INCOMPRESSIBILITY OF GENERALIZED SEVERI-BRAUER VARIETIES

NIKITA A. KARPENKO

Abstract. Let F be an arbitrary field. Let A be a central simple F -algebra. Let G be the algebraic group Aut A of automorphisms of A. Let 𝔛_{A} be the class of finite direct products of projective G-homogeneous F -varieties (the class 𝔛_{A} includes the generalized Severi-Brauer varieties of the algebra A).

Let p be a positive prime integer. For any variety in 𝔛_{A}, we determine its canonical dimension at p. In particular, we find out which varieties in 𝔛_{A} are p-incompressible. If A is a division algebra of degree p^{n} for some n ≥ 0, then the list of p-incompressible varieties includes the generalized Severi-Brauer variety X(p^{m}; A) of ideals of reduced dimension p^{m} for m = 0, 1, …, n.

We also determine the structure of the Chow motives with coefficients in 𝔽_{p} of the varieties in 𝔛_{A}. More precisely, it is known that the motive of any variety in 𝔛_{A} decomposes (in a unique way) into a sum of indecomposable motives, and we describe the indecomposable summands which appear in the decompositions. An application of the above results is a proof of the hyperbolicity conjecture on orthogonal involutions.

1. Introduction

A smooth complete irreducible variety X over a field F is said to be incompressible, if any rational map X −−→ X is dominant.

An important (at least by the amount of available applications, see, e.g., [16], [11], or [5]) example of an incompressible variety is as follows. Let p be a positive prime integer. Let D be a central division F -algebra of degree a power of p, say, p^{n} (where n ≥ 1). For any integer i, the generalized Severi-Brauer variety X(i; D) is the F -variety of right ideals in D of reduced dimension i. The variety X(1; D) (the usual Severi-Brauer variety of D) is incompressible.

In fact, the variety X = X(1; D) has a stronger property: it is p-incompressible, meaning that for any element α in the Chow group CH_{dim X}(X × X) of the product of two copies of X, the multiplicity of α over the first factor is divisible by p if and only if the multiplicity of α over the second factor is so. The original proof, given in [16] and [14], makes use of Quillen’s computation of the K-theory of Severi-Brauer varieties. A recent proof, given in [21], makes use of Steenrod operations on the modulo p Chow groups (it better explains the reason of the p-incompressibility but works only over fields of characteristic ≠ p). A third, particularly simple proof is given here (see Remark 2.19 and Corollary 2.22).

Date: March 2009.

Key words and phrases. Algebraic groups, projective homogeneous varieties, Chow groups.

2000 Mathematical Subject Classifications: 14L17; 14C25.

Supported by the Collaborative Research Centre 701 of the Bielefeld University.

The main result of the present paper is Theorem 3.7 which says that the variety X(p^{m}; D) is p-incompressible also for m = 1, …, n−1 (in the case of p = 2 and m = n−1 this was shown earlier by Bryant Mathews [19] using a different method). The remaining results of this paper are either obtained on the way to the main result or are quite immediate consequences of it.

We start by associating to each integer m = 0, 1, …, n an indecomposable 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.

Then we show (see Theorem 3.3) that the motive of any finite product of generalized Severi-Brauer varieties of the algebra D decomposes into a sum of twists of the motives M_{m,D} (with various m). In fact, we prove this for more general related to D varieties: for products of projective (Aut D)-homogeneous varieties (that is, for products of varieties of flags of ideals in D).

Even more generally, for an arbitrary central simple F -algebra A, we consider the class 𝔛_{A} of finite direct products of projective (Aut A)-homogeneous F -varieties. Taking as D the p-primary component of a central division algebra Brauer-equivalent to A, we decompose the motive (still with coefficients in 𝔽_{p}) of any variety in 𝔛_{A} into a sum of twists of the motives M_{m,D} (see Corollary 3.4). Therefore we solve the case of the inner type A_{n} of the following general problem.

Let G be a semisimple affine algebraic group over a field. According to [9] (see also §2a here), the motive (still with 𝔽_{p} coefficients, p a fixed prime) of any product of projective G-homogeneous F -varieties decomposes and in a unique way in a finite direct sum of indecomposable motives. For simple groups G of any given type, the problem is to describe the indecomposable summands which appear this way.

The situation with the types B_{n} and D_{n} however is very complicated: numerous examples of motivic decompositions of projective quadrics are produced (e.g., in [26]) which do not leave a hope of a simple general answer. Therefore a reasonable answer for the inner type A_{n} obtained here is a fortune.

With this in hand, we prove two structure results concerning the motives M_{m,D} (Theorem 3.6). We show that the d-dimensional Chow group of M_{m,D}, where d = dim X(p^{m}; D), is also non-zero. This result is equivalent to the p-incompressibility of the variety X(p^{m}; D), so that we get the main result of the paper (Theorem 3.7) at this point. The second structure result on the motive M_{m,D} is a computation of the p-adic valuation of its rank. In fact, we can not separate the proofs of these two structure results: we prove them simultaneously by induction on deg D (and using Theorem 3.3).

Now we start getting consequences of the main result. We recall (in §2e) the notion of canonical dimension at p (or canonical p-dimension) cd_{p}(X) of a smooth complete irreducible algebraic variety X. This is a certain non-negative integer satisfying cd_{p} X ≤ dim X; moreover, cd_{p} X = dim X if and only if X is p-incompressible. In particular, by our main result, cd_{p} X(p^{m}; D) = dim X(p^{m}; D) = p^{m}(p^{n} − p^{m}). The canonical dimension at p of any variety in 𝔛_{A} (where A is an arbitrary central simple F -algebra) can be easily computed in terms of cd_{p} X(p^{m}; D) (where D is the p-primary part of a division algebra Brauer-equivalent to A), see Corollary 3.8.

