CANONICAL p-DIMENSION OF ALGEBRAIC GROUPS

NIKITA A. KARPENKO AND ALEXANDER S. MERKURJEV

Abstract. We describe a way to compute the p-relative version of the Berhuy-Reichstein canonical dimension for an arbitrary split semisimple algebraic group over an arbitrary field of an arbitrary characteristic (p is any prime integer). The canonical p-dimension is computed for all split simple groups of classical types.

Contents

1. Agreements and preliminaries 2 1.1. Varieties 2 1.2. Chow groups 2 1.3. Places 3 1.4. Places and rational morphisms 3 2. Canonical dimension of determination functions 3 3. Notion of canonical p-dimension 4 4. Canonical (p-)dimension of regular complete varieties 6 5. Generically p-split varieties 8 6. Canonical p-dimension of algebraic groups 12 7. Remarks on C̃H(G/P) 14 8. Canonical p-dimension of split simple groups of classical types 16 8.1. Type A_{n−1} 16 8.2. Type B_{n} 17 8.3. Type C_{n} 18 8.4. Type D_{n} 19 8.5. Appendix: Type G_{2} 21 References 21

The notion of the canonical dimension of an algebraic structure was introduced by Berhuy and Reichstein in [1]. The canonical dimension measures the size of generic

Date: December, 2004. Key words and phrases. Algebraic groups, torsors, splitting fields, projective homogeneous varieties,

Chow groups. 2000 Mathematical Subject Classifications: 14L17; 14C25.

The James D. Wolfensohn Fund and The Ellentuck Fund support is acknowledged by the first author. The first author was partially supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00287, KTAGS.

The work of the second author has been supported by the NSF grant DMS #0355166.

splitting fields of the structure. The formal definition is given in Section 2. Here we present two basic examples:

• Let X be a scheme over a field F. A field extension L/F is called a splitting field of X, if X has a point over L. A splitting field L is called generic, if for any splitting field K of X there exists an F -place L → K. The canonical dimension of X is the minimum of the transcendence degree (over F) of all generic splitting fields of X.

• Let G be an algebraic group over F. The canonical dimension of G is the maximum of the canonical dimension of all principal homogeneous varieties (G-torsors), defined over field extensions of F.

When dealing with a given algebraic structure, we usually have finitely many “significant” prime integers involved. For example, such primes associated with an algebraic group G are the torsion prime integers of G (see Remark 6.7). In order to locate contribution of a prime integer p to the canonical dimension, we define canonical p-dimension in a similar fashion.

It turns out that canonical dimension and p-dimension of an arbitrary regular complete variety X is closely related to the algebraic cycles on X (see Corollaries 4.7 and 4.12). We express canonical p-dimension of a generically cellular variety in terms of its Chow group (see Theorem 5.8).

The main result of the paper is Theorem 6.9, giving a recipe to compute canonical p-dimension of an arbitrary split semisimple algebraic group over an arbitrary field (of arbitrary characteristic). The values of the canonical p-dimension are given for all split simple groups of classical type (see §8).

Acknowledgments: Authors thank V. Chernousov, I. Panin, and M. Rost for helpful remarks. The first author thanks the Princeton Institute for Advanced Study: the results were obtained and the paper written during his stay at the Institute; he was also inspired by the lecture course of A. A. Suslin on the Rost norm varieties.

1. Agreements and preliminaries

1.1. Varieties. We refer as schemes to separated schemes of finite type over a field (there is no restrictions on the field, its characteristic is arbitrary).

Variety in the paper is an integral scheme.

For a scheme X, the integer d(X) is defined as the g.c.d. of degree of all closed points on X; for a prime integer p, d_{p}(X) is the p-primary part of d(X).

1.2. Chow groups. Let X be a scheme over a field F. We write CH(X) for the integral Chow group of X (see [6]). Fixing a prime p, we write Ch(X) for the modulo p Chow group:

Ch(X) = CH(X)/p · CH(X) .

Furthermore, we write Ch(\overline{X}) (resp. CH(\overline{X})) for the colimit of Ch(X_{L}) (resp. CH(X_{L})) with L running over all field extensions L/F, and we write \overline{Ch}(X) (resp. \overline{CH}(X)) for the image of the restriction homomorphism res: Ch(X) → Ch(\overline{X}) (resp. CH(X) → CH(\overline{X})). The group \overline{CH}(X) is called the reduced Chow group of X; the group \overline{Ch}(X) is called the modulo p reduced Chow group of X. Note that

\overline{Ch}(X) = \overline{CH}(X)/( \overline{CH}(X) ∩ p CH(\overline{X}))

is not the same as \overline{CH}(X)/p \overline{CH}(X).

1.3. Places. Let K be a field. A valuation ring R of K is a subring R ⊂ K, satisfying K = R∪(R\{0})^{−1}. Any valuation ring is local; R = K is a trivial example of a valuation ring.

Given two fields K and L, a place K → L is a local ring homomorphism π: R → L of a valuation ring R ⊂ K (an embedding of fields is a trivial example of a place).

If K and L are extensions of a field F, an F -place (or place over F) is a place K → L with π defined and identical on F.

Places are composable: if K → L is a place, given by a ring homomorphism π, and L → E a place to a third field E, given by a homomorphism ρ of a ring S ⊂ L, then the composition is the place K → E, given the homomorphism ρ ∘ π: π^{−1}(S) → E, defined on the valuation ring π^{−1}(S). In particular, any place L → E can be restricted to any subfield K ⊂ L.

In this paper, a place K → L is said to be geometric, if it can be represented as a composition of places with valuation rings being discrete valuation rings.

1.4. Places and rational morphisms. Let X and Y be varieties over a field F. If X is complete, then for any valuation ring R of F (X) there exists an F -morphism Spec R → X [8, ch. II th. 4.7]; therefore an F -place F (X) → F (Y) produces a rational morphism Y → X.

Vice versa, if there is a rational morphism Y → X and X is regular, then there exists a geometric F -place F (X) → F (Y). Indeed, since X is regular at the image x of the generic point of Y, there exists a system of local parameters around x, which produces a geometric place F (X) → F (x); composing with the embedding F (x) ⊂ F (Y), we get a required place F (X) → F (Y).

2. Canonical dimension of determination functions

Let F be a field, Fields_{F} the category of all field extensions of F. Let 2^{0} be the category of the subsets of a 1-elemental set 0. A determination function D over F is a continuous functor Fields_{F} → 2^{0}, where by continuity we mean that D commutes with the filtered colimits. In other words, D is a rule assigning to each E ∈ Fields_{F} a value D(E) ∈ {∅, 0} such that

• if D(E) = 0 for some E, then D(E′) = 0 for any E′ admitting an F -embedding E → E′;

• (continuity property) if D(E) = 0 for some field E covered by a (possibly infinite)

filtered family of subfields E_{i}, then D(E_{i}) = 0 for some E_{i}. A field E ∈ Fields_{F} is called a splitting field of a determination function D, if D(E) = 0.

A splitting field E of D is called generic, if for any splitting field L there exists an F -place E → L. If D has at least one generic splitting field, canonical dimension cd(D) of D is defined as the minimum of the transcendence degree (over F) of all generic splitting fields of D; if D does not admit a generic splitting field, we set cd(D) = ∞.

Lemma 2.1. For a given determination function D, any its splitting field, which is a subfield of a generic splitting field, is also generic. Besides, any splitting field contains a finitely generated splitting field and cd(D) = ∞ only if D does not admit generic splitting.

Proof. If E is a generic splitting field and E′ a splitting field contained in E, then for any splitting field L, restricting a place E → L to E′, we get a place E′ → L; therefore E′ is also generic.

Any splitting field contains a finitely generated splitting field by the continuity of the determination function.

If D has a generic splitting field, then, taking a finitely generated splitting subfield, we get a finitely generated generic splitting field, showing that cd(D) is finite. □

A determination function D over F is split, if D(F) = 0. In this case, F is a generic splitting field of D and cd(D) = 0.

Our basic example of a determination function is the determination function associated with a scheme X over F:

L ↦ \begin{cases}∅, if X(L) = ∅; \\ 0, otherwise. \end{cases}

Canonical dimension cd(X) of an F -scheme X is defined as the canonical dimension of the associated determination function.

