INCOMPRESSIBILITY OF PRODUCTS OF WEIL TRANSFERS OF GENERALIZED SEVERI-BRAUER VARIETIES

NIKITA A. KARPENKO

Abstract. We generalize the result of [11] on incompressibility of Galois Weil transfer of generalized Severi-Brauer varieties, to direct products of varieties of such type; as shown in [11], this is needed to compute essential dimension of representations of finite groups. We also provide a generalization to non-Galois (separable) Weil transfer.

Contents

1. Introduction 1 2. Non-Galois extensions 2 3. Incompressibility of products 4 4. Particular products 8 4a. Products of Severi-Brauer varieties 8 4b. Weil transfer of products of Severi-Brauer varieties 10 4c. Weil transfer of products of generalized Severi-Brauer varieties 10 5. Quadratic extensions 11 References 12

1. Introduction

Let F be a field and let L/F be a finite Galois field extension. A central division Lalgebra D is balanced, if for every g ∈ Gal(L/F), the conjugate L-algebra g(D), obtained from D by the base change g : L → L, is Brauer-equivalent to a tensor power of D.

For a balanced D as above, we refer as RSB-variety to the Weil transfer R_{L/F} of a generalized Severi-Brauer variety SB(j, D) of D (j = 1, …, deg D). Let p be a prime integer. An RSB-variety is a pRSB-variety, if the integers [L : F], deg D, and j are p-powers.

The following problem has been raised in [11] for needs of computing essential pdimension of a representation of a finite group: compute canonical p-dimension of an arbitrary finite direct product of RSB-varieties. This problem has been solved in [11] in the case of a single RSB-variety. The core result producing the solution was the following statement on p-incompressibility of pRSB-varieties (we refer to [11] for an introduction and further references on p-incompressibility, canonical p-dimension, Weil transfer, etc.):

Date: 26 June 2014.

Key words and phrases. Central simple algebras, algebraic groups, projective homogeneous varieties, Severi-Brauer varieties, Weil transfer, Chow groups and motives, canonical dimension and incompressibility. Mathematical Subject Classification (2010): 20G15; 14C25.

The author acknowledges a partial support of the French Agence Nationale de la Recherche (ANR)

under reference ANR-12-BL01-0005; his work has been also supported by the start-up grant of the University of Alberta and a Discovery Grant from the National Science and Engineering Board of Canada.

Theorem 1.1 ([11, Theorem 11.2]). Any pRSB-variety is p-incompressible.

In the present paper, we generalize Theorem 1.1 in two directions. First, we replace Galois field extensions in the definition of pRSB-varieties, by separable ones. Although this generalization is not motivated anymore by study of representations, it is a natural step to do. The statement remains the same (see Theorem 2.3), as does the main outline of the proof. The main change is in the definition of a balanced algebra over a separable extension. All this is done in §2.

As a second (and principal) generalization, we establish a criterion of p-incompressibility of a product of pRSB-varieties, see Corollary 3.6. The basic result here is a p-incompres-sibility criterion for a product of a pRSB-variety by an arbitrary projective homogeneous (under an action of a semisimple affine algebraic group) variety given in Theorem 3.1.

As a particular case of Corollary 3.6, we recover in §4a (with a new proof and a simplified statement) an old result [10, Theorem 2.1] on p-incompressibility of products of Severi-Brauer varieties. But now we can also determine canonical p-dimension of any p-primary Weil transfer of a product of this type, see §4b. In particular, we show that a p-primary Weil transfer of a p-incompressible product of this type is also p-incompressible. Moreover, the latter statement also holds for generalized Severi-Brauer varieties in place of the usual ones, see §4c.

In the last section (§5), we drop the balance assumption to do a complete analysis of a quadratic field extension. Note however, that the balanced case, motivated by the representations, also looks interesting (not only for a quadratic L/F) from the following viewpoint: for a pRSB-variety Y, the integer cd_{p}(Y) = dim Y turns out to be much bigger than cd_{p}(Y_{L}) = (dim Y)/[L : F] so that the role of the field extension and the Weil transfer shows up as crucial. In the imbalanced case however, the variety Y_{L} may, for instance, happen to be p-incompressible (c.f. Example 5.3), trivially implying p-incompressibility of Y itself.

Most of our terminology and notation being introduced “on the move”, we only mention here that smooth projective varieties X and Y over F are called equivalent if there exist rational maps X −−→ Y and Y −−→ X. Equivalent varieties have the same canonical (p-)dimension, [11, Lemma 3.3(a)].

Acknowledgements. I am grateful to Alexander Merkurjev and Zinovy Reichstein for useful comments on early versions of the preprint. Its current version has been prepared during my stay at the Universität Duisburg-Essen (research group of Marc Levine), under ideal work conditions.

