CONNECTEDNESS OF CLASSES OF FIELDS AND ZERO CYCLES ON PROJECTIVE HOMOGENEOUS VARIETIES

VLADIMIR CHERNOUSOV AND ALEXANDER MERKURJEV

1. Introduction

Let X be a proper scheme of finite type over a field F. A zero-cycle on X is the formal sum ∑ n_{i}[x_{i}] where n_{i} ∈ ℤ and x_{i} are closed (zero-dimensional) points of the variety X. The factor group of the group of zero-cycles modulo rational equivalence is called Chow group of dimension zero and is denoted by CH_{0}(X). The assignment x ↦ deg(x) extends to the degree homomorphism

deg : CH_{0}(X) → ℤ.

The image of deg coincides with n(X)ℤ where n(X) is the greatest common divisor of the degrees deg(x) = [F (x) : F] over all closed points x ∈ X. We denote the kernel of deg by \overline{CH}_{0}(X).

The main purpose of the paper is to present a characteristic free uniform method of computing the group \overline{CH}_{0}(X) for projective homogeneous varieties of semisimple algebraic groups. The method is based on the idea of parametrization of fields over which X has a point. We illustrate the method by proving that in many cases the group \overline{CH}_{0}(X) is trivial and give examples of varieties when this group is not trivial. The main results of the paper can be summarized as follows.

Let X be a scheme over F. We denote by A(X) the class of all field extensions L/F such that X(L) ≠ ∅. We say that two fields L_{0}, L_{1} ∈ A(X) of the same degree n over F are simply X-equivalent if they are members of a continuous family of fields L_{t} ∈ A(X), t ∈ 𝔸^{1}, of degree n over F (for precise definition see Section 6). We say that L and L′ are X-equivalent if they can be connected by a chain of fields L_{0} = L, L_{1}, …, L_{r} = L′ such that L_{i} and L_{i+1} are simply X-equivalent for i = 0, …, r − 1. Furthermore we say that the class A(X) is connected if every two fields in A(X_{E}) of degree n(X_{E}) over E are X_{E}-equivalent over any special extension E/F (see Section 6).

Our first result (Theorem 6.5) asserts that if X is an arbitrary proper scheme over F such that the class A(X) is connected and \overline{CH}_{0}(X_{L}) = 0 for any field L ∈ A(X), then \overline{CH}_{0}(X) = 0. Note that the condition \overline{CH}_{0}(X_{L}) = 0 always holds for projective homogeneous varieties X. Thus the connectedness of the class A(X) for such X implies \overline{CH}_{0}(X) = 0.

2000 Mathematics Subject Classification. Primary 20G15, 14M15, 14M17.

The first author was supported in part by Canada Research Chairs Program and NSERC Canada Grant G121210944.

The second author was supported in part by NSF Grant #0355166.

We prove the connectedness of A(X) for various classes of projective homogeneous varieties. These include: Severi-Brauer varieties, certain generalized Severi-Brauer varieties, quadrics, involution varieties, projective homogeneous varieties related to groups of exceptional types ^{3,6}D_{4}, G_{2}, F_{4},^{1,2}E_{6}, E_{7} with trivial Tits algebras. As an application we get that \overline{CH}_{0}(X) = 0, i.e. the Chow group CH_{0}(X) is infinite cyclic for all above mentioned varieties. We borrow from [11] the idea of using symplectic involutions in the case of generalized Severi-Brauer varieties and involution varieties.

Some of our results were known before under certain restrictions on characteristic of the ground field F. Triviality of \overline{CH}_{0}(X) in the case of the Severi-Brauer variety X = SB(A) was proven by Panin in [15] if char F does not divide ind(A). Quadrics over fields of characteristic ≠ 2 were considered by Swan [27] and Karpenko [8]. The cases of certain generalized Severi-Brauer varieties and involution varieties were treated by Krashen in [11] under the assumption char F = 0. Involution varieties of algebras of index at most 2 were considered in [13] by the second author under assumption char F ≠ 2. Petrov, Semenov and Zainoulline [19], independently, have recently shown that \overline{CH}_{0}(X) = 0 for projective homogeneous varieties X related to groups of types G_{2}, F_{2},^{1}E_{6} over fields of characteristic 0 with trivial Tits algebras.

The notion of X-equivalence used in the paper as the main technical tool is formulated in terms of field extensions and discrete valuations, so that we avoid symmetric power constructions used in [11] to describe closed points. Flexibility of the notion of X-equivalence allows us not to impose any characteristic restrictions on F (except for the trialitarian D_{4} and E_{6}, E_{7} where we assume that characteristic is ≠ 2, 3). Another advantage of our method is its transparency and shortness. Even for those varieties where the result was already known our proofs are simpler.

Finally we remark that most likely our results on algebraic groups of exceptional types are close to optimal. It looks hopeless to weaken restrictions on Tits algebras and prove that CH_{0}(X) is infinite cyclic for larger classes of projective homogeneous varieties of exceptional groups.

At the end of the paper we give two examples of projective homogeneous varieties X with \overline{CH}_{0}(X) ≠ 0 related to algebraic groups of types A_{1}+A_{1}+A_{1} and B_{3} with nontrivial Tits algebras. Note that our first result is minimal possible since \overline{CH}_{0}(X) = 0 for all projective homogeneous varieties X of dimension at most 2 (see Proposition 4.5).

2. Preliminary facts on algebraic groups

2.1. Parabolic subgroups. Let G be a simple simply connected algebraic group over a field F. Fix a maximal torus T ⊂ G over F and a basis Δ of the root system Σ = Σ(G, T) of G with respect to T. Recall that for each subset S ⊂ Δ one can associate the parabolic subgroup P_{S} in G, whose semisimple part is generated by the corresponding root subgroups U_{±α} of G for all roots α ∈ S. It is defined over a separable closure F_{sep} of F and is called the standard parabolic subgroup of type S. Every parabolic subgroup P in G over F_{sep} is conjugate to a unique standard parabolic subgroup P_{S}. We say that P is of type S.

Let X_{S} be the variety of all parabolic subgroups of G of type S. If S is stable with respect to the so-called ∗-action of the Galois group Gal (F_{sep}/F) (see [28]), the variety X_{S} is defined over F. It has a rational point if and only if G contains an F -defined parabolic subgroup of type S. If X_{S}(F) ≠ ∅, then we say that X_{S} is isotropic.

2.2. Tits indices. Recall [28] that given G one can associate a geometric picture called the Tits index of G. It consists of the corresponding Dynkin diagram of G with some vertices circled. The set S_{0} of uncircled vertices corresponds to an F -subgroup of G called a semisimple anisotropic kernel. This subgroup is the semisimple part of the centralizer C_{G}(T_{1}) where T_{1} ⊂ G is a maximal F -split subtorus in G. All maximal F -split subtori are conjugate over F. Hence a semisimple anisotropic kernel of G is defined uniquely up to conjugation.

All vertices in Tits index of G are circled if and only if G is quasi-split over F and none of them is circled if and only if G is F -anisotropic. The last occurs if and only if G contains no nontrivial F -split subtorus.

Let S be stable with respect to the ∗-action of Gal (F_{sep}/F). Then the variety X_{S} of parabolic subgroups in G of type S is defined over F and it has a rational point if and only if S contains the subset S_{0} of all uncircled vertices. Hence an F -defined parabolic subgroup P ⊂ G of type S is minimal if and only if S = S_{0}; in particular all minimal parabolic subgroups in G have the same type and they are conjugate over F.

2.3. Basic types. We say that S is a basic type for G if there is a form H of G over a field extension L/F such that H is L-isotropic and its minimal parabolic subgroups have type S. In other words, a type S is basic if in the classification table [28] of Tits indices there is one whose set of all uncircled vertices coincides with S.

3. Strongly inner forms

One knows that G is a form of a quasi-split simply connected group G_{0}, i.e. there exists a cocycle ξ ∈ Z^{1}(F, Aut G_{0}) such that G is isomorphic over F to the twisted group ^{ξ}G_{0}. One says that G is of the same inner type as G_{0} or an inner form of G_{0} if the class [ξ] is contained in Im [H^{1}(F, \overline{G}_{0}) → H^{1}(F, Aut G_{0})] where \overline{G}_{0} = G_{0}/Z is the corresponding adjoint group and Z ⊂ G_{0} is the center of G_{0}. Otherwise G is an outer form of G_{0}. One says that G is a strongly inner form of G_{0} if [ξ] is contained in Im [H^{1}(F, G_{0}) → H^{1}(F, Aut G_{0})].

For future reference we need some structure facts on strongly inner forms of a quasi-split simple simply connected group G_{0}. In what follows ξ ∈ Z^{1}(F, G_{0}) and G = ^{ξ}G_{0} is the corresponding twisted group.

3.1. Quasi-splitness criterion.

Lemma 3.1. G is quasi-split if and only if [ξ] = 1.

Proof. Let G be quasi-split. Since H^{1}(F, Aut G_{0}) classifies all forms of G_{0},

[ξ] ∈ Ker [H^{1}(F, G_{0}) → H^{1}(F, Aut G_{0})].

The group Aut G_{0} is the semi-direct product of the adjoint group \overline{G}_{0} and the automorphism group Aut D of the Dynkin diagram D of G_{0}. It follows that Aut G_{0}(F) → Aut D(F) is surjective, hence the mapping H^{1}(F, \overline{G}_{0}) → H^{1}(F, Aut G_{0}) has trivial kernel. This implies that [ξ] is contained in the kernel of H^{1}(F, G_{0}) → H^{1}(F, \overline{G}_{0}). Consider now an exact sequence 1 → Z → G_{0} → \overline{G}_{0} → 1. It gives rise to an exact sequence

H^{1}(F, Z) → H^{1}(F, G_{0}) → H^{1}(F, \overline{G}_{0}).

It follows that ξ up to equivalence takes values in Z. As G_{0} is simply connected and quasi-split, it contains a maximal quasi-split torus T ⊂ G_{0}. Since Z ⊂ T and H^{1}(F, T) = 1, by Hilbert Theorem 90, the result follows. □

3.2. Application of Steinberg’s Theorem. Let C_{1} ⊂ G be a maximal F - split torus. The centralizer C_{G}(C_{1}) is a reductive group over F whose central torus C contains C_{1}. Note that if G_{0} is split, we have C = C_{1}. However in general case C ≠ C_{1}. The semisimple part H = [C_{G}(C_{1}), C_{G}(C_{1})] of C_{G}(C_{1}) is an F -anisotropic simply connected group, not necessarily simple. Its Tits index coincides with the subindex of G generated by uncircled vertices. The group C_{G}(C_{1}) is an almost direct product of H and C, i.e H ∩ C is a finite central subgroup of H and C_{G}(S_{1}) = H · C.

Theorem 3.2. Assume that H^{1}(F, C/H ∩C) = 1. Then H is a strongly inner form of a quasi-split simply connected group H_{0} over F.

Proof. Let C_{2} ⊂ H be a maximal torus over F and let T = C·C_{2}. By dimension count, T is a maximal torus in G over F. It easily follows from arguments contained in [25, § 10] (see also [20, Propositions 6.18, 6.19, p. 338–339]) that there is an F -embedding ϕ : T → G_{0} such that the class [ξ] is contained in Im [H^{1}(F, T) → H^{1}(F, G_{0})]. Since C ∩ H = C ∩ C_{2} and H^{1}(F, C/C ∩ H) = 1, the mapping H^{1}(F, C_{2}) → H^{1}(F, T) is surjective. Thus, up to equivalence, we may assume that ξ takes values in C_{2}.

The centralizer C_{G_{0}}(C_{1}) is a reductive F -subgroup in G_{0} and from the construction of ϕ (see the proof of in [20, Proposition 6.18, p. 339]) it follows that C_{G_{0}}(C_{1}) and C_{G}(C_{1}) are isomorphic over F_{sep}. Hence their semisimple parts H_{0} and H are isomorphic over F_{sep}. As all maximal F -split tori in G_{0} are conjugate, C_{G_{0}}(C_{1}) (and hence H_{0} = [C_{G_{0}}(C_{1}), C_{G_{0}}(C_{1})]) is a quasi-split group. By construction H_{0} contains C_{2}, hence ξ can be viewed as an element in Z^{1}(F, H_{0}) and then clearly we have H ≃ ^{ξ}H_{0}. □

3.3. Groups isotropic over a quadratic extension. Assume that F is infinite and perfect. Let L/F be a quadratic extension making G isotropic. We denote the generator of Gal(L/F) by σ. If P ⊂ G is an L-defined standard parabolic subgroup of type S ⊂ Δ we denote by P^{−} the opposite parabolic subgroup (it is generated by root subgroups U_{−α}, α ∈ Δ \ S, and U_{±α}, α ∈ S). The class P of parabolic subgroups conjugate to P is called self-opposite if it contains P^{−}. Clearly, P ∩ P^{−} is a reductive part of P.