Example 2.2 ([12, th. 4.3]). Let F be a field of characteristic ≠ 2. Let X be an anisotropic smooth projective quadric over F. Then cd(X) = dim X − 𝔦_{1}(X) + 1, where 𝔦_{1}(X) is the first Witt index of X.

Let pSets be the category of pointed sets and let k be a field. A functor

F: Fields_{k} → pSets

is called continuous, if it commutes with filtered colimits. If F is a continuous functor, then for any F ∈ Fields_{k} and α ∈ F(F), we get a determination function D_{α} over F by setting

D_{α}(L) = \begin{cases}0, if α_{L} is the distinguished point of the set F(L); \\ ∅, otherwise. \end{cases}

Berhuy-Reichstein canonical dimension of a continuous functor F [1, §10] is the supremum of cd(D_{α}) for all F and α ∈ F(F). If G is an algebraic group over the field k, canonical dimension of G is defined as canonical dimension of the (continuous) functor Tors_{G}, taking a field F to the set of isomorphism classes Tors_{G}(F) of G-torsors over F. We note that canonical dimension of an algebraic group G/k is not the same as canonical dimension of the underlying variety of G (which is always 0 because G(k) ≠ ∅).

3. Notion of canonical p-dimension

Let us fix an arbitrary prime p and refer to a splitting field E of a determination function D as p-generic,^{1} if for any splitting field L of D there exists a finite field extension L′/L of degree prime to p admitting a place E → L′. Replacing generic splitting fields by the p-generic ones in the definitions of section 2, we get a modified notion of canonical dimension which we call canonical p-dimension and denote cd_{p}.

1Our notion of p-generic splitting is based on on the notion of p-generic splitting varieties of symbols in a modulo p Milnor’s K-group of a field, introduced in [16].

We refer to a finite field extension as p-coprime, if its degree is not divisible by p.

The following two Lemmas are useful when working with cd_{p}.

Lemma 3.1 (cf. [12, lemma 3.3]). Let K be an arbitrary field, p a prime, K′/K a pcoprime field extension, and L/K an arbitrary field extension. Then there exists a field L′, containing K′ and L, such that the extension L′/L is also p-coprime.

Proof. We almost repeat the proof of [12, lemma 3.3], where the case of p = 2 was treated. We may assume that K′ is generated over K by one element; let f(t) ∈ F [t] be its minimal polynomial. Since the degree of f is coprime with p, there exists an irreducible divisor g ∈ L[t] of f over L such that deg(g) is coprime with p as well. We set L′ = L[t]/(g). □

Lemma 3.2. Let K be a field extension of F of finite transcendence degree over F; let K → L be a geometric F -place and let K′ be a p-coprime field extension of K. Then there exists a p-coprime field extension L′/L such that the place K → L extends to a place K′ → L′.

Proof. By Lemma 3.1 it suffices to prove Lemma 3.2 in the case where the place K → L is surjective and its valuation ring R is a discrete valuation ring. Also it is clear, that is suffices to consider only two cases: (1) K′/K is purely inseparable and (2) K′/K is separable.

In the first case, the degree [K′ : K] is a power of a prime q ≠ p. We take an arbitrary valuation ring R′ of K′, lying over R, i.e., such that R′ ∩ K = R and the embedding R → R′ is local (such R′ exists in the case of an arbitrary field extension K′/K, [17, ch. VI th. 5′]). Let L′ be the residue field of R′ so that we have a surjective place K′ → L′. We show that L′ is also purely inseparable over L (and therefore [L′ : L], being a power of the same q, is coprime to p). For this, we take an element l ∈ L′ and show that l^{q^{n}} ∈ L for some n: let k ∈ R′ be a preimage of l; then k^{q^{n}} ∈ K for some n and consequently l^{q^{n}} ∈ L for the same n.

In the second case we consider all valuation rings R_{1}, …, R_{r} of K′, lying over R (the number of such valuation rings is finite by [17, ch. VI th. 12 cor. 4]). The residue field of each R_{i} is a finite extension of L. Moreover, ∑_{i=1}^{r} e_{i}n_{i} = [K′ : K] [17, ch. VI th. 20 and p. 63] (the discrete valuation ring assumption and the separability assumption are needed for this equality), where n_{i} is the degree over L of the residue field of R_{i}, and e_{i} is the reduced ramification index of R_{i} over R, [17, def. on pp. 52–53]. It follows that at least one of n_{i} is not divisible by p. □

Let us make some first general observations on cd_{p}. Clearly, a generic splitting field of a determination function is also p-generic; therefore we always have cd ≥ cd_{p}.

Also it is clear, that cd_{p} is not interesting, if the determination function in question has a p-coprime splitting field. More precisely, one has a simple

Lemma 3.3. If a determination function D is split by an algebraic field extension E/F of degree non-divisible by p, then cd_{p}(D) = 0.

Proof. Taking a finitely generated splitting subfield of E, we get a p-coprime splitting field. It follows by Lemma 3.1, that F is a p-generic splitting field of D. □

Example 3.4. The computation of cd(X) for an anisotropic smooth projective quadric X (over a field of characteristic ≠ 2), given in [12] (see Example 2.2), shows in fact also that cd_{2}(X) = cd(X).

Example 3.5. Let X be a Severi-Brauer variety. If d(X) = d_{p}(X) (that is, d(X) is a power of the prime p), then cd(X) = cd_{p}(X) = d_{p}(X) − 1, [1, th. 11.4]. Now if d(X) is not a power of a prime, the value of cd(X) is not known, while cd_{p}(X) is still d_{p}(X) − 1 (see Example 5.10).

Example 3.6. The computation of cd(SO_{n}), given in [11], also shows that cd_{2}(SO_{n}) = cd(SO_{n}) (see also Example 5.11 as well as (8.2) and (8.4)).

Remark 3.7. Let F and F′ be continuous functors Fields_{k} → pSets with a morphism f : F → F′. If the kernel of f is trivial, then for any F ∈ Fields and any α ∈ F(F) the determination function of α coincides with the determination function of f(α) (cf. [1, lemma 10.2(a)]); therefore cd(F) ≤ cd(F′) (cf. [1, lemma 10.2(b)]) and cd_{p}(F) ≤ cd_{p}(F′) (for any p). If moreover f is surjective (but may not be necessarily injective), then cd(F) = cd(F′) (cf. [1, lemma 10.2(c)]) and cd_{p}(F) = cd_{p}(F′).

4. Canonical (p-)dimension of regular complete varieties

Lemma 4.1. The function field of a regular variety X is a generic splitting field of X; in particular, cd(X) ≤ dim X for regular X.

Proof. The function field F (X) is a splitting field of X (even in the non-regular case). To check that it is generic, we take an arbitrary splitting field L of X, find a finitely generated splitting subfield of L, take a model Y/F of it, and consider a rational morphism Y → X, that exists because X has a point over the field F (Y). Since X is regular, there exists a geometric place F (X) → F (Y); composing it with the embedding F (Y) ⊂ L, we get the required place F (X) → L. □

Remark 4.2. The place F (X) → L, constructed in the end of proof of Lemma 4.1, is geometric (as defined in §1.3).

We have the following generalization of Lemma 4.1:

Lemma 4.3. If Y is a closed subvariety of a regular variety X, admitting a dominant rational morphism X → Y, then the function field of Y is a generic splitting field of X. In particular, cd(X) ≤ dim Y.

Proof. Clearly, F (Y) is a splitting field of X. The dominant rational morphism produces an embedding of F (Y) into the field F (X), which by Lemma 4.1 is a generic splitting field of X. It follows by Lemma 2.1 that F (Y) is a generic splitting field too. □

Lemma 4.4. Let Y be a scheme over a field F, X a variety over F.

(1) If Y admits a dominant rational morphism X → Y, then the F (X)-scheme Y_{F (X)} has a closed rational point.

(2) If the F (X)-scheme Y_{F (X)} has a closed rational point, then there exists a closed subvariety Y′ ⊂ Y, admitting a dominant rational morphism X → Y′.

Proof. Existence of a rational morphism X → Y is equivalent to existence of a closed rational point on Y_{F (X)}. To prove the second statement, we take as Y′ the closure of the image of the rational morphism X → Y. □

Proposition 4.5. Any regular complete variety X has a closed subvariety Y ⊂ X of dimension dim Y = cd(X), admitting a dominant rational morphism X → Y.

