INCOMPRESSIBILITY OF PRODUCTS

NIKITA A. KARPENKO

Abstract. We show that the conjectural criterion of p-incompressibility for products of projective homogeneous varieties in terms of the factors, previously known in a few special cases only, holds in general. We identify the properties of projective homogeneous varieties actually needed for the proof to go through. For instance, generically split (nonhomogeneous) varieties also satisfy these properties.

Let F be a field. A smooth complete irreducible F -variety X is incompressible, if every rational self-map X −−→ X is dominant. This means that cdim X = dim X, where the canonical dimension cdim X is defined as the minimum of dim Y for Y running over closed irreducible subvarieties of X admitting a rational map X −−→ Y.

For the whole exposition, let p be a fixed prime number. Canonical p-dimension cdim_{p} X is defined as the minimum of dim Y for Y running over closed irreducible sub-

varieties of X admitting a degree 0 correspondence X ⇝^{p′} Y of p-prime multiplicity. The variety X is p-incompressible, if every degree 0 self-correspondence X ⇝^{p′} X of p-prime multiplicity is dominant, i.e., if cdim_{p} X = dim X. The closure of the graph of a rational map is a degree 0 correspondence of multiplicity 1; therefore a p-incompressible (for at least one p) variety is incompressible.

Studying canonical p-dimension, instead of the integral Chow group CH, it is more appropriate to use the Chow group Ch with coefficients in 𝔽_{p} := ℤ/pℤ. Multiplicities of correspondences as well as degrees of 0-cycles take then values in 𝔽_{p}. We also consider the Chow motives with coefficients in 𝔽_{p}, see [2, Chapter XII].

Now we are going to introduce a class of varieties, called nice here, for which we can prove that the following criterion holds (see Theorem 9): the product X × Y of F - varieties X and Y is p-incompressible if and only if the varieties X_{F (Y)} and Y_{F (X)} are p-incompressible.

A smooth complete variety is split, if its motive decomposes into a finite direct sum of Tate motives. By Tate motive, we mean an arbitrary shift of the motive of the point Spec F. For instance, an (absolutely) cellular variety is split, [2, Corollary 66.4].

A smooth complete variety X is nice, if it has the following three properties:

(i) The variety X is geometrically split, that is, there exists a field extension L/F such that the L-variety X_{L} is split.

Date: January 31, 2015.

Key words and phrases. Algebraic groups; projective homogeneous varieties; Severi-Brauer varieties; Chow groups and motives; canonical dimension and incompressibility. Mathematical Subject Classification (2010): 20G15; 14C25.

This work has been supported by a Start-Up Grant of the University of Alberta and a Discovery Grant from the National Science and Engineering Board of Canada.

(ii) The variety X is A-trivial (cf. [11, Definition 2.3]), that is, for any field extension L/F with X(L) ≠ ∅, the degree homomorphism deg : Ch_{0}(X_{L}) → 𝔽_{p} is an isomorphism.

(iii) For any field extension L/F, one has cdim_{p} X ≥ d, where d is the minimal integer such that there exist an element a ∈ Ch_{d} X_{L} and an element b ∈ Ch^{d}(X_{L(X)}) with deg(a_{L(X)} · b) = 1 (see Remarks 3 and 4).

Remark 1. The definition of “nice” depends on the prime p. We should probably better say “p-nice”, but we keep saying “nice” for short. The same applies to “split” and “Atrivial”. On the other hand, we do not abbreviate “p-incompressible”.

Remark 2. A nice variety remains nice under any base field extension. On the other hand, it is not clear if the product of two nice varieties is necessarily nice.

Remark 3. Property (iii), referring to the function field of X_{L}, is well-defined because any A-trivial variety is geometrically integral, see [11, Remark 2.4]. In particular, any nice variety is geometrically integral.

Remark 4. The opposite to the inequality in (iii) always holds (cf. [9, Proof of Theorem 5.8, part “≤”]). Indeed, take the minimal d such that there exist a ∈ Ch_{d} X, and b ∈ Ch^{d}(X_{F (X)}) with deg(a_{F (X)} · b) = 1. We may assume that a = [Y] and b = [Z] for closed subvarieties Y ⊂ X and Z ⊂ X_{F (X)}. Since the product [Y_{F (X)}]·[Z] ∈ Ch(X_{F (X)}), which is a 0-cycle class of degree 1, can be represented by a 0-cycle with support on the intersection Y_{F (X)} ∩ Z (see [3, §8.1]), the variety Y_{F (X)} has a 0-cycle of degree 1, that is, there exists a degree 0 correspondence X ⇝ Y of multiplicity 1 (see [2, Page 328] concerning the relation between correspondences and 0-cycles). Therefore cdim_{p} X ≤ dim Y = d.

