GROTHENDIECK—SERRE CONJECTURE FOR GROUPS OF TYPE F_{4} WITH TRIVIAL f_{3} INVARIANT

V. PETROV AND A. STAVROVA

Abstract. Assume that R is a semi-local regular ring containing an infinite perfect field. Let K be the field of fractions of R. Let H be a simple algebraic group of type F_{4} over R such that H_{K} is the automorphism group of a 27-dimensional Jordan algebra which is a first Tits construction. If char K ≠ 2 this means precisely that the f_{3} invariant of H_{K} is trivial. We prove that the kernel of the map

H_{ét}^{1}(R, H) → H_{ét}^{1}(K, H)

induced by the inclusion of R into K is trivial.

This result is a particular case of the Grothendieck—Serre conjecture on rationally trivial torsors. It continues the recent series of papers [PaSV], [Pa], [PaPS] and complements the result of Chernousov [Ch] on the Grothendieck—Serre conjecture for groups of type F_{4} with trivial g_{3} invariant.

1. Introduction

In the present paper we address the Grothendieck—Serre conjecture [Se, p. 31, Remarque], [Gr, Remarque 1.11] on the rationally trivial torsors of reductive algebraic groups. This conjecture states that for any reductive group scheme G over a regular ring R, any G-torsor that is trivial over the field of fractions K of R is itself trivial; in other words, the natural map

H_{ét}^{1}(R, G) → H_{ét}^{1}(K, G)

has trivial kernel. It has been settled in a variety of particular cases, and we refer to [Pa] for a detailed overview. The most recent result in the area belongs to V. Chernousov [Ch] who has proved that the Grothendieck—Serre conjecture holds for an arbitrary simple group H of type F_{4} over a local regular ring R containing the field of rational numbers, given that H_{K} has a trivial g_{3} invariant. We prove that the Grothendieck—Serre conjecture holds for another natural class of groups H of type F_{4}, those for which H_{K} has trivial f_{3} invariant. In fact, since our approach is characteristic-free, we establish the following slightly more general result.

Theorem 1. Let R be a semi-local regular ring containing an infinite perfect field. Let K be the field of fractions of R. Let J be a 27-dimensional exceptional Jordan algebra over R such that J_{K} is a first Tits construction. Then the map

H_{ét}^{1}(R, Aut (J)) → H_{ét}^{1}(K, Aut (J))

induced by the inclusion of R into K has trivial kernel.

Corollary. Let R be a semi-local regular ring containing an infinite perfect field k such that char k ≠ 2. Let K be the field of fractions of R. Let H be a simple group scheme of type F_{4} over R such that H_{K} has trivial f_{3} invarint. Then the map

H_{ét}^{1}(R, H) → H_{ét}^{1}(K, H)

induced by the inclusion of R into K has trivial kernel.

V. Petrov is supported by Max-Plank-Institut für Mathematik, RFBR 08-01-00756, RFBR 09-01-90304 and RFBR 09-01-91333; V. Petrov and A. Stavrova are supported by RFBR 09-01-00878.

2. Isotopes of Jordan algebras

In the first two sections R is an arbitrary commutative ring.

A (unital quadratic) Jordan algebra is a projective R-module J together with an element 1 ∈ J and an operation

J × J → J
(x, y) ↦ U_{x}y,

which is quadratic in x and linear in y and satisfies the following axioms:

• U_{1} = id_{J};
• {x, y, U_{x}z} = U_{x}{y, x, z};
• U_{U_{x}y} = U_{x}U_{y}U_{x},

where {x, y, z} = U_{x+z}y − U_{x}y − U_{z}y stands for the linearization of U. It is well-known that the split simple group scheme of type F_{4} can be realized as the automorphism group scheme of the split 27-dimensional exceptional Jordan algebra J_{0}. This implies that any other group scheme of type F_{4} is the automorphism group scheme of a twisted form of J_{0}.

Let v be an invertible element of J (that is, U_{v} is invertible). An isotope J^{(v)} of J is a new Jordan algebra whose underlying module is J, while the identity and U-operator are given by the formulas

1^{(v)} = v^{−1};
U_{x}^{(v)} = U_{x}U_{v}.

