A (MOD 3) INVARIANT FOR EXCEPTIONAL JORDAN ALGEBRAS

MARKUS ROST

Abstract — We sketch the construction of an invariant in H^{3}(F, Z/3) for simple exceptional Jordan algebras over a field F with Char F ≠ 2, 3. This invariant vanishes if and only if the algebra has zero divisors. In case μ_{3} ⊂ F^{∗} the invariant can be lifted to Milnor’s K-group K_{3}^{M}F/3.

Un invariant (mod 3) pour les algb̀res de Jordan exceptionelles

Résumé — Nous décrivons la construction d’une invariante dans H^{3}(F, Z/3) pour des algèbres de Jordan simples exceptionelles sur un corps F de caracteristique ≠ 2, 3. Cette invariante est 0 si et seulement si l’algèbre a des diviseurs des zero. Lorsque F contient les racine cubiques de l’unité, cet invariante provient d’un invariant à valeurs dans le groupe de Milnor K_{3}^{M}F/3.

Version Française Abrégée — Soit F un corps de caractéristique ≠ 2, 3. Soit J une F -algèbre de Jordan centrale simple exceptionnelle, de dimension 27. Nous décrivons la construction d’un invariant (à isomorphisme près) g(J) ∈ H^{3}(F, Z/3). Cet invariant est 0 si et seulement si J a des diviseurs de zéro. Le foncteur J ↦ g(J) définit une application

H^{1}(F, G) → H^{3}(F, Z/3),

oú G est le groupe des automorphismes de l’algèbre de Jordan exceptionnelle décomposée (i.e. un groupe de type F_{4} déployé). L’éxistence d’une telle application avait été conjecturée par J.-P. Serre [3].

Cet invariant a des relations intéressantes avec le symbole galoisien

h: K_{3}^{M}F/3 → H^{3}(F, μ_{3}^{⊗3}).

Lorsque F^{∗} contient μ_{3} on peut relever g à K_{3}^{M}F/3, et en déduire que l’application h ci-dessus est injective sur le sous-ensemble des symboles de K_{3}^{M}F/3.

Dans ce qui suit, nous donnons le principe de la construction de g; un exposé plus détaillé est en préparation.

On considère d’abord des couples (J, B), oú B est une sous-algèbre de Jordan de J, de dimension 9, isomorphe à M_{3}(F)^{+} sur une extension convenable de F. Par une analyse de la ^{≪} première construction de Tits^{≫}, on obtient un invariant de couples à isomorphisme près :

g(J, B) ∈ H^{3}(F, Z/3).

Soit alors

g(J) = g(J ⊗ H, B_{0}) ∈ H^{3}(H, Z/3),

The text appeared as: C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 12, 823–827.

oú B_{0} est une sous-algèbre générique de dimension 9 ayant le corps H pour corps de définition. On montre que ∂_{v}(g(J)) = 0 pour tous les homomorphismes bords ∂_{v} attachés aux valuations discrètes v de H triviales sur F. Comme on peut supposer que H est une extension transcendante pure de F, cette propriété entraîne que g(J) appartient au sous-groupe H^{3}(F, Z/3) de H^{3}(H, Z/3). Par spécialisation, on en déduit que g(J) = g(J, B) pour tout B, d’oú l’invariant cherché.

1. Introduction

Let F be a field of characteristic ≠ 2, 3. Let J be an exceptional Jordan algebra over F (we assume throughout J to be central simple and finite dimensional, so that dim J = 27). We sketch the construction of an invariant up to isomorphism:

g(J) ∈ H^{3}(F, Z/3).

Its basic property is that g(J) = 0 if and only if J has zero divisors. g gives rise to a functorial map

H^{1}(F, G) → H^{3}(F, Z/3)

where G is the automorphism group of the split exceptional Jordan algebra. G is of type F_{4} and is isomorphic to the automorphism group of the cubic form in 26 variables

Ñ(x, y, z) = det x + det y + det z − trace xyz

where x, y, z are 3 × 3 matrices with trace x = 0 ([1], Chapt. VI, Theorem 7). The invariant is also of interest for questions related with the Galois symbol

h: K_{3}^{M}F/3 → H^{3}(F, μ_{3}^{⊗3}).

In case μ_{3} ⊂ F^{∗} the invariant can be lifted to K_{3}^{M}F/3 and one can use this to show that h is injective when restricted to the subset of symbols in K_{3}^{M}F/3.

In this note we just describe briefly the principle of the construction of g; a more detailed paper is in preparation.