Lemma 3.3. If P is self-opposite, then it contains an L-parabolic subgroup Q ∈ P such that Q ∩ σ(Q) is a reductive part of Q.

Proof. See [20, Lemma 17′, p. 383]. □

4. Comparison of two homogeneous varieties

Proposition 4.1. Let X be a scheme over F and Y be a projective homogeneous variety over F such that Y_{F (x)} is isotropic for every x ∈ X. Then the projection f : X × Y → X induces an isomorphism f_{∗} : CH_{0}(X × Y) → CH_{0}(X).

Proof. Consider the spectral sequence [23, §8]

E_{p,q}^{1} = ∐_{x∈X_{(p)}} A_{q}(Y_{F (x)}, K_{p}) ⇒ A_{p+q}(X × Y, K_{0})

for the projection f, where A_{∗}(X × Y, K_{∗}) are the K-homology groups defined in [23]. It gives an exact sequence

E_{1,0}^{1} ⟶^{∂} E_{0,0}^{1} → CH_{0}(X × Y) → 0.

We shall identify the terms E_{n,0}^{1}. It is proven in [3, Th. 7.5] that the Chow motive of the isotropic variety Y_{F (x)} is isomorphic over F (x) to a direct sum of the motive ℤ and twisted motives of the form Z(r) for some projective homogeneous variety Z and r > 0. By the dimension consideration, A_{0}(Z(r), K_{n}) = A_{−r}(Z, K_{n+r}) = 0, hence

A_{0}(Y_{F (x)}, K_{n}) ≃ A_{0}(Spec F (x), K_{n}) = K_{n}F (x)

where the isomorphism is the push-forward map with respect to the structure morphism Y_{F (x)} → Spec F (x). Thus

E_{n,0}^{1} = ∐_{x∈X_{(0)}} K_{n}F (x),

and therefore CH_{0}(X × Y) = Coker(∂) ≃ CH_{0}(X). □

Corollary 4.2. Let X and Y be two projective homogeneous varieties over F such that each of them is isotropic over residue field of every point of another. Then CH_{0}(X) ≃ CH_{0}(Y) and \overline{CH}_{0}(X) ≃ \overline{CH}_{0}(Y).

Taking Y = Spec F in Corollary 4.2 we get

Corollary 4.3. Let X be an isotropic projective homogeneous variety. Then \overline{CH}_{0}(X) = 0.

Let X be a variety over a field F. Consider the Brown-Gersten-Quillen spectral sequence

(4.4)  E_{2}^{p,q} = A^{p}(X, K_{−q}) ⇒ K_{−p−q}(X),

where A^{p}(X, K_{−q}) is the K-cohomology group and K_{n}(X) is the K-group of X (see [21, §7]). If X is projective, the group E_{2}^{0,−1} = A^{0}(X, K_{1}) of regular invertible functions on X is equal to K_{1}(F) = F^{×}. Since the composite K_{1}(F) → K_{1}(X) ⟶^{e} A^{0}(X, K_{1}) is the identity, the edge homomorphism e is surjective. In particular, all the differentials going from E_{∗}^{0,−1} are trivial. If d = dim X ≤ 2, all differentials coming to E_{∗}^{d,−d} are trivial, in particular, the edge homomorphism e′ : CH_{0}(X) → K_{0}(X) is injective.

If X is projective, the degree homomorphism for CH_{0}(X) is the composition

CH_{0}(X) ⟶^{e′} K_{0}(X) ⟶^{f_{∗}} K_{0}(Spec F) = ℤ,

where the f_{∗} is the push-forward homomorphism with respect to the structure morphism f : X → Spec(F). Therefore, Ker(e′) ⊂ \overline{CH}_{0}(X). If X is a projective homogeneous variety, the group K_{0}(X) is torsion free (see [16]), hence \overline{CH}_{0}(X) ⊂ Ker(e′) since \overline{CH}_{0}(X) is a torsion group. Thus \overline{CH}_{0}(X) = Ker(e′). In particular we have proven

Proposition 4.5. Let X be a projective homogeneous variety of dimension at most 2. Then \overline{CH}_{0}(X) = 0.

4.1. Reduction to a basic type. Let G be a simple simply connected group over F. We fix a maximal torus T ⊂ G over F, the corresponding root system Σ = Σ(G, T), a basis Δ ⊂ Σ and a subset S ⊂ Δ. Consider the variety X_{S} of parabolic subgroups in G of type S. Let L = F (X_{S}) be the field of rational functions of X_{S}. Since X_{S}(L) ≠ ∅, there is an L-defined parabolic subgroup P in G of type S. It contains a minimal parabolic subgroup P′ ⊂ P of G over L whose type S′ is a subset in S.

Proposition 4.6. There are canonical isomorphisms CH_{0}(X_{S}) ≃ CH_{0}(X_{S′}) and \overline{CH}_{0}(X_{S}) ≃ \overline{CH}_{0}(X_{S′}).

Proof. We show that X_{S} and X_{S′} satisfy conditions of Corollary 4.2. As S′ ⊂ S, there is a natural map f : X_{S′} → X_{S} over F. Therefore X_{S} is isotropic over F (x′) for every x′ ∈ X_{S′}.

Conversely, let x ∈ X_{S} and set E = F (x). The set S′ consists of all uncircled vertices of the Tits index of G over F (X). Therefore S′ contains all uncircled vertices of the Tits index of G over E(X). Since E(X)/E purely transcendental, and the Tits index does not change under such extensions, the set S′ contains all uncircled vertices of the Tits index of G over E, i.e., X_{S′}(E) ≠ ∅. □

Remark 4.7. The proposition allows us to consider basic subsets S ⊂ Δ only while computing CH_{0}(X_{S}). In fact we proved even more: it is sufficient to compute CH_{0}(X_{S}) for those basic subsets S that appear as sets of uncircled vertices of the Tits indices of G over all field extensions L/F. Thus we can narrow the set of basic types by considering Tits indices of strongly inner forms of G_{0} only.

5. Special fields

Let p be a prime integer. A field F is called p-special if the degree of every finite field extension of F is a power of p. We say that F is special if F is p-special for some p. The following properties of special fields follow directly from the definition.

(1) An algebraic extension of a p-special field is p-special.

(2) For every field F and every prime p there is an algebraic field extension F_{p}/F such that the field F_{p} is p-special and the degree of any finite sub-extension L/F of F_{p}/F is prime to p.

If X is a scheme over a p-special field F, then n(X) is a power of p and there is a closed point x ∈ X such that deg x = n(X).

The following statement reduces the problem of computing CH_{0}(X) to the case of schemes over p-special fields.

Lemma 5.1. Let X be a proper scheme over F. Assume that for any p-special field extension L/F, the group \overline{CH}_{0}(X_{L}) is trivial. Then \overline{CH}_{0}(X) = 0.

Proof. Let a ∈ \overline{CH}_{0}(X) and fix a prime integer p. As the image of a in \overline{CH}_{0}(X_{F_{p}}) is trivial, a is trivial over some finite sub-extension L/F in F_{p}/F. Applying the push-forward homomorphism with respect to the morphism X_{L} → X, we get [L : F] · a = 0. In particular, the order of a is not divisible by p. Since this holds for every prime p, we have a = 0. □

The following statement enables us to consider only closed points of minimal degree.

Lemma 5.2. Let X be a proper scheme over a p-special field F. Assume that for any finite field extension L/F, every two closed points of X_{L} of the smallest degree n(X_{L}) are rationally equivalent. Then \overline{CH}_{0}(X) = 0.

Proof. Let x be a closed point of X of the smallest degree p^{a} = n(X) and let y be a closed point of X of degree p^{b}. We claim that [y] = p^{b−a}[x] in CH_{0}(X).

We shall prove the claim by induction on p^{b} = deg(y). We may assume that p^{b} > p^{a}. Choose an intermediate field F ⊂ L ⊂ F (y) such that [L : F] = p and a point y′ ∈ X_{L} over y of degree p^{b−1}. Note that n(X_{L}) divides n(X) and n(X) divides p · n(X_{L}). Therefore we have the following two cases.

Case 1: n(X_{L}) = n(X) = p^{a}. Then there is a point x′ ∈ X_{L} over x of degree p^{a}. By the induction hypothesis applied to X_{L}, [y′] = p^{b−1−a}[x′] in CH_{0}(X_{L}). Taking the push-forward homomorphism with respect to f : X_{L} → X we get

[y] = f_{∗}([y′]) = p^{b−1−a}f_{∗}([x′]) = p^{b−1−a} · p [x] = p^{b−a}[x].

Case 2: n(X_{L}) = p^{a−1}. Choose a closed point z′ ∈ X_{L} of degree p^{a−1} and denote by z its image in X. The degree of z is p^{a}, in particular, [z] = [x] in CH_{0}(X). By the induction hypothesis applied to X_{L}, [y′] = p^{(b−1)−(a−1)}[z′] in CH_{0}(X_{L}). Therefore

[y] = f_{∗}([y′]) = p^{(b−1)−(a−1)}f_{∗}([z′]) = p^{b−a}[z] = p^{b−a}[x].

The claim is proven. It follows that every 0-cycle α on X is equivalent to m[x] for some m ∈ ℤ. If deg α = 0, then m = 0, i.e., α = 0 in CH_{0}(X). □

6. Equivalences of field extensions

6.1. X-equivalence. Let F be a field and let A be the class of all field extensions of F. For any subclass B ⊂ A and every field L ∈ A we write B_{L} for the subclass of B of all fields containing L. Let n(B) denote the greatest common divisor of the degrees [E : F] over all fields E ∈ B of finite degree over F. (We set n(B) = 0 if B has no finite degree extensions.) In particular, n(A) = 1. If F is p-special then n(B) is a power of p or 0 and if n(B) > 0 then there is a field E ∈ B such that [E : F] = n(B).

For every a ∈ F, let v_{a} be the discrete valuation of the rational function field F (t) over F corresponding to the irreducible polynomial f(t) = t − a.

Let B ⊂ A be a subclass of field extensions of F. Two fields L and L′ in B of the same degree n over F are called simply B-equivalent if there is a degree n field extension E/F (t) such that E ∈ B and two discrete valuations of E over v_{0} and v_{1} with residue fields isomorphic to L and L′ respectively over F. Two fields L and L′ in B are called B-equivalent if there is a chain of fields L_{0} = L, L_{1}, …, L_{r} = L′ in B such that L_{i} is simply B-equivalent to L_{i+1} for all i = 0, 1, …, r − 1.

Note that if two fields L and L′ in B contain a common subfield K ∈ A, i.e., L, L′ ∈ B_{K}, then L ∼_{B_{K}} L′ implies L ∼_{B} L′.

Example 6.1. Let X be a scheme of finite type over F. Denote by A(X) the class of all fields E ∈ A such that X(E) ≠ ∅. The number n(A(X)) coincides with n(X), and A(X)-equivalent fields will be called X-equivalent.

Example 6.2. In the notation of Proposition 4.6 the classes A(X_{S}) and A(X_{S′}) coincide.

We shall need the following

Lemma 6.3. Let X be a proper scheme of finite type over F such that \overline{CH}_{0}(X_{L}) = 0 for any field L ∈ A(X) (e.g., X is a projective homogeneous variety by Corollary 4.3). Let x and x′ be two closed points of X such that residue fields F (x) and F (x′) are isomorphic over F. Then [x] = [x′] in CH_{0}(X).

Proof. Let L = F (x). By assumption there are rational points y and y′ of X_{L} over x and x′ respectively. As [y] = [y′] in CH_{0}(X_{L}), taking the push-forward homomorphism with respect to X_{L} → X we get [x] = f_{∗}([y]) = f_{∗}([y′]) = [x′]. □

Lemma 6.4. Let X be a proper scheme of finite type over F such that \overline{CH}_{0}(X_{L}) = 0 for any field L ∈ A(X). Let x and x′ be two rational points of X of degree n(X) such that residue fields F (x) and F (x′) are X-equivalent. Then [x] = [x′] in CH_{0}(X).

