COHOMOLOGICAL INVARIANTS AND R-TRIVIALITY OF ADJOINT CLASSICAL GROUPS

GRÉGORY BERHUY, MARINA MONSURRÒ, AND JEAN-PIERRE TIGNOL

Abstract. Using a cohomological obstruction, we construct examples of absolutely simple adjoint classical groups of type ^{2}A_{n} with n ≡ 3 mod 4, C_{n} or 1D_{n} with n ≡ 0 mod 4, which are not R-trivial hence not stably rational.

Introduction

For an algebraic group G defined over a field F, let G(F)/R be the group of R-equivalence classes introduced by Manin in [6]. The algebraic group G is called R-trivial if G(L)/R = 1 for every field extension L/F. It was established by Colliot-Thélène and Sansuc in [2] (see also [7, Proposition 1]) that the group G is R-trivial if the variety of G is stably rational.

In this paper, we focus on the case where G is an absolutely simple classical group of adjoint type. Adjoint groups of type ^{1}A_{n} or B_{n} are easily seen to be rational (see [7, pp. 199, 200]). Voskresenskiĭ and Klyachko [11, Cor. of Th. 8] proved that adjoint groups of type ^{2}A_{n} are rational if n is even, and Merkurjev [7, Prop. 4] showed that adjoint groups of type C_{n} are stably rational for n odd. On the other hand, Merkurjev also produced in [7] examples of adjoint groups of type 2A_{3} (= ^{2}D_{3}) and of type ^{2}D_{n} for any n ≥ 4 which are not R-trivial, hence not stably rational. Examples of adjoint groups of type ^{1}D_{4} which are not R-trivial were constructed by Gille in [3].

The goal of the present paper is to construct examples of adjoint groups of type 2A_{n} with n ≡ 3 mod 4 and of adjoint groups of type C_{n} or ^{1}D_{n} with n ≡ 0 mod 4 which are not R-trivial. Our constructions are based on Merkurjev’s computation in [7] of the group of R-equivalence classes of adjoint classical groups, which we now recall briefly. According to Weil (see [4, §26]), every absolutely simple classical group of adjoint type over a field F of characteristic different from 2 can be obtained as the connected component of the identity in the automorphism group of a central^{1} simple algebra with involution (A, σ) over F. Let Sim(A, σ) be the algebraic group of similitudes of (A, σ), defined (as a group scheme) by

Sim(A, σ)(E) = {u ∈ A ⊗_{F} E | (σ ⊗ Id)(u)u ∈ E^{×}}

2000 Mathematics Subject Classification. 11E57, 20G15.

The authors gratefully acknowledge support from the TMR network “K-theory and linear algebraic groups” (ERB FMRX CT97-0107). The third author is also grateful to the National Fund for Scientific Research (Belgium) for partial support.

1We use the same terminology as in [4]. In particular, the center of A is F if σ is of the first kind; it is a quadratic étale extension of F if σ is of the second kind.

for every commutative F -algebra E, and let PSim(A, σ) be the group of projective similitudes,

PSim(A, σ) = Sim(A, σ)/R_{K/F} (G_{m,K} )

where K is the center of A. The connected component of the identity in these groups is denoted by Sim^{+}(A, σ) and PSim^{+}(A, σ) respectively. We let Sim(A, σ), PSim(A, σ), Sim^{+}(A, σ) and PSim^{+}(A, σ) denote the corresponding groups of F - rational points:

Sim(A, σ) = Sim(A, σ)(F), PSim(A, σ) = PSim(A, σ)(F), etc.

The group PSim^{+}(A, σ) is canonically isomorphic (under the map which carries every similitude g to the induced inner automorphism Int(g)) to the connected component of the identity in the automorphism group of (A, σ). To describe the group of R-equivalence classes of PSim^{+}(A, σ), consider the homomorphism

μ: Sim(A, σ) → G_{m}

which carries every similitude to its multiplier

μ(g) = σ(g)g.