2. Non-Galois extensions

In this section we generalize Theorem 1.1 to the case of an arbitrary finite separable (not necessarily Galois) field extension L/F of degree a power of p.

Here is the “Galois to separable” generalization of [11, Lemma 11.1] (with practically the same proof). Here and everywhere else in the paper, the motive we are talking about are Chow motives with coefficients in 𝔽_{p} (the finite field of p elements), see e.g. [2, §64]. Weil transfer of motives has been introduced in [5]. Corestriction of motives is from [7, §3].

Lemma 2.1. Let L/F be an arbitrary finite separable field extension and let E/F be a finite Galois field extension containing L. For and any m ≥ 1, let M_{1}, …, M_{m} be m motives over L. Then the motive R_{L/F} (M_{1} ⊕ ⋯ ⊕ M_{m}) decomposes into a direct sum

R_{L/F} (M_{1} ⊕ ⋯ ⊕ M_{m}) ≃ R_{L/F} M_{1} ⊕ ⋯ ⊕ R_{L/F} M_{m} ⊕ N,

where N is a direct sum of corestrictions to F of motives over fields K with F ⊊ K ⊂ E. □

Let L/F be a finite separable field extension. We generalize the notion of a balanced central simple L-algebra D (which we already have in the Galois case, see §1). A central simple L-algebra D is balanced, if for a Galois field extension E/F containing L and for any F -imbedding of L into E with an image L_{0}, the LL_{0}-algebra obtained from D by the base change L → L_{0} ⊂ LL_{0} is Brauer-equivalent to a tensor power of D_{LL_{0}}. This definition does not depend on the choice of E; E can be taken to be a Galois closure of L/F.

From now on, we extend the notion of a pRSB-variety, introduced in §1, by allowing separable (not necessarily Galois) Weil transfer:

Definition 2.2. Let p be a prime number. An F -variety Y is a pRSB-variety, if there exist a finite p-primary separable field extension L/F and a balanced division L-algebra D of degree p^{n} such that X ≃ R_{L/F} SB(p^{i}, D) for some i = 0, …, n.

Theorem 2.3. For any prime number p, any pRSB-variety is p-incompressible.

Proof. Let p be a prime number, L/F a finite separable field extension of degree a power of p, D a balanced central division L-algebra of degree p^{n} for some n ≥ 0, and X the generalized Severi-Brauer variety SB(p^{i}, D) of D with some i = 0, 1, …, n. Let us prove that the variety R_{L/F} X, given by the Weil transfer of X, is p-incompressible.

Let E be a finite Galois field extension of F containing L (for instance, E can be a Galois closure of L/F). The question on canonical p-dimension of R_{L/F} X easily reduces to the case where [E : F] is also a power of p. Indeed, let G be the Galois group of E/F and let H be its subgroup corresponding to L. Let G′ be a Sylow p-subgroup of G containing a Sylow p-subgroup H′ of H. We get the following diagram of subgroups and the corresponding diagram of subfields, where the sign p^{?} marks p-primary while the sign p|̸ marks p-coprime indexes/degrees:

[figure: {1}; p^{?}; H′; p^{?}; p |̸; H; p^{?}; G′; p |̸; G; E; p^{?}; L′; p^{?}; p |̸; L; p^{?}; F′; p |̸; F]

Since the degree [F′ : F] is not divisible by p, canonical p-dimension of R_{L/F} X does not change under the base change F′/F. Moreover, (R_{L/F} X)_{F′} ≃ R_{L′/F′}(X_{L′}) because L′ ≃ L ⊗_{F} F′. Finally, observe that the index of a p-primary central simple L-algebra is not changed under the base change L′/L. In particular, D_{L′} is still a division algebra. Moreover, this central division L′-algebra is still balanced (now with respect to the subfield F′ ⊂ L′).

For the rest of the proof we will assume that the degree [E : F] is a p-power. We follow the proof of Theorem 1.1, given in [11, Theorem 11.2], and only indicate the changes.

The first place where a change is needed is the place where we compute the index of D over the field L′ = F′ ⊗_{F} L, where F′ is the function field of the variety R_{L/F} SB(p^{n−1}, D). In the case of Galois L/F, the Weil transfer disappears because R_{L/F} SB(p^{n−1}, D)_{L} is the product of the generalized Severi-Brauer varieties given by the conjugate algebras. This happens because of the isomoprhism

L ⊗_{F} L ≃ L × ⋯ × L

of L-algebras, where L acts on L ⊗ L on the left. Note that this isomorphism is also L-linear for the right action of L on the tensor product if one lets L act on the factors of L × ⋯ × L by means of the F -automorphisms of L.