Proof. We may assume that F (x) and F (x′) are simply X-equivalent. Let E be the field extension of F (t) of degree n = n(X) in the definition of the simple equivalence between F (x) and F (x′). We have the dominant morphism Spec E → Spec F (t) → ℙ_{F}^{1} . Since E ∈ A(X), there is a morphism Spec E → X over F. Denote by C the closure of the image of the diagonal morphism Spec E → X × ℙ_{F}^{1} . Thus C is an integral proper curve with a surjective morphism f : C → ℙ_{F}^{1} . By assumption there are two discrete valuations v and v′ of E over the valuations v_{0} and v_{1} of F (t) respectively. Let v and v′ dominate points c and c′ on C respectively, so that f(c) = 0 and f(c′) = 1. Note that the residue field of c is isomorphic to a subfield of the residue field of v, which in its turn is isomorphic to the field F (x) of degree n over F. On the other hand, C ⊂ X × ℙ_{F}^{1} , hence n(C) ≥ n(X) = n. Therefore, c is a closed point of degree n, f^{−1}(0) = Spec F (c) and F (c) ≃ F (x). Similarly, f^{−1}(1) = Spec F (c′) and F (c′) ≃ F (x′).

The morphism f is flat, hence [c] = f^{∗}([0]) = f^{∗}([1]) = [c′] in CH_{0}(C). Consider the projection g : C → X. Let z = g(c) and z′ = g(c′). As F (z) is isomorphic to a subfield of F (c) and n(X) = n, we have F (z) ≃ F (c) ≃ F (x). Similarly F (z′) ≃ F (c′) ≃ F (x′). Therefore, [z] = g_{∗}([c]) = g_{∗}([c′]) = [z′]. By Lemma 6.3, [x] = [z] = [z′] = [x′]. □

A class B ⊂ A is called connected if for any special field E ∈ A every two fields in B_{E} of degree n(B_{E}) over E are B_{E}-equivalent. For example, the whole class A is connected, and therefore for every X with X(F) ≠ 0 the class A(X) = A is connected.

Theorem 6.5. Let X be a proper scheme of finite type over F such that \overline{CH}_{0}(X_{L}) = 0 for any field L ∈ A(X). Suppose that the class A(X) is connected. Then \overline{CH}_{0}(X) = 0.

Proof. By Lemma 5.1 we may assume that F is p-special for some prime p. It follows from Lemma 6.4 that for any finite field extension L/F, every two closed point of X_{L} of degree n(X_{L}) are rationally equivalent. By Lemma 5.2, \overline{CH}_{0}(X) = 0. □

7. Severi-Brauer varieties

Let A be a central simple algebra of degree m over a field F and let X be the corresponding Severi-Brauer variety of right ideals of A of dimension deg(A) (see [10, §1]). The variety X is projective homogeneous of the group SL_{1}(A) of type S = {α_{2}, …, α_{m−1}} where simple roots α_{1}, …, α_{m−1} of the Dynkin diagram A_{m−1} are numbered as follows:

[figure: …; α_{1}; α_{2}; α_{m−1}]

Here m − 1 is the rank of SL_{1}(A). A field extension L/F belongs to A(X) if and only if the algebra A_{L} = A ⊗_{F} L is split. In particular n(X) = ind(A). If A is split, i.e., A = End(V) for some vector space V, the Severi-Brauer variety X is the projective space ℙ(V).

Let D be a (unique up to isomorphism) central division algebra F -algebra Brauer equivalent to A. Every maximal subfield of D has degree n = n(X) over F and it is a splitting field for D and A and it belongs to A(X). Conversely, every field in A(X) of the smallest degree n is isomorphic over F to a maximal subfield of D [6, §4.4].

For every element d ∈ D denote by Prd_{d} = Prd_{d}(s) ∈ F [s] the reduced characteristic polynomial of d. The polynomial Prd_{d} is monic of degree n and Prd_{d} is irreducible if and only if d generates a maximal subfield of D. In this case F (d) ≃ F [s]/(Prd_{d}).

Recall that a finite field extension L/F is called simple if L is generated over F by one element. In particular every separable finite extension is simple.

Lemma 7.1. Let L ⊂ D be a simple maximal subfield and let d ∈ D. Then there is a maximal subfield K ⊂ D containing d such that L ∼_{X} K.

Proof. We have L = F (a) for some a ∈ D. In particular the polynomial Prd_{a} is irreducible. Consider the element c = td + (1 − t)a ∈ D[t] and its reduced characteristic polynomial Prd_{c} ∈ F [s, t]. Note that Prd_{c}(s, 0) = Prd_{a} and Prd_{c}(s, 1) = Prd_{d}. In particular Prd_{c} is irreducible.

Set R = F [s, t]/(Prd_{c}). Since Prd_{c} is irreducible, R is a domain. The quotient field E of R is the maximal subfield of D_{F (t)} generated by c and therefore belongs to A(X). Let Q be the principal ideal of R generated by t. The factor ring R/Q is isomorphic to F [s]/(Prd_{a}) ≃ L. In particular, Q is a maximal ideal of R. As the localization R_{Q} is a Noetherian local domain of dimension one with principal maximal ideal, R_{Q} is a discrete valuation ring. The corresponding discrete valuation of E with residue field L restricts to v_{0} on F (t).

Let v be a discrete valuation of E restricting to v_{1}. Since R is integral over F [t], R is contained in the valuation ring O of v. As R/(1 − t) = F [s]/(Prd_{d}) and Prd_{d} is a power of the minimal polynomial h = h(s) of d, the intersection of R with the maximal ideal of O is generated by 1 − t and h. In particular residue field K of O contains F [s]/(h) = F (d).

On the other hand, since X is proper, a morphism Spec E → X extends to Spec O → X. Therefore K contains F (x) for some x ∈ X. But [K : F] ≤ [E : F (t)] = n, therefore K = F (x) and [K : F] = n. Thus K is isomorphic to a maximal subfield of D containing d and L ∼_{X} K. □

Theorem 7.2. Let A be a central simple F -algebra and let X be the Severi-Brauer variety SB(A). Then the class A(X) is connected.

Proof. It is sufficient to show that every two maximal subfields of D are Xequivalent. We proceed by induction on deg D and may assume that deg(D) > 1. Choose a separable maximal subfield L of D. It is sufficient to show that L is X-equivalent to any maximal subfield E ⊂ D. Choose an element d ∈ E \ F. By Lemma 7.1, L ∼_{X} K for some maximal subfield K containing d. Let M = F (d) and consider the centralizer D′ of M in D. The fields E and K are maximal subfields of D′. By the induction hypothesis, K ∼_{X_{M}} E and therefore, K ∼_{X} E. Finally, L ∼_{X} K ∼_{X} E. □

Theorem 6.5 yields

Corollary 7.3. (Cf., [15]) Let X be a Severi-Brauer variety. Then \overline{CH}_{0}(X) = 0.

8. Generalized Severi-Brauer varieties

Let A be a central simple F -algebra. Denote by X = SB(2, A) the generalized Severi-Brauer variety of right ideals of A of dimension 2 deg(A) (see [10, §1]). It is a projective homogeneous variety of the algebraic group SL_{1}(A) of type S = {α_{1}, α_{3}, …, α_{m}} where simple roots are numbered as in Section 7. A field extension L/F belongs to A(X) if and only if ind A_{L} ≤ 2. If A is split, i.e., A = End(V) for some vector space V, the variety SB(2, A) is the Grassmannian variety of planes in V.

If index of A is odd, then a field extension L/F belongs to A(X) if and only if L splits A. Therefore the class A(X) coincides with A(SB(A)). Hence the class A(X) is connected, by Theorem 7.2.

Suppose now that ind A = 2n is even. We have n(X) = n. Let D be a central division F -algebra of degree 2n Brauer equivalent to A. In particular A(X) = A(SB(2, D)). Every finite extension of F of degree n in A(X) is isomorphic to a subfield of D and conversely every subfield of D of degree n is contained in A(X).

Suppose that D has a symplectic involution σ. Denote by Symd(D, σ) the subspace of D of all elements d+σ(d) for d ∈ D. The characteristic polynomial of an element a ∈ Symd(D, σ) is the square of the monic pfaffian characteristic polynomial Prp_{a}(s) of degree n [10, Prop. 2.9]. In particular, [F (a) : F] ≤ n, moreover the polynomial Prp_{a}(s) is irreducible if and only if [F (a) : F] = n.

Lemma 8.1. For any subfield K of D of degree n there is a symplectic involution τ of D such that K ⊂ Symd(D, τ).

Proof. The centralizer Q of K in D is a quaternion algebra over K. By [10, Th. 4.14], the canonical involution of Q can be extended to a symplectic involution τ on D. Choose an element y ∈ Q such that y+τ(y) = 1. Then x = xy+τ(xy) for every x ∈ K, i.e., K ⊂ Symd(D, τ). □

Lemma 8.2. Let σ and τ be two distinct symplectic involutions on D and let L be a simple subfield of Symd(D, σ) of degree n. Then

(1) There exists a subfield K of D of degree n such that L ∼_{X} K and K ∩ Symd(D, τ) contains a non-constant element.

(2) For every d ∈ Symd(D, σ) there exists a subfield K of D of degree n such that d ∈ K and L ∼_{X} K.

Proof. Let L = F (a) for some a ∈ Symd(D, σ).

(1) Choose b ∈ D^{×} such that σ = Int(b) ∘ τ [10, Prop. 2.7]. We have ba ∈ Symd(D, τ). Replacing a by a + 1 if necessary we assume that ba ≠ F.

Consider the element c = t + (1 − t)b ∈ D[t] and the involution σ_{t} = Int(c) ∘ σ_{F (t)} on D_{F (t)}. Then ca ∈ Symd(D_{F (t)}, σ_{t}). The pfaffian characteristic polynomial Prp_{ca} belongs to F [s, t]. We have Prp_{ca}(s, 0) = Prp_{ba} and Prp_{ca}(s, 1) = Prp_{a}, in particular Prp_{ca} is irreducible. Considering the quotient field E of the factor ring F [s, t]/(Prp_{ca}) as in the proof of Lemma 7.1 we find two discrete valuations of E over v_{1} and v_{0} respectively with residue fields isomorphic to L and a subfield K of D containing the nonconstant element ba. Thus L ∼_{X} K.

(2) Let E be the degree n subfield of D_{F (t)} generated by ta + (1 − t)d. As in the proof of Lemma 7.1 we see that there are two discrete valuations of E with residue fields L = F (a) and K respectively, where K is a subfield of D of degree n and d ∈ K. □

Theorem 8.3. Let X be the generalized Severi-Brauer variety SB(2, A) of a central simple algebra A of exponent not divisible by 4. Then the class A(X) is connected.

Proof. By definition of connectedness we may assume that F is a p-special field. If p is odd and we already know that the class A(X) is connected. Thus we may assume that p = 2, exp(A) = 2 and ind(A) = 2n is even. By [10, Th. 3.8], the algebra A has a symplectic involution.

We use the notation above. It is sufficient to show that every two subfields of D of degree n are X-equivalent.

Let L and E be subfields of D of degree n. We may assume that L/F is separable (see [29, Prop. 5]) and therefore is simple. We prove that L ∼_{X} E by induction on n. By Lemma 8.1, we can choose symplectic involutions σ and τ of D such that L ⊂ Symd(A, σ) and E ⊂ Symd(A, τ).

Case 1: σ = τ. Choose an element d ∈ E \ F. By Lemma 8.2(2), there is a subfield K of D of degree n containing d such that L ∼_{X} K. Let D′ be the centralizer of M = F (d) in D. By the induction hypothesis applied to D′ with the involution τ|_{D′} we have K ∼_{X_{M}} E, therefore K ∼_{X} E and L ∼_{X} K ∼_{X} E.

Case 2: σ ≠ τ. By Lemma 8.2(1), there exists a subfield K of D of degree n such that L ∼_{X} K and K ∩ Symd(D, τ) contains a non-constant element d. Choose a subfield H ⊂ Symd(D, τ) of degree n containing d. Let D′ be the centralizer of M = F (d) in D. By the induction hypothesis applied to D′, K ∼_{X_{M}} H and therefore K ∼_{X} H. By the first case, H ∼_{X} E, hence L ∼_{X} K ∼_{X} H ∼_{X} E. □

Corollary 8.4. (Cf., [11]) Let X be the generalized Severi-Brauer variety SB(2, A) of a central simple algebra A of exponent not divisible by 4. Then \overline{CH}_{0}(X) = 0.