An isotopy between two Jordan algebras J and J′ is an isomorphism g: J → J^{′(v)}; it follows that v = g(1)^{−1}. We are particularly interested in autotopies of J; one can see that g is an autotopy if and only if

U_{g(x)} = gU_{x}g^{−1}U_{g(1)}

for all x ∈ J. In particular, transformations of the form U_{x} are autotopies. The group scheme of all autotopies is called the structure group of J and is denoted by Str(J). Obviously it contains G_{m} acting on J by scalar transformations.

It is convenient to describe isotopies as isomorphisms of some algebraic structures. This was done by O. Loos who introduced the notion of a Jordan pair. We will not need the precise definition, see [Lo75] for details. It turns out that every Jordan algebra J defines a Jordan pair (J, J), and the isotopies between J and J′ bijectively correspond to the isomorphims of (J, J) and (J′, J′) ([Lo75, Proposition 1.8]). In particular, the structure group Str(J) is isomorphic to Aut ((J, J)). We use this presentation of Str(J) to show that, if J is a 27-dimensional exceptional Jordan algebra, Str(J) can be seen as a Levi subgroup of a parabolic subgroup of type P_{7} (with the enumeration of roots as in [B]) in an adjoint group of type E_{7}. See also Garibaldi [Ga].

Lemma 1. Let J be a 27-dimensional exceptional Jordan algebra over a commutative ring R. There exists an adjoint simple group G of type E_{7} over R such that Str(J) is isomorphic to a Levi subgroup L of a maximal parabolic subgroup P of type P_{7} in G.

Proof. By [Lo78, Theorem 4.6 and Lemma 4.11] for any Jordan algebra J the group Aut ((J, J)) is isomorphic to a Levi subgroup of a parabolic subgroup P of a reductive group PG(J) (not necessarily connected; the definition of a parabolic subgroup extends appropriately). Moreover, PG(J) ≅ Aut (PG(J)/P). If J is a 27-dimensional exceptional Jordan algebra, i.e., an Albert algebra, the group PG(J) is of type E_{7} and P is a parabolic subgroup of type P_{7}. Let G be the corresponding adjoint group of type E_{7}. Then by [Dem, Thérème 1] we have Aut (PG(J)/P) ≅ Aut (G) ≅ G. Hence Aut ((J, J)) is isomorphic to a Levi subgroup of a parabolic subgroup P of type P_{7} in G. □

3. Cubic Jordan algebras and the first Tits construction

A cubic map on a projective R-module V consists of a function N: V → R and its partial polarization ∂N: V × V → R such that ∂N(x, y) is quadratic in x and linear in y, and N is cubic in the following sense:

• N(tx) = t^{3}N(x) for all t ∈ R, x ∈ V;

• N(x + y) = N(x) + ∂N(x, y) + ∂N(y, x) + N(y) for all x, y ∈ V.

These data allow to extend N to V_{S} = V ⊗_{R} S for any ring extension S of R.

A cubic Jordan algebra is a projective module J equipped with a cubic form N, quadratic map #: J → J and an element 1 ∈ J such that for any extension S/R

• (x^{#})^{#} = N(x)x for all x ∈ J_{S};

• 1^{#} = 1; N(1) = 1;
• T (x^{#}, y) = δN(x, y) for all x, y ∈ J_{S};
• 1 × x = T (x)1 − x for all x ∈ J_{S},

where × is the linearization of #, T (x) = ∂N(1, x), T (x, y) = T (x)T (y) − N(1, x, y), N(x, y, z) is the linearization of ∂N.

There is a natural structure of a quadratic Jordan algebra on J given by the formula

U_{x}y = T (x, y)x − x^{#} × y.

Any associative algebra A of degree 3 over R (say, commutative étale cubic algebra or an Azumaya algebra of rank 9) can be naturally considered as a cubic Jordan algebra, with N being the norm, T being the trace, and x^{#} being the adjoint element to x.