Here is our basic example of nice varieties:

Example 5. Any projective homogeneous (under an action of a semi-simple affine algebraic group) variety over a p-special field is nice: see [13] for (i), [11, Example 2.5] for (ii), and [6, Proposition 6.1] for (iii). A field F is p-special, if it has no finite extension fields of degree prime to p. The condition that F is p-special is only needed for (iii).

A smooth complete geometrically irreducible F -variety is generically split, if for any field extension L/F with X(L) ≠ ∅, the L-variety X_{L} is split.

Example 6. Any generically split variety is nice. Indeed, (i) holds for L = F (X), (ii) holds by [9, discussion after Remark 5.6], and (iii) holds by [9, Theorem 5.8 with Remark 5.6].

The direct product of two projective homogeneous varieties is also projective homogeneous and therefore – over a p-special field – nice. Similarly, the direct product of two generically split varieties is generically split (and nice). The mixed product (over a p-special field) turns out to be nice as well:

Example 7. Over a p-special field, the direct product X of a projective homogeneous variety by a generically split one is nice. Indeed, X is, clearly, geometrically split and A-trivial. Property (iii) can be obtained for X in the same way as it is obtained for a projective homogeneous variety in [6, Proposition 6.1]. The upper motive U(X), used in the proof of [6, Proposition 6.1], is defined for X in [7]; [5, Theorem 5.1 and Proposition 5.2], also used in the proof of [6, Proposition 6.1], can be proved for X by almost literal repetition of their proofs; the same is valid for [7, Theorem 1.1], used in the proof of [5, Proposition 5.2].

The following well-known criterion of p-incompressibility for projective homogeneous varieties actually holds for arbitrary A-trivial varieties:

Lemma 8. An A-trivial variety X is p-incompressible if and only if mult ρ = mult ρ^{t} for any degree 0 correspondence ρ : X ⇝ X, where ρ^{t} is the transpose of ρ. In particular, this criterion holds for any nice variety X.

Proof. We almost repeat the proof of [5, Lemma 2.7].

If X is p-compressible, there exists a correspondence α : X ⇝ Y of degree 0 and multiplicity 1 to a proper closed subvariety Y ⊂ X. Considering α as a correspondence X ⇝ X, we have mult α = 1 and mult α^{t} = 0. Therefore the “only if” part of Lemma 8 holds for arbitrary smooth complete irreducible varieties X, not only for A-trivial ones.

The other way round, suppose that we are given a degree 0 correspondence α : X ⇝ X with mult α ≠ mult α^{t}. Adding a multiple of the diagonal class and multiplying by an element of 𝔽_{p}, we may achieve that mult α = 1 and mult α^{t} = 0. In this case the pull-back of α with respect to the morphism X_{F (X)} → X × X induced by the generic point of the second factor of the product X × X, is a 0-cycle class of degree 0. Since X is A-trivial, the degree homomorphism Ch_{0}(X_{F (X)}) → 𝔽_{p} is an isomorphism. Therefore the pull-back of α is 0. By the continuity property of Chow groups [2, Proposition 52.9], there exists a non-empty open subset U ⊂ X such that the pull-back of α to X ×U is already 0. By the localization sequence [2, Proposition 57.9], it follows that α is the push-forward of some degree 0 correspondence β : X ⇝ Y ∈ Ch_{dim X}(X × Y), where Y is the proper closed subset Y := X \ U of X. Since mult β = mult α = 1, the variety X is p-compressible. □

The main result of this note is the “≥” part of equality (10) in the following theorem:

Theorem 9. Let X and Y be nice F -varieties such that the product X × Y is also nice. The variety X × Y is p-incompressible if and only if the varieties X_{F (Y)} and Y_{F (X)} are p-incompressible. Moreover

(10)  cdim_{p}(X × Y) = cdim_{p} X_{F (Y)} + cdim_{p} Y_{F (X)}

provided that at least one of the three varieties X_{F (Y)}, Y_{F (X)}, X × Y is p-incompressible.