In spite of a big number of obtained results, one may say that (the motivic part of) this paper raises more questions that it answers. Indeed, although we show that the motives of the varieties in 𝔛_{D} decompose into sums of twists of M_{m,D} (and find a restriction on m in terms of a given variety), we do not precisely determine this decomposition: neither we know how many copies of M_{m,D} (for a given m and a given variety) do really appear in the decomposition, nor we determine the twisting numbers. Moreover, the understanding of the structure of the motives M_{m,D} themselves is not satisfactory. It may happen that M_{m,D} is always the whole motive of the variety X(p^{m}, D) (that is, the motive of this variety probably is indecomposable): we do not possess a single counter-example. In fact, the variety X(p^{m}, D) is indecomposable for certain values of p, n, and m. Two cases are known for a long time: m = 0 (the Severi-Brauer case, see Remark 2.19) and m = 1 with p = 2 = n (reducing the exponent of D to 2, we come to the case of an Albert quadric here, where we can refer to [26], [10], or [12]). The Albert case is generalized in Example 3.5. The other values of p, n, m should be studied in this regard.

But the qualitative analysis is done (for instance, the properties of M_{m,D} we establish show that this motive behaves essentially like the whole motive of the variety X(p^{m}, D) even if it is “smaller”). And the proofs are not complicated. This is a study of generalized Severi-Brauer varieties which are twisted forms of grassmannians, and there is no single Young diagram in the text! Combinatorics or complicated formulas do not show up at all, in particular, because we neglect the twisting numbers of motivic summands. The results we are getting this way are less precise but, as we believe, they contain the essential piece of information. They can be (and are) applied (in [13]) to prove the hyperbolicity conjecture on orthogonal involutions, which was the main motivation to study the problems of this paper.

We conclude the introduction by some remarks on the motivic category we are using. First of all, the category of Chow motives with coefficients in 𝔽_{p}, in which we are working in this paper, can be replaced by a simpler category. This simpler category is constructed in exactly the same way as the category of Chow motives with the only difference that one kills the elements of Chow groups which vanish over extensions of the base field (see Remark 2.7). Working with this simpler category, we do not need the nilpotence tricks (the nilpotence theorem and its standard consequences, cf. §2a) anymore. This simplification of the motivic category does not harm to any external application of our motivic results. So, this is more a question of taste than a question of necessity that we stay with the usual Chow motives.

On the other hand, somebody may think that our category of usual Chow motives is not honest or usual enough because these are Chow motives with coefficients in 𝔽_{p} and not in ℤ. Well, there are at least three arguments here. First, decompositions into sums of indecomposables are not unique for coefficients in ℤ, even in the case of projective homogenous varieties of inner type A_{n} (see [9, Example 32] or [7, Corollary 2.7]). Therefore the question of describing the indecomposables does not seem so reasonable for the integral motives. Second, any decomposition with coefficients in 𝔽_{p} lifts (and in a unique way) to the coefficients ℤ/p^{n}ℤ for any n ≥ 2, [23, Corollary 2.7]. Moreover, in the case of varieties in 𝔛_{A}, where A is a central simple algebra of a p-primary index, it also lifts to ℤ (nonuniquely this time), [23, Theorem 2.16]. And third, may be the most important argument is that the results on motives with coefficients in 𝔽_{p} are sufficient for the applications.

2. Preliminaries

This section is long because it also includes some non-standard (but simple) staff.

2a. Chow motives with finite coefficients. Our basic reference for Chow groups and Chow motives (including notation) is [10]. We fix an associative unital commutative ring Λ (most frequently Λ will be the finite field 𝔽_{p} of p elements, where p is a prime) and for a variety (i.e., a separated scheme of finite type over a field ) X we write CH(X; Λ) for its Chow group with coefficients in Λ. We use a simplified notation CH(X) for the integral Chow group CH(X; ℤ), and a simplified notation Ch(X) for the modulo p Chow group CH(X; 𝔽_{p}) = CH(X)/p. Our category of motives is the category CM(F, Λ) of graded Chow motives with coefficients in Λ, [10, definition of §64]. By a sum of motives we always mean the direct sum. We also write Λ for the motive M(Spec F) ∈ CM(F, Λ). A Tate motive is the motive Λ(i) with i an integer (which may differ from ±1).

We shall often assume that our coefficient ring Λ is finite. This simplifies significantly the situation (and is sufficient for most applications). For instance, for a finite Λ, the endomorphism rings of finite sums of Tate motives are also finite and the following easy statement applies:

Lemma 2.1. An appropriate power of any element of any finite associative (not necessarily commutative) ring is idempotent.

Proof. Since the ring is finite, any its element x satisfies x^{a} = x^{a+b} for some a ≥ 1 and b ≥ 1. It follows that x^{ab} is idempotent. □

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. 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 variety is geometrically split and satisfies the nilpotence principle, [10, Theorem 92.4 with Remark 92.3].

Corollary 2.2. Assume that the coefficient ring Λ is finite. Let X be a geometrically split variety satisfying the nilpotence principle. Then an appropriate power of any endomorphism of the motive of X is a projector.

Proof. Let F̄/F be a splitting field of the motive M(X), that is, M(X)F̄ is a sum of Tate motives. Let f be an endomorphism of M(X). Since Λ is finite, the ring End(M(X)F̄) is finite. Therefore a power of fF̄ is idempotent by Lemma 2.1, and (replacing f by an appropriate power of f) we may assume that fF̄ is idempotent. Since X satisfies the nilpotence principle, the element ε := f^{2} − f is nilpotent. Let n be a positive integer such that ε^{n} = 0 = nε. Then (f + ε)^{n^{n}} = f^{n^{n}} because the binomial coefficients \binom{n^{n}}{i} for i < n are divisible by n. Therefore f^{n^{n}} is a projector. □

Lemma 2.3 (cf. [9, Theorem 28]). Assume that the coefficient ring Λ is finite. Let X be a geometrically split variety satisfying the nilpotence principle and let π ∈ End(M(X)) be a projector. Then the motive (X, π) decomposes into a finite sum of indecomposable motives.

Proof. If (X, π) does not decompose this way, we get an infinite sequence

π_{0} =π, π_{1}, π_{2}, … ∈ End(M(X))

of pairwise distinct projectors such that π_{i} ∘ π_{j} = π_{j} = π_{j} ∘ π_{i} for any i < j.