9. Quadrics

Let q be a quadratic form on an F -vector space V of finite dimension. Denote by X the projective quadric hypersurface in ℙ(V) defined by the equation q = 0. If X is smooth, it is a projective homogeneous variety of the special orthogonal group O^{+}(q) of type S = {α_{2}, …, α_{m}} where simple roots α_{1}, … α_{m} of the Dynkin diagrams D_{m} and B_{m} are numbered as follows:

[figure: …; α_{1}; α_{2}; α_{m−2}; α_{m−1}; α_{m}; …; >; α_{1}; α_{2}; α_{m−1}; α_{m}]

Here the first and second diagrams correspond to the cases dim V = 2m and dim V = 2m + 1 respectively.

The quadric X has a rational point if and only if q is isotropic. If q is anisotropic, then n(X) = 2 by Springer’s theorem.

Example 9.1. Let q be a nonzero 2-dimensional quadratic form. Then we have dim X = 0, moreover X = Spec C_{0}(q), where C_{0}(q) is the even Clifford algebra of q. The form q is non-degenerate (resp. anisotropic) if and only if C_{0}(q) is an étale quadratic F -algebra (resp. a quadratic field extension of F).

Theorem 9.2. The class A(X) for a projective quadric X is connected.

Proof. We may assume that q is anisotropic. Let L/F be a quadratic field extension making q isotropic and let x ∈ X be a point with F (x) ≃ L. Let {1, α} be a basis of L over F and let w = u + αv, u, v ∈ V, be an isotropic vector in V_{L}. Then w ∈ U_{L}, where U is the 2-dimensional subspace of V spanned by v and u. Moreover, by Example 9.1, F (x) ≃ C_{0}(q|_{U}).

Conversely, if U ⊂ V is a 2-dimensional subspace, then

X ∩ ℙ(U) = Spec F (x) ≃ C_{0}(q|_{U}),

where x is a closed point of degree 2. Thus it is sufficient to show that for every two 2-dimensional subspaces U and U′, the quadratic field extensions L = C_{0}(q|_{U}) and L′ = C_{0}(q|_{U′}) are X-equivalent. We may assume that U ∩ U′ ≠ 0.

Let {v, u} and {v, u′} be bases of U and U′ respectively. Consider the F [t]- submodule W of V_{F [t]} generated by v and tu + (1 − t)u′. Let R = C_{0}(q_{F [t]}|_{W}). The quotient field E of R belongs to A(X). Since R/(t) = L′ is a field, the principal ideal Q generated by t is maximal. Therefore the localization R_{Q} is a Noetherian local domain of dimension one with principal maximal ideal, i.e., R_{Q} is a discrete valuation ring. The corresponding discrete valuation of E with residue field L′ restricts to v_{0} on F (t). Similarly, there is a discrete valuation of E with residue field L restricting to v_{1}, i.e., L and L′ are simply X-equivalent. □

Corollary 9.3. (Cf., [8], [27]) Let X be a smooth projective quadric. Then \overline{CH}_{0}(X) = 0.

10. Involution varieties

Let A be a central simple algebra of degree 2m with a quadratic pair (σ, f), where σ is an involution on A of the first kind and f : Sym(A, σ) → F is a linear map satisfying certain properties [10, §5.B]. If A is split, i.e., A ≃ End(V) for a 2n-dimensional vector space V over F, the space Sym(A, σ) is identified with the subspace of symmetric elements in V ⊗_{F} V and f(v ⊗ v) = q(v) for a non-degenerate quadratic form q on V.

Consider the morphism

φ : SB(A) → ℙ(Sym(A, σ))

taking an ideal I in A to the (1-dimensional) subspace I · σ(I) of Sym(A, σ) [10, p. 122]. Let W = Ker(f). The inverse image X = φ^{−1}ℙ(W) is called the involution variety of the quadratic pair. Thus X is a hypersurface in the Severi-Brauer variety SB(A). In the split case as above, X is a smooth projective quadric hypersurface in ℙ(V) given by the quadratic form q. In other words, X is a twisted form of a projective quadric. It is a projective homogeneous variety of the special orthogonal group O^{+}(A, σ, f) [10, §26] of type S = {α_{2}, …, α_{m}}, where S is the subset of vertices of the Dynkin diagram D_{m}. In fact, any twisted form of a smooth projective quadric is isomorphic to an involution variety.

If A is not split, then ind A is even. Consider the case ind(A) = 2. Let Q be a quaternion division algebra Brauer equivalent to A. Then A ≃ End_{Q}(V) for some right Q-module V. The Severi-Brauer variety SB(A) can be identified with the variety ℙ_{Q}(V) of reduced rank 1 Q-submodules in V. Note that for any Q-submodule U ⊂ V of rank 1, the subvariety ℙ_{Q}(U) of ℙ_{Q}(V) is a conic curve. Then either the conic ℙ_{Q}(U) is contained in X or the intersection of ℙ_{Q}(U) with X is a point of degree 2. Therefore n(X) = 2.

Theorem 10.1. If ind(A) = 2 then the class A(X) is connected.

Proof. As n(X) = 2 it is sufficient to show that every two closed points x and x′ of degree 2 in X are X-equivalent. By [13, Lemma 1], there are Q-submodules U ⊂ V and U′ ⊂ V of rank 1 such that x ∈ ℙ_{Q}(U) and x′ ∈ ℙ_{Q}(U′). In fact F (x) and F (x′) are isomorphic to the even Clifford algebras of the restrictions of the quadratic pair to U and U′ respectively. Let v and v′ be nonzero vectors of U and U′ respectively. Consider w = tv+(1−t)v′ ∈ V_{F [t]} and let W = Q[t]w. Let R be the even Clifford algebra of the restriction of the quadratic pair (σ, f)⊗F [t] on W. As in the proof of Theorem 9.2 we see that the quotient field E of R is a quadratic extension of F (t) equipped with two discrete valuations over v_{0} and v_{1} respectively and residue fields isomorphic to F (x) and F (x′), i.e., F (x) and F (x′) are simply X-equivalent. □

Consider the general case when ind(A) = 2n. Assume that F is a 2-special field. We claim that n(X) = 2n. Indeed, n(X) ≥ 2n since X is a hypersurface in SB(A) and n(SB(A)) = 2n. On the other hand, choose a field extension K/F of degree n such that ind A_{K} = 2. As n(X_{K}) = 2, we conclude that n(X) ≤ 2n.

Set Y = SB(2, A). We know that n(Y) = n.

Lemma 10.2. Let y_{0} and y_{1} be two closed points of Y of degree n such that F (y_{0}) and F (y_{1}) are simply Y -equivalent. Then there exist points x_{0} and x_{1} of X of degree 2n such that F (x_{0}) and F (x_{1}) are simply X-equivalent and F (y_{i}) is isomorphic to a subfield of F (x_{i}) for i = 0, 1.

Proof. Let E/F (t) be an extension of degree n such that Y (E) ≠ ∅ and let w_{i}, i = 0, 1, be two discrete valuations of E over the valuations v_{i} of F (t) with F (w_{i}) ≃ F (y_{i}).

Since ind A_{E} = 2, there is a quadratic field extension K/E such that X(K) ≠ ∅. Let u_{i} be a discrete valuation of K extending w_{i} and let L_{i} be the residue field of u_{i}. As X is projective, a morphism Spec K → X factors through a morphism Spec O_{i} → X, where O_{i} is the valuation ring of u_{i}. In particular X(L_{i}) ≠ ∅, therefore

2n = [K : F (t)] ≥ [L_{i} : F] ≥ n(X) = 2n.

Hence there is a closed point x_{i} ∈ X of degree 2n with F (x_{i}) ≃ L_{i}. Note that L_{0} and L_{1} are simply X-equivalent and F (y_{i}) ≃ F (w_{i}) ⊂ L_{i}. □

Theorem 10.3. Let A be a central simple algebra of even degree with a quadratic pair and let X be the corresponding involution variety. Then the class A(X) is connected.

Proof. We may assume that F is p-special. If p ≠ 2, then X(F) ≠ ∅ and the class A(X) is connected. Let F be a 2-special field. If A splits, then X is a smooth projective quadric, and the result follows from Theorem 9.2. Thus we may assume that ind A = 2n for some n ≥ 1.

Let x and x′ be two closed points of X of degree 2n. We would like to show that F (x) and F (x′) are X-equivalent. As F is 2-special, we can choose closed points y and y′ ∈ Y of degree n such that F (y) ⊂ F (x) and F (y′) ⊂ F (x′). By Theorem 8.3, F (y) and F (y′) are Y -equivalent, i.e., there are closed points y_{0} = y, y_{1}, …, y_{r} = y′ of degree n such that F (y_{i}) and F (y_{i+1}) are simply Y - equivalent for i = 0, …, r − 1. By Lemma 10.2 applied to each pair (y_{i}, y_{i+1}) there are closed points x′_{i}, x_{i+1} of X of degree 2n such that F (x′_{i}) and F (x_{i+1}) are simply X-equivalent and F (y_{i}) ⊂ F (x′_{i}), F (y_{i+1}) ⊂ F (x_{i+1}).

Set for convenience x_{0} = x and x′_{r} = x′. For every i = 0, …, r, the fields F (x_{i}) and F (x′_{i}) contain F (y_{i}). The index of A over F (y_{i}) is equal to 2. By Theorem 10.1, the fields F (x_{i}) and F (x′_{i}) are X_{F (y_{i})}-equivalent, and therefore are X-equivalent. Finally we have

F (x) = F (x_{0}) ∼ F (x′_{0}) ∼ F (x_{1}) ∼ F (x′_{1}) ∼ ⋯ ∼ F (x_{r}) ∼ F (x′_{r}) = F (x′). □

Corollary 10.4. (Cf., [11]) Let A be a central simple algebra of even degree with a quadratic pair and let X be the corresponding involution variety. Then \overline{CH}_{0}(X) = 0.

11. Cayley-Dickson and Jordan algebras

Let C be a Cayley-Dickson algebra over F. Denote by N the norm form of C given by N(x) = xx̄, where x ↦ x̄ is the canonical involution of C. It is known [10, Prop. 33.18, Th. 33.19] that N is a 3-fold Pfister form, and C is split if and only if N is split. Therefore, Theorem 9.2 applied to the quadric given by N yields

Theorem 11.1. The class of splitting fields of C is connected.

Remark 11.2. One can give a direct proof of Theorem 11.1 along the lines of the proof of Lemma 7.1 using the fact that a quadratic field extension of L/F splits C if and only if L can be embedded into C.

Let J be an exceptional simple 27-dimensional Jordan algebra over a field F of degree 3 arising from the first Tits construction [7, Ch. IX, §12]. Recall that there is an associated central simple F -algebra A of degree 3 and an element a ∈ F^{×}. The algebra J is split if and only if a is a reduced norm for A.

Theorem 11.3. Let J be an exceptional simple 27-dimensional Jordan algebra over a 3-special field F of degree 3 arising from the first Tits construction. Then the class of all splitting fields of J is connected.

Proof. We may assume that J is not split. Therefore for the class B of all splitting fields of J we have n(B) = 3. Let L and L′ be two cubic extensions of F splitting J. We shall show that L ∼_{B} L′. By [17, Cor. 3] we may assume that L and L′ are subfields of J. Choose generators x and x′ of L and L′ over F respectively and let E be the subfield of J_{F (t)} generated by tx + (1 − t)x′. As in the proof of Lemma 7.1 one shows using the cubic minimal polynomials of x and x′ that there are two discrete valuations of E over v_{0} and v_{1} with residue fields L and L′ respectively, i.e., L and L′ are simply B-equivalent. □

12. Groups of exceptional types

In what follows G_{0} denotes split or quasi-split simple simply connected algebraic group over F of one of the following types: G_{2}, F_{4}, D_{4}, E_{6}, E_{7}. Let ξ ∈ Z^{1}(F, G_{0}) be a cocycle and let G = ^{ξ}G_{0} be the corresponding strongly inner form of G_{0}.

Theorem 12.1. Let X = X_{S} be a projective homogeneous variety of the group G as above. If G is an outer form of type E_{6} of of type E_{7} we assume that S ≠ ∅. If G is a trialitarian group or of types E_{6}, E_{7} we assume that char(F) ≠ 2, 3. Then the class A(X) is connected. In particular, we have \overline{CH}_{0}(X) = 0.