The existence of the invariant g has been suggested by J.-P. Serre [3]. I am indebted to J.-P. Serre for pointing out the relation between exceptional Jordan algebras and 3-symbols mod 3 and to T. A. Springer for introducing me to the theory of Jordan algebras.

2. Tits’ first construction

As general reference for Jordan algebras within this Note see [1], [4]. For an associative algebra A one denotes by A^{+} the Jordan algebra with the same underlying space with Jordan product

a . b = \frac{1}{2}(ab + ba).

Let F be a field (Char F ≠ 2, 3), let A be a central simple algebra over F with dim_{F} A = 9 and let μ ∈ F^{∗}. In Tits’ first construction one defines an exceptional central simple Jordan algebra (A, μ) with underlying space A_{0} ⊕ A_{1} ⊕ A_{2} by an explicit multiplication table, where the A_{i} are three copies of A (see [1], Ch. IX, Sect. 12). This multiplication provides A_{0} with the Jordan product of A^{+}, so that A^{+} is a canonical Jordan subalgebra of (A, μ). The trace T_{J} and the norm N_{J} on J = (A, μ) are given by

T_{J}(a_{0}, a_{1}, a_{2}) = T_{A}(a_{0}),
N_{J}(a_{0}, a_{1}, a_{2}) = N_{A}(a_{0}) + μN_{A}(a_{1}) + μ^{−1}N_{A}(a_{2}) − T_{A}(a_{0}a_{1}a_{2}),

where T_{A} and N_{A} are the (reduced) trace and norm of A respectively ([4], 5.10).

The algebra (A, μ) is an exceptional Jordan algebra over F. It has zero divisors if and only if its normform has a nontrivial zero and if and only if μ ∈ N_{A}(A^{∗}) ([1], p. 227 and p. 416).

We consider pairs (J, B) where J is an exceptional Jordan algebra over F and B ⊂ J is a Jordan subalgebra which is isomorphic to M_{3}(F)^{+} over the algebraic closure of F. Such subalgebras always exist ([1], Ch. IX, Lemma 2). One knows that over a separably closed field all such pairs are isomorphic ([1], Ch. IX, in particular Theorem 3). In the following we give an explicit recipe how to identify any (J, B) with some ((A, μ), A^{+}) at least after replacing the base field by a quadratic extension. Let

B^{⊥} = { x ∈ J | T_{J}(b . x) = 0 for b ∈ B }.

Then B . B^{⊥} = B^{⊥} and J = B ⊕ B^{⊥} ([1], p. 227 and p. 240). We define C ⊂ End_{F} (B^{⊥}) as the subalgebra generated by all endomorphisms

B^{⊥} → B^{⊥}, x ↦ b . x

with b ∈ B.

Proposition 1. The centre K of C is a separable quadratic extension of F and C is an Azumaya algebra over K of rank 9.

We omit a proof here. However note that it suffices to consider a separably closed field F in which case (J, B) is isomorphic to the explicit model

((M_{3}(F), 1), M_{3}(F)^{+}).

A similar remark holds for proposition 2 below.

Let p_{K} ∈ K ⊗ K be the canonical idempotent, that is (1 − p_{K})K ⊗ K is the kernel of the multiplication map K ⊗K → K. Then p_{K} ∈ K ⊗K ⊂ End_{K}(B^{⊥} ⊗K) and there is a decomposition

J ⊗ K = B ⊗ K ⊕ p_{K}(B^{⊥} ⊗ K) ⊕ (1 − p_{K})(B^{⊥} ⊗ K).

This can be identified with the decomposition occuring in Tits’ first construction as follows.

Proposition 2. There exist Z ∈ p_{K}(B^{⊥} ⊗ K) such that λ = N_{J⊗K}(Z) ∈ K is a unit. For any such Z one has

(i) p_{K}(B^{⊥} ⊗ K) = C . Z and N_{J⊗K}(p_{K}(B^{⊥} ⊗ K)) = N_{C}(C^{∗}) . λ. (ii) There is a unique isomorphism

ψ: J ⊗ K → (C, λ)

of Jordan algebras over K with ψ(B ⊗ K) = C^{+} such that ψ(Z) = (0, 1, 0) and ψ(b) = φ(b) for b ∈ B, where

φ: B → C ⊂ End_{F} (B^{⊥})

is given by φ(b)(x) = (T_{J}(b) − 2b) . x.

3. The invariant g(J)

As usual let

(c) ∈ H^{1}(F, μ_{3}) = F^{∗}/(F^{∗})^{3} and [A] ∈ H^{2}(F, μ_{3}) ⊂ Br(F)

be the classes of c ∈ F^{∗} and a 9-dimensional algebra A respectively. Note that μ_{3} ⊗ μ_{3} = Z/3 as Galois module.