Proof. Let us take a generic splitting field of X, having the transcendence degree cd(X) over F, replace it by a finitely generated splitting subfield (Lemma 2.1), and consider a projective model T/F of it. Since X is regular, F (X) is a generic splitting field of X (Lemma 4.1) and there exists a place F (X) → F (T). Since the variety X is complete, such a place produces a rational morphism T → X. Replacing T by the closure in T × X of the graph of such a rational morphism, we get another model of the same field, but now we have a regular morphism T → X instead of the rational one we had before.

Since F (T) is a generic splitting field of X, there exists a place F (T) → F (X). By completeness of T, it produces a rational morphism X → T. Composing it with the regular morphism T → X, we get a rational morphism X → X; let Y be the closure of its image. Clearly, dim Y ≤ dim T = cd(X). On the other hand, by Lemma 4.3, dim Y ≥ cd(X). Therefore, dim Y = cd(X). □

Combining Lemma 4.3 and Proposition 4.5, we get

Corollary 4.6. Canonical dimension of a regular complete variety X is the minimum of dimension of closed subvarieties Y ⊂ X, admitting a dominant rational morphism X → Y. □

Taking into account Lemma 4.4, we get the following variation of Corollary 4.6:

Corollary 4.7. Canonical dimension of a regular complete variety X is the minimum of dimension of closed subvarieties Y ⊂ X, satisfying Y (F (X)) ≠ ∅. □

Now we establish variations of Lemma 4.3, Proposition 4.5, and Corollaries 4.6 and 4.7, related to the canonical p-dimension.

Let Y be a scheme over a field F and let X be a variety over F. We say that Y satisfies condition (∗) (with respect to X), if there exists an F -variety X′, a dominant rational morphism X′ → Y, and a dominant rational morphism X′ → X such that the field extension F (X′)/F (X) is p-coprime.

Lemma 4.8. If Y is a closed subvariety of a regular variety X, satisfying condition (∗), then the function field of Y is a p-generic splitting field of X. In particular,

cd_{p}(X) ≤ dim Y .

Proof. Clearly, F (Y) is a splitting field of X. Let now L/F be any splitting field of X. Then we can find a geometric place F (X) → L (see Lemma 4.1 with Remark 4.2). Applying Lemma 3.2 to this place and the field extension F (X′)/F (X), we get a place F (X′) → L′ for some p-coprime field extension L′/L. Restricting the latter place to the subfield F (Y) ⊂ F (X′), we get a place F (Y) → L′; therefore, the splitting field F (Y) is p-generic. □

Lemma 4.9. Let Y be a scheme over a field F, X a variety over F.

(1) If Y satisfies condition (∗), then d_{p}(Y_{F (X)}) = 1 (see §1.1 for definition of d_{p}).

(2) If d_{p}(Y_{F (X)}) = 1, then there exists a closed subvariety Y′ ⊂ Y ⊂ X, satisfying condition (∗). □

Proposition 4.10. Any regular complete variety X has a closed subvariety Y ⊂ X of dim Y = cd_{p}(X), satisfying condition (∗).

Proof. Let us take a finitely generated p-generic splitting field of X, having the transcendence degree cd_{p}(X) over F, and consider a projective model T/F of it. Since F (X) is a generic splitting field of X, while F (T) is a splitting field, there exists a place F (X) → F (T). Since the variety X is complete, such a place produces a rational morphism T → X. Replacing T by the closure in T × X of the graph of this morphism, we get another model of the same field, but now we have a regular morphism T → X instead of the rational one.

Since F (T) is a p-generic splitting field of X, there exists a place F (T) → L′, where L′ ⊃ F (X) is a field, p-coprime over F (X). Let X′/F be a model of the field L′. By completeness of T, the place F (T) → F (X′) produces a rational morphism X′ → T. Composing it with the regular morphism T → X, we get a rational morphism X′ → X; let Y be the closure of its image. Clearly, dim Y ≤ dim T = cd_{p}(X). On the other hand, by Lemma 4.8, dim Y ≥ cd_{p}(X). Therefore, dim Y = cd_{p}(X). □

Lemma 4.8 and Proposition 4.10 together produce

Corollary 4.11. Canonical p-dimension of a regular complete variety X is the minimum of dimension of the closed subvarieties Y ⊂ X, satisfying (∗). □

By Lemma 4.9, the following variation of Corollary 4.11 also holds:

Corollary 4.12. Canonical p-dimension of a regular complete variety X is the minimum of dimension of the closed subvarieties Y ⊂ X with d_{p}(Y_{F (X)}) = 1. □

5. Generically p-split varieties

In this section X stands for a smooth complete absolutely irreducible variety over a field F.

Lemma 5.1. Degree homomorphism deg : Ch_{0}(\overline{X}) → 𝔽_{p} is an isomorphism if and only if dim_{𝔽_{p}} Ch_{0}(\overline{X}) = 1.

Proof. The degree homomorphism is non-zero and therefore surjective. □

Lemma 5.2. Assume that dim_{𝔽_{p}} Ch_{0}(\overline{X}) = 1. Let T be an arbitrary F -scheme, let E_{1} and E_{2} be field extensions of F, and let f_{1} : Spec E_{1} → X and f_{2} : Spec E_{2} → X be F -morphisms. Then the diagram

[figure: \overline{Ch}(T × X); \overline{Ch}(T_{E_{1}}); \overline{Ch}(T_{E_{2}}); Ch(\overline{T}); (id_{T} ×f_{1})^{∗}; (id_{T} ×f_{2})^{∗}]

is commutative.

Proof. Let E be a field extension of F, containing E_{1} and E_{2}. Replacing T and X by T_{E} and X_{E}, we come to the following situation: E = E_{1} = E_{2} = F and for some closed rational points x_{1}, x_{2} ∈ X, f_{i} is the embedding T = T × {x_{i}} ↪ T × X. We want to show that f_{1}^{∗} = f_{2}^{∗}: \overline{Ch}(T × X) → \overline{Ch}(T). Since pr_{∗} ∘f_{i∗} = id for i = 1, 2, where pr is the projection T × X → T, it suffices to show that

f_{1∗} ∘ f_{1}^{∗} = f_{2∗} ∘ f_{2}^{∗}: \overline{Ch}(T × X) → \overline{Ch}(T × X) .

The composition f_{i∗} ∘ f_{i}^{∗} coincides with the multiplication by [T × x_{i}]. Since by the assumption on dim_{𝔽_{p}} Ch_{0}(\overline{X}) and Lemma 5.1, the degree homomorphism deg: Ch(\overline{X}) → 𝔽_{p} is an isomorphism, we have [x_{1}] = [x_{2}] ∈ \overline{Ch}(X), and therefore [T × x_{1}] = [T × x_{2}] in \overline{Ch}(T × X). The required assertion follows. □

Let i: Y ↪ X be a closed subvariety of X. The closed embedding

(id_{Y} × i × id_{X}): Y × X → Y × X × X

is regular, and we define a paring

\overline{Ch}(Y) ⊗ \overline{Ch}(X × X) → \overline{Ch}(Y × X) ,

by the formula α ⊗ β ↦ (id_{Y} × i × id_{X})^{∗}(α × β).

Proposition 5.3. Let Y be a closed subvariety of X. Assume that \overline{Ch}(X_{F (X)}) = Ch(\overline{X}) and dim_{𝔽_{p}} Ch_{0}(\overline{X}) = 1. Then the above paring is surjective.