If S = Δ, then X_{S} = Spec F and there is nothing to prove. Thus, by Proposition 4.6 we may assume that S is a proper basic subset of Δ. By Lemma 5.1, we may also assume that F is p-special.

If p ≠ 2, 3, then we have H^{1}(F, G_{0}) = 1 for all types under the consideration. Indeed, by Steinberg’s theorem [25], ξ is equivalent to a cocycle with coefficients in a maximal torus T ⊂ G_{0} and, by [20, Proposition 6.21, p. 375], we have H^{1}(F, T) = 1 for p-special fields with p ≠ 2, 3. It follows that G is split or quasi-split and hence X_{S}(F) ≠ ∅. Thus we may assume that p = 2, 3.

To finish the proof we shall either show directly that A(X) is connected or construct a projective homogeneous variety Y (under some algebraic simple group) such that

(i) The class A(Y) is connected; (ii) A(X_{S}) = A(Y).

We shall construct Y using case-by-case consideration.

13. Types G_{2} and F_{4}

By [28], each isotropic group of type G_{2} is split. Hence the only basic type for G is S = ∅. Since X_{∅} is a variety of Borel subgroups of G, the class A(X_{∅}) consists of all field extensions making G split.

Recall that the correspondence C ↦ G = Aut(C) gives rise to a bijection between the set of isomorphism classes of Cayley-Dickson algebras and the set of isomorphism classes of simple groups of type G_{2} [10, Theorem 26.19]. In particular the classes of splitting fields for C and G coincide. The connectedness of A(X_{∅}) follows from Theorem 11.1.

We pass to type F_{4}. According to [28], if G is isotropic over a field extension L/F but not L-split, then its Tits index is of the form

[figure: >; α_{1}; α_{2}; α_{3}; α_{4}]

It follows that there exist two basic types only:

S_{1} = ∅ and S_{2} = {α_{1}, α_{2}, α_{3}}.

The class A(X_{S_{1}}) consists of all field extensions L/F making G split and A(X_{S_{2}}) consists of all field extensions L/F making G isotropic.

The correspondence J ↦ G = Aut(J) gives rise to a bijection between the set of isomorphism classes of exceptional simple 27-dimensional Jordan algebra of degree 3 and the set of isomorphism classes of simple groups of type F_{4} [10, Theorem 26.18]. In particular the classes of splitting fields for J and G coincide.

If p = 3, then J arises from the first Tits construction (cf. [7, Ch. IX, §12]). We claim that the diagram above cannot appear as the Tits index of G over any extension L/F of F. Indeed, assume the contrary. Since G is not split over L, so is J. The anisotropic kernel of G over L has type B_{3}, hence there is a finite extension E/L of degree [E : L] = 2^{a} for some positive integer a making G (and hence J) split. On the other side, the extension E/L does not split J since it arises from the first Tits construction – a contradiction. Thus we showed that the classes A(X_{S_{1}}), A(X_{S_{2}}) and the class of all splitting fields of J coincide and we are done by Theorem 11.3.

If p = 2, then J is reduced. Recall (cf. [4], [18]) that given J one can associate a 3-Pfister form f_{3}(J) and a 5-Pfister form f_{5}(J) with the following properties (cf. [26], [22]):

(a) The group G is split over a field extension L/F if and only if f_{3}(J) is hyperbolic over L;

(b) The group G is isotropic over a field extension L/F if and only if f_{5}(J) is hyperbolic over L.

Let Y_{1} and Y_{2} be the projective quadric defined by the equations f_{3}(J) = 0 and f_{5}(J) = 0 respectively. By (a) and (b), we have A(X_{S_{1}}) = A(Y_{1}) and A(X_{S_{2}}) = A(Y_{2}). The connectedness of A(X_{S_{1}}) and A(X_{S_{2}}) follows from Theorem 9.2.

14. Type D_{4}

Suppose first that G is anisotropic of type ^{1}D_{4} or ^{2}D_{4}. Then G ≃ Spin(f), where f is an 8-dimensional quadratic form with trivial even Clifford algebra. Let dF^{×2} be the discriminant of f. The form f ⊥ ⟨1, −d⟩ has trivial discriminant and Clifford algebra, therefore it is represented by a 3-Pfister form g in the Witt ring W (F).

As g can have Witt index 0 or 4 over a field extension L/F, it follows that if f becomes isotropic over L/F, then its Witt index over L is either 1 or f is quasi-split (i.e., its Witt index is either 3 or 4). Hence for the Tits index of G over L there are the following possibilities only:

[figure: α_{1}; α_{2}; α_{3}; α_{4}; α_{1}; α_{2}; α_{3}; α_{4}; α_{1}; α_{2}; α_{3}; α_{4}]

By Remark 4.7, we need to consider the following basic types only:

S_{1} = {α_{2}, α_{3}, α_{4}} and S_{2} = ∅.

The variety X_{S_{1}} is isomorphic to the quadric Y defined by f = 0. The class A(Y) is connected by Theorem 9.2.

The second case S_{2} corresponds to the variety of Borel subgroups of G. A field extension L/F makes X_{∅} isotropic if and only if it splits g. Let Z be the projective quadric defined by g = 0. Then the classes A(X_{∅}) and A(Z) coincide, and we are done again by Theorem 9.2.

We now turn to trialitarian cases ^{3}D_{4} and ^{6}D_{4}. We need to consider the case p = 3 only. Let E/F be a cubic extension over which G is a classical group.

Lemma 14.1. Let F be 3-special. If L/F is an extension such that G is L-isotropic, then G is quasi-split over L.

Proof. If G is not quasi-split over L, then, by [28], its Tits index over L is of the form

[figure]

Hence the semisimple L-anisotropic kernel of G is isomorphic to R_{E′/L}(SL_{1}(D)), where D is a quaternion algebra over E′ and E′ = E · L. Note that such a group is still anisotropic over E′. On the other side, since F is 3-special, G is quasi-split over E and hence over E′ = E · L – a contradiction. □

Lemma 14.1 implies that we need to consider the basic type S = ∅ only. Let R_{G_{0}} : H^{1}(F, G_{0}) → H^{3}(F, ℤ/3) be the Rost invariant. We refer to [4] for the definition and properties of the Rost invariant. It takes values in ℤ/3 since F is 3-special. By [2, Theorem 6.14], Ker R_{G_{0}} = 1 and R_{G_{0}}([ξ]) is a 3-symbol, say (a) ∪ (b) ∪ (c), where a, b, c ∈ F^{×}.

Let J be the Jordan algebra arising from the first Tits construction corresponding to the central simple algebra A = (a, b) over F of degree 3 and element c. Recall that it is split over a field extension L/F of F if and only if c ∈ Nrd A_{L} or equivalently if and only if R_{G_{0}}([ξ]) is trivial over L, by [14, Th. 12.2]. As Ker R_{G_{0}} = 1, the last holds if and only if [ξ_{L}] = 1. Thus, by Lemma 3.1, the classes of splitting fields of J and A(X_{∅}) coincide and the connectedness of X_{∅} follows from Theorem 11.3.

15. Inner forms of type ^{1}E_{6}

According to [28] all admissible Tits’ indices of type ^{1}E_{6} corresponding to isotropic (not split) groups are as follows:

[figure: (a); α_{1}; α_{3}; α_{4}; α_{5}; α_{6}; α_{2}; (b); α_{1}; α_{3}; α_{4}; α_{5}; α_{6}; α_{2}]

In case (b) the corresponding anisotropic kernel has type A_{2} +A_{2}. Since SL_{3} × SL_{3} has no nontrivial strongly inner forms, Theorem 3.2 implies that diagram (b) can not appear as the Tits index of a strongly inner form of type ^{1}E_{6} (note that in the notation of that theorem the condition H^{1}(F, C/C ∩ H) = 1 holds, since C = C_{1} is an F -split torus). Thus, by Remark 4.7, the only basic types to be consider are

S_{1} = ∅ and S_{2} = {α_{2}, α_{3}, α_{4}, α_{5}}.

Lemma 15.1. Assume that G has diagram (a) over an extension L/F. Then the anisotropic L-kernel of G is isomorphic to Spin(f) where f is a 3-Pfister form over L.

Proof. The anisotropic L-kernel is a strongly inner form of type ^{1}D_{4}, by Theorem 3.2. So the result is clear. □

Lemma 15.2. Let F be 3-special. Then A(X_{S_{1}}) = A(X_{S_{2}}).

Proof. The inclusion A(X_{S_{1}}) ⊂ A(X_{S_{2}}) is clear. Conversely, let L ∈ A(X_{S_{2}}). If X_{S_{1}}(L) = ∅, then G has Tits index (a) over L. The corresponding anisotropic kernel of G over L is isomorphic to Spin(f) where f is a 3-Pfister form, by Lemma 15.1. Hence it can be split by a quadratic extension E/L. On the other side, the Rost invariant R_{G_{0}} has trivial kernel (cf. [2], [5]), takes values in H^{3}(F, ℤ/3) and R_{G_{0}}([ξ]) is a symbol, by [2]. Let J be a Jordan algebra corresponding to the symbol R_{G_{0}}([ξ]). As Ker R_{G_{0}} = 1, J is nontrivial over L and remains so over its quadratic extension E/L since J arises from the first Tits construction. However J, [ξ] (and hence G) have the same splitting fields – a contradiction. □

If p = 3, then by Lemma 15.2, A(X_{S_{1}}) = A(X_{S_{2}}) and the classes of splitting fields of J and A(X_{S_{1}}) coincide where J is the Jordan algebra constructed in Lemma 15.2. The connectedness of A(X_{S_{1}}), A(X_{S_{2}}) follows from Theorem 11.3.

Let now p = 2.

Lemma 15.3. If F is 2-special, then G is F -isotropic.

Proof. There exists the canonical embedding H_{0} → G_{0} where H_{0} is a split group of type F_{4}. It induces the mapping H^{1}(F, H_{0}) → H^{1}(F, G_{0}) which is surjective, by [4, page 51, Exercise 22.9], since F is 2-special. Each class in H^{1}(F, H_{0}) is split over a quadratic extension E/F, by property (a) in Section 13, hence so is [ξ]. By [2, Lemma 6.4], G is isotropic over F. □

Lemma 15.3 implies that X_{S_{2}}(F) ≠ ∅ and hence A(X_{S_{2}}) is connected. Furthermore, if G is not split, then, by Lemma 15.1, its anisotropic kernel is isomorphic to Spin(f) where f is a 3-Pfister form f. Hence A(X_{S_{1}}) = A(Y) where Y is the quadric defined by f = 0.

16. Outer forms of type E_{6}

If p ≠ 2, then G_{0} and G are inner forms of type E_{6}. So we may assume that p = 2. Then F is perfect since by our assumption char(F) ≠ 2.

16.1. Basic types. Let K = F (\sqrt{d}) be the quadratic extension which makes G_{0} (and hence G) an inner form of type E_{6}. We denote by σ a unique nontrivial automorphism of K/F. According to Tits’ classification [28] if G is F -isotropic but not quasi-split, then its F -index is one of the following:

[figure: (a); α_{2}; α_{4}; α_{3}; α_{1}; α_{6}; α_{5}; (b); α_{2}; α_{4}; α_{3}; α_{1}; α_{6}; α_{5}; (c); α_{2}; α_{4}; α_{3}; α_{1}; α_{6}; α_{5}; (d); α_{2}; α_{4}; α_{3}; α_{1}; α_{6}; α_{5}]

Lemma 16.1. The Tits index of G is not of the form (a).

Proof. In case (a) the anisotropic kernel of G has type A_{2} + A_{2} and a maximal F -split torus C_{1} has dimension 2. By dimension count C_{1} coincides with the center C of its centralizer; in particular C and all its quotients have trivial cohomology in dimension 1. Hence, by Theorem 3.2, the anisotropic kernel of G is a strongly inner form of R_{K/F} (SL_{3}). As H^{1}(F, R_{K/F} (SL_{3})) = 1, the group R_{K/F} (SL_{3}) has no nontrivial strongly inner forms. □

Thus there are four basic types only:

S_{1} = {α_{2}, α_{3}, α_{4}, α_{5}}, S_{2} = {α_{1}, α_{3}, α_{4}, α_{5}, α_{6}}, S_{3} = {α_{3}, α_{4}, α_{5}}, S_{4} = ∅.

According to Lemma 15.3, G is isotropic over K. If it is not K-split, then its Tits index is given by diagram (a) in Section 15 and hence its anisotropic kernel over K is isomorphic to Spin(f) where f is a 3-Pfister form over K. Since f can be split by a quadratic extension of K, we obtain that n(X_{S_{4}}) divides 4.

