1, D), Y = SB(4, D). Let us assume that the motive M(1) is a summand of M(SB(4, D)). Then there exist two correspondences α ∈ Ch_{dimX+1}(X × Y) and β ∈ Ch_{dimY −1}(Y × X), such that β ∘ α = 1, where 1 means the diagonal class in Ch_{dimX}(X × X). Let f be a projection Y × X → Y. Then the cycle f_{∗}(β_{E}) ∈ Ch^{1}(X_{E}) is F -rational and non-zero. The contradiction follows from [3, Proposition 5.1] and [9, Corollary 2.7]. So M(1) and by duality M(8) could not be the summands of M(Y).

Assume now that there are more than 6 motives M (with some shifts) in the decomposition of M(SB(4, D)). Then by duality there are at least 8 such summands. But the decomposition of any of these 8 summands M into the sum of Tate motives over the splitting field E contains 𝔽_{2}(7). We have a contradiction with rk_{7} M(SB(4, D)) = 7. Therefore

M(SB(4, D)) = M_{2,D} ⊕ M(2) ⊕ M(3) ⊕ M(4) ⊕ M(5) ⊕ M(6) ⊕ M(7) .  (1)

We can write the decomposition of M_{2,D} over the function field L = F (SB(4, D)):

(M_{2,D})_{L} = 𝔽_{2} ⊕ M(SB(1, C))(1) ⊕ M(SB(2, C))(4) ⊕ M(SB(2, C))(8) ⊕ M(SB(1, C))(12) ⊕ 𝔽_{2}(16) ,

where C is a central division L-algebra (of degree 4) Brauer-equivalent to D_{L}.

Remark 2.10. We have the same decomposition as (1) for the integral motive of the variety SB(4, D). To show this one can apply [10, Corollary 2.7] and then [10, Theorem 2.16].

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 and Sergey Tikhonov for very useful discussions.

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