Let G^{+}(A, σ) = μ(Sim^{+}(A, σ)) ⊂ F^{×} and NK^{×} = μ(K^{×}) ⊂ F^{×} (so NK^{×} = F^{×2} if K = F). Let also Hyp(A, σ) be the subgroup of F^{×} generated by the norms of the finite extensions L of F such that (A, σ) becomes hyperbolic after scalar extension to L. In [7, Theorem 1], Merkurjev shows that the multiplier map μ induces a canonical isomorphism

(1)  PSim^{+}(A, σ)/R ≃ G^{+}(A, σ)/(NK^{×} · Hyp(A, σ)).

For any positive integer d, let H^{d}(F, μ_{2}) be the degree d cohomology group of the absolute Galois group of F with coefficients μ_{2} = {±1}. In Section 3 we consider the case where σ is of the first kind. If it is orthogonal, we assume further that its discriminant is trivial. Assuming the index of A divides \frac{1}{2} deg A, we construct a homomorphism

Θ_{1} : PSim^{+}(A, σ)/R → H^{4}(F, μ_{2}),

and give examples where this homomorphism is nonzero, hence PSim^{+}(A, σ)/R ≠ 1. Similarly, if σ is of the second kind and the exponent of A divides \frac{1}{2} deg A, we construct in Section 4 a homomorphism

Θ_{2} : PSim^{+}(A, σ)/R → H^{3}(F, μ_{2})

and show that this map is nonzero in certain cases. In all the examples where we show Θ_{1} ≠ 0 or Θ_{2} ≠ 0, the algebra with involution has the form (A, σ) = (B, ρ) ⊗ (C, τ) where ρ is an orthogonal involution which admits improper similitudes.

Throughout the paper, the characteristic of the base field F is different from 2.

1. Improper similitudes

Let (A, σ) be a central simple F -algebra with orthogonal involution of degree n = 2m. The group of similitudes Sim(A, σ) is denoted GO(A, σ). This group is not connected. Its connected component of the identity GO^{+}(A, σ) is defined by the equation

Nrd_{A}(g) = μ(g)^{m},

where Nrd_{A} is the reduced norm. We denote by GO(A, σ) and GO^{+}(A, σ) the group of F -rational points

GO(A, σ) = GO(A, σ)(F), GO^{+}(A, σ) = GO^{+}(A, σ)(F).

The elements in GO^{+}(A, σ) are called proper similitudes, and those in the nontrivial coset

GO^{−}(A, σ) = {g ∈ GO(A, σ) | Nrd_{A}(g) = −μ(g)^{m}}

are called improper similitudes.

For example, if m = 1 (i.e. A is a quaternion algebra), then every orthogonal involution has the form σ = Int(q) ∘ γ, where γ is the canonical involution, q is an invertible pure quaternion and Int(q) is the inner automorphism induced by q, mapping x ∈ A to qxq^{−1}. It is easily checked that

GO^{+}(A, σ) = F (q)^{×}  and  GO^{−}(A, σ) = q′F (q)^{×},

where q′ is a unit which anticommutes with q. Therefore, GO^{−}(A, σ) ≠ ∅.

If m > 1, the existence of improper similitudes is an important restriction on A and σ, since it implies that A is split by the quadratic étale F -algebra F [\sqrt{disc σ}], where disc σ is the discriminant of σ, see [9, Theorem A] or [4, (13.38)]. In particular, the index of A satisfies ind A ≤ 2, i.e. A is Brauer-equivalent to a quaternion algebra. Moreover, if m is even, then −1 ∈ Nrd_{A}(A), see [9, Corollary 1.13]. There is no other restriction on A, as the following proposition shows.

1. Proposition. Let H be an arbitrary quaternion F -algebra and let m be an arbitrary integer. If m is even, assume −1 ∈ Nrd_{H}(H^{×}). Then the algebra M_{m}(H) carries an orthogonal involution which admits improper similitudes.

Proof. Suppose first m is odd. Let i, j be elements in a standard quaternion basis of H. We set

σ = t ⊗ (Int(i) ∘ γ) on M_{m}(H) = M_{m}(F) ⊗_{F} H,

where γ is the canonical involution on H. It is readily verified that 1 ⊗ j is an improper similitude of σ.

Suppose next m is even, and q ∈ H satisfies Nrd_{H}(q) = −1. We pick a quaternion basis 1, i, j, k = ij such that i commutes with q, and set

σ = Int diag(j, i, …, i) ∘ (t ⊗ γ)  and  g = diag(j, qj, …, qj).