To compute the numbers n(X_{S}) we need information on groups of type D_{n}.

16.2. Classical groups of type D_{n}. Recall that a simple simply connected F -group of type ^{1}D_{n} can be realized as a spinor group Spin(A, τ) associated to a central simple algebra A over F equipped with an involution τ of the first kind, orthogonal type and trivial discriminant. If A is represented by a skew field D in the Brauer group Br (F), then A ≃ End_{D}(V) where V is a right D-module, and τ corresponds to a non-degenerate skew-hermitian form h of trivial discriminant on V with respect to a suitable involution τ′ on D of symplectic type.

Spin(A, τ) is isotropic over F if and only if h is isotropic, i.e. there is a vector v ∈ V such that h(v, v) = 0. Tits index of Spin(A, τ) over F is of the form

[figure: …; …; …; α_{1}; α_{2}; α_{d}; α_{id}; α_{rd−1}; α_{rd}]

Here d = deg(D), i is the Witt index of h and rd = n.

16.3. Trialitarian effect. We keep the above notation. Assume that we are given a pair (A, τ) such that A has index 2 and degree 8. Consider the even Clifford algebra of (A, τ). It is of the form A_{1} × A_{2} where A_{1}, A_{2} are central simple algebras of degree 8 equipped with involutions τ_{1}, τ_{2} of orthogonal type and A · A_{1} · A_{2} = 1 in the Brauer group Br (F) (see [10]). The center Z of Spin(A, τ) is isomorphic to Z ≃ ℤ/2 × ℤ/2. Three nontrivial elements in Z give rise to the quotient morphisms ϕ_{i} : Spin(A, τ) → O^{+}(A_{i}, τ_{i}), i = 1, 2 and ϕ : Spin(A, τ) → O^{+}(A, τ). If Spin(A, τ) is isotropic over F, there are two possibilities for its Tits index:

[figure: α_{1}; α_{2}; α_{3}; α_{4}; α_{1}; α_{2}; α_{3}; α_{4}]

In the first case all algebras A, A_{1}, A_{2} have index 2 and in the second case one of the algebras A_{1}, A_{2} is split and a quadratic form f corresponding to its involution is isotropic. Note that according to the above picture isotropy of f implies automatically that its Witt index is 2.

Lemma 16.2. Assume that A, A_{1}, A_{2} have indices at most 2. Then there is a quadratic extension L/F such that Spin(A, τ) has rank at least 2 over L.

Proof. If all algebras A, A_{1}, A_{2} are matrix algebras, then G ≃ Spin(f) where f is a 3-Pfister form. Such a form can be split by a quadratic extension.

Assume that A has index 2. Consider the pair (A_{1}, τ_{1}) and the corresponding involution variety Y. As n(Y) = 2, there is a quadratic extension L/F belonging to A(Y). The remark above the lemma shows that L-rank of Spin(A, τ) is at least 2. □

16.4. Computing n(X_{S}).

Proposition 16.3. We have n(X_{S_{1}}) = n(X_{S_{2}}) = n(X_{S_{3}}) = 2.

Proof. We consider the most difficult case when G is anisotropic over F. Isotropic cases are reduced to groups of classical types and can be treated much easier.

We may assume that G is not split over K, since otherwise X_{S_{i}}(K) ≠ ∅, i = 1, …, 4 and we are done. Then its Tits K-index is given by diagram (a) in Section 15 and so the variety of parabolic subgroups in G of type S_{1} = {α_{2}, α_{3}, α_{4}, α_{5}} contains a K-point. By Lemma 3.3, there is a parabolic subgroup P in G over K of type S_{1} such that H_{1} = P ∩ σ(P) is a reductive part of P.

H_{1} is defined over F and is an almost direct product of a simple simply connected F -group of classical type D_{4} and a 2-dimensional central torus T_{1}. Since T_{1} is F -anisotropic and split over K, σ acts on the character lattice of T_{1} by multiplication −1. Hence each 1-dimensional subtorus in T_{1} is F -defined.

Let α̃ be the highest root in Σ with respect to the ordering given on diagram (a) in Section 15. The restriction mapping α̃|_{T_{1}} : T_{1} → G_{m} is a nontrivial homomorphism. Let T_{2} be its kernel. It has dimension 1 and, by construction, T_{2} commutes with U_{±α̃}. As T_{2} is defined over F, so is H_{2} = C_{G}(T_{2}) which is a reductive group whose semisimple part [H_{2}, H_{2}] is a simple simply connected group of classical type D_{5} generated by roots α_{2}, α_{3}, α_{4}, α_{5}, α̃.

Let us describe the structure of [H_{2}, H_{2}]. Note first that it has K-rank 1 (and hence K-isotropic), since T_{1} is contained in H_{2} = C_{G}(T_{2}) and T_{1} is a maximal K-split torus in G of dimension 2. Furthermore, up to isogeny, we have [H_{2}, H_{2}] ≃ SU(D, h) where D is a skew field over F equipped with a symplectic involution τ and h is a skew hermitian form over D with respect to τ. As [H_{2}, H_{2}] is K-isotropic and its anisotropic semisimple kernel over K coincides with that of G (it is a group of type D_{4} generated by {α_{2}, α_{3}, α_{4}, α_{5}}), Theorem 3.2 applied to G over K shows that this anisotropic kernel has trivial Tits algebras over K. It follows that the discriminant of h is trivial over K, the algebra D splits over K and hence D is either F or a quaternion skew field.

We claim that the discriminant of h is d. Indeed, we know that it is trivial over K, hence we need only to exclude the possibility for it to be trivial over F. Let Z be the variety of Borel subgroups of [H_{2}, H_{2}]. If the discriminant of h is trivial over F, then [H_{2}, H_{2}] is a group of inner type and hence it has rank 5 over a field extension F (Z)/F. This implies automatically that G is split over F (Z). But F is algebraically closed in F (Z), hence K is not contained in F (Z). It follows that G is still a group of outer type over F (Z)

– a contradiction.

Assume now that D is a quaternion skew field. By [13, Lemma 1], h can be written in the form h = h_{1} ⊕ h_{2} where h_{2} is a 1-dimensional skew-hermitian form over D isotropic over K and h_{1} is a 4-dimensional form with trivial discriminant. Lemma 16.2 applied to (D, h_{1}) shows that there is a quadratic extension L/F such that L-rank of SU(D, h_{1}) is at least 2. Note that the conditions of this lemma are satisfied since the anisotropic kernel of SU(D, h_{1}) over K is a group of classical type D_{4} with trivial Tits algebras. An inspection of possible Tits indices of G over L shows that G is either quasi-split over L or has Tits index (b) in the above list of Tits indices of outer type E_{6}. In all cases we have X_{S_{1}}(L) ≠ ∅, X_{S_{2}}(L) ≠ ∅, X_{S_{3}}(L) ≠ ∅ and hence n(X_{S_{1}}) = n(X_{S_{2}}) = n(X_{S_{3}}) = 2.

If D = F and h is a 10-dimensional quadratic form over F, we can apply the same argument as above. Namely, since h is isotropic over K, we can write it in the form h = h_{1} ⊕ h_{2} where h_{2} is a 2-dimensional quadratic form splitting over K and h_{1} is an 8-dimensional quadratic form with trivial discriminant. Arguing as in Lemma 16.2 we easily see that h_{1} has rank at least 2 over a proper quadratic extension L/F. So the result follows. □

16.5. Basic type S_{1}. Let L ∈ A(X_{S_{1}}) and [L : F] = 2. As K ∈ A(X_{S_{1}}), it suffices to show that L and K are X_{S_{1}}-equivalent. We shall construct an F -subgroup H ⊂ G of classical type D_{5} with the properties:

1) there is an F -embedding Y → X_{S_{1}}, where Y is the involution variety corresponding to H;

2) L, K ∈ A(Y).

Property 2) implies that L and K are Y -equivalent, by Theorem 10.3, and property 1) implies that L and K are X_{S_{1}}-equivalent.

To construct H with the required properties we repeat the argument in 16.4 by verbatim. Namely, G contains a parabolic subgroup P of type S_{1} over L. We may assume that P is standard in the corresponding ordering of the root system of G. Let τ be the nontrivial automorphism of L/F. Without lost of generality we may assume that H_{1} = P ∩ τ(P) is a reductive part of P, by Lemma 3.3. It is an F -reductive group whose semisimple part is a simple simply connected group of type D_{4} generated by roots α_{2}, α_{3}, α_{4}, α_{5} and whose central torus T_{1} has dimension 2. This torus is isotropic over L and splits over K · L.

Let T_{2} ⊂ T_{1} be the kernel of the restriction mapping α̃|_{T_{1}} where α̃ is the highest root in the chosen ordering of the root system of G. Below we shall show that T_{2} is F -defined. Then the centralizer H_{2} = C_{G}(T_{2}) is a reductive F -group whose semisimple part H = [H_{2}, H_{2}] is a simple simply connected group of type D_{5} over F. We claim that it has the required properties.

Let us check property 1). The Dynkin diagram of H is of the form

[figure: −α̃; α_{2}; α_{4}; α_{3}; α_{5}]

The corresponding involution variety Y is the variety of parabolic subgroups in H of type S_{1} = {α_{2}, α_{4}, α_{3}, α_{5}}. The subgroup P_{H} = P ∩H of H is parabolic of type S_{1}. Then the required embedding is given by H/P_{H} → G/P, hP_{H} → hP. It is well defined and it is easy to check that it is F -defined using the twisted argument.

To check 2) we need information about the structure of T_{1}. It is proven in [1, Theorem 2.12], that T_{1} viewed over L is isomorphic to T_{1} ≃ R_{E/L}(G_{m,E}) where E = L · K. Thus T_{1} viewed over L is an almost direct product of a 1dimensional split L-subtorus, say V_{1}, and a 1-dimensional L-anisotropic torus V_{2} = R_{E/L}^{(1)}(G_{m,E}). It follows that Gal(L/F) preserves V_{1}, V_{2} and hence both of them are F -defined. Note that computation in [1] shows that V_{1} is contained in H and commutes with the subgroup of H generated by α_{2}, α_{4}, α_{3}, α_{5}. The last implies that H is L-isotropic and L ∈ A(Y).

Lemma 16.4. We have V_{2} = T_{2} and T_{2} splits over K.

Proof. The 2-torsion part of T_{1} is of the form R_{E/L}(μ_{2}). One easily checks the centralizer of −1 ∈ R_{E/L}(μ_{2}) in G is H · T_{2}, hence T_{2} is L-defined. But T_{1} contains only two L-defined subtori, namely V_{1} and V_{2}. As V_{1} ⊂ H, we get T_{2} = V_{2}.

Assume that T_{2} is not split over K. Let Z be the variety Z of Borel subgroups of H over K. The group H being quasi-split over K(H/B) contains a split torus over K(H/B), say T_{3}, of dimension at least 4. One easily checks that C_{G}(T_{3}) is a maximal torus in G containing T_{2}. As K is algebraically closed in K(Z), the torus T_{2} is still anisotropic over K(Z) and hence C_{G}(T_{3}) does not split over K(Z).

On the other side, as rank of G over K(H/B) is at least 4, Tits classification implies that G is split over K(H/B) (see the diagrams in Section 15) and hence so is the torus C_{G}(T_{3}) because any K(Z)-split torus in G is contained in a maximal K(Z)-split torus of G. This contradicts our assumption that T_{2} is anisotropic over K(H/B). □

As T_{2} is K-split, the K-anisotropic kernel of G is contained in H_{2} = C_{G}(T_{2}), hence G and H have the same K-anisotropic kernel. Then Tits classification of groups of inner type E_{6} shows that K ∈ A(Y).

16.6. Basic type S_{2}. Assume that L ∈ A(X_{S_{2}}) and [L : F] = 2. Let τ be the nontrivial automorphism of L/F. Choose a parabolic subgroup P ⊂ G over L of type S_{2} such that H′ = P ∩ τ(P) is a reductive part of P. It is a reductive subgroup of G over F whose semisimple part [H′, H′] has type A_{5} and is generated by roots α_{1}, α_{3}, …, α_{6} in a proper ordering of the corresponding root system of G. The centralizer H′′ = C_{G}([H′, H′]) is an F -defined subgroup of G of type A_{1} containing the central torus of H′. It is generated by the root subgroups U_{±α̃} where α̃ is the highest root.