Let F̄/F be a splitting field of X. Since the ring End(M(X)F̄) is finite, we have (π_{i})F̄ = (π_{j})F̄ for some i < j. The difference π_{i} − π_{j} is nilpotent and idempotent, therefore π_{i} = π_{j}. □

A (non necessarily commutative) ring is called local, if the sum of any two non-invertible elements differs from 1 in the ring. Since the sum of two nilpotents is never 1, we have

Lemma 2.4. A ring, where each non-invertible element is nilpotent, is local. In particular, by Corollary 2.2, so is the ring End(M(X)), if Λ is finite and X is a geometrically split variety satisfying the nilpotence principle and such that the motive M(X) is indecomposable. □

A complete decomposition of an object in an additive category is a finite direct sum decomposition with indecomposable summands.

Theorem 2.5 ([2, Theorem 3.6 of Chapter I]). Let M be an object of a pseudo-abelian category which is a direct sum of a finite number of indecomposable objects having local endomorphism rings. Then any finite direct sum decomposition of M can be refined to a complete one, and there is only one (up to a permutation of the summands) complete decomposition of M.

To be precise, the uniqueness part of Theorem 2.5 states that if

M = M_{1} ⊕ ⋯ ⊕ M_{m} = N_{1} ⊕ ⋯ ⊕ N_{n}

are two complete decompositions of M, then m = n and there exists a permutation σ of the set {1, 2, …, n} such that M_{i} ≃ N_{σ(i)} for any i. The isomorphism here is meant to be an isomorphism of abstract objects: in general, there is no such isomorphism respecting the embeddings into M. Later on, when we speak of “isomorphism of summands” of a motive, we always mean an isomorphism of abstract motives between the 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. In the sequel, we are constantly using the following statement which is an immediate consequence of Lemmas 2.3 and 2.4 and Theorem 2.5:

Corollary 2.6 (cf. [9, Corollary 35]). Assume that the coefficient ring Λ is finite. The Krull-Schmidt principle holds for any twist of any summand of the motive of any geometrically split F -variety satisfying the nilpotence principle. In other words, the Krull-Schmidt principle holds for the objects of the pseudo-abelian Tate subcategory in CM(F, Λ) generated by the motives of the geometrically split F -varieties satisfying the nilpotence principle. □

Remark 2.7. Replacing the Chow groups CH(−; Λ) by the reduced Chow groups \overline{CH}(−; Λ) (cf. [10, §72]) in the definition of the category CM(F, Λ), we get a “simplified” motivic category \overline{CM}(F, Λ) (which is still sufficient for the main purpose of this paper). Working within this category, we do not need the nilpotence principle any more. In particular, the Krull-Schmidt principle holds (with a simpler proof) for the twists of the summands of the motives of the geometrically split F -varieties.

2b. Outer summands. We assume here that the coefficient ring Λ is connected. We shall often assume that Λ is finite.

The following definition is extending some terminology of [25].

Definition 2.8. Let M ∈ CM(F, Λ) be a summand of the motive of a smooth complete irreducible variety of dimension d. The summand M is called left-outer, if CH^{0}(M; Λ) ≠ 0. The summand M is called right-outer, if CH_{d}(M; Λ) ≠ 0. The summand M is called outer, if it is left-outer and right-outer simultaneously.

For instance, the whole motive of a smooth complete irreducible variety is an outer summand of itself.

Given a correspondence, an element α ∈ CH_{dim X}(X × Y; Λ) of the Chow group of the product of smooth complete irreducible varieties X and Y, we write mult α ∈ Λ for the multiplicity (or multiplicity over the first factor) of α, [10, definition of §75]. Multiplicity of a composition of two correspondences is the product of multiplicities of the composed correspondences (cf. [16, Corollary 1.7]). In particular, multiplicity of a projector is idempotent and therefore ∈ {0, 1} because the coefficient ring Λ is connected.

Lemma 2.9. Let X be a smooth complete irreducible variety. The motive (X, π) given by a projector π ∈ CH_{dim X}(X × X; Λ) is left-outer if and only if mult π = 1. The motive (X, π) is right-outer if and only if mult π^{t} = 1, where π^{t} is the transpose of π.

Proof. The group CH^{0} ((X, π); Λ), defined as Hom ((X, π), Λ), is the image of the endomorphism of the group CH^{0}(X; Λ) = Λ · [X] given by the multiplication by mult π. The group CH_{d} ((X, π); Λ), defined as Hom (Λ(d), (X, π)), is the image of the endomorphism of the same group CH_{d}(X; Λ) = CH^{0}(X; Λ) = Λ · [X] given by the multiplication by mult π^{t}. □

Lemma 2.10. Let d be a non-negative integer and let X be a smooth complete irreducible variety of dimension d such that the degree homomorphism deg : CH^{d}(X; Λ) → Λ is an isomorphism. The following three statements on a summand M of M(X) ∈ CM(F, Λ) are equivalent:

• M is right-outer;

• the subgroup CH^{d}(M) of the group CH^{d}(X; Λ) coincides with CH^{d}(X; Λ);

• CH^{d}(M) ≠ 0.

Proof. Let π ∈ CH^{d}(X × X; Λ) be the projector giving M. Let x be an element of CH^{d}(X; Λ) with deg x = 1 ∈ Λ. In particular, x is a generator of the Λ-module CH^{d}(X; Λ).

Then the Λ-module CH^{d}(M) = Hom(M, Λ(d)) coincides with the submodule of CH^{d}(X; Λ) generated by (mult π^{t}) · x. □

Lemma 2.11. Assume that a summand M of the motive of a smooth complete irreducible variety of dimension d decomposes into a sum of Tate motives. Then M is left-outer if and only if the Tate motive Λ is present in the decomposition; it is right-outer if and only if the Tate motive Λ(d) is present in the decomposition.

Proof. For any i ∈ ℤ we have: CH^{0}(Λ(i); Λ) ≠ 0 if and only if i = 0; CH_{d}(Λ(i); Λ) ≠ 0 if and only if i = d. □

Remark 2.12. Assume that the coefficient ring Λ is finite. Let X be an irreducible geometrically split variety satisfying the nilpotence principle. Then the complete motivic decomposition of X contains precisely one left-outer summand and it follows by Corollary 2.6 that a left-outer indecomposable summand of M(X) is unique up to an isomorphism (of motives, not of summands). Of course, the same is true for the right-outer summands.