Again, computation shows that g is an improper similitude of σ. □

Necessary and sufficient conditions for the existence of improper similitudes for a given involution σ are not known if m ≥ 4. For m = 2 (resp. m = 3), Corollary (15.9) (resp. (15.26)) in [4] shows that GO^{−}(A, σ) ≠ ∅ if and only if the Clifford algebra C(A, σ) has outer automorphisms (resp. outer automorphisms which commute with its canonical involution). (For m = 2 another equivalent condition is that A is split by the center of C(A, σ), see [4, (15.11)] or [9, Prop. 1.15].) We use this fact to prove the following result:

2. Proposition. Let (A, σ) be a central simple F -algebra with orthogonal involution of degree 4. Assume that A is not split and disc σ ≠ 1. Then there exists a field extension L/F such that A_{L} is not split and GO^{−}(A_{L}, σ_{L}) ≠ ∅.

Proof. By hypothesis, F (\sqrt{disc σ}) is a quadratic field extension of F. We denote it by K for simplicity and let ι be its nontrivial F -automorphism. The Clifford algebra C = C(A, σ) is a quaternion K-algebra. Let X be the Severi-Brauer variety of C ⊗_{K} ^{ι}C and let L be the function field of its Weil transfer:

L = F (R_{K/F} (X)).

Then (C ⊗_{K} ^{ι}C) ⊗_{K} KL splits, so C_{KL} is isomorphic to ^{ι}C_{KL}, which means that C_{KL} has outer automorphisms. By [4, (15.9)], it follows that GO^{−}(A_{L}, σ_{L}) ≠ ∅.

On the other hand, by [9, Corollary 2.12], the kernel of the scalar extension map Br(F) → Br(L) is generated by the corestriction of C⊗_{K}^{ι}C. Since this corestriction is trivial, A_{L} is not split. □

2. Trace forms

In this section, A is a central simple F -algebra of even degree with an involution σ of the first kind. We consider the quadratic forms T_{A} and T_{σ} on A defined by

T_{A}(x) = Trd_{A}(x^{2}),  T_{σ}(x) = Trd_{A}(σ(x)x)  for x ∈ A,

where Trd_{A} is the reduced trace on A. We denote by T_{σ}^{+} (resp. T_{σ}^{−}) the restriction of T_{σ} to the space Sym(σ) of symmetric elements (resp. to the space Skew(σ) of skew-symmetric elements), so that

(2)  T_{A} = T_{σ}^{+} ⊥ −T_{σ}^{−}  and  T_{σ} = T_{σ}^{+} ⊥ T_{σ}^{−}.

Recall that if σ is orthogonal the (signed) discriminant disc T_{σ}^{+} is equal to the discriminant disc σ up to a factor which depends only on the degree of A, see for instance [4, (11.5)]. In the following, we denote by I^{n}F the n-th power of the fundamental ideal IF of the Witt ring W F.

3. Lemma. Let σ, σ_{0} be two involutions of the first kind on A.

• If σ and σ_{0} are both symplectic, then T_{σ}^{+} − T_{σ_{0}}^{+} ∈ I^{3}F.

• If σ and σ_{0} are both orthogonal, then disc(T_{σ}^{+} −T_{σ_{0}}^{+}) = disc σ disc σ_{0}. Moreover, if disc σ = disc σ_{0}, then T_{σ}^{+} − T_{σ_{0}}^{+} ∈ I^{3}F.

Proof. The symplectic case has been considered in [1, Theorem 4]. For the rest of the proof, we assume that σ and σ_{0} are both orthogonal. By [4, (11.5)], there is a factor c ∈ F^{×} such that

disc T_{σ}^{+} = c disc σ  and  disc T_{σ_{0}}^{+} = c disc σ_{0},

hence

disc(T_{σ}^{+} − T_{σ_{0}}^{+}) = disc T_{σ}^{+} disc T_{σ_{0}}^{+} = disc σ disc σ_{0}.