Under the assumptions of Theorem 2.3 however, L⊗_{F} L is identified with L_{1} ×⋯×L_{r}, where each L_{i} is LL_{0} for an appropriate (depending on i) choice of L_{0} as in the definition of balanced algebra. Since [L_{i} : L] > 1 for at least one value of i, we cannot avoid the Weil transfer here and may want to use the index reduction formulas for the Weil transfer of generalized Severi-Brauer varieties from [15] (see also [14]). But this is not really necessary. Indeed, since D is balanced, for any i, the L_{i}-algebra in question is Brauer-equivalent to a tensor power of D_{L_{i}}, its index will divide p^{n−1} if we extend L to the function field of SB(p^{n−1}, D). The Weil transfer becomes rational over this extension of L so that the index of D won’t be affected by passing to the function field of the Weil transfer.

We have explained why the index of D_{L′} is indeed p^{n−1}. The final adjustment we need to make, in order to adopt the proof of Theorem 1.1 to the setting of Theorem 2.3, is in the choice of the degree p Galois field subextension L̃/F of L/F. Recall that the Galois group G of E/F is a p-group. Let H be its subgroup corresponding to the field L. By [3, Theorem 4.2.1], H is contained in a normal subgroup H̃ of G such that [G : H̃] = p. We take for L̃ the field corresponding to H̃ and we have to compute canonical p-dimension of the variety R_{L/F} (X)L̃.

The variety R_{L/F} (X)L̃ is isomorphic to the product ∏_{g̃∈G̃} g̃(R_{L/L̃}X), where G̃ = G/H̃ is the Galois group of L̃/F. Since D is balanced, this product is equivalent to its factor R_{L/L̃}X given by g̃ = 1. In particular, the canonical p-dimension of R_{L/F} (X)L̃ is given by the dimension of R_{L/L̃}X. After this is established, the remainder of the proof of Theorem 1.1 goes through unchanged. □

3. Incompressibility of products

Here is our basic result on p-incompressibility of products:

Theorem 3.1. Let Y be a pRSB-variety (for a given prime number p) and let Z be a projective homogeneous F -variety. The product Y × Z is p-incompressible provided that the varieties Y_{F (Z)} and Z_{F (Y)} are so.

Proof. Let Y := R_{L/F} SB(p^{i}, D), where L/F is a finite separable field extension of degree p^{r} with r ≥ 0, D is a balanced central division L-algebra of degree p^{n} with n ≥ 0, and i = 0, …, n. Let Z be a projective homogeneous F -variety such that the varieties Y_{F (Z)} and Z_{F (Y)} are p-incompressible. We are going to prove that the product Y × Z is p-incompressible.

By Theorem 2.3, the assumption requiring that Y_{F (Z)} is p-incompressible is equivalent to the assumption that D_{L(Z)} is a division algebra.

Since canonical p-dimension is not changed under base field extensions of degree prime to p (see [12, Proposition 1.5(2)]), we may assume that there exists a finite p-primary Galois field extension E/F containing L and such that Z_{E} is of inner type. This assumption allows us to apply results of [7] and [8, §6].

For any j = 0, …, n, we set X^{j} := SB(p^{j}, D) and Y^{j} := R_{L/F} X^{j}. We also set X := X^{i} so that Y = R_{L/F} X.

We induct on n. For n = 0 the statement is trivial. From now on, we assume that n > 0.

For i = n the statement is trivial. From now on, we assume that i < n.

Let F′ be the function field of the variety Y^{n−1}. Let L′ := F′⊗_{F} L. Since D is balanced, the index of the central simple L′-algebra D_{L′} := D ⊗_{L} L′ = D ⊗_{F} F′ is p^{n−1} so that there exists a central division L′-algebra D′ such that the algebra of (p × p)-matrices over D′ is isomorphic to D_{L′}. We set X′^{j} := SB(p^{j}, D′), Y′^{j} := R_{L′/F′}X′^{j}, X′ := X′^{i}, and Y′ := Y′^{i}. By [4] (see also [1, Theorem 7.5]) and [9, Theorem 3.8], the motive of the variety X_{L′} decomposes into a direct sum

M(X_{L′}) ≃ M(X′) ⊕ M(X′)(p^{i+n−1})⊕

M(X′)(2p^{i+n−1}) ⊕ ⋯ ⊕ M(X′)((p − 1)p^{i+n−1}) ⊕ N,

where N is a direct sum of some shifts of the upper motives U(X′^{j}) of some varieties X′^{j} with j < i. Therefore, by Lemma 2.1 and [5, Theorem 5.4] (which we use just to determine the shifts), the motive of the variety Y_{F′} = R_{L′/F′}(X_{L′}) decomposes into a direct sum