Corollary 11. The product X × Y of projective homogeneous F -varieties X and Y is p-incompressible if and only if the varieties X_{F (Y)} and Y_{F (X)} are p-incompressible. Moreover, (10) holds provided that at least one of the varieties X_{F (Y)}, Y_{F (X)}, X × Y is pincompressible.

Proof. Since canonical p-dimension of a variety does not change under any base field extension of degree prime to p (see [16, Proposition 1.5]), we may assume that F is p-special. By Example 5, X, Y, and X × Y are nice in this case so that Theorem 9 applies. □

Partial cases of Corollary 11, dealing with some special types of projective homogeneous varieties, have been recently proved in [8] and [4]. For an older result in this direction see Example 13 below.

The p-incompressibility criterion, given in Theorem 9 for nice products of two nice varieties, immediately generalizes to finite products of arbitrary length:

Corollary 12. For n ≥ 1, let X_{1}, …, X_{n} be F -varieties such that every sub-product of the product X := X_{1}×⋯×X_{n} is nice. Then X is p-incompressible if and only if for every i = 1, …, n the variety (X_{i})_{F (X_{1}×···×X_{i−1}×X_{i+1}×X_{n})} is p-incompressible. The criterion also holds if for any i = 1, …, n the variety X_{i} is projective homogeneous or generically split.

Proof. Assuming that the statement holds for some n ≥ 1, we prove it for n + 1. Set X := X_{1} × ⋯ × X_{n} and Y := X_{n+1}. If X × Y = X_{1} × ⋯ × X_{n+1} is p-incompressible, X_{F (Y)} and Y_{F (X)} are p-incompressible, and it follows by induction hypothesis that the variety (X_{i})_{F (X_{1}×···×X_{i−1}×X_{i+1}×X_{n+1})} is p-incompressible for any i = 1, …, n + 1.

The other way round, if (X_{i})_{F (X_{1}×···×X_{i−1}×X_{i+1}×X_{n+1})} is p-incompressible for any i, then, in particular, Y_{F (X)} is p-incompressible and – by induction hypothesis – X_{F (Y)} is pincompressible. It follows that X × Y is p-incompressible. The first statement is proved.

Since any finite direct product of projective homogeneous or generically split varieties over a p-special field is nice (see Example 7), the second statement follows. □

Example 13. For purpose of computing the essential dimension of finite groups, Corollary 12 for Severi-Brauer varieties X_{1}, …, X_{n} has been obtained in [10]. A second and simpler proof has been given in [8]. The third proof, given here (see Proof of Theorem 9), is particularly simple. The result has numerous further applications, see, e.g., [14, 15].

Example 14. For purpose of computing the essential dimension of representations of finite groups, introduced in [12], Corollary 12 for Weil transfers of generalized Severi-Brauer varieties has been obtained in [8] under assumption that the corresponding central simple algebras are balanced. Corollary 12 shows that this assumption is superfluous. Another area of applications for this result is provided in [1].

Proof of Theorem 9. We start by introducing some notation and by making some preliminary observations.

We fix a field extension F̄/F splitting both X and Y. For any finite direct product T of copies of X and Y, we write T̄ for TF̄. We work with the Chow group Ch T̄ with coefficients in 𝔽_{p}. Note that for any field extension E/F̄, the change of field homomorphism Ch T̄ = Ch TF̄ → Ch T_{E} is an isomorphism, so that we may identify Ch T_{E} with Ch T̄. For a geometrically integral F -variety S (e.g., S = X, S = Y, or S = X × Y), an element c ∈ Ch T̄ = Ch TF̄ = Ch T_{F̄(S)} is F (S)-rational, if it lies in the image of the change of field homomorphism Ch T_{F (S)} → Ch T_{F̄(S)}.

Since the varieties X̄ and Ȳ are split, any correspondence λ : T̄ ⇝ T̄′, where T and T′ are finite direct products of copies of X and Y, decomposes in a finite sum of external products c × c′ with homogeneous c ∈ Ch T̄ and c′ ∈ Ch T̄′. This makes it easy to perform computations with correspondences. For instance, the composition of composable correspondences (e′ × e′′) ∘ (c × c′) is equal to

(15)  (e′ × e′′) ∘ (c × c′) = deg(c′ · e′) · (c × e′′).