Moreover, given an invertible scalar λ ∈ R^{×}, one can equip the direct sum A ⊕ A ⊕ A with the structure of a cubic Jordan algebra in the following way (which is called the first Tits construction):

1 = (1, 0, 0);
N(a_{0}, a_{1}, a_{2}) = N(a_{0}) + λN(a_{1}) + λ^{−1}N(a_{2}) − T (a_{0}a_{1}a_{2});
(a_{0}, a_{1}, a_{2})^{#} = (a_{0}^{#} − a_{1}a_{2}, λ^{−1}a_{2}^{#} − a_{0}a_{1}, λa_{1}^{#} − a_{2}a_{0}).

Now we state a transitivity result (borrowed from [PeR, Proof of Theorem 4.8]) which is crucial in what follows.

Lemma 2. Let E be a cubuc étale extension of R, A is the cubic Jordan algebra obtained by the first Tits construction from E, y be an invertible element of E considered as a subalgebra of A. Then y lies in the orbit of 1 under the action of subgroup of Str(A)(R) generated by G_{m}(R) and elements of the form U_{x}, x is an invertible element of A.

Proof. As an element of A y equals (y, 0, 0). Now a direct calculation shows that

U_{(0, 0, 1)}U_{(0, y, 0)}y = N(y)1.
□

Over a field, Jordan algebras that can be obtained by the first Tits construction can be characterized in terms of cohomological invariants. Namely, to each J one associates a 3-fold Pfister form π_{3}(J), and J is of the first Tits construction if and only if π_{3}(J) is hyperbolic (see [Pe, Theorem 4.10]). Another equivalent description is that J splits over a cubic extension of the base field. If the characteristic of the base field is distinct from 2, π_{3} is also known as the cohomological f_{3} invariant,

f_{3} : H_{ét}^{1}(−, F_{4}) → H^{3}(−, μ_{2}).

4. Springer form

From now on J is a 27-dimensional cubic Jordan algebra over R.

Let E be a cubic étale subalgebra of J. Denote by E^{⊥} the orthogonal complement to E in J with respect to the bilinear form T (it exists for the restriction of T to E is non-degenerate); it is a projective R-module of rank 24. It is shown in [PeR, Proposition 2.1] that the operation

E × E^{⊥} → E^{⊥};
(a, x) ↦ −a × x

equips E^{⊥} with a structure of E-module compatible with its R-module structure. Moreover, if we write

x^{#} = q_{E}(x) + r_{E}(x), q_{E}(x) ∈ E^{⊥}, r_{E}(x) ∈ E,

then q_{E} is a quadratic form on E^{⊥}, which is nondegenerate as one can check over a covering of R splitting J. This form is called the Springer form with respect to E.

The following lemma relates the Springer form and subalgebras of J.

Lemma 3. Let v be an element of E^{⊥} such that q_{E}(v) = 0 and v is invertible in J. Then v is contained in a subalgebra of J obtained by the first Tits construction from E.

Proof. It is shown in [PeR, Proof of Proposition 2.2] that the embedding

(a_{0}, a_{1}, a_{2}) ↦ a_{0} − a_{1} × v − N(v)^{−1}a_{2} × v^{#}

defines a subalgebra desired. □

Recall that the étale algebras of degree n are classified by H^{1}(R, S_{n}), where S_{n} is the symmetric group in n letters. The sign map S_{n} → S_{2} induces a map

H^{1}(R, S_{n}) → H^{1}(R, S_{2})

that associates to any étale algebra E a quadratic étale algebra δ(E) called the discriminant of E. The norm N_{δ(E)} is a quadratic form of rank 2.

Over a field, the Springer form can be computed explicitly in terms of π_{3}(J) and δ(E). We will need the following particular case:

Lemma 4. Let J be a Jordan algebra over a field K with π_{3}(J) = 0. Then

q_{E} = N_{δ(E)}_{E} ⊥ h_{E} ⊥ h_{E} ⊥ h_{E},

h stands for the hyperbolic form of rank 2.

Proof. Follows from [PeR, Theorem 3.2]. □

We will also use the following standard result.

Lemma 5. Let J be a Jordan algebra over an algebraically closed field F. Then any two cubic étale subalgebras E and E′ of J are conjugate by an element of Aut (J)(F).