Consider H = H′′ · [H′, H′]. It is an F -defined semisimple subgroup of G of type A_{1} + A_{5} whose component of type A_{1} is split over L and is of the form SL_{1}(D) where D is a quaternion algebra over F. It follows that L ∈ A(SB(D)). Conversely, any splitting field of D is contained in A(X_{S_{2}}).

Let L_{1} be another quadratic extension of F such that L_{1} ∈ A(X_{S_{2}}). We are going to show that L_{1} ∈ A(SB(D)). Theorem 7.2 then implies L_{1} ∼_{SB(D)} L and hence L_{1} ∼_{X_{S_{2}}} L.

Applying the above construction to L_{1} we can construct an F -subgroup H_{1} of G of type A_{1} + A_{5}. Its component of type A_{1} correspond to quaternion algebras D_{1} splitting over L_{1} and it suffices to show that D_{1} = D.

H_{1} and H are related to subsystems of type A_{1} + A_{5} in the root system E_{6}. Since every two roots in the root system of type E_{6} are conjugate by an element in the corresponding Weyl group W (E_{6}), so are every two root subsystems of type A_{1} + A_{5}. Then H_{1}, H are conjugate over F_{sep}. Let H_{1} = gHg^{−1} where g ∈ G(F_{sep}). Fixing F -defined maximal tori T_{1}, T in H_{1}, H respectively, we may assume additionally that T_{1} = gT g^{−1}. As H_{1}, H, T_{1}, T are F -defined, for each σ ∈ Gal(F_{sep}/F) we have a_{σ} = g^{−σ+1} ∈ N_{G}(H) ∩ N_{G}(T). It follows that H_{1} = ^{λ}H where λ = (a_{σ}); in particular we have D_{1} = ^{λ}D.

The torus T can be decomposed as T = T′T′′ where T′, T′′ are intersections of T with simple components of H of types A_{1} and A_{5} respectively. Since groups of type A_{1} have no nontrivial outer automorphisms, one checks that a_{σ} can be decomposed as a_{σ} = a′_{σ}a′′_{σ} where a′_{σ} and a′′_{σ} are contained in the normalizers of T′ and T′′ in the simple components of H respectively. Thus twisting of D is given actually by the cocycle λ′ = (a′_{σ}). This cocycle takes values in SL_{1}(D) and hence twisting does not change D since any cocycle in Z^{1}(F, SL_{1}(D)) is equivalent to a cocycle with coefficients in the center μ_{2} of SL_{1}(D).

16.7. Basic type S_{3}. Let L_{1}, L_{2} ∈ A(X_{S_{3}}) be two quadratic extensions of F contained in A(X_{S_{3}}). As A(X_{S_{3}}) ⊂ A(X_{S_{2}}), the above construction applied to L_{1} or L_{2} gives us an F -defined subgroup H of G of type A_{1} + A_{5}. Its simple components, say H′ and H′′, are of the form H′ = SL_{1}(D) and H′′ = SU(A, τ) where D is a quaternion algebra over F splitting over L_{1}, L_{2} and A is a central simple algebra of degree 6 over K equipped with an involution τ of the second kind.

Any field extension L/F splitting D makes H′′ a strongly inner form of a quasi-split group of type A_{5}, by Theorem 3.2, hence it also splits A. Then [A] = [D ⊗_{F} K] in the Brauer group Br(K) and hence H′′ ≃ SU(T, h) where T = D ⊗_{F} K and h is a skew-hermitian form of dimension 3 given on a vector space V over T equipped with the standard involution, say σ, of the second kind.

We are also given that H′′ is isotropic over L_{1}, L_{2}, i.e. h represents zero over L_{1}, L_{2}. Since L_{1}, L_{2} splits T and have degree 2 over F, there are 1-dimensional T -submodules U_{1}, U_{2} in V such that h restricted to U_{1}, U_{2} is isotropic over L_{1} and L_{2} respectively. This fact can be reformulated as follows. Let v_{1} ∈ U_{2} and v_{2} ∈ U_{2} be two nontrivial vectors. Let σ_{1}, σ_{2} be involutions on T corresponding to the elements a_{1} = h(v_{1}, v_{1}) and a_{2} = h(v_{2}, v_{2}), i.e. σ_{1}, σ_{2} are compositions of σ with the inner conjugation given by a_{1}, a_{2}. Then SU(T, σ_{1}) and SU(T, σ_{2}) are split over L_{1} and L_{2} respectively.

Clearly SU(T, σ_{1}) and SU(T, σ_{2}) are subgroups in H′′ = SU(T, h) and being groups of type A_{1} they are of the form SL_{1}(D_{1}) and SL_{1}(D_{2}) where D_{1}, D_{2} are quaternion algebras over F contained in T such that D_{1} ⊗ K = D_{2} ⊗ K = T. Recall that the pure part D_{i}^{∘} of D_{i} is given by

D_{i}^{∘} = { x ∈ T | σ_{i}(x) = −x} = { x ∈ D^{∘} ⊗ K | a_{i}σ(x)a_{i}^{−1} = −x}.

We now note that v_{i} is defined up to t_{i} ∈ T, hence D_{i} is defined up to conjugation in T. Since by our construction, D_{i} and D are split over L_{i}, we can modify v_{i}, a_{i} and D_{i} (by replacing them with t_{i}v_{i}, t_{i}a_{i}σ(t_{i}) and t_{i}D_{i}t_{i}^{−1} for a proper element t_{i} ∈ T) such that D ∩ D_{i} contains a common subfield isomorphic to L_{i}.

More precisely, this common subfield can be obtained as follows. As a_{i} is skew with respect to σ we can write it in the form a_{i} = a′_{i} + a′′_{i} where a′_{i} ∈ K and a′′_{i} ∈ D^{∘}. Modifying a_{i}, if necessary, we may assume that a′′_{i} ≠ 0. Then we have

D^{∘} ∩ D_{i}^{∘} = { d ∈ D^{∘} | σ_{i}(d) = −d} = { d ∈ D^{∘} | a_{i}da_{i}^{−1} = d}.

It follows that D^{∘} ∩ D_{i}^{∘} has dimension 1 and is generated by a′′_{i}.

We are ready to finish the proof. Consider a vector v(t) = (1 − t)v_{1} + tv_{2} where t is an indeterminate. Let a(t) = h(v(t), v(t)). Write a(t) = a(t)′ +a(t)′′ as above. Here a(t)′ ∈ K ⊗_{F} F (t) and a(t)′′ ∈ D^{∘} ⊗_{F} F (t). Clearly, a(t)′′ generate a maximal subfield L(t) in D ⊗_{F} F (t) splitting D ⊗_{F} F (t) and making H′′ isotropic. This implies that L(t) ∈ A(X_{S_{3}}). Since v(0) = v_{1} and v(1) = v_{2}, arguing as in Theorem 9.2 we see that L(t) has two discrete valuations with residue fields isomorphic to L_{1}, L_{2}.

17. Type E_{7}

By Tits classification [28], if G is isotropic over an extension L/F but not split, then its Tits index is one of the following:

[figure: a); α_{7}; α_{6}; α_{5}; α_{4}; α_{3}; α_{1}; α_{2}; b); α_{7}; α_{6}; α_{5}; α_{4}; α_{3}; α_{1}; α_{2}; c); α_{7}; α_{6}; α_{5}; α_{4}; α_{3}; α_{1}; α_{2}]

Thus we need to consider the following basic types only:

S_{1} = {α_{1}, …, α_{6}}, S_{2} = {α_{2}, …, α_{7}}, S_{3} = {α_{2}, …, α_{5}}, S_{4} = ∅.

If p = 3, then we can repeat the same argument as in the case of type ^{1}E_{6}. Consider the case p = 2.

Lemma 17.1. Let F be 2-special. Then we have n(S_{1}) = n(S_{2}) = n(S_{3}) = 2.

Proof. By [30], G contains an F -defined subgroup H of type E_{6}. It is a strongly inner form of a quasi-split group of type E_{6}, since F is 2-special. We know that any such a group (and hence G) has rank at least 2 over a proper quadratic extension E/F. The above diagrams show that X_{S_{i}}(E) ≠ ∅ for i = 1, 2, 3. □

By Lemma 15.3, if E/F is a quadratic extension such that X_{S_{1}}(E) ≠ ∅, then X_{S_{3}}(E) ≠ ∅. Hence it suffices to consider types S_{2}, S_{3} only.

17.1. Basic type S_{2}. Repeating verbatim the argument in 16.6 we find that G contains an F -defined subgroup H of type A_{1} + D_{6} such that A(X_{S_{2}}) = A(SB(D)) where D is a quaternion algebra over F corresponding to the simple component of H of type A_{1}. The connectedness of A(S_{2}) follows from Theorem 7.2.

17.2. Basic type S_{3}. Let L/F be a quadratic extension belonging to A(X_{S_{3}}). Let H_{1} and H_{2} be the simple components of types A_{1} and D_{6} respectively of the group H constructed in 17.1. We have H_{1} ≃ SL_{1}(D) where D is a quaternion algebra over F. The above Tits diagrams show that the anisotropic L-kernel of H_{2} is of the form Spin(f) where f is a 4-Pfister form. It follows that up to isogeny H is of the form SU(T, h) where T is a quaternion algebra over F equipped with the standard symplectic involution τ and h is a skew hermitian form with respect to τ defined over a 6-dimensional vector space V over T.

Lemma 17.2. If T is nontrivial, then we have D = T.

Proof. If E/F is an extension splitting D, then H_{2} viewed over E is a strongly inner form of a split group of type D_{6}. Hence H_{2} is of the form Spin(g) where g is a 6-dimensional quadratic form from I^{3}. It follows that T is split over E.

As this is true for all splitting fields of D, the quaternion algebras D and T coincide. □

Let Y be the involution variety corresponding to SO(D, h). By our construction, L ∈ A(Y). Conversely, let E ∈ A(Y). We noted above that H_{2} viewed over E is related to the quadratic form g of dimension 12 from I^{3}. Since g is isotropic over E and since 10-dimensional quadratic forms from I^{3} are isotropic, we conclude that E-rank of H_{2} is at least 2. This implies that E-rank of G is at least 2. An inspection of the above Tits diagrams shows that E ∈ A(X_{S_{3}}). Thus we showed that A(Y) = A(X_{S_{3}}) and the connectedness of A(Y) follows from Theorem 10.3.

18. Two examples of a nontrivial \overline{CH}_{0}(X)

In this section we give two examples of projective homogeneous varieties X with nontrivial group \overline{CH}_{0}(X).

18.1. First Example. We owe to Vishik the observation that [9] essentially contains an example. Let F be a field of characteristic different from 2 and let a, b ∈ F^{×}. Let Q_{a,b} be the quadric of the 2-fold Pfister form ⟨⟨a, b⟩⟩ = ⟨1, −a, −b, ab⟩. In the split case (over a quadratic extension E/F), Q_{a,b} is isomorphic to ℙ_{F}^{1} × ℙ_{F}^{1} , in particular, there are two families of lines on Q_{a,b}. Let l_{a,b} be one of these lines and denote by u_{a,b} ∈ CH_{1}(Q_{a,b}) the push-forward of the class of l_{a,b} under the morphism (Q_{a,b})_{E} → Q_{a,b}.

Let X_{a,b} be the variety of isotropic planes of ⟨⟨a, b⟩⟩. Over the field E ∈ A(X_{a,b}), the variety X_{a,b} is the disjoint union of two copies of the projective line. We can view l_{a,b} as a rational point of X_{a,b} over E. Denote the push-forward of the class of l_{a,b} in CH_{0}(X_{a,b}) by x_{a,b}, so x_{a,b} is a 0-cycle of degree 2. The incidence correspondence between Q_{a,b} and X_{a,b} induces a map CH_{0}(X_{a,b}) → CH_{1}(Q_{a,b}) taking u_{a,b} to x_{a,b}.

Let q = ⟨1, −a, −b, ab, −c, −d, cd⟩ for some a, b, c, d ∈ F^{×} and let Q be the corresponding 5-dimensional quadric. We have natural embeddings Q_{a,b} ↪ Q and Q_{c,d} ↪ Q. Denote the images of u_{a,b} and u_{c,d} in CH_{1}(Q) by the same symbols.

It is shown in [9, Cor. 5.2] that in the case −1 ∈ F^{×2}, the classes u_{a,b} and u_{c,d} are equal if and only if the Pfister form ⟨⟨a, b, c, d⟩⟩ is isotropic.