Lemma 2.13. Assume that the coefficient ring is finite. Let X be an irreducible geometrically split variety satisfying the nilpotence principle. Let M be a motive. Assume that there exist morphisms α : M(X) → M and β : M → M(X) such that mult(β ∘ α) = 1. Then the irreducible left-outer summand of M(X) is isomorphic to a summand of M.

Proof. By Corollary 2.2, the composition π := (β ∘ α)^{∘n} ∈ End M(X) is a projector for some integer n ≥ 1. Therefore τ := (α ∘ β)^{∘2n} ∈ End M is also a projector and the summand (X, π) of M(X) (which is left-outer by Lemma 2.9) is isomorphic to the summand (M, τ) of M given by the image of τ: mutually inverse isomorphisms are, say,

α ∘ (β ∘ α)^{∘(2n)} : (X, π) → (M, τ) and β ∘ (α ∘ β)^{∘(4n−1)} : (M, τ) → (X, π) .

Consequently, any (in particular, an irreducible left-outer) summand of (X, π) is isomorphic to a summand of M. □

2c. Rank of a motive. We are still assuming that the coefficient ring Λ is connected.

Definition 2.14. Let M be a geometrically split motive. Over an extension of the base field the motive M becomes isomorphic to a finite sum of Tate motives. The rank rk M of M is defined as the number of summands in this decomposition.

Remark 2.15. The number of summands in the above definition does not depend on the choice of the extension or of the decomposition. This is simply the rank of the free Λ-module colim_{L/F} CH(M_{L}; Λ), where the colimit is taken over all field extensions L/F.

Example 2.16. Let n be a positive integer and let i be an integer in the interval [0, n]. Let A be a central simple F -algebra of degree n. Since the variety X = X(i, A) (see §2d) is a twisted form of the grassmannian of i-planes in an n-dimensional vector space, the rank of the motive of X coincides with the rank of the motive of the grassmannian and is equal to \binom{n}{i}.

Remark 2.17. Let M be a direct summand of a geometrically split variety X satisfying the nilpotence principle. Then we have: rk M = 0 if and only if M = 0; rk M = rk M(X) if and only if M = M(X).

For any integer l, we write v_{p}(l) for the exponent of the highest power of p dividing l.

Lemma 2.18. Let M be a direct summand of the motive with coefficients in 𝔽_{p} of a geometrically split equi-dimensional variety X. Let d be the greatest common divisor (gcd) of the degrees of the closed points on X. Then v_{p}(d) ≤ v_{p}(rk M).

Proof. Let π ∈ End M(X) = CH_{dim X}(X × X)/p be the projector defining the summand M. Let n = v_{p}(d). Let π′ ∈ CH_{dim X}(X×X)/p^{n} be a lifting of π. By Lemma 2.1, replacing π′ by its appropriate power, we may assume that π′ is a projector. The rank of the motive (X, π′) ∈ CM(F, ℤ/p^{n}) coincides with rk M. Let L/F be a splitting field of the motive (X, π′). Mutually inverse isomorphisms between (X, π′)_{L} and a sum of m = rk M Tate motives are given by two sequences of homogeneous elements a_{1}, …, a_{m} and b_{1}, …, b_{m} in CH(X_{L}; ℤ/p^{n}) satisfying π′_{L} = a_{1} ×b_{1} +⋯+a_{m} ×b_{m} and such that for any i, j = 1, …, m the degree deg(a_{i}b_{j}) is 0 for i ≠ j and 1 ∈ ℤ/p^{n} for i = j. The pull-back of π′ via the diagonal morphism of X is therefore a 0-cycle class on X of degree m (modulo p^{n}). □

Remark 2.19. Lemma 2.18 gives a new, particularly simple proof of the fact that the motive with coefficients in 𝔽_{p} of the Severi-Brauer variety X of a central division algebra of degree p^{n} (n ≥ 0 an integer) is indecomposable: since rk M(X) = p^{n} and the gcd of the degrees of the closed points on X is also p^{n}, the rank of any summand of M(X) is 0 or p^{n} by Lemma 2.18. The original proof of indecomposability of M(X), given in [14], makes use of Quillen’s computation of the K-theory of X and actually gives a more precise statement: the reduced Chow group \overline{Ch}_{dim X}(X × X) is generated by the diagonal class (while the above argument only shows that the diagonal class is the only non-zero projector).

2d. Varieties (of flags) of ideals. Let A be a central simple F -algebra. The F - dimension of any right ideal in A is divisible by the degree deg A of A; the quotient is the reduced dimension of the ideal. For any integer i, we write X(i; A) for the generalized Severi-Brauer variety of the right ideals in A of reduced dimension i. In particular, X(0; A) = Spec F = X(deg A; A) and X(i, A) = ∅ for i outside of the interval [0, deg A]. The variety X(1, A) is the usual Severi-Brauer variety of A studied in [1].

For a finite sequence of integers i_{1}, …, i_{r}, we write X(i_{1}, …, i_{r}; A) for the variety of flags of right ideals in A of reduced dimensions i_{1}, …, i_{r} (non-empty if and only if 0 ≤ i_{1} ≤ ⋯ ≤ i_{r} ≤ deg A).

By an index reduction formula of [22], we have

ind A_{F (X(i_{1},...,i_{r};A))} = gcd(i_{1}, …, i_{r}, ind A).

Another classical property of the variety X(i_{1}, …, i_{r}; A) which we are using frequently is that the greatest common divisor of the degrees of its closed points is equal to

ind A/ gcd(i_{1}, …, i_{r}, ind A).

The varieties introduced above are projective homogeneous under the natural action of the algebraic group Aut A. As in Introduction, we write 𝔛_{A} for the class of finite direct products of such varieties.

Lemma 2.20. Let A be a central simple F -algebra. Let n be a positive integer. Let M_{n}(A) be the F -algebra of (n × n)-matrices with entries in A. Let X be a variety in 𝔛_{M_{n}(A)}. Let m = v_{p}(ind A_{F (X)}).

Then the motive M(X) (with integral and therefore also with arbitrary coefficients) decomposes into a sum of twists of motives of projective (Aut A)-homogeneous F -varieties Y satisfying v_{p}(ind A_{F (Y)}) ≤ m.