Proof. We proceed by induction on dim Y. We have a commutative diagram

⨁_{Y′} \overline{Ch}(Y′) ⊗ \overline{Ch}(X × X) ⟶ \overline{Ch}(Y) ⊗ \overline{Ch}(X × X) ⟶ \overline{Ch}(X × X) ⟶ 0
↓  ↓  ↓
⨁_{Y′} \overline{Ch}(Y′ × X)  ⟶  \overline{Ch}(Y × X)  ⟶ \overline{Ch}(X_{F (Y)}) ⟶ 0

where Y′ runs over closed subvarieties of codimension 1 in Y (the upper row is the obvious exact sequence ⨁_{Y′} \overline{Ch}(Y′) → \overline{Ch}(Y) → ℤ → 0, tensored by \overline{Ch}(X × X) over 𝔽_{p}). The rows are exact and the left vertical map is surjective by the induction hypothesis. The right vertical map is surjective because the rhombus

[figure: \overline{Ch}(X × X); \overline{Ch}(X_{F (Y)}); \overline{Ch}(X_{F (X)}); Ch(\overline{X})]

is commutative (as guaranteed by the assumption on Ch_{0}(\overline{X}) and Lemma 5.2 applied to T = X). □

Corollary 5.4. Under condition of Proposition 5.3, if the push-forward

(i × id_{X})_{∗}: \overline{Ch}(Y × X) → \overline{Ch}(X × X)

is non-zero, then the push-forward i_{∗}: \overline{Ch}(Y) → \overline{Ch}(X) is also non-zero and, in particular, \overline{Ch}_{i}(X) ≠ 0 for at least one i ≤ dim Y.

Proof. The square

[figure: \overline{Ch}(Y) ⊗ \overline{Ch}(X × X); \overline{Ch}(Y × X); \overline{Ch}(X) ⊗ \overline{Ch}(X × X); \overline{Ch}(X × X); i_{∗}⊗id; (i×id)_{∗}]

is commutative. □

Definition 5.5. We say that a (complete smooth absolutely irreducible) variety X over F as p-balanced, if the symmetric bilinear form

Ch(\overline{X}) × Ch(\overline{X}) → 𝔽_{p} , (α, β) ↦ deg(α · β)

is non-degenerate (in the sense that its radical is trivial; note that dim_{𝔽_{p}} Ch(\overline{X}) can be infinite).

A variety X over F is called cellular, if there is a filtration

∅ = X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{n−1} ⊂ X_{n} = X

by closed subschemes such that for every i = 0, 1, …, n − 1 the scheme X_{i+1} \ X_{i} is isomorphic to an affine space over F.

Remark 5.6. Let X be a geometrically cellular variety, that is, X_{E} is cellular for some field extension E/F. We claim that X is p-balanced (for any p). Indeed, the Chow-motive of the cellular variety X_{E} decomposes into a finite direct sum of twists of the motive of the point (see, e.g., [10, th. 6.5]). Therefore CH(X_{E}) = CH(\overline{X}). Moreover, the mutually inverse isomorphisms of the motive of X_{E} with the above direct sum are given by certain sequences e_{0}, …, e_{n} and e′_{0}, …, e′_{n} of homogeneous elements in CH(X_{E}), that are bases of CH(X_{E}) mutually dual with respect to the ℤ-bilinear form (α, β) ↦ deg(α · β) (simply because they define mutually inverse isomorphisms of motives).

Note that for any p-balanced X and any integer i, one has dim_{𝔽_{p}} Ch^{i}(\overline{X}) = dim_{𝔽_{p}} Ch_{i}(\overline{X}), if at least one of these dimensions is finite. Since dim_{𝔽_{p}} Ch^{0}(\overline{X})=1, the above relation, taken with i = 0, implies that dim_{𝔽_{p}} Ch_{0}(\overline{X}) = 1 for a p-balanced X.

Definition 5.7. A p-balanced variety X over F is called p-split, if \overline{Ch}(X) = Ch(\overline{X}).

A cellular variety is p-split.

We say that a variety X has a property generically, if X over its own function field has this property. This way we get a notion of generically p-split variety. According to above remarks, a generically cellular variety is generically p-split.

We are ready to prove the main result of the first half of the paper, interpreting the canonical p-dimension of a generically p-split variety in terms of its modulo p reduced Chow group:

Theorem 5.8. If X is a generically p-split variety (see Definitions 5.7 and 5.5), then

cd_{p}(X) = min\{i | \overline{Ch}_{i}(X) ≠ 0\} .

In particular, the formula holds for a generically cellular X.

Proof. Two inequalities are proved separately. ≤ Let i be an integer such that the group \overline{Ch}_{i}(X) is non-zero. Then [Y] ≠ 0 for a closed i-dimensional subvariety Y ⊂ X. We are going to show that d_{p}(Y_{F (X)}) = 1 for such Y, what is enough for our aim by Corollary 4.12.

Since the variety X_{F (X)} is p-split, there exists a prime cycle Z ⊂ X_{F (X)} such that deg([Y_{F (X)}] · [Z]) ≠ 0. Since the product [Y_{F (X)}] · [Z] can be represented by a cycle on the intersection Y_{F (X)} ∩ Z (see [6, §8.1]), the scheme Y_{F (X)} has a closed p-coprime point, meaning that d_{p}(Y_{F (X)}) = 1. ≥ Let now Y be a closed subvariety of X, satisfying condition (∗) We will show that \overline{Ch}_{i}(X) ≠ 0 for some i ≤ dim Y. The desired inequality will then follow by Proposition 4.10.

Let Z be the closure of the image of the rational morphism X′ → Y ×X (the variety X′ with the rational morphisms X′ → Y and X′ → X comes from condition (∗)). The cycle (i × id_{X})_{∗}([Z]) ∈ \overline{Ch}(X^{2}) is non-zero, because for the second projection pr : X^{2} → X, we have

pr_{∗}(i × id_{X})_{∗}[Z] = [F (X′) : F (X)] · [X] ≠ 0 ∈ \overline{Ch}(X) .

It follows by Corollary 5.4 that \overline{Ch}_{i}(X) ≠ 0 for some i ≤ dim Y. □

Remark 5.9. If we take Y with dim Y = cd_{p}(X) in the beginning of the (≥)-part of the proof of Theorem 5.8, then, since we have already proved the (≤)-part of the theorem, we come to the conclusion that the (dim Y)-dimensional component of the homomorphism i_{∗}: \overline{Ch}(Y) → \overline{Ch}(X) is non-zero. Since the (dim Y)-dimensional component of the image of i_{∗} is generated by [Y] ∈ \overline{Ch}(X), we see that in fact the class in \overline{Ch}(X) of Y itself is non-zero.

Example 5.10. Let X be the Severi-Brauer variety of a central simple F -algebra A. Since by Theorem 5.8, cd_{p}(X) = cd_{p}(X_{L}) for any p-coprime field extension L/F, cd_{p}(X) = cd_{p}(Y), where Y is the Severi-Brauer variety of a division algebra, Brauer-equivalent to the p-primary part of A. Furthermore, \overline{Ch}(Y) = \overline{Ch}^{0}(Y) by [9, prop. 2.1.1]. Therefore, by Theorem 5.8, cd_{p}(Y) = dim Y, so that we get

cd_{p}(X) = cd_{p}(Y) = dim Y = d_{p}(X) − 1 .

Example 5.11. In this example p = 2. Let X/F be the orthogonal grassmannian of n-dimensional totally isotropic subspaces of a (2n + 1)-dimensional quadratic form. If d_{2}(X) = 2^{n}, then \overline{Ch}(X) = \overline{Ch}^{0}(X) by [11, prop. 1.4] and therefore

cd_{2}(X) = dim(X) = n(n + 1)/2 .

Without any restriction on d_{2}(X), canonical 2-dimension of X can be expressed as the sum of all i such that the i-th special Schubert class e_{i} ∈ CH^{i}(\overline{X}) is non-rational, i.e, does not lie in \overline{CH}(X) (cf. [18, conj. 6.6]): indeed, by [18, main th. 5.7], the product of all rational e_{i}, reduced modulo 2, is a non-zero element of \overline{Ch}(X) of the smallest possible dimension.

6. Canonical p-dimension of algebraic groups

If P is an algebraic group over a field F, we write CH(BP) for the P -equivariant Chow ring CH_{P} (Spec F) of the point Spec F (see [5]).

Let G be a connected algebraic group over F and let P be a subgroup of G. Consider the homomorphism

φ_{G} = φ_{G,P} : CH(BP) = CH_{P} (Spec F) ⟶^{q^{∗}} CH_{P} (G) = CH(G/P),

where q : G → Spec F is the structure morphism.

Remark 6.1. If G is a subgroup of a group G′, then φ_{G} = i^{∗}∘φ_{G′}, where i : G/P → G′/P is the morphism, induced by embedding of the groups.