For e ∈ Ch T̄, (c × c′)_{∗}(e) = deg(c · e) · c′, where (c × c′)_{∗} : Ch T̄ → Ch T̄′ is the induced by c × c′ homomorphism, see [2, §62]. If b ∈ Ch T′_{F̄(T)} = Ch T̄′ is the image of λ under the pull-back with respect to the morphism T̄′_{F̄(T)} → T̄ × T̄′, given by the generic point of T, then λ decomposes as

(16)  λ = [T̄] × b + …,

where … stands for a sum of c × c′ with codim c > 0 (and dim c′ > dim b). If the correspondence λ has degree 0, then it decomposes as

λ = (mult λ) · ([T̄] × [pt′]) + …,

where pt′ is a rational point on T̄′ and where … stands for a sum of c×c′ with codim c = dim c′ > 0; if moreover dim T = dim T′, then

(17)  λ = (mult λ) · ([T̄] × [pt′]) + (mult λ^{t}) · ([pt] × [T̄′]) + …,

where pt is a rational point on T̄ and where … stands for a sum of c × c′ with codim c = dim c′ > 0 and dim c = codim c′ > 0.

In order to prove Theorem 9 in whole, we only need to prove equality (10). We start the prove of its (more difficult) “≥” part now. If the variety X × Y is p-incompressible, the “≥” part is however trivial. We therefore assume that the F (X)-variety Y_{F (X)} is p-incompressible, that is, cdim_{p} Y_{F (X)} = dim Y.

Let d be an integer such that there exist F -rational a ∈ Ch_{d}(X̄ × Ȳ) and F (X × Y)- rational b ∈ Ch^{d}(X̄ × Ȳ) with deg(a · b) = 1. Since the product X × Y is nice, we have cdim_{p}(X × Y) ≥ d. Our aim is to show that d ≥ cdim_{p} X_{F (Y)} + dim Y.

Let α ∈ Ch(X̄ × Ȳ × X̄ × Ȳ) be the push-forward of a under the diagonal morphism of X̄ × Ȳ. The element α is F -rational. Note that α = (a × [X̄] × [Ȳ]) · Δ, where Δ ∈ Ch(X̄ × Ȳ × X̄ × Ȳ) is the diagonal class.

Let β be a homogeneous F -rational preimage of b under the flat pull-back

Ch ((X̄ × Ȳ) × (X̄ × Ȳ)) → Ch(X̄ × Ȳ)_{F̄(X×Y)},

along the morphism induced by the generic point of the first factor of the product (X̄ × Ȳ) × (X̄ × Ȳ). For existence of β, see [2, Corollary 57.11].

Let δ ∈ Ch(Ȳ × X̄ × Ȳ) be the image of the diagonal class of Y under the push-forward with respect to the closed imbedding Ȳ × Ȳ ↪ Ȳ × X̄ × Ȳ induced by a closed rational point ptX̄ on X̄. Since the element [ptX̄] ∈ Ch X̄ is F (X)-rational, the element δ is also F (X)-rational.

Finally, let γ ∈ Ch(X̄ × Ȳ × Ȳ) be the class of the graph of the projection X̄ × Ȳ → Ȳ. The element γ is F -rational.

We consider the elements α, β, γ, δ as correspondences and take their composition ρ in the following order:

ρ : Ȳ ⇝^{δ} X̄ × Ȳ ⇝^{β} X̄ × Ȳ ⇝^{α} X̄ × Ȳ ⇝^{γ} Ȳ .

The correspondence ρ : Ȳ ⇝ Ȳ is F (X)-rational.

Let ptȲ be a rational point on Ȳ. Since the variety Y is A-trivial, the class [ptȲ] does not depend on the choice of ptȲ. A direct computation shows that

ρ_{∗}([ptȲ]) = [ptȲ],

where ρ_{∗} : Ch Ȳ → Ch Ȳ is the homomorphism induced by ρ. Indeed,

[ptȲ] ↦^{δ_{∗}} [ptX̄] × [ptȲ] ↦^{β_{∗}} b ↦^{α_{∗}} [ptX̄] × [ptȲ] ↦^{γ_{∗}} [ptȲ],

where the image under β_{∗} is computed via the formulae (16) and (15).