Proof. By the assumption on X we have X = X_{1} × ⋯ × X_{r}, where X_{1}, …, X_{r} are some projective (Aut M_{n}(A))-homogeneous F -varieties. Since m = min_{i=1}^{r}{v_{p}(ind A_{F (X_{i})})}, we may assume that m = v_{p}(ind A_{F (X_{1})}). According to [9], the motive of X decomposes into a sum of twists of motives of projective (Aut M_{n}(A))-homogeneous varieties Y admitting a morphism to X_{1}. In particular, X_{1}(F (Y)) ≠ ∅ and consequently v_{p}(ind A_{F (Y)}) ≤ m. Therefore we may assume that X is a projective (Aut M_{n}(A))-homogeneous F -variety, that is, X = X(n_{1}, …, n_{r}; M_{n}(A)) for some positive integer r and some integers n_{1}, …, n_{r} satisfying 0 ≤ n_{1} ≤ ⋯ ≤ n_{r} ≤ deg M_{n}(A).

According to [15] (see also [8]), the motive of X decomposes into a sum of twists of motives of products of n varieties

Y = X(i_{11}, …, i_{1r}; A) × ⋯ × X(i_{n1}, …, i_{nr}; A)

where i_{11}, …, i_{nr} run over the integers satisfying 0 ≤ i_{j1} ≤ ⋯ ≤ i_{jr} ≤ deg A for any j = 1, …, n and i_{1k}+⋯+i_{nk} = n_{k} for any k = 1, …, r. Since min_{jk} v_{p}(i_{jk}) ≤ min_{k} v_{p}(n_{k}), it follows that

v_{p}(ind A_{F (Y)}) =

min{ind A, min_{jk} v_{p}(i_{jk})} ≤ min{ind A, min_{k} v_{p}(n_{k})} =
v_{p}(ind A_{F (X)}) = m,

and we are done. □

2e. Canonical dimension. The notion of canonical dimension was introduced in [4], of canonical dimension at p in [17], of essential dimension in [6], and of essential dimension at p in [24]. We refer to [17] and [20] for proofs of the statements cited below.

Let X be a smooth complete variety over F. We associate to X a functor

F_{X} : Fields/F → Sets,

where Fields/F is the category of field extensions of F and Sets is the category of sets. For a field L/F such that X(L) = ∅ the value F_{X}(L) is empty. For a field L/F such that X(L) ≠ ∅ the value F_{X}(L) is the singleton {L}. Canonical dimension cd X of X is defined as the essential dimension (see [3, Definition 1.2] or [18, §1.3] or [20, §1.1]) of the functor F_{X}. For a positive prime integer p, canonical dimension at p, or canonical p-dimension cd_{p} X of X is defined as the essential dimension at p (see [18, §1.3] or [20, §1.1]) of the functor F_{X}.

If L/F is a finite field extension of degree prime to p, then cd_{p} X_{L} = cd_{p} X.

If two smooth complete irreducible F -varieties X_{1} and X_{2} are such that there exist rational maps X_{1} −−→ X_{2} and X_{2} −−→ X_{1}, then cd X_{1} = cd X_{2} and cd_{p} X_{1} = cd_{p} X_{2} for any p.

Assume that X is irreducible.

One has cd_{p} X ≤ cd X ≤ dim X.

The variety X is called incompressible (minimal in [20]) if cd X = dim X and pincompressible (p-minimal in [20]) if cd_{p} X = dim X. A p-incompressible (for some prime p) variety is incompressible.

The variety X is incompressible if and only if any rational map X −−→ X is dominant. The variety X is p-incompressible if and only if for any element α ∈ CH_{dim X}(X × X)

the multiplicity mult(α) is divisible by p if and only if the multiplicity mult(α^{t}) of the transpose α^{t} of α is divisible by p.

Lemma 2.21. A smooth complete irreducible variety X is p-incompressible if and only if any left-outer summand of the motive of X with coefficients in 𝔽_{p} is outer.

Proof. A left-outer summand of M(X) is given by a projector

π ∈ CH_{dim X}(X × X; 𝔽_{p}).

Since the summand is left-outer, mult(π) = 1 ∈ 𝔽_{p}. If X is p-incompressible, then mult(π^{t}) ≠ 0; therefore mult(π^{t}) = 1 and the summand is right-outer.

Conversely, let α ∈ CH_{dim X}(X × X; 𝔽_{p}) be an element with mult(α) ≠ 0. A power (with respect to the composition of correspondences) of α is a projector which determines a left-outer summand of M(X). If this summand is also right-outer, then mult(α^{t}) ≠ 0. □

Corollary 2.22. If the motive with coefficients in 𝔽_{p} of a smooth complete variety X is indecomposable, then the variety X is p-incompressible. □

3. Basic Theorems

Let p be a positive prime integer. The coefficient ring Λ is 𝔽_{p} in this section.

Let n be a non-negative integer. Let F be a field. Let D be a central division F -algebra of degree p^{n}.

Definition 3.1. For any integer l satisfying 0 ≤ l ≤ n, we write M_{l,D} for the indecomposable left-outer (see Definition 2.8 and Remark 2.12) summand of the Chow motive (with coefficients in 𝔽_{p}) of the generalized Severi-Brauer variety X(p^{l}, D).

Remark 3.2. It can be seen using Lemma 2.13 that the motive M_{l,D} is also the indecomposable left-outer motivic summand of the variety X(i, D) for any integer i satisfying 0 ≤ i ≤ deg D and v_{p}(i) = l.

The first basic theorem of the present paper is as follows:

Theorem 3.3 (First basic theorem). Any indecomposable summand of the Chow motive with coefficients in 𝔽_{p} of any variety X in the class 𝔛_{D} (see §2d) is isomorphic to a twist of M_{l,D} for some l ≤ v_{p}(ind D_{F (X)}).

Proof. We proof Theorem 3.3 by induction on n. The base of the induction is n = 0 and trivial (the class 𝔛_{D} consists of one variety Spec F whose motive is equal to 𝔽_{p} = M_{n,D}).