Theorem. There is an invariant up to isomorphism

g(J) ∈ H^{3}(F, Z/3)

which is natural with respect to base change and such that

g((A, μ)) = [A] ∪ (μ).

By a result of A. S. Merkuriev and A. A. Suslin [2] one has [A] ∪ (μ) = 0 if and only if μ ∈ N_{A}(A^{∗}). Moreover a cubic form has a nontrivial zero if this holds over a quadratic extension. From this and the remarks above it follows that g(J) = 0 if and only if J has zero divisors.

A proof of the theorem can be briefly sketched as follows. One first defines a relative invariant for the pairs (J, B) by the formula

g(J, B) = − cor_{K/F} ([C] ∪ (λ)) ∈ H^{3}(F, Z/3)

where C, K, λ are as in proposition 1 and proposition 2. This invariant is welldefined by proposition 2 (i). One can show that [C]∪(λ) is invariant under Aut(K/F) and therefore

[C] ∪ (λ) = g(J, B)_{K}.

For a pair obtained from Tits’ first construction it turns out that

g((A, μ), A^{+}) = [A] ∪ (μ).

The invariant g(J, B) does not depend on the choice of B. To prove this one uses the following two lemmas.

Lemma 1. For A, μ as above let Σ be the norm variety over F with coordinates x ∈ A and equation N_{A}(x) = μ. Then the restriction map

r: H^{2}(F, μ_{3}) → H^{2}(F (Σ), μ_{3})

is injective.

Proof (see also [5]). We may assume

μ_{3} ⊂ F^{∗} and [A] = (λ) ∪ (ρ)

in H^{2}(F, μ_{3}) ≃ H^{2}(F, Z/3) for some λ, ρ ∈ F^{∗}. Let ν = λ^{k}ρ^{ℓ} with (k, ℓ) ∉ 3Z × 3Z and let Ω be the Brauer-Severi variety for the algebra with invariant (ν)∪(μ). Then Σ has a F (Ω)-rational point. Hence the kernel of r is contained in the kernel of

H^{2}(F, μ_{3}) → H^{2}(F (Ω), μ_{3})

which is generated by (ν) ∪ (μ). However the intersection of these groups over all ν is trivial.

Lemma 2. Let H/F be an extension of fields and let v: H^{∗} → Z be a discrete valuation trivial on F. Then for J over F and a pair (J ⊗ H, B) over H the element g(J ⊗ H, B) vanishes under the boundary map

∂_{v}: H^{3}(H, Z/3) → H^{2}(κ(v), μ_{3}).

Proof. We may assume that J = (A, μ) after replacing F by some quadratic extension, see proposition 2. By lemma 1 we may replace F by F (Σ). But then J has zero divisors and g(J ⊗ H, B) = 0.

Now let X, Y be two independent generic elements of J; their field of definition H is rational in a doubled set of coordinate functions on J. Let B_{0} ⊂ J ⊗ H be the Jordan subalgebra generated by X and Y. Then (J ⊗ H, B_{0}) is a pair over H as above ([1], Ch. IX, Lemma 2). Since H is rational over F it follows from lemma 2 that g(J ⊗ H, B_{0}) is in the image of the injective restriction map

H^{3}(F, Z/3) → H^{3}(H, Z/3).

Now take for g(J) the preimage of g(J⊗H, B_{0}). By construction g(J) is an invariant of J up to isomorphism. Using an appropriate specialization one finds g(J) = g(J, B) for any choice of B.

References

[1] N. Jacobson, Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications XXXIX, Amer. Math. Soc., Providence, R. I., 1968.

[2] A. S. Merkurjev and A. A. Suslin, K-Cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk S.S.S.R., 46, 1982, pp. 1011–1046 (=Math. U.S.S.R. Izvestiya, 21, 1983, pp. 307-340).

[3] J.-P. Serre, Résumé des cours de l’année 1990–1991, Collège de France.

[4] T. A. Springer, Jordan algebras and algebraic groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 75, Springer-Verlag, 1973.

[5] A. A. Suslin, K-Theory and K-cohomology of certain group varieties, Algebraic K-theory, Adv. Soviet Math. 4, Amer. Math. Soc., Providence, R. I., 1991, pp. 53–74

NWF I - Mathematik, Universität Regensburg, D-93040 Regensburg, Germany

   E-mail address: markus.rost@mathematik.uni-regensburg.de
   URL: http://www.physik.uni-regensburg.de/~rom03516