Proposition 6.2. Let G = GL_{n}. Then the map φ_{G} is surjective and the left G-action on G/P induces the trivial action on CH(G/P).

Proof. The group G is embedded into the affine space of End(F^{n}) as a G-equivariant open subset. The map q^{∗} factors as the composite

CH_{P} (Spec F) → CH_{P} (End(F^{n})) → CH_{P} (G),

where the first pull-back map is an isomorphisms by the homotopy invariance property and the second restriction map is surjective by the localization. Hence φ_{G} is surjective.

For a rational point g of G, let λ_{g} : G → G is the morphism of the left multiplication by g. It follows from q ∘ λ_{g} = q that λ_{g}^{∗} ∘ q^{∗} = q^{∗}. Since q^{∗} is surjective, λ_{g}^{∗} is the identity, i.e., G acts trivially on CH(G/P). □

Recall that we write CH(\overline{G/P}) for the colimit of CH(G_{L}/P_{L}) over all field extensions L/F. We define a homomorphism \overline{φ}_{G} as the composite

\overline{φ}_{G} : CH(BP) ⟶^{φ_{G}} CH(G/P) ⟶^{res} CH(\overline{G/P}) .

Let E be a (right) G-torsor over a field extension F′ of F. Set K = F′(E). Let ψ_{E}: CH(E/P) → CH(G_{K}/P_{K}) be pull-back map with respect to the morphism G_{K}/P_{K} → E/P, induces by the G-equivariant morphism G_{K} → E, taking the identity of G to the generic point of E. We define a homomorphism \overline{ψ}_{E} as the composite

\overline{ψ}_{E} : CH(E/P) ⟶^{ψ_{E}} CH(G_{K}/P_{K}) ⟶^{res_{K}} CH(\overline{G/P}) .

We identify G with a subgroup of S = GL_{n} for some n.

Lemma 6.3. Suppose that there is a G-equivariant morphism E → S over F and let h : E/P → S/P be the induced morphism. Then \overline{φ}_{G} = \overline{ψ}_{E} ∘ h^{∗} ∘ φ_{S}.

Proof. The composition G_{K} → E_{K} → S_{K} of the morphisms induced by G_{K} → E and E → S, differs from the inclusion G_{K} ↪ S_{K} by a left multiplication by an element of S(K). By Proposition 6.2, the induced pull-back homomorphisms CH(S_{K}/P_{K}) → CH(G_{K}/P_{K}) coincide. Composing with the restriction homomorphism CH(S/P) → CH(S_{K}/P_{K}), we get ψ_{E} ∘ h^{∗} = res_{K/F} ∘ i^{∗}, where i : G/P → S/P is the morphisms, induced by the embedding G ↪ S. We have:

\overline{φ}_{G} = res ∘ φ_{G} = res_{K} ∘ res_{K/F} ∘ i^{∗} ∘ φ_{S} = res_{K} ∘ψ_{E} ∘ h^{∗} ∘ φ_{S} = \overline{ψ}_{E} ∘ h^{∗} ∘ φ_{S}

(for the second equality, see Remark 6.1). □

Theorem 6.4. 1. For any G-torsor E (over any field extension of F) we have

Im(\overline{φ}_{G}) ⊂ Im(\overline{ψ}_{E}) .

2. There exists a G-torsor E (over a field extension of F) such that Im(\overline{φ}_{G}) = Im(\overline{ψ}_{E}).

Proof. 1. We may assume that E is a G-torsor over F. By the Hilbert theorem 90, the S-torsor (E × S)/G is trivial (where (E × S)/G stands for the quotient of E × S by the action (e, s)·g = (eg, g^{−1}s) of G; the action of G on this quotient is defined by the formula (e, s) · g = (e, sg), so that the embedding E = E × 1 ↪ E × S induces a G-equivariant morphism E → (E × S)/G). In particular, there is a G-equivariant morphism E → S. By Lemma 6.3, Im(\overline{φ}_{G}) ⊂ Im(\overline{ψ}_{E}).

2. Let X = S/G and K = F (X). Denote by E → Spec K the generic fiber of the projection S → X. Clearly, E is a G-torsor over K. Denote by h : E/P_{K} → S/P the morphism induced by the canonical G-equivariant morphism E → S. Since E/P_{K} is a localization of S/P, the pull-back homomorphism h^{∗} is surjective. By Proposition 6.2, φ_{S} is also surjective. It follows from Lemma 6.3 that Im(\overline{φ}_{G}) = Im(\overline{ψ}_{E}). □

Let G be an algebraic group over a field F and let Tors_{G} be the functor Fields_{F} → pSets, taking a field K to the set of isomorphism classes Tors_{G}(K) of G-torsors over K. For a G-torsor E/K, the determination function of E ∈ Tors_{G}(K) coincides with the determination function of the K-variety E.

Let P ⊂ G be a subgroup. We assume that P is a special group, that is, the functor Tors_{P} is trivial.

Lemma 6.5. The determination functions of the varieties E and E/P coincide.

Proof. Suppose E/P has a point over K. We need to show that E(K) ≠ ∅. The fiber of the natural morphism E → E/P over the point is a P -torsor. Since P is special, this torsor is trivial, i.e., the fiber has a point over K. □

Remark 6.6. Let G be a split semisimple algebraic group and let P be a parabolic subgroup. The variety G/P is cellular (see, e.g., [2]), therefore CH(\overline{G/P}) = CH(G/P).

Remark 6.7. Suppose further that P is a Borel subgroup of G. The image of the composite

CH(BP) ⟶^{φ_{G}} CH(G/P) ⟶^{deg} ℤ

is a subgroup t_{G}ℤ with a positive integer t_{G} known as the torsion index of G (see [7]). It follows from Theorem 6.4 and Lemma 6.5 that t_{G} is the l.c.m. of the numbers d(E) over all G-torsors over all field extension. This statement is known as Grothendieck’s theorem [7, th. 2]. The prime divisors of the torsion index t_{G} are called the torsion primes of G.

It follows that the canonical dimension of G (resp. cd_{p}(G)) is the supremum of the canonical dimension of E/P (resp. cd_{p}(E/P)) over all G-torsors over all field extensions of F.

Let G be a split semisimple group and let P ⊂ G be a parabolic subgroup. Suppose P is a special group (for example, P is a Borel subgroup of G). Let E be a G-torsor. Note that the variety E/P is projective. In order to apply Theorem 5.8 to the variety E/P, we need the following

Corollary 6.8. The variety E/P is generically cellular.

Proof. By Lemma 6.5, the torsor E is split over the function field L = F (E/P), hence E_{L} ≃ G_{L} and therefore (E/P)_{L} ≃ (G/P)_{L}. The latter variety is cellular. □

Theorems 5.8 and 6.4 yield

Theorem 6.9. Let G be a split semisimple group and let P ⊂ G be a special parabolic subgroup (for example, a Borel subgroup). Denote by C̃H(G/P) the image of the graded ring homomorphism φ_{G} : CH(BP) → CH(G/P). Then cd_{p}(G) for a prime p, is equal to the smallest integer i such that C̃H_{i}(G/P) is not contained in p CH_{i}(G/P). □

Remark 6.10. The canonical dimension cd_{p}(G) is positive if and only if p is a torsion prime of G (see Remark 6.7). Indeed, by Theorem 6.9, cd_{p}(G) = 0 if and only if C̃H_{0}(G/P) is not divisible by p in CH_{0}(G/P), where P is a Borel subgroup of G. Since CH_{0}(G/P) is an infinite cyclic group generated by the class of a rational point, the latter is equivalent to the condition that p does not divide t_{G}, i.e., p is not a torsion prime of G.

7. Remarks on C̃H(G/P)

Let P be an arbitrary subgroup of an algebraic group G. Let P → GL(V) be a finitedimensional representation. The group P acts (on the right) on the product G × V by (g, v) · p = (g · p, p^{−1} · v). The factor variety (G × V)/P is a vector bundle over G/P, we denote it by Bun(V).

We can view V as a P -equivariant vector bundle over the point Spec F. For any n ≥ 0, the n-th P -equivariant Chern class c_{n}(V) is an element of CH^{n}(BP) (see [5]).