M(Y_{F′}) ≃ M(Y′) ⊕ M(Y′)(p^{r+i+n−1})⊕

M(Y′)(2p^{r+i+n−1}) ⊕ ⋯ ⊕ M(Y′)((p − 1)p^{r+i+n−1}) ⊕ N ⊕ N′,

where now N is a direct sum of shifts of U(Y′^{j}) with j < i, and N′ is a direct sum of corestrictions of motives over fields K with F ⊊ K ⊂ E. It follows that

M(Y × Z)_{F′} ≃ M(Y′ × Z_{F′}) ⊕ M(Y′ × Z_{F′})(p^{r+i+n−1})⊕

M(Y′ × Z_{F′})(2p^{r+i+n−1}) ⊕ ⋯ ⊕ M(Y′ × Z_{F′})((p − 1)p^{r+i+n−1}) ⊕ N ⊕ N′,

with N′ of the same shape as before and with N being a direct sum of shifts of

U(Y′^{j}) ⊗ M(Z), j < i.

We claim that the variety Y′ × Z_{F′} is p-incompressible by the induction hypothesis. To check the claim, we check that the varieties Z_{F′(Y′)} and Y′_{F′(Z)} are p-incompressible. To check that Z_{F′(Y′)} is p-incompressible, we check that Z over a lager field F′(Y′ × Y_{F′}) is so. The field F′(Y′ × Y_{F′}) = F′(Y)(Y′) is purely transcendental over F (Y) and Z_{F (Y)} is p-incompressible (this is the place in the proof of Theorem 3.1, where the assumption on Z_{F (Y)} is used). Therefore Z over F′(Y′ × Y_{F′}) is p-incompressible. (To see that canonical p-dimension of projective homogeneous varieties does not change under a purely transcendental base field extension, one may use the characterization [8, Corollary 6.2] of the canonical p-dimension in terms of algebraic cycles together with the fact that a purely transcendental base field extension does not affect the Chow groups.)

To check that Y′_{F′(Z)} is p-incompressible, we check that D′_{L′(Z)} is a division algebra. Since L′ = L(Y^{n−1}), we have L′(Z) = L(Y^{n−1} × Z) = L(Z)(Y^{n−1}). Since D_{L(Z)} is a balanced (over F (Z)) division algebra, D_{L(Z)(Y^{n−1})} has index p^{n−1}. Therefore D′_{L′(Z)} is a division algebra.

The claim being proved, it follows that

(3.2) M(Y × Z)_{F′} ≃ U(Y′ × Z_{F′}) ⊕ U(Y′ × Z_{F′})(p^{r+i+n−1})⊕

U(Y′ × Z_{F′})(2p^{r+i+n−1}) ⊕ ⋯ ⊕ U(Y′ × Z_{F′})((p − 1)p^{r+i+n−1}) ⊕ N,

with N having the property that no summand of its complete decomposition is isomorphic to a shift of U(Y′ × Z_{F′}).

We want to show that the variety Y × Z is p-incompressible. Let l be the number of those summands in the complete decomposition of the motive U(Y × Z)_{F′}, which are isomorphic to a shift of U(Y′ × Z_{F′}). We have 1 ≤ l ≤ p and it suffices to show that l = p. Indeed, l = p implies that the complete decomposition of U(Y × Z)_{F′} contains the summand U(Y′ × Z_{F′}) of (3.2) with the maximal shift (p − 1)p^{r+i+n−1}. Therefore

cd_{p}(Y × Z) = dim U(Y × Z) = dim U(Y × Z)_{F′} ≥

dim U(Y′ × Z_{F′}) + (p − 1)p^{r+i+n−1} = dim(Y × Z).

(We refer to [6, Theorem 5.1] for the relation between canonical p-dimension and dimension of the upper motive of a projective homogeneous variety, used here.)

Our next claim is: l divides p (therefore l = 1 or l = p, and we will only need to show that l ≠ 1). To prove the claim, we consider the complete motivic decomposition of Y × Z. It contains several shifts of U(Y × Z) (it contains one non-shifted U(Y × Z) and - as we hope - no other shifts of this motive, but we do not know whether the hope comes true by now). Let N be any of the remaining (indecomposable) summands. We affirm that no summand of the complete decomposition of N_{F′} is isomorphic to a shift of U(Y′ × Z_{F′}). Clearly, this affirmation implies the claim that l divides p.

To prove the affirmation, let us note that N can be of two alternative types. The first type is given by corestriction to F of a motive over a field K with F ⊊ K ⊂ E. For such N, any indecomposable summand of N_{F′} is a corestriction to F′ of a motive over K′ := K ⊗_{F} F′ (see [7, Proposition 3.1]) which is never isomorphic to a shift of the upper motive of a projective homogeneous F′-variety.