To complete the proof, observe that the Witt-Clifford invariant e_{2}(T_{σ}^{+}) (or, equivalently, the Hasse invariant w_{2}(T_{σ}^{+})) depends only on disc σ and on the Brauer class of A, as was shown by Quéguiner [10, p. 307]. Therefore, if disc σ = disc σ_{0}, then e_{2}(T_{σ}^{+}) = e_{2}(T_{σ_{0}}^{+}), hence T_{σ}^{+} − T_{σ_{0}}^{+} ∈ I^{3}F by a theorem of Merkurjev. □

We next compute the Arason invariant e_{3}(T_{σ}^{+} − T_{σ_{0}}^{+}) ∈ H^{3}(F, μ_{2}) in the special case where σ and σ_{0} decompose. We use the following notation: [A] ∈ H^{2}(F, μ_{2}) is the cohomology class corresponding to the Brauer class of A under the canonical isomorphism H^{2}(F, μ_{2}) = _{2} Br(F). For a ∈ F^{×} we denote by (a) the cohomology class corresponding to the square class of a under the canonical isomorphism H^{1}(F, μ_{2}) = F^{×}/F^{×2}.

4. Lemma. Suppose A = B ⊗_{F} C for some central simple F -algebras B, C of even degree. Let ρ and ρ_{0} be orthogonal involutions on B and let τ be an involution of the first kind on C. Let also σ = ρ ⊗ τ and σ_{0} = ρ_{0} ⊗ τ.

If τ (hence also σ and σ_{0}) is symplectic, then

e_{3}(T_{σ}^{+} − T_{σ_{0}}^{+}) = \begin{cases}0 & if deg C ≡ 0 mod 4, \\ (disc ρ disc ρ_{0}) ∪ [C] & if deg C ≡ 2 mod 4. \end{cases}

If τ (hence also σ and σ_{0}) is orthogonal, then

e_{3}(T_{σ}^{+} − T_{σ_{0}}^{+}) = \begin{cases}(disc ρ disc ρ_{0}) ∪ (disc τ) ∪ (−1) & if deg C ≡ 0 mod 4, \\ (disc ρ disc ρ_{0}) ∪ ((disc τ) ∪ (−1) + [C]) & if deg C ≡ 2 mod 4. \end{cases}

Proof. The decomposition

Sym(σ) = (Sym(ρ) ⊗ Sym(τ)) ⊕ (Skew(ρ) ⊗ Skew(τ))

yields

T_{σ}^{+} = T_{ρ}^{+}T_{τ}^{+} + T_{ρ}^{−}T_{τ}^{−} in W F .

Since T_{B} = T_{ρ}^{+} − T_{ρ}^{−} we may eliminate T_{ρ}^{−} in the equation above to obtain

T_{σ}^{+} = T_{ρ}^{+}T_{τ}^{+} + (T_{ρ}^{+} − T_{B})T_{τ}^{−}.

Similarly,

T_{σ_{0}}^{+} = T_{ρ_{0}}^{+}T_{τ}^{+} + (T_{ρ_{0}}^{+} − T_{B})T_{τ}^{−}

and subtracting the two equalities yields

T_{σ}^{+} − T_{σ_{0}}^{+} = (T_{ρ}^{+} − T_{ρ_{0}}^{+})T_{τ}^{+} + (T_{ρ}^{+} − T_{ρ_{0}}^{+})T_{τ}^{−} = (T_{ρ}^{+} − T_{ρ_{0}}^{+})T_{τ} .

Since deg C is even, we have T_{τ} ∈ I^{2}F (see [4, (11.5)]), hence

e_{3}(T_{σ}^{+} − T_{σ_{0}}^{+}) = (disc(T_{ρ}^{+} − T_{ρ_{0}}^{+})) ∪ e_{2}(T_{τ}) in H^{3}(F, μ_{2}).

By Lemma 3 we have

disc(T_{ρ}^{+} − T_{ρ_{0}}^{+}) = disc ρ disc ρ_{0}.

The computation of e_{2}(T_{τ}) in [10, Theorem 1] or [5] completes the proof. □

Remark. If σ and σ_{0} are symplectic, the Arason invariant e_{3}(T_{σ}^{+} − T_{σ_{0}}^{+}) is the discriminant Δ_{σ_{0}}(σ) investigated in [1].