Let T be a split torus. There is a canonical isomorphism

S(T̂) →^{∼} CH(BT),

(where T̂ is the character group of T, S stands for the symmetric algebra) defined by the property that the image of a character χ is the first Chern class c_{1}(χ) where χ is considered as a 1-dimensional representation of T [5, 3.2].

Let P be a special parabolic subgroup of a split semisimple algebraic group G. Let T be a maximal split torus contained in P and let W_{P} be the Weyl group of P. Since P is special, the canonical homomorphism

CH(BP) → CH(BT)^{W_{P}} = S(T̂)^{W_{P}}

is an isomorphism [5, prop. 6]. Identifying CH(BP) with S(T̂)^{W_{P}}, we get a homomorphism

φ_{G} : S(T̂)^{W_{P}} → CH(G/P)

with the image the subring C̃H(G/P).

Lemma 7.1. Let χ_{1}, χ_{2}, …, χ_{m} ∈ T̂ be all characters (with multiplicities) of a representation P → GL(V). Let s_{n} ∈ S^{n}(T̂)^{W_{P}} be the standard symmetric functions on the characters χ_{i}. Then φ_{G}(s_{n}) = c_{n}(Bun(V)).

Proof. By naturality of the Chern classes, we have φ_{G}(c_{n}(V)) = c_{n}(Bun(V)). On the other hand, c_{n}(V) is the n-th standard symmetric functions on the characters of V. □

Remark 7.2. Let G be a split semisimple group over an arbitrary field (of an arbitrary characteristic), B ⊂ G a Borel subgroup, T ⊂ B a split maximal torus, W the Weyl group of G. The closures X_{w} of the cells BwB/B of the cellular variety G/B are indexed by the elements w ∈ W and called generalized Schubert varieties of G/B; moreover, dim X_{w} = l(w), where l: W → ℤ_{≥0} is the length function. Taking the unique maximal length element w_{0} ∈ W and setting X^{w} = X_{w_{0}w}, we get a different (preferable for us) indexation of the same varieties, for which codim X^{w} = l(w). The group CH(G/B) is free and the classes [X^{w}], called generalized Schubert classes, form its basis.

The following formula for the product of a 1-codimension Schubert class with an arbitrary Schubert class is given in [4, §4.4 cor.2]:

[X^{s_{α}}] · [X^{w}] = ∑_{β}⟨β^{∨}, ω_{α}⟩ · [X^{w·s_{β}}] ,

where α is a simple root, ω_{α} its fundamental weight, s_{α} ∈ W the reflection with respect to α; β runs over the set of positive roots such that l(w · s_{β}) = l(w) + 1, and β^{∨} is the dual to β root. Note that the coefficients of this formula depend only on the root system; in particular, they do not depend on the base field and its characteristic. Moreover, this formula completely determines the multiplication table of the basis [X^{w}], w ∈ W, because the ℚ-algebra CH(G/B) ⊗ ℚ is generated by CH^{1}(G/B) [4].

Remark 7.3. Let P = B be a Borel subgroup of G. We have W_{B} = 1 and therefore the subring C̃H(G/B) is generated by ϕ_{G}(T̂). In the case of simply connected G, for the weight ω_{α} of a simple root α, one has the formula

ϕ_{G}(ω_{α}) = −[X^{s_{α}}] , [4, §4 formula (7)] ,

which determines ϕ_{G} also in the non simply connected case. This formula also shows that if the group G is simply connected, then ϕ_{G}(T̂) = CH^{1}(G/B), and therefore C̃H(G/B) is the subring of CH(G/B), generated by CH^{1}(G/B).

Remark 7.4. From Theorem 6.9 and Remark 7.3, we see that:

(1) if G_{1} and G_{2} are split semisimple groups, then cd_{p}(G_{1} × G_{2}) = cd_{p}(G_{1}) + cd_{p}(G_{2});

(2) if G′ → G is a central isogeny of split semisimple groups, then cd_{p}(G′) ≤ cd_{p}(G).

Remark 7.5. Let us consider pairs (Φ, A), consisting of a root system Φ and a subgroup of the quotient of the weight lattice of Φ by its root lattice. An isomorphism of pairs (Φ, A) → (Φ′, A′) is an isomorphism of the root systems Φ → Φ′ such that the induced isomorphism of the lattice quotients maps A to A′. To any split semisimple algebraic group G one attaches an isomorphism class of above pairs, to which we refer as extended type of G. Theorem 6.9 with Remarks 7.2 and 7.3 shows that cd_{p}(G) (for any p) depends only on the extended type of G; in particular, it does not depend on the base field and especially on its characteristic (so that computing cd_{p}(G) one may always assume that G is defined over ℂ).

8. Canonical p-dimension of split simple groups of classical types

In this section we compute canonical p-dimension of all split simple groups of classical types. We will need the following

Lemma 8.1. Let R be a commutative ring, r ∈ R, and let A be the factor ring of the polynomial ring R[x_{1}, x_{2}, …, x_{n}] modulo the ideal generated by the polynomial x_{1} + x_{2} + ⋯+x_{n} −r. The symmetric group W = S_{n} acts on A by permuting the x_{i}. If R has trivial ℤ-torsion, then A^{W} = R[s_{2}, s_{3}, …, s_{n}], where s_{i} are the standard symmetric functions.

Proof. Consider the natural W -action on the ring R[x] = R[x_{1}, x_{2}, …, x_{n}]. We have the exact sequence 0 → R[x] ⟶^{f} R[x] → A → 0, where the first map is the multiplication by f = x_{1} + x_{2} + ⋯ + x_{n} − r. Applying the W -invariants we get an exact sequence

0 → R[x]^{W} ⟶^{f} R[x]^{W} → A^{W} → H^{1}(W, R[x]).

The ring R[x]^{W} coincides with R[s] = R[s_{1}, s_{2}, …, s_{n}]. The monomials in the variables x_{i} form a permutation basis of the R-module R[x]. By the Faddeev-Shapiro lemma, the group H^{1}(W, R[x]) is a direct sum of the groups H^{1}(W′, R) = Hom(W′, R) for certain subgroups W′ ⊂ W. Since R has trivial ℤ-torsion, the latter group is trivial. Therefore,

A^{W} = R[s]/(f) = R[s_{2}, s_{3}, …, s_{n}].  □

8.1. Type A_{n−1}. A split simple group of type A_{n−1} is isomorphic to G = SL(n)/μ_{l}, where l is a divisor of n. Let P ⊂ SL(n) be the stabilizer of the line U = [1 : 0 : …: 0] ∈ ℙ^{n−1} with respect to the natural action of SL(n) on ℙ^{n−1}. The semisimple part of P is SL(n−1) and it intersects μ_{l} trivially. Hence the parabolic subgroup P_{l} = P/μ_{l} of G is special. We have G/P_{l} = ℙ^{n−1}.

The intersection T of the group of diagonal matrices D(n) of GL(n) with SL(n) is a maximal torus of SL(n). The character group T̂ is identified with the factor group of ℤ^{n} = { \atop D(n)} with the standard basis x_{1}, x_{2}, …, x_{n} by the subgroup generated by x_{1}+x_{2}+⋯+x_{n}.

The character group of the maximal torus T_{l} = T/μ_{l} of G is the subgroup of T̂ consisting of all sums ∑ a_{i}x_{i} such that ∑ a_{i} is divisible by l. Hence, T̂_{l} is generated by lx_{1} and x_{i} − x_{1} for all i = 2, …, n with the relation ∑_{i≥2}(x_{i} − x_{1}) = −nx_{1}.

The Weyl group W = W_{P_{l}} is the symmetric group S_{n−1}, permuting x_{2}, …, x_{n}. Applying Lemma 8.1 to the ring R = ℤ[lx_{1}], the element r = −nx_{1}, the variables x_{i} − x_{1} and the group W, we get S(T̂_{l})^{W} = ℤ[lx_{1}, s_{2}, s_{3}, …, s_{n−1}], where the s_{i} are the standard symmetric functions on the x_{i} − x_{1}, i ≥ 2.