The second type of N is a shift of U(T), where T is a projective homogeneous variety with deg Ch_{0}(T_{F (Y ×Z)}) = 0 (and with deg Ch_{0}(Y × Z)_{F (T)} = 𝔽_{p}). Over F′ we still have deg Ch_{0}(T_{F′(Y ×Z)}) = 0 (and deg Ch_{0}(Y × Z)_{F′(T)} = 𝔽_{p}) because the field extension

F′(Y × Z)/F (Y × Z)

is purely transcendental. Any indecomposable summand of N_{F′}, which is not corestriction from some F ⊊ K ⊂ E, is a shift of U(S), where S is a projective homogeneous F′-variety with deg Ch_{0}(T_{F′(S)}) = 𝔽_{p}. It follows that deg Ch_{0}(S_{F′(Y ×Z)}) = 0 (otherwise we would be in contradiction with deg Ch_{0}(T_{F′(Y ×Z)}) = 0) implying that no shift of U(S) (and consequently no shift of U(S)) is isomorphic to U(Y′ × Z_{F′}), see [9, Corollary 2.15] for the criterion of isomorphism for upper motives.

We proved the affirmation and the claim. This means that we only need to show that l ≠ 1 to finish the proof of Theorem 3.1. So, we assume that l = 1 and we look for a contradiction. By [8, Proposition 2.4], the complete decomposition of U(Y ×Z)_{F′} contains as a summand the motive U(Y′ × Z_{F′}) shifted by the difference

dim U(Y × Z) − dim U(Y′ × Z_{F′}).

Therefore, l = 1 implies that the above difference is 0, and we come to

cd_{p}(Y × Z) = dim U(Y × Z) = dim U(Y′ × Z_{F′}) = d,

where d := dim(Y′ × Z_{F′}) = dim Y′ + dim Z.

By [8, Proposition 6.1], there exist α ∈ Ch^{d}(Y × Z)_{F (Y ×Z)} and β ∈ Ch_{d}(Y × Z) with deg(α · β) ≠ 0 ∈ 𝔽_{p}. In the last formula, we consider both cycles over a common field extension of their fields of definition, before we multiply them. We use this convention below (in similar formulas on degree of products) as well.

Since cd_{p}(Y_{F′}) = dim Y′ =: d′, we can find α′ ∈ Ch^{d′}(Y_{F′(Y)}) and β′ ∈ Ch_{d′}(Y_{F′}) with deg(α′ · β′) ≠ 0. Using these α′ and β′ and a rational point pt ∈ Z_{F (Z)}, we get the cycles

α′ × [pt] ∈ Ch^{d}(Y × Z)_{F′(Y ×Z)} and β′ × [Z] ∈ Ch_{d}(Y × Z)_{F′},

having the same property as α and β:

deg ((α′ × [pt]) · (β′ × [Z])) ≠ 0.

It follows by [8, Lemma 6.5] that one can ”mix up” the old cycles with the new ones and get the relation

deg ((α′ × [pt]) · β) ≠ 0.

Since α′ × [pt] = (α′ × [Z]) · ([Y] × [pt]), the last degree relation can be rewritten as deg(α′ · β′′) ≠ 0, where β′′ ∈ Ch_{d′}(Y_{F (Z)}) is the push-forward of the product ([Y] × [pt]) · β along the projection (Y ×Z)_{F (Z)} → Y_{F (Z)}. Since the field extension F′(Y)/F (Y) is purely transcendental, there exists α′′ ∈ Ch^{d′}(Y_{F (Y)}) mapped to α′ under the change of field homomorphism. Changing notation, we write α′′ for the image of α′′ in Ch^{d′}(Y_{F (Z)(Y)}).

The cycles α′′ ∈ Ch^{d′}(Y_{F (Z)(Y)}) and β′′ ∈ Ch_{d′}(Y_{F (Z)}) thus constructed have the property deg(α′′ · β′′) ≠ 0. It follows by [8, Corollary 6.2] that cd_{p}(Y_{F (Z)}) ≤ d′. Since

(3.3)  d′ = dim Y′ = p^{i}(p^{n−1} − p^{i}) < p^{i}(p^{n} − p^{i}) = dim Y,

the relation cd_{p}(Y_{F (Z)}) ≤ d′ obtained contradicts the assumption on p-incompressibility of the variety Y_{F (Z)}. □

The necessary condition for p-incompressibility of a product of projective homogeneous varieties, showing up in the following lemma, turns out to be sufficient in the case of the special varieties we are interested in (see Corollary 3.5):

Lemma 3.4. For any projective homogeneous F -varieties X and Y,

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

In particular, a necessary condition for p-incompressibility of X ×Y is p-incompressibility of X_{F (Y)} and Y_{F (X)}.