Denote by X the variety of isotropic planes of q. The images of x_{a,b} and x_{c,d} in CH_{0}(X) will be denoted by the same symbols. We have the incidence correspondence between Q and X that induces a homomorphism CH_{0}(X) → CH_{1}(Q) taking x_{a,b} − x_{c,d} to u_{a,b} − u_{c,d}.

If F is a field such that −1 ∈ F^{×2} and the Pfister form ⟨⟨a, b, c, d⟩⟩ is anisotropic, the class u_{a,b} − u_{c,d} is nontrivial and therefore x_{a,b} − x_{c,d} ≠ 0. As deg(x_{a,b} − x_{c,d}) = 0, the class x_{a,b} − x_{c,d} represents a nontrivial element of \overline{CH}_{0}(X). Note that X corresponds to the subset \{α_{1}, α_{3}\} of the set of vertices of the Dynkin diagram B_{3}.

18.2. Second Example. We shall give an example of a 3-dimensional projective homogeneous variety X with a nontrivial group \overline{CH}_{0}(X). Note that this is an example of the smallest possible dimension in view of Proposition 4.5.

Let C_{i}, i = 1, 2, 3, be three conic curves corresponding to quaternion algebras Q_{i} over F and let X = C_{1} × C_{2} × C_{3}. For every subset S ⊂ {1, 2, 3} let Q_{S} be the tensor product of Q_{i} for all i ∈ S (in particular, Q_{∅} = F). We define the complex

(18.1)  ∐_{S≠∅} K_{1}(Q_{S}) ⟶^{α} (F^{×})^{3} ⟶^{β} H^{3}(F, ℤ/2ℤ)

by the following formulas:

α(x) = (a_{1}, a_{2}, a_{3}), a_{i} = ∏_{S∋i} Nrd_{S}(x_{S}),

β(a_{1}, a_{2}, a_{3}) = (a_{1}) ∪ [Q_{1}] + (a_{2}) ∪ [Q_{2}] + (a_{3}) ∪ [Q_{3}],

where Nrd_{S} : K_{1}(Q_{S}) → K_{1}(F) = F^{×} is the reduced norm homomorphism, (a) denotes the class in H^{1}(F, ℤ/2ℤ) corresponding to a ∈ F^{×} under the Kummer isomorphism and [Q] denotes the class of Q in H^{2}(F, ℤ/2ℤ).

Proposition 18.2. The group \overline{CH}_{0}(X) is isomorphic to the homology group of the complex (18.1).

Proof. In the spectral sequence (4.4) for X the only possibly non-trivial differential coming to E_{∗}^{3,−3} is d_{2} : A^{1}(X, K_{2}) = E_{2}^{1,−2} → E_{2}^{3,−3} = CH_{0}(X) and all the differentials coming to E_{∗}^{1,−2} are trivial. Therefore, the sequence

K_{1}(X)^{(1)} → A^{1}(X, K_{2}) ⟶^{d_{2}} CH_{0}(X) ⟶^{e′} K_{0}(X)

is exact. As the image of d_{2} coincides with Ker(e′) = \overline{CH}_{0}(X), it is sufficient to identify the cokernel of the first homomorphism in the sequence with the homology group of the complex (18.1).

The group A^{1}(X, K_{2}) was computed in [12]. If all Q_{i} are split, we have

A^{1}(X, K_{2}) = K_{1}(F) ⊗ CH^{1}(X) ≃ F^{×} ⊗ ℤ^{3} = (F^{×})^{3},

where we identify CH^{1}(X) with Pic(X) = ℤh_{1} ⊕ ℤh_{2} ⊕ ℤh_{3} ≃ ℤ^{3} (here h_{i} is the class of a rational point in C_{i}). In the general case the group A^{1}(X, K_{2}) is canonically isomorphic to a subgroup of (F^{×})^{3}, namely to the kernel of the homomorphism β.

The group K_{1}(X) was computed in [21, §8, Th. 4.1]:

(18.3)  K_{1}(X) = ∐_{S} K_{1}(Q_{S}).

The first term of the topological filtration K_{1}(X)^{(1)} is the kernel of the natural homomorphism K_{1}(X) → K_{1}F (X). We claim that for every S, the corresponding homomorphism K_{1}(Q_{S}) → K_{1}F (X) is the reduced norm map Nrd_{S} followed by the natural homomorphism K_{1}F → K_{1}F (X). Since the group K_{1} of a field injects when the field gets extended, we may assume that all Q_{i} are split. In this case the reduced norm Nrd_{S} is an isomorphism and (18.3) identifies K_{1}(X) with the direct sum of K_{1}(F)L_{S}, where L_{S} is the tensor product of the canonical line bundles L_{i} on the C_{i} ≃ ℙ_{F}^{1} such that i ∈ S. As L_{S} maps to 1 in K_{0}F (X), the claim follows.

We have shown that K_{1}(X)^{(1)} consists of all x = (x_{S}) such that ∑ Nrd_{S}(x_{S}) = 0. We claim that the image of x in A^{1}(X, K_{2}) ⊂ (F^{×})^{3} is equal to (a_{1}, a_{2}, a_{3}), where a_{i} is the product of Nrd_{S}(x_{S}) for all S ∋ i. We can assume that all Q_{i} are split. Under the identification of K_{1}(Q_{S}) with K_{1}(F), the element x corresponds to ∑ b_{S}L_{S} = ∑_{S≠∅} b_{S}(L_{S} − 1), where b_{S} = Nrd_{S}(x_{S}). Note that the image of L_{S} − 1 under the natural map K_{0}(X)^{(1)} → CH^{1}(X) is equal to the first Chern class c_{1}(L_{S}) = ∑_{i∈S} c_{1}(L_{i}) = ∑_{i∈S} h_{i}. Therefore, the image of x in A^{1}(X, K_{2}) = ℤh_{1} ⊕ ℤh_{2} ⊕ ℤh_{3} is equal to a_{1}h_{1} ⊕ a_{2}h_{2} ⊕ a_{3}h_{3}, where a_{i} is the product of b_{S} for all S ∋ i.

It follows from the claim that the image of the homomorphism K_{1}(X)^{(1)} → A^{1}(X, K_{2}) ⊂ (F^{×})^{3} is equal to Im α. □

Example 18.4. Let F = k((t)) be the Laurent power series field over a field k of characteristic different from 2, Q_{i} = (b_{i}, −t)_{F} for some b_{i} ∈ k^{×}. Let b_{S} be the product of b_{i} for all i ∈ S, k_{S} = k(b_{S}^{1/2}) and Norm_{S} : k_{S}^{×} → k^{×} be the norm homomorphism. Since (t^{n}, t^{m}, t^{p}) ∈ Im(α) for all n, m, p, H^{p}(F, ℤ/2ℤ) is equal to H^{p}(k, ℤ/2ℤ)⊕H^{p−1}(k, ℤ/2ℤ)∪(−t) and Im Nrd_{S} = Im Norm_{S} ·F^{×2} ∪ Im Norm_{S} ·t·F^{×2}, the homology group of (18.1) is isomorphic to the homology group of the complex

∐_{S≠∅} k_{S}^{×} ⟶^{α′} (k^{×})^{3} ⟶^{β′} H^{2}(k, ℤ/2ℤ),

where

α′(x) = (a_{1}, a_{2}, a_{3}), a_{i} = ∏_{S∋i} Norm_{S}(x_{S}),

β′(a_{1}, a_{2}, a_{3}) = (a_{1}) ∪ (b_{1}) + (a_{2}) ∪ (b_{2}) + (a_{3}) ∪ (b_{3}).

There is a field k and the elements b_{i} so that the complex is not acyclic (see [24, §5]) and therefore \overline{CH}_{0}(X) ≠ 0 in this case.

References

[1] V. Chernousov. L. Timoshenko, On the group of R-equivalence classes of semisimple groups over arithmetic fields, Algebra and Analis, 11 (1999), N 6, 191–221.

[2] V. Chernousov, The kernel of the Rost invariant, Serre’s Conjecture II and the Hasse principle for quasi-split groups ^{3,6}D_{4}, E_{6}, E_{7}, Math. Ann. 326 (2003), 297–330.

[3] V. Chernousov, S. Gille, and A. Merkurjev, Motivic decomposition of isotropic projective homogeneous varieties, Duke Math. J. 126 (2005), no. 1, 137–159.

[4] S. Garibaldi, A. Merkurjev and J.-P. Serre, Cohomological Invariants in Galois Cohomology, A.M.S., Univ. Lectures Series 28, Providence, R.I., 2003.

[5] S. Garibaldi, The Rost invariant has trivial kernel for quasi-split groups of low rank, Comment. Math. Helv. 76 (2001), no. 4, 684–711.

[6] I. N. Herstein, Noncommutative rings, Mathematical Association of America, Washington, DC, 1994, Reprint of the 1968 original, With an afterword by Lance W. Small.

[7] N. Jacobson, Structure and representations of Jordan algebras, Amer. Math. Soc. Colloq. Publ., vol. 39, American Mathematical Society, Providence, RI, 1968

[8] N. Karpenko, Algebro-geometric invariants of quadratic forms, Leningrad (St. Petersburg) Math. J. 2 (1991), 119–138.

[9] N. A. Karpenko and A. S. Merkurjev, Chow groups of projective quadrics, Algebra i Analiz 2 (1990), no. 3, 218–235.

[10] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, American Mathematical Society, Providence, RI, 1998, With a preface in French by J. Tits.

[11] D. Krashen, Zero cycles on homogeneous varieties, http://front.math.ucdavis.edu/math.AG/0501399 (2006).

[12] A. S. Merkurjev, The group H^{1}(X, K_{2}) for projective homogeneous varieties, Algebra i Analiz 7 (1995), no. 3, 136–164.

[13] A. S. Merkurjev, Zero-dimensional cycles on some involutive varieties, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 227 (1995), no. Voprosy Teor. Predstav. Algebr i Grupp. 4, 93–105, 158.

[14] A. S. Merkurjev and A. A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011– 1046, 1135–1136.

[15] I. Panin, Application of K-theory in algebraic geometry, Thesis, LOMI, Leningrad (1984).

[16] I. A. Panin, On the algebraic K-theory of twisted flag varieties, K-Theory 8 (1994), no. 6, 541–585.

[17] H. P. Petersson and M. L. Racine, Cubic subfields of exceptional simple Jordan algebras, Proc. Amer. Math. Soc. 91 (1984), no. 1, 31–36.

[18] H. Petersson, M. Racine, On the invariants mod 2 of Albert Algebras, Journal of Algebra 174 (1995), 1049–1072.

[19] V. Petrov, N. Semenov, K. Zainoulline, Zero-cycles on twisted Cayley plane, preprint (2005), http://www.math.uni-bielefeld.de/lag/

[20] V. Platonov, A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press 1994.

[21] D. Quillen, Higher algebraic K-theory. I, (1973), 85–147. Lecture Notes in Math., Vol. 341.

[22] M. Racine, A note on Quadratic Jordan Algebras of degree 3, Trans. Amer. Math. Soc. 164 (1972), 93–103.

[23] M. Rost, Chow groups with coefficients, Doc. Math. 1 (1996), No. 16, 319–393 (electronic).

[24] Daniel B. Shapiro, Jean-Pierre Tignol, and Adrian R. Wadsworth, Witt rings and Brauer groups under multiquadratic extensions. II, J. Algebra 78 (1982), no. 1, 58– 90.

[25] R. Steinberg, Regular elements of semisimple groups, Publ. Math. I.H.E.S. 25 (1965), 281–312.

[26] T. A. Springer, The classification of reduced exceptional simple Jordan algebras, Indag. Math. 22 (1960), 414–422.

[27] R. G. Swan, Zero cycles on quadratic hypersurfaces, Proc. Amer. Math. Soc. 107 (1989), 43–46

[28] J. Tits, Classification of algebraic semisimple groups, In Algebraic Groups and discontinuous Subgroups, (eds. A.Borel and G.D.Mostow), Proc. Symp. Pure Math. 9 (1966), 33–62.

[29] J. Tits, Formes quadratiques, groupes orthogonaux et algèbres de Clifford, Invent. Math. 5 (1968), 19–41.

[30] J. Tits, Sur les degrés des extensions de corps déployant les groupes algébriques simples, C. R. Acad. Sci. 315 (1992), 1131-1138.

Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

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

E-mail address: chernous@math.ualberta.ca
E-mail address: merkurev@math.ucla.edu