From now on we are assuming that n ≥ 1 and that Theorem 3.3 is already proven for p-primary central division algebras (over all fields) of degree less than deg D = p^{n}.

Let X be a variety in 𝔛_{D} and let m = v_{p}(ind D_{F (X)}). We have 0 ≤ m ≤ n. The statement of Theorem 3.3 is trivial in the case of m = n (because X = Spec F and M(X) = 𝔽_{p} = M_{n,D} in this case). We assume that m < n in the sequel.

Let M be an indecomposable summand of M(X). We have to show that M is isomorphic to a twist of M_{l,D} for some l satisfying 0 ≤ l ≤ m.

Let L be the function field of the variety X(p^{n−1}, D). Therefore ind D_{L} = p^{n−1}. Let C be a central division L-algebra Brauer-equivalent to D_{L}. By Lemma 2.20, the motive M(X)_{L} decomposes into a sum of twists of motives of projective (Aut C)-homogeneous L-varieties Y satisfying ind C_{L(Y)} ≤ p^{m}. It follows by the induction hypothesis (applied to C), that the summands of the complete decomposition of M(X)_{L} are twists of M_{l,C} with 0 ≤ l ≤ m. The complete decomposition of M_{L} is a part of the above decomposition (in the sense of the Krull-Schmidt principle, see §2a).

Each summand of the complete decomposition of M_{L} decomposes over an algebraic closure L̄ of L into a sum of Tate motives. This gives a decomposition of M̄ = ML̄ into a sum of Tate motives. Let us choose a Tate summand 𝔽_{p}(i) with the smallest i in the decomposition of M̄. This summand comes from the decomposition of the ith twist of some M̄_{l,C} for some integer l with 0 ≤ l ≤ m. We shall show that M ≃ M_{l,D}(i) for these l and i.

By Lemma 2.13 and since M is indecomposable, it suffices to construct morphisms

α : M(X(p^{l}, D))(i) → M and β : M → M(X(p^{l}, D))(i)

satisfying mult(β ∘ α) = 1.

The fixed above summand 𝔽_{p}(i) of M̄ is produced by two elements b ∈ Ch^{i}(X̄) and a ∈ Ch_{i}(X̄) such that deg(a · b) = 1 ∈ 𝔽_{p}. Let M_{l,C}^{t} be the summand of M(X(p^{l}, C)) obtained by transposing the projector giving the summand M_{l,C}(i). Since M_{l,C} is the indecomposable left-outer motivic summand of the variety X(p^{l}, C), a twist of M_{l,C}^{t} is the indecomposable right-outer motivic summand of the same variety. (Later on we shall prove that the left-outer indecomposable motivic summand of X(p^{l}, C) coincides with the right-outer one, but this fact not needed here.) The element a is the image under the embedding Ch(M̄_{l,C}^{t}) ↪ Ch(X̄) of the class of a rational point in Ch (X̄(p^{l}, C)) (see Lemma 2.10). Therefore the element a_{L̄(X(p^{l},D))} is L(X(p^{l}, D))-rational. The field extension L(X(p^{l}, D))/F (X(p^{l}, D)) is purely transcendental. Consequently, the element a_{L̄(X(p^{l},D))} is F (X(p^{l}, D))-rational and lifts to an element α_{1} ∈ Ch (X(p^{l}, D) × X). We mean here a lifting with respect to the composition

Ch (X(p^{l}, D) × X) →→ Ch(X_{F (X(p^{l},D))}) ⟶^{res_{L̄(X(p^{l},D))/F (X(p^{l},D))}} Ch(X_{L̄(X(p^{l},D))})

where the first map is the epimorphism given by the pull-back with respect to the morphism X_{F (X(p^{l},D))} → X(p^{l}, D) × X induced by the generic point of the variety X(p^{l}, D).

We define the morphism α as the composition

M(X(p^{l}, D))(i) ⟶^{α_{1}} M(X) ⟶ M.

The element b is the image of the class [X(p^{l}, C)] under the embedding Ch(M̄_{l,C}) ↪ Ch(X̄). Let β_{1} ∈ Ch (X(p^{l}, C) × X(p^{l}, D_{L})) be the class of the closure of the graph of a rational map (of L-varieties) X(p^{l}, C) → X(p^{l}, D_{L}). Let β_{2} be the image of β_{1} under the composition of the homomorphisms

Ch(X(p^{l}, C) × X(p^{l}, D_{L})) → Ch(M_{l,C} ⊗ M(X(p^{l}, D_{L}))) → Ch(X_{L} × X(p^{l}, D_{L})).

Since the field extension L(X)/F (X) is purely transcendental, the element (β_{2})_{L(X)} is F (X)-rational. Consequently, it lifts to an element β_{3} ∈ Ch(X ×X ×X(p^{l}, D)) (where we use the generic point of the second factor in this triple direct product). Let π ∈ Ch(X×X) be the projector defining the summand M of M(X). Considering β_{3} as a correspondence from X to X × X(p^{l}, D), we define β_{4} ∈ Ch(X × X × X(p^{l}, D)) as the composition β_{3} ∘ π. We get β_{5} ∈ Ch(X × X(p^{l}, D)) as the image of β_{4} under the pull-back with respect to the diagonal of X. Finally, we define the morphism β as the composition

M ⟶ M(X) ⟶^{β_{5}} M(X(p^{l}, D))(i).

We finish the proof by checking that mult(β ∘ α) = 1. Since the multiplicity is not changed under extension of scalars, we may do the computation over the field L̄. Decompositions of the motives of the varieties X̄(p^{l}, D) = X(p^{l}, D)L̄ and X̄ = XL̄ into sums of Tate motives give certain homogeneous 𝔽_{p}-bases of their Chow groups with coefficients in 𝔽_{p}. Let us use an arbitrary motivic decomposition of X̄(p^{l}, D) and we use a decomposition of X̄ which is a refinement of the decomposition into the sum of M̄ and the complementary summand. Then the elements a and b are in the basis of Ch(X̄). The basis of Ch (X̄(p^{l}, D)) contains the class 1 = [X̄(p^{l}, D)] and the class x of a rational point.