Proof. Again, we are using the characterization of canonical p-dimension of projective homogeneous varieties given in [8, Corollary 6.2]. Since canonical p-dimension is not changed under base field extensions of degree prime to p (see [12, Proposition 1.5(2)]), we may assume that the condition of [8, Corollary 6.2] on the projective homogeneous variety is satisfied for X and Y: both of them become of inner type over a finite p-primary extension of F. We set x := cd_{p}(X) and y := cd_{p}(Y_{F (X)}). We find α_{X} ∈ Ch^{x}(X_{F (X)}) and β_{X} ∈ Ch_{x}(X) with deg(α_{X} · β_{X}) ≠ 0. Similarly, we find α_{Y} ∈ Ch^{y}(Y_{F (X)(Y)}) and β_{Y} ∈ Ch_{y}(Y_{F (X)}) with deg(α_{Y} · β_{Y}) ≠ 0. Let β′_{Y} ∈ Ch_{dim X+y}(X × Y) be an element mapped to β_{Y} under the surjection

Ch_{dim X+y}(X × Y) →→ Ch_{y}(Y_{F (X)})

given by the pull-back along the morphism Y_{F (X)} → X × Y induced by the generic point of X. We set

α := α_{X} × α_{Y} ∈ Ch^{x+y}(X × Y)_{F (X×Y)} and β := (β_{X} × [Y]) · β′_{Y} ∈ Ch_{x+y}(X × Y).

We have the relation deg(α · β) ≠ 0 showing that cd_{p}(X × Y) ≤ x + y. □

Corollary 3.5. For two products X and Y of pRSB-varieties over F, their product X ×Y is p-incompressible if and only if the varieties X_{F (Y)} and Y_{F (X)} are p-incompressible.

Proof. The “only if” part being served by Lemma 3.4, we only prove the “if” part. We write X as product X_{1} × ⋯ × X_{r} of pRSB-varieties and induct on r. The case of r = 1 follows from Theorem 3.1. For r > 1, set X′ := X_{2} × ⋯ × X_{r}. To show that X × Y = X_{1} × (X′ × Y) is p-incompressible, by Theorem 3.1, it suffices to check that (X_{1})_{F (X′×Y)} and (X′ × Y)_{F (X_{1})} are p-incompressible. The assumption on p-incompressibility of X_{F (Y)} implies (by Lemma 3.4) p-incompressibility of (X_{1})_{F (X′×Y)}. And the variety (X′ ×Y)_{F (X_{1})} is p-incompressible by induction hypothesis. □

Here is the most convenient statement to check p-incompressibility of a general product of pRSB-varieties. Basically, it reduces the problem to application of index reduction formulas.

Corollary 3.6. Product X_{1} × ⋯ × X_{r} of pRSB-varieties over F is p-incompressible if and only if (X_{i})_{F (X_{1}×···×X_{i−1}×X_{i+1}×···×X_{r})} for every i = 1, …, r, is p-incompressible. □

4. Particular products

4a. Products of Severi-Brauer varieties. Applying Corollary 3.6 to a product of Severi-Brauer varieties (Weil transfer do not show up here), we get a version of [10, Theorem 2.1]. Note that the proof is different from the original one: it does not involve K-theory.

Corollary 4.1. Let D_{1}, …, D_{r} be p-primary central division F -algebras. The product of their Severi-Brauer varieties is p-incompressible if and only if each D_{i} remains division algebra over the function field of the product of the Severi-Brauer varieties of the remaining algebras. □

Remark 4.2. By the index reduction formula for Severi-Brauer varieties, the condition on D_{i} means that the index of every product of D_{i} by tensor powers of the remaining algebras is ≥ ind D_{i}. With this, one may see that D_{1}, …, D_{r} satisfy the condition of Corollary 4.1 if and only if their nonzero Brauer classes (put in some/any order increasing the degrees) form a minimal basis in the sense of [10, Remark 2.9].

Remark 4.3. In general, the property of being p-incompressible for a projective homogeneous variety X is weaker than the property of having indecomposable motive. However, for a generically split X (i.e., for X such that the motive of X_{F (X)} is split, i.e., is a direct sum of shifts of the motive of a point), the above two properties are equivalent. Indeed, M(X) for a generically split X is a direct sum of shifts of U(X) so that M(X) = U(X) is and only if dim U(X) = dim X.

A product of Severi-Brauer varieties is a generically split projective homogeneous variety. Therefore the motive of a p-incompressible product of Severi-Brauer varieties is indecomposable.

There are numerous simplification in the proof of Theorem 3.1 when we adopt it to products of Severi-Brauer varieties. It might be therefore helpful for the reader to look at the simplified proof below before going through the actual proof of Theorem 3.1.

Proposition 4.4. Let X be the product of Severi-Brauer varieties X_{1}, …, X_{r} over F such that for every i = 1, …, r, the variety (X_{i})_{F (X_{1}×···×X_{i−1}×X_{i+1}×···×X_{r})} is p-incompressible. Then the motive of X is p-incompressible.