3. Involutions of the first kind

In this section, A is a central simple F -algebra of even degree, and σ is an involution of the first kind on A. We assume ind A divides \frac{1}{2} deg A, i.e. A ≃ M_{2}(A_{0}) for some central simple F -algebra A_{0}, so that A carries a hyperbolic involution σ_{0} of the same type as σ. If σ is orthogonal, we assume disc σ = 1 (= disc σ_{0}), so that in all cases T_{σ}^{+} − T_{σ_{0}}^{+} ∈ I^{3}F, by Lemma 3.

5. Proposition. The map θ_{1} : Sim(A, σ) → H^{4}(F, μ_{2}) defined by

θ_{1}(g) = (μ(g)) ∪ e_{3}(T_{σ}^{+} − T_{σ_{0}}^{+})

induces a homomorphism

Θ_{1} : PSim^{+}(A, σ)/R → H^{4}(F, μ_{2}).

Moreover, for all g ∈ Sim(A, σ), we have

θ_{1}(g) ∪ (−1) = 0 in H^{5}(F, μ_{2}).

Proof. In view of the isomorphism (1), it suffices to show that for every finite field extension L/F such that (A, σ) ⊗_{F} L is hyperbolic and for every x ∈ L^{×},

(N_{L/F} (x)) ∪ e_{3}(T_{σ}^{+} − T_{σ_{0}}^{+}) = 0 in H^{4}(F, μ_{2}).

The projection formula yields

(N_{L/F} (x)) ∪ e_{3}(T_{σ}^{+} − T_{σ_{0}}^{+}) = cor_{L/F} ((x) ∪ e_{3}(T_{σ}^{+} − T_{σ_{0}}^{+})_{L}).

Since σ_{L} is hyperbolic, the involutions σ_{L} and (σ_{0})_{L} are conjugate, hence

e_{3}(T_{σ}^{+} − T_{σ_{0}}^{+})_{L} = 0.

For the last equality, observe that (2) yields the following equations in W F:

T_{σ} + T_{A} = ⟨1, 1⟩T_{σ}^{+}  and  T_{σ_{0}} + T_{A} = ⟨1, 1⟩T_{σ_{0}}^{+},

hence

T_{σ} − T_{σ_{0}} = ⟨1, 1⟩(T_{σ}^{+} − T_{σ_{0}}^{+}).

Since σ_{0} is hyperbolic, we have T_{σ_{0}} = 0. Moreover, for g ∈ Sim(A, σ) the map x ↦ gx is a similitude of T_{σ} with multiplier μ(g), hence

⟨1, −μ(g)⟩T_{σ} = ⟨1, −μ(g)⟩⟨1, 1⟩(T_{σ}^{+} − T_{σ_{0}}^{+}) = 0.

Since

e_{5}(⟨1, −μ(g)⟩⟨1, 1⟩(T_{σ}^{+} − T_{σ_{0}}^{+})) = θ_{1}(g) ∪ (−1),

the proposition follows. □

6. Proposition. Let (A, σ) = (B, ρ) ⊗ (C, τ), where B and C are central simple F -algebras of even degree and ρ, τ are involutions of the first kind. Suppose ind B divides \frac{1}{2} deg B and ρ is orthogonal. For g ∈ GO^{−}(B, ρ), we have g⊗1 ∈ Sim^{+}(A, σ) and

θ_{1}(g ⊗ 1) = \begin{cases}0 & if deg C ≡ 0 mod 4, \\ [B] ∪ [C] & if deg C ≡ 2 mod 4. \end{cases}

Proof. For g ∈ GO(B, ρ), we have

σ(g ⊗ 1)g ⊗ 1 = ρ(g)g = μ(g)

and

Nrd_{A}(g ⊗ 1) = Nrd_{B}(g)^{deg C},

so g ⊗ 1 ∈ Sim^{+}(A, σ).

Since ind B divides \frac{1}{2} deg B, we may find a hyperbolic orthogonal involution ρ_{0} on B, and set σ_{0} = ρ_{0} ⊗ τ, a hyperbolic involution on A of the same type as σ.