The group P acts naturally on the space V = F^{n}. The characters of this representation are x_{1}, x_{2}, …, x_{n}. The corresponding vector bundle Bun(V) over ℙ^{n−1} = SL(n)/P = G/P_{l} is the trivial vector bundle of rank n. The line U can be viewed as a 1-dimensional representation of P given by the character x_{1}. We have Bun(U) = L^{∨}, where L is the canonical line bundle on ℙ^{n−1} (with the sheaf of sections O(1)). Consider the representation M = (V/U) ⊗ U^{∨} of the group P with the characters x_{i} − x_{1} for all i = 2, …, n. Note that the group μ_{l} is contained in the kernel of the representation, hence M is a representation of P_{l}.

By Lemma 7.1, we have φ_{G}(lx_{1}) = lc_{1}(L^{∨}) = −lh, where h ∈ CH_{1}(ℙ^{n−1}) is the class of a hyperplane, and also φ_{G}(s_{i}) = c_{i}(Bun(M)) for all i. Hence the subring C̃H(ℙ^{n−1}) of CH(ℙ^{n−1}) = ℤ[h]/(h^{n}) is generated by lh and the Chern classes c_{i}(Bun(M)). Since

Bun(M) = (Bun(V)/ Bun(U)) ⊗ Bun(U^{∨}) = (Bun(V)/L^{∨}) ⊗ L,

the class [Bun(M)] is equal to n[L] − 1 in K_{0}(ℙ^{n−1}). Hence, c_{•}(Bun(M)) = c_{•}(L)^{n} = (1 + h)^{n}. Thus the subring C̃H(ℙ^{n−1}) is generated by lh and \binom{n}{i}h^{i} for i = 2, …, n − 1.

Let p a be prime integer and let p^{k} be the largest power of p dividing n. Note that the binomial coefficient \binom{n}{i} is divisible by p unless i is divisible by p^{k}. The largest value of i < n such that \binom{n}{i} is not divisible by p is n − p^{k}. By Theorem 6.9,

cd_{p}(SL(n)/μ_{l}) = \{ \begin{matrix}p^{k} − 1, & if p divides l; \\ 1, & otherwise. \end{matrix}

Denote by CSA_{n,l}(K) the set of isomorphism classes of central simple K-algebras of degree n and exponent dividing l. The exact sequence 1 → μ_{l} → SL(n) → SL(n)/μ_{l} → 1 yields a surjective map Tors_{SL(n)/μ_{l}}(K) → CSA_{n,l}(K) with trivial kernel. By Remark 3.7,

cd_{p}(CSA_{n,l}) = \{ \begin{matrix}p^{k} − 1, & if p divides l; \\ 1, & otherwise. \end{matrix}

8.2. Type B_{n}. The only torsion prime is p = 2.

Taking a (2n + 1)-dimensional vector space, endowed with a completely split quadratic form, let a vector g together with vectors e_{i}, f_{i}, i = 1, 2, …, n form a basis such that {e_{i}, f_{i}} are pairwise orthogonal hyperbolic pairs, while g is orthogonal to all e_{i}, f_{i}. Let G = SO(2n+1) be the corresponding special orthogonal group. The inclusion of D(n) into SO(2n + 1) given by t(e_{i}) = t_{i}e_{i}, t(f_{i}) = t_{i}^{−1}f_{i} and t(g) = g, where t = diag(t_{1}, …, t_{n}), identifies D(n) with a maximal torus T of SO(2n + 1). In particular, the group T̂ is identified with ℤ^{n} = { \atop D(n)}. We write x_{1}, x_{2}, …, x_{n} for the standard basis of ℤ^{n}.

Let V be the totally isotropic subspace of dimension n generated by all the e_{i}. Denote by P the stabilizer of V in G, so that X = G/P is the variety of all dimension n totally isotropic subspaces. The characters of the natural representation P → GL(V) are x_{1}, x_{2}, …, x_{n}. The vector bundle Bun(V) over X is the tautological vector bundle.

The group W = W_{P} is the symmetric group S_{n} permuting the x_{i}. The semisimple part of P is SL(n), so that P is special.

We have S(T̂)^{W} = ℤ[s_{1}, s_{2}, …, s_{n}], where s_{k} are the standard symmetric functions on the x_{i}. By Lemma 7.1, the subring C̃H(X) of CH(X) is generated by the Chern classes of Bun(V). These Chern classes are divisible by 2 in CH(X) [15, ch. III th. 6.11]. Thus, C̃h^{j}(X) = 0 if j > 0. We conclude by Theorem 6.9 that

cd_{2} SO(2n + 1) = \frac{n(n + 1)}{2}

(see also Examples 3.6 and 5.11). The set Tors_{SO(2n+1)}(K) is identified with the set of similarity classes Q_{2n+1}(K) of non-degenerate quadratic forms of dimension 2n + 1 over K. Thus,

cd_{2} Q_{2n+1} = \frac{n(n + 1)}{2}.

Let G = Spin(2n + 1) be the spinor group. There is an exact sequence

1 → μ_{2} → T′ → T → 1,

where T′ is a maximal torus of Spin(2n + 1). We have T̂′ = T̂ + ℤy = ℤ^{n} + ℤy, where y = (x_{1} + ⋯ + x_{n})/2. By Lemma 8.1 applied to the ring R = ℤ[y], the element r = 2y and the group W, the ring S(T̂)^{W} is the polynomial ring ℤ[y, s_{2}, s_{3}, …, s_{n}].

By Lemma 7.1, φ_{G}(s_{1}) = c_{1}(Bun(V)). The latter class coincides with 2e where e is a generator of CH^{1}(X) [15, ch. III th. 6.11]. Since s_{1} = 2y and CH^{1}(X) is torsion free, we have φ_{G}(y) = e.

As noted above, the images of the s_{i} in CH(X) are divisible by 2. Hence the image of C̃H(X) in Ch(X) = CH(X)/2 is the subring generated by e mod 2. Let m be the smallest integer such that 2^{m} > n. Then e^{2^{m}} = 0 and e^{2^{m}−1} ≠ 0 in Ch(X) [15, ch. III th. 6.11]. Thus,

cd_{2} Spin(2n + 1) = \frac{n(n + 1)}{2} − 2^{m} + 1.

Let \overline{Q}_{2n+1}(K) be the subset of Q_{2n+1}(K) consisting of all classes of forms with trivial even Clifford invariant. The exact sequence 1 → μ_{2} → Spin(2n + 1) → SO(2n + 1) → 1 yields a surjective map Tors_{Spin(2n+1)}(K) → \overline{Q}_{2n+1}(K) with trivial kernel. In particular,

cd_{2} \overline{Q}_{2n+1} = \frac{n(n + 1)}{2} − 2^{m} + 1.

8.3. Type C_{n}. The group Sp(2n) is special, so that cd_{p} Sp(2n) = 0 for all p.

Let G = PGSp(2n) be the projective symplectic group. The number p = 2 is the only torsion prime of G. Instead of applying the general method, we proceed as follows.

The set Tors_{PGSp(2n)}(K) is identified with the set of isomorphism classes ASI_{2n}(K) of central simple K-algebras A of degree 2n with a symplectic involution [13, §29.22].

The forgetful functor ASI_{2n} → CSA_{2n,2} has trivial kernel and is surjective. Therefore, by Remark 3.7 and (8.1),

cd_{2} PGL(2n) = cd_{2} ASI_{2n} = cd_{2} CSA_{2n,2} = 2^{k} − 1,

where 2^{k} is the largest power of 2 dividing 2n.

8.4. Type D_{n}. Let {e_{i}, f_{i}}, i = 1, 2, …, n be pairwise orthogonal hyperbolic pairs of a hyperbolic quadratic form of dimension 2n. The inclusion of D(n) into SO(2n) given by t(e_{i}) = t_{i}e_{i} and t(f_{i}) = t_{i}^{−1}f_{i}, where t = diag(t_{1}, …, t_{n}), identifies D(n) with a maximal torus T′ of SO(2n). In particular, the group T̂′ is identified with ℤ^{n} = { \atop D(n)}. We write x_{1}, x_{2}, …, x_{n} for the standard basis of ℤ^{n}.

Let V be the totally isotropic subspace of dimension n generated by all the e_{i} and let U be the line F e_{1}. Denote by P the stabilizer of the flag U ⊂ V in G = Spin(2n) and set X = G/P. The semisimple part of P is isomorphic to SL(n − 1) and intersects trivially the center of G. Hence the image of P in any simple group of type D_{n} (under a central isogeny of G) is a special group.

