<< . .

. 68
( : 75)



. . >>

“Freudenthal algebras,” a class of Jordan algebras connected with Hurwitz algebras
and we describe the Springer construction, which ties twisted compositions with cu-
bic Jordan algebras. On the other hand, cubic Jordan algebras are also related to
cubic associative algebras through the Tits constructions (§??). Of special interest,
and the main object of study of this chapter, are the exceptional simple Jordan al-
gebras of dimension 27, whose automorphism groups are of type F4 . The di¬erent
constructions mentioned above are related to interesting subgroups of F4 . For ex-
ample, the automorphism group of a split twisted composition is a subgroup of F4
and outer actions on Spin8 (triality!) become inner over F4 . Tits constructions
are related to the action of the cyclic group Z/3Z on Spin 8 which yields invariant
subgroups of classical type A2 , and Freudenthal algebras are related to the action
of the group S3 on Spin8 which yields invariant subgroups of exceptional type G2 .
Cohomological invariants of exceptional simple Jordan algebras are discussed
in the last section.

§37. Jordan algebras
We assume in this section that F is a ¬eld of characteristic di¬erent from 2. A
Jordan algebra J is a commutative ¬nite dimensional unitary F -algebra such that
the multiplication (a, b) ’ a q b satis¬es
(37.1) (a q a) q b q a = (a q a)(b q a)
for all a, b ∈ J. For any associative algebra A, the product
a q b = 1 (ab + ba)
2

gives A the structure of a Jordan algebra, which we write A+ . If B is an associative
algebra with involution „ , the set Sym(B, „ ) of symmetric elements is a Jordan
subalgebra of B + which we denote H(B, „ ).
Observe that A+ H(B, „ ) if B = A — Aop and „ is the exchange involution.
A Jordan algebra A is special if there exists an injective homomorphism A ’
+
D for some associative algebra D and is exceptional otherwise.
A Jordan algebra is strictly power-associative and we write an for the nth power
of an element a. Hence it admits a generic minimal polynomial
PJ,x (X) = X m ’ s1 (x)X m’1 + · · · + (’1)m sm (x)1,
511
512 IX. CUBIC JORDAN ALGEBRAS


where TJ = s1 is the generic trace and NJ = sm the generic norm. The bilinear
trace form T (x, y) = TJ (xy) is associative (see Corollary (??)). By Dieudonn´™s
e
theorem (??), a Jordan algebra is separable if T is nonsingular. The converse is a
consequence of the following structure theorem:
(37.2) Theorem. (1) Any separable Jordan F -algebra is the product of simple
Jordan algebras whose centers are separable ¬eld extensions of F .
(2) A central simple Jordan algebra is either
(a) the Jordan algebra of a nondegenerate quadratic space of dimension ≥ 2,
(b) a Jordan algebra H(B, „ ) where B is associative and K-central simple as
an algebra with involution „ , and where K is either quadratic ´tale and „
e
is unitary with respect to K or K = F and „ is F -linear, or
(c) an exceptional Jordan algebra of dimension 27.
Reference: (??) is [?, Theorem 4, p. 239], and (??) (which goes back to Albert [?])
follows from [?, Corollary 2, p. 204] and [?, Theorem 11, p. 210]. We de¬ne and
discuss the di¬erent types occurring in (??) in the following sections.
Let Sepjord n (m) be the groupoid of separable Jordan F -algebras of dimension n
and degree m with isomorphisms as morphisms.
37.A. Jordan algebras of quadratic forms. Let (V, q) be a nonsingular
¬nite dimensional quadratic space with polar bq (x, y) = q(x + y) ’ q(x) ’ q(y). We
de¬ne a multiplication on J(V, q) = F • V by setting
1
(», v) q (µ, w) = »µ + 2 bq (v, w), »w + µv
for v, w ∈ V and », µ ∈ F . The element (1, 0) is an identity and the canonical
embedding of J(V, q) = F q 1•V into the Cli¬ord algebra C(V, q) shows that J(V, q)
is a Jordan algebra (and is special). The generic minimal polynomial of J(V, q) is
PJ,a (X) = X 2 ’ 2ξX + ξ 2 ’ q(v) 1
where a = (ξ, v) ∈ F q 1 • V , hence J(V, q) has degree 2, the trace is given by
TJ (ξ, v) = 2ξ and the norm by NJ (ξ, v) = ξ 2 ’ q(v). Thus NJ is a nonsingular
quadratic form. The bilinear trace form T : (x, y) ’ TJ (x q y) is isomorphic to
2 ⊥ bq , furthermore T is associative, hence by (??) J is separable if and only if
q is nonsingular. We set J : Qn ’ Sepjord n+1 (2) for the functor (V, q) ’ J(V, q).
Let J be a separable Jordan algebra of degree 2, with generic minimal polynomial
PJ,a (X) = X 2 ’ TJ (a)X + NJ (a)1.
Linearizing and taking traces shows that
2TJ (x q y) ’ 2TJ (x)TJ (y) + 2bNJ (x, y) = 0,
with bNJ the polar of NJ ; hence
(37.3) bNJ (x, y) = TJ (x)TJ (y) ’ TJ (x q y).
For J 0 = { x ∈ J | TJ (x) = 0 }, we have an orthogonal decomposition
J = F q 1 ⊥ J0
with respect to the bilinear trace form T as well as with respect to NJ and, in view
of (??), NJ is nonsingular on J 0 if and only if T is nonsingular on J 0 if and only
if J is separable. Let
Q : Sepjord n+1 (2) ’ Qn
§37. JORDAN ALGEBRAS 513