If τ is symplectic, Lemma 4 yields

e_{3}(T_{σ}^{+} − T_{σ_{0}}^{+}) = \begin{cases}0 & if deg C ≡ 0 mod 4, \\ (disc ρ) ∪ [C] & if deg C ≡ 2 mod 4. \end{cases}

The proposition follows by taking the cup-product with (μ(g)), since (μ(g)) ∪ (disc ρ) = [B] by [9, Theorem A] (see also [4, (13.38)]).

Suppose next τ is orthogonal. By Lemma 4,

e_{3}(T_{σ}^{+} − T_{σ_{0}}^{+}) = \begin{cases}(disc ρ) ∪ (disc τ) ∪ (−1) & if deg C ≡ 0 mod 4, \\ (disc ρ) ∪ ((disc τ) ∪ (−1) + [C]) & if deg C ≡ 2 mod 4. \end{cases}

Using again the equation (μ(g)) ∪ (disc ρ) = [B] and taking into account the equation (−1) ∪ [B] = 0, which follows from [9, Corollary 1.13], we obtain the formula for θ_{1}(g ⊗ 1). □

Using Proposition 6, it is easy to construct examples where θ_{1} ≠ 0. For these examples, the map Θ_{1} of Proposition 5 is not trivial, hence PSim^{+}(A, σ) is not R-trivial.

7. Corollary. Let Q, H be quaternion F -algebras satisfying

(−1) ∪ [H] = 0 in H^{3}(F, μ_{2})  and  [H] ∪ [Q] ≠ 0 in H^{4}(F, μ_{2}).

Let A = M_{2r}(H)⊗M_{s}(Q), where r is arbitrary and s is odd. Let ρ be an orthogonal involution on M_{2r}(H) which admits improper similitudes (see Lemma 1), and let τ be any involution of the first kind on M_{s}(Q). Then PSim^{+}(A, ρ ⊗ τ) is not R-trivial.

To obtain explicit examples, we may take for F the field of rational fractions in four indeterminates F = ℂ(x_{1}, y_{1}, y_{2}, y_{2}) and set H = (x_{1}, y_{1})_{F}, Q = (x_{2}, y_{2})_{F}. Note that the degree of A can be any multiple of 8 and that the conditions on Q and H in Corollary 7 imply ind A = 4. Indeed, if there is a quadratic extension of F which splits Q and H, then [H] ∪ [Q] is a multiple of (−1) ∪ [H].

Other examples can be obtained from Proposition 2.

8. Corollary. Let (B, ρ) be a central simple algebra of degree 4 and index 2 with orthogonal involution of nontrivial discriminant over a field F_{0}. Let F = F_{0}(x, y) be the field of rational fractions in two indeterminates x, y over F_{0}, and let (C, τ) be a central simple F -algebra with involution of the first kind such that

deg C ≡ 2 mod 4  and  [C] = (x) ∪ (y) ∈ H^{2}(F, μ_{2}).

Then PSim^{+}(B ⊗ C, ρ ⊗ τ) is not R-trivial.

Proof. Proposition 2 yields an extension L_{0}/F_{0} such that ρ_{L_{0}} admits an improper similitude g and B_{L_{0}} is not split. Set L = L_{0}(x, y). By Proposition 6,

g ⊗ 1 ∈ Sim^{+}(B ⊗ C, ρ ⊗ τ)(L)  and  θ_{1}(g ⊗ 1) = [B_{L}] ∪ (x) ∪ (y).

Since [B_{L_{0}}] ≠ 0, taking successive residues for the x-adic and the y-adic valuations shows that θ_{1}(g ⊗ 1) ≠ 0. Therefore, PSim^{+}(B ⊗ C, ρ ⊗ τ)(L)/R ≠ 1, hence PSim^{+}(B ⊗ C, ρ ⊗ τ) is not R-trivial. □

4. Involutions of the second kind

We assume in this section that (A, σ) is a central simple algebra with unitary involution over F. In this case, the group of similitudes is connected,

Sim^{+}(A, σ) = Sim(A, σ)  and  PSim^{+}(A, σ) = PSim(A, σ).

We denote by K the center of A and write K = F [X]/(X^{2} − α). We assume the degree of A is even, deg A = n = 2m, and denote by D(A, σ) the discriminant algebra of (A, σ) (see [4, §10] for a definition).