Proof. Present E as F e_{1} ⊕ F e_{2} ⊕ F e_{3}, where e_{i} are idempotents whose sum is 1; do the same with E′. By [Lo75, Theorem 17.1] there exists an element g ∈ Str(J)(F) such that ge_{i} = e′_{i}. But then g stabilizes 1, hence belongs to Aut (J)(F). □

5. Proof of Theorem 1

Proof of Theorem 1. Set H = Aut (J). It is a simple group of type F_{4} over R. We may assume that H_{K} is not split, otherwise the result follows from [Pa, Theorem 1.0.1]. Let J be the Jordan algebra corredponding to H; we have to show that if J′ is a twisted form of J such that J′_{K} ≃ J_{K} then J′ ≃ J. Set L = Str(J); then L is a Levi subgroup of a parabolic subgroup of type P_{7} of an adjoint simple group scheme G of type E_{7} by Lemma 1. By [SGA, Exp. XXVI Cor. 5.10 (i)] the map

H_{ét}^{1}(R, L) → H_{ét}^{1}(K, G)

is injective. Since G is isotropic, by [Pa, Theorem 1.0.1] the map

H_{ét}^{1}(R, G) → H_{ét}^{1}(K, G)

has trivial kernel, and so does the map

H_{ét}^{1}(R, L) → H_{ét}^{1}(K, L).

But (J′_{K}, J′_{K}) ≃ (J_{K}, J_{K}), therefore (J′, J′) ≃ (J, J), that is J′ is isomorphic to J^{(y)} for some invertible y ∈ J. It remains to show that y lies in the orbit of 1 under the action of Str(J)(R).

Present the quotient of R by its Jacobson radical as a direct product of the residue fields ∏ k_{i}. An argument in [PeR, Proof of Theorem 4.8] shows that for each i one can find an invertible element v_{i} ∈ J_{k_{i}} such that the discriminant of the generic polynomial of U_{v_{i}}y_{k_{i}} is nonzero. Lifting v_{i} to an element v ∈ J and changing y to U_{v}y we may assume that the generic polynomial f(T) ∈ R[T] of y has the property that R[T]/(f(T)) is an étale extension of R. In other words, we may assume that y generates a cubic étale subalgebra E in J.

Note that E_{K} is a cubic field extension of K; otherwise J_{K} is reduced, hence split, for π_{3}(J_{K}) = 0 (see [Pe, Theorem 4.10]). Consider the form

q = N_{δ(E)}_{E} ⊥ h_{E} ⊥ h_{E} ⊥ h_{E};

then by Lemma 4 q_{K} = q_{E_{K}}. By the analog of the Grothendieck—Serre conjecture for étale quadratic algebras (which follows, for example, from [EGA2, Corollaire 6.1.14]), q and q_{E} have the same discriminant. So q_{E} is a twisted form of q given by a cocycle ξ ∈ H^{1}(E, SO(q)). Now ξ_{K} is trivial, and [Pa, Theorem 1.0.1] imply that ξ is trivial itself, that is q_{E} = q. In particular, q_{E} is isotropic. Let us show that there is an invertible element v in J such that q_{E}(v) = 0.

The projective quadric over E defined by q_{E} is isotropic, hence has an open subscheme U ≃ 𝔸_{E}^{n}. Denote by U′ the open subscheme of R_{E/R}(U) consisting of invertible elements. It suffices to show that U′(k_{i}) is non-empty for each i, or, since the condition on R implies that k_{i} is infinite, that U′(k̄_{i}) is non-empty.

But Jk̄_{i} splits, and, in particular, it is obtained by a first Tits construction from a split Jordan algebra of 3 × 3 matrices over k̄_{i}. The diagonal matrices in this matrix algebra constitute a cubic étale subalgebra of Jk̄_{i}. By Lemma 5 we may assume that this étale subalgebra coincides with Ek̄_{i}. By [PeR, Proposition 2.2] there exists an invertible element v_{i} ∈ E⊥ ; k̄_{i} such that q_{Ek̄_{i}} (v_{i}) = 0. Thus the scheme of invertile elements intersects the quadric over k̄_{i}, hence, U′(k̄_{i}) is non-empty.

Finally, Lemma 3 and Lemma 2 show that y belongs to the orbit of 1 under the group generated by G_{m}(R) and elements of the form U_{x}. So J′ ≃ J^{(y)} ≃ J, and the proof is completed. □

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