We consider the Chow group of the products

X̄(p^{l}, D) × X̄, X̄ × X̄(p^{l}, D), and X̄(p^{l}, D) × X̄(p^{l}, D)

together with the bases given by the external products of the elements of the bases of the Chow groups of the factors. We have ᾱ_{1} = 1 × a + …, where “… ” stands for a linear combination of basis elements whose first factor has codimension > codim 1 = 0. The projector π which determines the summand M of the motive of X looks over L̄ as π̄ = b × a + …, where “… ” stands for a linear combination of basis elements whose first factor has codimension > codim b = i. Since α = π ∘ α_{1}, it follows that ᾱ = 1 × a + …, where “… ” still stands for a linear combination of basis elements whose first factor is of positive codimension.

Now let us go through the construction of β. To describe β̄_{1}, we fix a homogeneous basis of Ch (X̄(p^{l}, C)) and use it to built up a basis of Ch (X̄(p^{l}, C)×X̄(p^{l}, D)). Abusing notation we write 1 also for the unit class in Ch (X̄(p^{l}, C)).

We have β̄_{1} = 1 × x + … with the usual convention on the meaning of “… ”. Then we have β̄_{2} = b×x+. . .. For β_{3} we have β̄_{3} = b×1×x+…, where 1 is the unit of Ch(X̄) and “… ” stands for a linear combination of basis elements of Ch (X̄ × X̄ × X̄(p^{l}, D)) which have a second factor of positive codimension or the first factor of dimension < dim b = i. The element β̄_{4} has the same shape with the additional property that the dimension of the first factor in each basis element appearing in the linear combination is ≤ i. By this reason β̄_{5} and also β̄ look as b × x + . . .. Therefore β̄ ∘ ᾱ = 1 × x + …, that is, mult(β̄ ∘ ᾱ) = 1. □

Corollary 3.4. For an arbitrary central simple F -algebra A, let D be the p-primary component of a central division F -algebra Brauer-equivalent to A. The motive with coefficients in 𝔽_{p} of any variety in 𝔛_{A} decomposes into a sum of twists of motives M_{l,D} (with various l).

Proof. We may replace F by a finite field extension of prime to p degree over which A is Brauer-equivalent to D. Then the motive of any variety in 𝔛_{A} decomposes into a sum of twists of motives of varieties in 𝔛_{D} and we may apply Theorem 3.3. □

Using Theorem 3.3, we give a new example of a generalized Severi-Brauer variety with indecomposable motive:

Example 3.5. Let F be a field. 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 (this is trivial for n = 1, well known for n = 2, recently proved using K-theory by Maksim Zykhovich for n = 3, new for n ≥ 4).

Proof. Let us prove it by induction on n. Assume that n ≥ 2 and that the statement is already proved for algebras (over all fields) of degree < 2^{n}. By Theorem 3.3 and Remark 2.19, if the motive of X(2; D) is decomposable (for a given D), then some twist of the motive of the Severi-Brauer variety X(1; D) is a summand of M(X(2; D)). We can check however that no twist of M(X(1; D))_{L} is a summand of M(X(2; D))_{L} where L/F is a field extension such that ind D_{L} = 2^{n−1}.

Indeed, let C be a central division L-algebra Brauer-equivalent to D_{L}. The complete decompositions of the motives of these two varieties over L are:

M(X(1; D))_{L} = M(X(1; C)) ⊕ M(X(1; C))(2^{n−1}) = M_{0,C} ⊕ M_{0,C}(2^{n−1})

and (we apply the induction hypothesis to C)

M(X(2; D))_{L} =

M(X(2; C)) ⊕ (M(X(1; C)) ⊗ M(X(1; C)))(2^{n−1} − 1) ⊕ M(X(2; C))(2^{n}) =

M_{1,C} ⊕ M_{0,C}(2^{n−1} − 1) ⊕ M_{0,C}(2^{n−1}) ⊕ ⋯ ⊕ M_{0,C}(2^{n} − 2) ⊕ M_{1,C}(2^{n}).

Note that the motives M_{0,C} and M_{1,C} are not isomorphic because, for instance, they have different ranks (see Example 2.16): rk M_{0,C} = rk M(X(1, C)) = \binom{deg C}{1} = 2^{n−1} and

rk M_{1,C} = rk M(X(2, C)) = \binom{deg C}{2} = 2^{n−2}(2^{n−1} − 1).  □

We come back to an arbitrary central division F -algebra of degree p^{n}. The second basic theorem describes some properties of the motives M_{m,D}, 0 ≤ m ≤ n:

Theorem 3.6 (Second basic theorem). The summand M_{m,D} of the motive of the variety X(p^{m}, D) is outer and

v_{p}(rk M_{m,D}) = n − m.

Proof. We prove Theorem 3.6 by induction on n. The base of the induction is the case of n = 0 which is trivial. Below we are assuming that n ≥ 1.

The statement is trivial for m = n. Below we are assuming that m < n.

We are proving Theorem 3.6 for a given n by induction on m. The base of the induction is the case of m = 0 where we know (see Remark 2.19) that M_{0,D} = M(X(1, D)) and rk M(X(1, D)) = p^{n}. Below we are assuming that m ≥ 1.

We ask the reader to note that in this proof we do not pay attention to the twisting numbers of summands in motivic decompositions: when we say that a motive W is a summand of a motive V, we mean that a twist of W is a summand of V.

Let L be the function field of the variety X(p^{n−1}, D). Let C be a central division Lalgebra (of degree p^{n−1}) Brauer-equivalent to D_{L}. The motive M(X(p^{m}, D))_{L} decomposes into the sum (of some twists) of the motives

(i_{1}, …, i_{p}) := M(X(i_{1}, C) × X(i_{2}, C) × ⋯ × X(i_{p}, C)),

where i_{1}, …, i_{p} run over the non-negative integers satisfying i_{1} + i_{2} + ⋯ + i_{p} = p^{m}. The left-outer summand in this decomposition is (p^{m}, 0, …, 0) = M(X(p^{m}, C)).