9. Lemma. D(A, σ) is split if (A, σ) is hyperbolic.

Proof. The lemma is clear if A is split, for then σ is adjoint to a hyperbolic hermitian form h and [D(A, σ)] = (α) ∪ (disc h) by [4, (10.35)]. The general case is reduced to the case where A is split by scalar extension to the field of functions L = F (R_{K/F} (SB(A))) of the Weil transfer of the Severi-Brauer variety of A. Indeed, A ⊗_{F} L is split and the scalar extension map Br(F) → Br(L) is injective by [9, Corollary 2.12]. □

10. Proposition. Suppose A^{⊗m} is split. The map θ_{2} : Sim(A, σ) → H^{3}(F, μ_{2}) defined by

θ_{2}(g) = (μ(g)) ∪ [D(A, σ)]

induces a homomorphism

Θ_{2} : PSim(A, σ)/R → H^{3}(F, μ_{2}).

Moreover, for any g ∈ Sim(A, σ),

θ_{2}(g) ∪ (α) = 0 in H^{4}(F, μ_{2}).

Proof. In view of the isomorphism (1), it suffices to show that for every finite field extension L/F such that (A, σ) ⊗_{F} L is hyperbolic and for every x ∈ L^{×},

(N_{L/F} (x)) ∪ [D(A, σ)] = 0 in H^{3}(F, μ_{2}),

and that for every λ ∈ K^{×},

(N_{K/F} (λ)) ∪ [D(A, σ)] = 0 in H^{3}(F, μ_{2}).

As in the proof of Proposition 5, we are reduced by the projection formula to proving that D(A, σ) is split by K and by every extension L/F such that (A, σ)⊗L is hyperbolic. The latter assertion follows from the lemma. On the other hand, by [4, (10.30)] and by the hypothesis on B we have

[D(A, σ)_{K}] = [λ^{m}A] = m[A] = 0.

To prove the last part, we use the trace form T_{σ} defined as in Section 2,

T_{σ}(x) = Trd_{A}(σ(x)x) for x ∈ A,

and its restrictions T_{σ}^{+}, T_{σ}^{−} to Sym(A, σ) and Skew(A, σ) respectively. In the case of involutions of unitary type we have

T_{σ} = T_{σ}^{+} ⊥ T_{σ}^{−} = ⟨1, −α⟩T_{σ}^{+}.

The computation of the Clifford algebra of T_{σ}^{+} in [4, (11.17)] shows that T_{σ} ∈ I^{3}F and

e_{3}(T_{σ}) = (α) ∪ [D(A, σ)].

Now, for g ∈ Sim(A, σ) the map x ↦ gx is a similitude of T_{σ} with multiplier μ(g), hence ⟨1, −μ(g)⟩T_{σ} = 0 in W F. Taking the image under e_{4} yields

0 = (μ(g)) ∪ e_{3}(T_{σ}) = θ_{2}(g) ∪ (α).
□

11. Remarks. (1) If ind A divides \frac{1}{2} deg A, so that A carries a hyperbolic unitary involution σ_{0}, then [4, (11.17)] and Lemma 9 yield

[D(A, σ)] = e_{2}(T_{σ}^{+} − T_{σ_{0}}^{+}).

This observation underlines the analogy between θ_{2} and the map θ_{1} of Proposition 5. Note however that no hypothesis on the index of A is required in Proposition 10.

(2) For g ∈ Sim(A, σ), the equation θ_{2}(g) ∪ (α) = 0 implies that θ_{2}(g) lies in the image of the corestriction map cor_{K/F} : H^{3}(K, μ_{2}) → H^{3}(F, μ_{2}), by [4, (30.12)]. On the other hand, if the characteristic does not divide m, Corollary 1.18 of [8] yields an explicit element ξ ∈ H^{3}(K, μ_{m}^{⊗2}) such that cor_{K/F} (ξ) = θ_{2}(g). In particular, if m is odd it follows that θ_{2} = 0.

The following explicit computation yields examples where θ_{2} ≠ 0.