Let Y be the connected component of the scheme of maximal (n-dimensional) totally isotropic subspaces such that V is a point of Y. The natural morphism f : X → Y is the projective bundle associated with the tautological vector bundle E over Y of rank n. In particular,

dim X = dim Y + (n − 1) = \frac{n(n − 1)}{2} + (n − 1).

Note that Y is isomorphic to the projective homogeneous variety of the group Spin(2n−1) considered in the type B_{n−1}. The Chern classes of E in CH(Y) are divisible by 2 (see the type B_{n}), hence Ch(X) = Ch(Y)[h]/(h^{n}), where h = c_{1}(L) for the canonical line bundle L over X.

Similar to the case B_{n}, the character group of the maximal torus T of Spin(2n) is equal to ℤ^{n} + ℤy, where y = (x_{1} + x_{2} + ⋯ + x_{n})/2. Set x′_{i} = x_{i} − x_{1} for i = 2, …, n, so that x′_{2} + ⋯ + x′_{n} = 2y − nx_{1}. The symmetric group W = W_{P} permutes the x′_{i} and acts trivially on y and x_{1}. Applying Lemma 8.1 to the variables x′_{i}, the ring R = ℤ[y, x_{1}] and the element r = 2y − nx_{1} we see that S(T̂)^{W} = ℤ[y, x_{1}, s_{2}, … s_{n−1}], where the s_{i} are the standard symmetric functions on the x′_{i}.

Consider the homomorphism (reduced modulo 2)

φ_{G} : ℤ[y, x_{1}, s_{2}, … s_{n−1}] → Ch(X) = Ch(Y)[h]/(h^{n}).

As in the case A_{n−1}, we have Bun(U) = L^{∨} and therefore φ_{G}(x_{1}) = c_{1}(L^{∨}) = −h. Similar to the case B_{n}, the class e = φ_{G}(y) is a generator of Ch^{1}(Y). Recall that e^{2^{m}−1} ≠ 0 and e^{2^{m}} = 0 where m is the smallest integer such that 2^{m} ≥ n.

Similar to the case A_{n−1}, we observe by Lemma 7.1 that the images of the s_{i} in Ch(X) are the Chern classes of the vector bundle (f^{∗}(E)/L^{∨}) ⊗ L. The class of this bundle in K_{0}(X) is equal to [f^{∗}(E) ⊗ L] − 1. Since the Chern classes of E are divisible by 2, we can replace E by the trivial bundle of rank n and replace [f^{∗}(E) ⊗ L] by n[L]. As in the case A_{n−1}, we see that φ_{G}(s_{i}) = \binom{n}{i}h^{i}.

The subring C̃h(X) = Im(φ_{G}) is generated by h and e. The largest degree nontrivial monomial in h and e is h^{n−1}e^{2^{m}−1}. By Theorem 6.9,

cd_{2} Spin(2n) = dim X − (n − 1) − (2^{m} − 1) = \frac{n(n − 1)}{2} − 2^{m} + 1.

Let \overline{Q}_{2n}(K) be the subset of the set Q_{2n}(K) of isomorphism classes of non-degenerate quadratic forms of dimension 2n consisting of all classes of forms with trivial discriminant and Clifford invariant. The exact sequence 1 → μ_{2} → Spin(2n) → SO(2n) → 1 yields a surjective map Tors_{Spin(2n)}(K) → \overline{Q}_{2n}(K) with trivial kernel. In particular,

cd_{2} \overline{Q}_{2n} = \frac{n(n − 1)}{2} − 2^{m} + 1.

Now let G = SO(2n). Recall that the character group T̂′ of the maximal torus T′ of G is the subgroup of T̂ generated by all the x_{i}. Thus we have S(T̂′) = ℤ[x_{1}, x_{2}, …, x_{n}] and therefore, S(T̂′)^{W} = ℤ[x_{1}, s_{1}, …, s_{n−1}]. The subring C̃h(X) is then generated by h. The largest degree nontrivial monomial in h is h^{n−1}. By Theorem 6.9,

cd_{2} SO(2n) = dim X − (n − 1) = \frac{n(n − 1)}{2}.

Let Q′_{2n}(K) be the subset of the set Q_{2n}(K) consisting of all classes of forms with trivial discriminant. There is a canonical bijection Tors_{SO(2n)}(K) →^{∼} Q′_{2n}(K). Therefore,

cd_{2} Q′_{2n} = \frac{n(n − 1)}{2}.

Let G = PGO^{+}(2n) be the projective orthogonal group. Let \overline{T} be the image of the maximal torus T under the canonical isogeny Spin(2n) → G. The character group \overline{T} is the subgroup of T̂ generated by all the simple roots. Thus we have S(ˆ\overline{T}) = ℤ[2x_{1}, x′_{2}, …, x′_{n}] and therefore, S(ˆ\overline{T})^{W} = ℤ[2x_{1}, s_{1}, …, s_{n−1}]. The subring C̃h(X) is then generated by \binom{n}{i}h^{i}. Let 2^{k} be the largest power of 2 dividing n. Note that the binomial coefficient \binom{n}{i} is even unless i is divisible by 2^{k}. The largest value of i < n such that \binom{n}{i} is odd is n − 2^{k}. The largest degree nontrivial monomial in h is \binom{n}{n−2^{k}}h^{n−2^{k}}. By Theorem 6.9,

cd_{2} PGO^{+}(2n) = dim X − (n − 2^{k}) = \frac{n(n − 1)}{2} + 2^{k} − 1.

Let AQP_{2n}(K) be the set of isomorphism classes of central simple algebras of degree 2n with a quadratic pair with trivial discriminant [13, §29.F]. The exact sequence 1 → PGO^{+}(2n) → PGO(2n) → ℤ/2ℤ → 1 yields a surjective map Tors_{PGO^{+}(2n)}(K) → AQP_{2n}(K) with trivial kernel. In particular,

cd_{2} AQP_{2n} = \frac{n(n − 1)}{2} + 2^{k} − 1.

Suppose now that n is even. There are two isomorphic semispinor groups. We set Spin^{∼}(2n) = Spin(2n)/H, where H is the intersection of Ker(y) with the center of Spin(2n). Let T′′ be the image of the maximal torus T under the canonical isogeny Spin(2n) → G. The character group of T′′ is the subgroup of T̂ generated by all the simple roots and y. Thus we have S(T̂′′) = ℤ[y, 2x_{1}, x′_{2}, …, x′_{n}].

Applying Lemma 8.1 to the elements x′_{i}, the ring R = ℤ[y, 2x_{1}], and the element r = 2y − nx_{1}, we see that S(T̂′′)^{W} = ℤ[y, 2x_{1}, s_{2}, … s_{n−1}].

The subring C̃h(X) is then generated by e and \binom{n}{i}h^{i}. The largest degree nontrivial monomial in h and e is \binom{n}{n−2^{k}}h^{n−2^{k}}e^{2^{m}−1}. By Theorem 6.9,

cd_{2} Spin^{∼}(2n) = dim X − (n − 2^{k}) − (2^{m} − 1) = \frac{n(n − 1)}{2} + 2^{k} − 2^{m}.

Let AQP′_{2n}(K) be the set of isomorphism classes of central simple algebras of degree 2n with a quadratic pair with trivial discriminant and trivial component of the Clifford algebra. The exact sequence 1 → μ_{2} → Spin^{∼}(2n) → PGO^{+}(2n) → 1 yields a surjective map Tors_{Spin^{∼}(2n)}(K) → AQP′_{2n}(K) with trivial kernel. In particular,

cd_{2} AQP′_{2n} = \frac{n(n − 1)}{2} + 2^{k} − 2^{m}.

8.5. Appendix: Type G_{2}. The only torsion prime is 2. Since Tors_{G} ≃ \overline{Q}_{8} for a split simple G of type G_{2}, we have cd_{2}(G) = cd_{2}(\overline{Q}_{8}) = 3 (see §8.4).

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Laboratoire de Mathématiques de Lens, Faculté des Sciences Jean Perrin, Université d’Artois, rue Jean Souvraz SP 18, 62307 Lens Cedex, FRANCE

  E-mail address: karpenko@euler.univ-artois.fr

Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

  E-mail address: merkurev@math.ucla.edu