Let M = M_{m,D}. Since the summand M of M(X(p^{m}, D)) is left-outer, the L-motive M_{L} contains the summand M_{m,C} of the summand (p^{m}, 0, …, 0). We claim that in fact M_{L} contains the p summands M_{m,C} coming from each of the p summands

(p^{m}, 0, …, 0) = M(X(p^{m}, D)), (0, p^{m}, 0, …, 0) = M(X(p^{m}, D)),

…, (0, …, 0, p^{m}) = M(X(p^{m}, D)).

Indeed, since the degree of any closed point on the variety X(p^{m}, D) is divisible by p^{n−m}, we have v_{p}(rk M) ≥ n − m by Lemma 2.18. On the other hand, by the induction hypothesis, v_{p}(rk M_{m,C}) = n − 1 − m. The remaining summands of the complete motivic decomposition of X(p^{m}, D)_{L} are M_{l,C} with l ≤ m − 1; v_{p} of their ranks are at least n − m (by the induction hypothesis once again).

We have in particular proved that M_{L} contains the summand M_{m,C} coming from the right-outer summand (0, …, 0, p^{m}) of M(X(p^{m}, D_{L})). Since M_{m,C} is a right-outer summand of X(p^{m}, C) (by the induction hypothesis), it follows that M is right-outer.

It remains to prove the statement on the rank of M = M_{m,D}.

To do this, we look at ranks of the summands in the complete decomposition of M_{L}. We have p summands M_{m,C} with v_{p}(rk M_{m,C}) = n − 1 − m. So, we have v_{p}(rk M_{m,C}^{⊕p}) = n − m for this part of the complete decomposition of M_{L}.

The summands M_{l,C} with l ≤ m − 2 have v_{p}(rk M_{l,C}) ≥ n − m + 1; so, we do not care about the number of such summands.

In order to show that v_{p}(rk M) = n − m, it suffices to show that the number of the summands M_{m−1,C} (which have v_{p}(rk M_{m−1,C}) = n − m) in the complete decomposition of M_{L} is divisible by p.

Let us first count the number of summands M_{m−1,C} in the complete motivic decomposition of X(p^{m}, D)_{L}. We use the complete decomposition of X(p^{m}, D)_{L} which is a refinement of the decomposition into a sum of (i_{1}, …, i_{p}) considered above. There are two types of such summands M_{m−1,C}: those which appear as summands of M(X(p^{m}, C)) (if any) and all the others. Since the number of the summands M(X(p^{m}, C)) is p, the number of the summands M_{m−1,C} of the first type is divisible by p.

The summands of the second type are summands of the summands (i_{1}, …, i_{p}) with min_{j} v_{p}(i_{j}) = m − 1. There is precisely one such summand (i_{1}, …, i_{p}) with i_{1} = ⋯ = i_{p}, namely, the summand (p^{m−1}, …, p^{m−1}). All the other such summands (i_{1}, …, i_{p}) can be divided into disjoint groups which are orbits of the action of the cyclic group ℤ/p on the indices: the group containing a given (i_{1}, …, i_{p}) consists of

(i_{1}, …, i_{p}), (i_{2}, …, i_{p}, i_{1}), …, (i_{p}, i_{1}, …, i_{p−1})

(note that these are p different summands because the integer p is prime). So, the number of the summands M_{m−1,C} coming from the summands (i_{1}, …, i_{p}) different from (p^{m−1}, …, p^{m−1}) is divisible by p.

As to the remaining summand (p^{m−1}, …, p^{m−1}) = M(X(p^{m−1}, C)^{×p}), we can show (using the induction hypothesis) that the number of the summands M_{m−1,C} in its complete motivic decomposition is also divisible by p. More generally, we can show that the number of the summands M_{m−1,C} in the complete motivic decomposition of X(p^{m−1}, C)^{×r} is divisible by p as far as r ≥ 2. Indeed,

v_{p}( rk M(X(p^{m−1}, C)^{×r})) = r(n − m) > n − m

(we recall that m < n). The complete motivic decomposition of X(p^{m−1}, C)^{×r} consists of the motives M_{l,C} with l ≤ m − 1. Finally, v_{p}(rk M_{l,C}) > n − m for l < m − 1 and v_{p}(rk M_{m−1,C}) = n − m.

We have shown that the number of summands M_{m−1,C} in the complete motivic decomposition of X(p^{m}, D)_{L} is divisible by p. We finish by showing that the number of those summands M_{m−1,C} which are not in the complete decomposition of M_{L} is also divisible by p.

If a summand M_{m−1,C} is not in M_{L} = (M_{m,D})_{L}, then it is in (M_{m−1,D})_{L}. However the number of summands M_{m−1,C} in M_{m−1,D} is divisible by p because v_{p}(rk M_{m−1,C}) = n − m and v_{p}(d) = n − m + 1, where d is the gcd of the degrees of the closed points on X(p^{m−1}, D). □

Theorem 3.6 together with Lemma 2.21 give the main theorem:

Theorem 3.7 (Main theorem). Let p be a positive prime integer. Let n be a non-negative integer. Let F be a field. Let D be a central division F -algebra of degree p^{n}. Let m be an integer in the interval [0, n]. Then the variety X(p^{m}; D) is p-incompressible. □

For an arbitrary central simple F -algebra A, canonical p-dimension of the varieties in 𝔛_{A} is computed as follows:

Corollary 3.8. Let X be a variety in 𝔛_{A}. Let n = v_{p}(ind A) and m = v_{p}(ind A_{F (X)}). Then cd_{p} X = p^{m}(p^{n} − p^{m}).

Proof. Let D be the p-primary part of a central division F -algebra Brauer-equivalent to A. Let L/F be a finite field extension of prime to p degree such that the L-algebra A_{L} is Brauer-equivalent to D_{L}. There exist rational maps X_{L} −−→ X(p^{m}, D_{L}) and X(p^{m}, D_{L}) −−→ X_{L}. It follows that

cd_{p} X = cd_{p} X_{L} = cd_{p} X(p^{m}, D_{L}) = dim X(p^{m}, D) = p^{m}(deg D − p^{m}) = p^{m}(p^{n} − p^{m}). □

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UPMC Univ Paris 06, Institut de Mathématiques de Jussieu, F-75252 Paris, FRANCE

Web page: www.math.jussieu.fr/~karpenko
E-mail address: karpenko at math.jussieu.fr