12. Proposition. Let ι be the nontrivial automorphism of K/F, and assume

(A, σ) = (B, ρ) ⊗_{F} (K, ι)

for some central simple F -algebra with orthogonal involution (B, ρ) of degree n = 2m. Assume m is even. For g ∈ GO^{−}(B, ρ) we have g ⊗ 1 ∈ Sim(A, σ) and

θ_{2}(g ⊗ 1) = (α) ∪ [B].

Proof. For g ∈ GO^{−}(B, ρ),

σ(g ⊗ 1)g ⊗ 1 = ρ(g)g = μ(g),

so g ⊗ 1 ∈ Sim(A, σ). By [4, (10.33)], we have

[D(A, σ)] = m[B] + (α) ∪ (disc ρ).

Since m is even, the first term on the right side vanishes. The proposition follows by taking the cup-product with (μ(g)), since [B] = (μ(g))∪(disc ρ) by [9, Theorem A] (see also [4, (13.38)]). □

Remark. If m is odd in Proposition 12, then the definition of θ_{2} requires the extra hypothesis that B is split by K. Computation then shows that θ_{2}(g ⊗ 1) = 0 for all g ∈ GO^{−}(B, ρ), as follows also from Remark 11.2 above.

13. Corollary. Let r be an arbitrary integer. Let H be a quaternion F -algebra, α ∈ F^{×}, K = F [X]/(X^{2} − α), and let ι be the nontrivial automorphism of K/F. Assume

(−1) ∪ [H] = 0 in H^{3}(F, μ_{2})  and  (α) ∪ [H] ≠ 0 in H^{3}(F, μ_{2}).

Let ρ be an orthogonal involution on M_{2r}(H) which admits improper similitudes (see Lemma 1). Then PSim(M_{2r}(H) ⊗_{F} K, ρ ⊗ ι) is not R-trivial.

As in the previous section (see Corollary 8), alternative examples can be constructed from Proposition 2:

14. Corollary. Let (B, ρ) be a central simple algebra of degree 4 with orthogonal involution over a field F_{0}. Assume B is not split and disc ρ ≠ 1. Let F = F_{0}(x) be the field of rational fractions in one indeterminate over F_{0}, let K = F (\sqrt{x}) and let ι be the nontrivial automorphism of K/F. The group PSim(B ⊗_{F_{0}} K, ρ ⊗ ι) is not R-trivial.

Note that this corollary also follows from [7, Theorem 3].

References

[1] G. Berhuy, M. Monsurrò, J.-P. Tignol, The discriminant of a symplectic involution. To appear in Pacific J. of Math.

[2] J.-L. Colliot-Thélène, J.-J. Sansuc, La R-équivalence sur les tores. Ann. scient. Éc. Norm. Sup., 4^{e} série, 10, 175–230 (1997).

[3] P. Gille, Examples of non-rational varieties of adjoint groups. J. Algebra 193, 728–747 (1997).

[4] M.-A. Knus, A.S. Merkurjev, M. Rost, J.-P. Tignol, The Book of Involutions. Amer. Math. Soc. Coll. Pub. 44, AMS, Providence, RI, 1998.

[5] D.W. Lewis, A note on trace forms and involutions of the first kind. Exposition. Math. 15, 265–272 (1997).

[6] Y. Manin, Cubic forms. North-Holland, Amsterdam, 1974.

[7] A. Merkurjev, R-equivalence and rationality problem for semisimple adjoint classical algebraic groups. Pub. Math. IHES 46, 189–213 (1996).

[8] A. Merkurjev, R. Parimala, J.-P. Tignol, Invariants of quasi-trivial tori and the Rost invariant. Preprint.

[9] A. Merkurjev, J.-P. Tignol, The multipliers of similitudes and the Brauer group of homogeneous varieties. J. reine angew. Math. 461, 13–47 (1995).

[10] A. Quéguiner, Cohomological invariants of algebras with involution. J. Algebra 194, no 1, 299–330 (1997).

[11] V.E. Voskresenkiĭ, A.A. Klyachko, Toroidal Fano varieties and root system. Math. USSR Izvestija 24, 221–244 (1985).

Département de mathématiques, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

E-mail address: gregory.berhuy@epfl.ch, marina.monsurro@epfl.ch

Institut de Mathématique Pure et Appliquée, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

E-mail address: tignol@math.ucl.ac.be