The general formula ρ_{∗}([ptȲ]) = (mult ρ)[ptȲ] implies that mult ρ = 1. Since the A-trivial F (X)-variety Y_{F (X)} is p-incompressible while ρ is F (X)-rational, it follows by Lemma 8 that mult ρ^{t} = 1. The general formula ρ_{∗}([Ȳ]) = (mult ρ^{t})[Ȳ] shows now that ρ_{∗}([Ȳ]) = [Ȳ]. We therefore have

[Ȳ] ↦^{δ_{∗}} [ptX̄] × [Ȳ] ↦^{β_{∗}} b′ ↦^{α_{∗}} [ptX̄] × [Ȳ] + … ↦^{γ_{∗}} [Ȳ]

for some b′ ∈ Ch(X̄ × Ȳ), where … stands for a sum of c × c′ with dim c = codim c′ > 0 so that the whole sum is an arbitrary element of Ch_{dim Y} (X̄ × Ȳ) mapped to [Ȳ] under γ_{∗}.

The diagonal class Δ ∈ Ch(X̄×Ȳ ×X̄×Ȳ) is the external product of the diagonal classes Δ_{X} ∈ Ch(X̄ × X̄) and Δ_{Y} ∈ Ch(Ȳ × Ȳ). Multiplying decompositions (17) of Δ_{X} and Δ_{Y}, we get a decomposition of Δ. This decomposition of Δ possesses a unique summand ending with [ptX̄] × [Ȳ]. This unique summand starts with [X̄] × [ptȲ]. Moreover, any other summand ends with c × c′ such that dim c > 0 or codim c′ > 0. The resulting decomposition of α = (a × [X̄] × [Ȳ]) · Δ also possesses a unique summand ending with [ptX̄] × [Ȳ]. This unique summand starts now with a′ := a · ([X̄] × [ptȲ]). Any other summand still ends with c × c′, where dim c > 0 or codim c′ > 0. Therefore, by (15), we must have deg(a′ · b′) = 1 in order to get the right image of b′ under α_{∗}.

Let pr be the projection X̄ × Ȳ → X̄. It follows that deg(a′′ · b′′) = 1, where

a′′ := pr_{∗}(a′) ∈ Ch X̄ and b′′ := pr_{∗} (([X̄] × [ptȲ]) · b′) ∈ Ch X̄.

Since a′′ is F (Y)-rational and b′′ is F (X × Y)-rational, it follows by Remark 4 that dim a′′ ≥ cdim_{p} X_{F (Y)}. Since dim a′′ = dim a′ = dim a − dim Y = d − dim Y, we get that d ≥ cdim_{p} X_{F (Y)} + dim Y. The “≥” part of equality (10) is proved.

The proof of the “≤” part, given in [8, Lemma 3.4] for projective homogeneous X and Y, also works in our current settings. For reader’s convenience, let us reproduce it. As in [8, Lemma 3.4], we prove the more general inequality

cdim_{p}(X × Y) ≤ cdim_{p} X + cdim_{p} Y_{F (X)}

without any p-incompressibility assumption (on X_{F (Y)}, on Y_{F (X)}, or on X × Y).

We set x := cdim_{p} X and y := cdim_{p} Y_{F (X)}. Since the variety X is nice, we can find F -rational a_{X} ∈ Ch_{x} X̄ and F (X)-rational b_{X} ∈ Ch^{x} X̄ with deg(a_{X} · b_{X}) = 1. Similarly, since the variety Y_{F (X)} is nice, we can find F (X)-rational a_{Y} ∈ Ch_{y} Ȳ and F (X)(Y)- rational b_{Y} ∈ Ch^{y} Ȳ with deg(a_{Y} · b_{Y}) = 1. Let α_{Y} ∈ Ch_{dim X+y}(X × Y) be an F -rational preimage of a_{Y} under the pull-back along the morphism Ȳ_{F̄(X)} → X̄ × Ȳ induced by the generic point of X. We set

a := (a_{X} × [Ȳ]) · α_{Y} ∈ Ch_{x+y}(X̄ × Ȳ) and b := b_{X} × b_{Y} ∈ Ch^{x+y}(X̄ × Ȳ).

The element a is F -rational, the element b is F (X × Y)-rational. We have the relation deg(a · b) = deg(a_{X} · b_{X}) · deg(a_{Y} · b_{Y}) = 1 showing by Remark 4 that cdim_{p}(X × Y) ≤ x + y. □

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Mathematical & Statistical Sciences, University of Alberta, Edmonton, CANADA

E-mail address: karpenko at ualberta.ca, web page:  www.ualberta.ca/~karpenko