Proof. Using induction on r, we are reduced to prove the following statement (which looks more like the statement of Theorem 3.1): the product Y × Z of a Severi-Brauer variety Y by a product of Severi-Brauer varieties Z is p-incompressible provided that the varieties Y_{F (Z)} and Z_{F (Y)} are so.

Actually, instead of being a product of Severi-Brauer varieties, Z can be any generically split projective homogeneous variety: we will only use this property of Z in the proof.

Now we go along the lines of the proof of Theorem 3.1, removing everything superfluous. The variety Y is the Severi-Brauer variety of a p-primary central division algebra D,

say, deg D = p^{n}. Note that D remains division over F (Z).

We induct on n. For n = 0 the statement we are proving is trivial. We assume that n ≥ 1 below.

Let F′ be the function field of the variety SB(p^{n−1}, D). The index of the central simple F′-algebra D_{F′} is p^{n−1} so that there exists a central division F′-algebra D′ such that the algebra of (p × p)-matrices over D′ is isomorphic to D_{F′}. We set Y′ := SB(D′). The motive of the variety Y_{F′} decomposes into the direct sum of p summands

M(Y_{F′}) ≃ M(Y′) ⊕ M(Y′)(p^{n−1}) ⊕ M(Y′)(2p^{n−1}) ⊕ ⋯ ⊕ M(Y′)((p − 1)p^{n−1}).

It follows that

(4.5) M(Y × Z)_{F′} ≃ M(Y′ × Z_{F′}) ⊕ M(Y′ × Z_{F′})(p^{n−1})⊕

M(Y′ × Z_{F′})(2p^{n−1}) ⊕ ⋯ ⊕ M(Y′ × Z_{F′})((p − 1)p^{n−1}).

We claim: the decomposition (4.5) is complete, i.e., the motive of the variety Y′ × Z_{F′} is indecomposable (see Remark 4.3), and that – by the induction hypothesis. To check the claim, we check that the varieties Z_{F′(Y′)} and Y′_{F′(Z)} are p-incompressible. This is done precisely as in the proof of Theorem 3.1.

Now we know that (4.5) is the complete decomposition and we want to show that the motive of Y × Z is indecomposable. In other words, we want to show that

U(Y × Z) = M(Y × Z),

or – equivalently – that U(Y × Z)_{F′} contains all the p indecomposable summands of the decomposition (4.5).

Let l be the number of summand in the complete decomposition of the motive U(Y × Z)_{F′} (all of them are automatically isomorphic to a shift of M(Z_{F′})). We have 1 ≤ l ≤ p and all we want to show is l = p.

It is now very easy to see that l divides p. Indeed, since the variety Y × Z is generically split, every summand of its complete motivic decomposition is a shift of U(Y × Z). Therefore l divides p, as claimed, and we only need to show that l ≠ 1.

So, we assume that l = 1 and we look for a contradiction. Clearly, l = 1 implies that

cd_{p}(Y × Z) = dim U(Y × Z) = dim M(Y′ × Z_{F′}) = d,

where d := dim Y′ + dim Z = p^{n−1} − 1 + dim Z. By [8, Proposition 6.1], there exist α ∈ Ch^{d}(Y × Z)_{F (Y ×Z)} and β ∈ Ch_{d}(Y × Z) with deg(α · β) ≠ 0 ∈ 𝔽_{p}. Starting from this point, the proof of Proposition 4.4 ends precisely as the proof of Theorem 3.1. Note that in (3.3) we will have i = 0 here. □

4b. Weil transfer of products of Severi-Brauer varieties.

Corollary 4.6. Let L/F be a p-primary separable field extension and let X be a product over L of Severi-Brauer varieties of some balanced p-primary division L-algebras. Then cd_{p}(R_{L/F} X) = [L : F] · cd_{p}(X). In particular, R_{L/F} X is p-incompressible provided that X is so.

Proof. Taking a minimal basis of the subgroup in Br(L) generated by the algebras (note that any L-algebra representing an element of this subgroup is balanced), consider the product X′ of their Severi-Brauer varieties. The variety X′ is p-incompressible and equivalent to X. Moreover, its Weil transfer R_{L/F} X′ is equivalent to R_{L/F} X. Therefore we reduced the proof of the first statement of Corollary 4.6 to the proof of the second one. The second statement follows directly from Corollary 3.6. □

4c. Weil transfer of products of generalized Severi-Brauer varieties. The second conclusion of Corollary 4.6 also holds for generalized Severi-Brauer varieties. Again, the result is an immediate consequence of Corollary 3.6:

Corollary 4.7. Let L/F be a p-primary separable field extension and let X be a product over L of some generalized Severi-Brauer varieties of some balanced p-primary division L-algebras. Then R_{L/F} X is p-incompressible if (and only if) X is so. □

5. Quadratic extensions

Let L/F be a separable quadratic field extension and let D be a central division Lalgebra of a 2-primary index. In this section, we determine canonical 2-dimension of the variety R_{L/F} SB(D) without imposing any restrictions on the conjugate algebra g(D), where g is the non-trivial element of G := Gal(L/F). We will provide two different recipes; which of them has to be applied depends on a property of the group A generated by the Brauer classes of D and g(D). Since the case of cyclic A has been already treated, we assume that A is not cyclic. Therefore A/2A ≃ ℤ/2ℤ⊕ℤ/2ℤ with g acting by exchanging the summands.

Let α ∈ A \ 2A be an element of the smallest index. Let D_{α} be a central division L-algebra representing α. If the image of α in A/2A is not invariant under the action of G, then the images of α and g(α) form a minimal basis of A/2A in the sense of [10, Remark 2.9]. It follows that the variety (R_{L/F} SB(D_{α}))_{L} ≃ SB(D_{α}) × SB(g(D_{α})) is 2incompressible. Therefore the variety R_{L/F} SB(D_{α}) is 2-incompressible as well. Since α and g(α) generate A, the varieties R_{L/F} SB(D) and R_{L/F} SB(D_{α}) are equivalent. So, cd_{2} R_{L/F} SB(D) = cd_{2} R_{L/F} SB(D_{α}) and we get that

(5.1)  cd_{2} R_{L/F} SB(D) = dim R_{L/F} SB(D_{α}) = 2(ind α − 1).

Now we assume that the image of α in A/2A is invariant under G. This means that this image is equal to the element (1, 1) ∈ ℤ/2ℤ ⊕ ℤ/2ℤ. Let β ∈ A be an element of the smallest index with the property that its image in A/2A is outside of the subgroup generated by the image of α. We claim that in this case

(5.2)  cd_{2} R_{L/F} SB(D) = dim R_{L/F} SB(D_{β}) = 2(ind β − 1).

Since the image of β in A/2A is (1, 0) or (0, 1), it is not invariant under G. Moreover, A is generated by β and g(β). It follows that the variety R_{L/F} SB(D) is equivalent to R_{L/F} SB(D_{β}), where D_{β} is a central division L-algebra representing β. To prove the claim, it suffices to prove that the variety R_{L/F} SB(D_{β}) is 2-incompressible. It is so because the F (R_{L/F} SB(D_{α}))-variety

R_{L/F} SB(D_{β})_{F (R_{L/F} SB(D_{α}))} ≃ R_{L(R_{L/F} SB(D_{α}))/F (R_{L/F} SB(D_{α}))} SB((D_{β})_{L(R_{L/F} SB(D_{α}))})

is 2-incompressible. Indeed, g(D_{β})_{L(R_{L/F} SB(D_{α}))} is Brauer-equivalent to a tensor power of (D_{β})_{L(R_{L/F} SB(D_{α}))} so that Theorem 1.1 applies. On the other hand, (D_{β})_{L(R_{L/F} SB(D_{α}))} is still a division algebra by the Schofield–van den Bergh index reduction formula [16] (see also [13]), because

ind(D_{β} ⊗ D_{α}^{⊗i}) ≥ ind D_{β}

for any i by the minimality of ind β = ind D_{β}.

We finish this section by examples where the above recipes apply.

Example 5.3. Let l be a field of characteristic ≠ 2, let L be the rational function field over l in variables x, y, x′, y′, let g be the l-automorphism of L exchanging x with x′ and y with y′, and let F be the subfield of L consisting of the elements fixed by g. The variety R_{L/F} SB(D), where D is the quaternion L-algebra (x, y), is 2-incompressible by formula (5.1) with D_{α} = D. Note that in this example the algebras D and g(D) are, informally speaking, “completely independent”.

In the next example, D and g(D) will be “partially dependent”. To get it, we replace F and L by the rational function fields

F (s_{1}, s_{2}, t_{1}, t_{2}) and L(s_{1}, s_{2}, t_{1}, t_{2})

in some variables s_{1}, s_{2}, t_{1}, t_{2}. For D := C ⊗ (x, y) with C := (s_{1}, t_{1}) ⊗ (s_{2}, t_{2}), the variety R_{L/F} SB(D) is 2-incompressible by formula (5.2) with D_{α} = (x, y) ⊗ (x′, y′) and D_{β} = D. Note that the non-zero elements of the group A in this situation are the Brauer classes of the index 8 conjugate algebras C ⊗ (x, y) and C ⊗ (x′, y′) and the index 4 invariant algebra (x, y) ⊗ (x′, y′); besides, 2A = 0.

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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