be the functor given by J ’ (J 0 , ’NJ ).
(37.4) Proposition. The functors J and Q de¬ne an equivalence of groupoids
Qn ≡ Sepjord n+1 (2).
In particular we have Autalg J(V, q) = O(V, q), so that Jordan algebras of type
Sepjord n+1 (2) are classi¬ed by H 1 (F, On ).
Proof : The claim follows easily from the explicit de¬nitions of J and Q.

(37.5) Remark. If dim V ≥ 2, J(V, q) is a simple√ Jordan algebra. If dim V = 1,
J(V, q) is isomorphic to the quadratic algebra F ( ») = F (X)/(X 2 ’ ») where
q ».
We next consider Jordan algebras of degree ≥ 3 and begin with Jordan algebras
associated to central simple algebras with involution.

37.B. Jordan algebras of classical type. Let K be an ´tale quadratic
e
algebra over F with conjugation ι or let K = F and ι = 1. Let (B, „ ) be a
K-central simple algebra with „ an ι-linear involution. As in Chapter ?? we denote
the groupoids corresponding to di¬erent types of involutions by An , Bn , Cn , and Dn .
We set A+ , Bn , Cn , resp. Dn for the groupoids of Jordan algebras whose objects
+ + +
n
are sets of symmetric elements H(B, „ ) for (B, „ ) ∈ A, B, C , resp. D. For each of
these categories A, B, C , D, we have functors S : A ’ A+ , . . . , D ’ D + induced
by (B, „ ) ’ H(B, „ ).
(37.6) Proposition. Let B, B be K-central simple with involutions „ , „ , of de-

gree ≥ 3. Any isomorphism H(B, „ ) ’ H(B , „ ) of Jordan algebras extends to a


unique isomorphism (B, „ ) ’ (B , „ ) of K-algebras with involution. In particular

H(B, „ ) and H(B , „ ) are isomorphic Jordan algebras if and only if (B, „ ) and
(B , „ ) are isomorphic as K-algebras with involution and the functor S induces an
isomorphism of corresponding groupoids.
Reference: See Jacobson [?, Chap. V, Theorem 11, p. 210].

Thus, in view of Theorem (??), the classi¬cation of special central simple
Jordan algebras of degree ≥ 3 is equivalent to the classi¬cation of central simple
associative algebras with involution of degree ≥ 3.
If (B, „ ) is a central simple algebra with a unitary involution over K, we have
an exact sequence of group schemes
Z/2Z) ’ 1
1 ’ AutK (B, „ ) ’ Aut(B, „ ) ’ Autalg (K)(
Thus there is a sequence
1 ’ AutK (B, „ ) ’ Aut H(B, „ ) ’ Z/2Z ’ 1.
If B = A — Aop and „ is the exchange involution, we obtain
1 ’ Aut(A) ’ Aut(A+ ) ’ Z/2Z ’ 1
(37.7)
and the sequence splits if A admits an anti-automorphism. The group scheme
Aut H(B, „ ) is smooth in view of Proposition (??), (??), since its connected
component PGU(B, „ ) = AutK (B, „ ) is smooth. Thus Aut(A+ ) is smooth too.
514 IX. CUBIC JORDAN ALGEBRAS


37.C. Freudenthal algebras. Let C be a Hurwitz algebra with norm NC
and trace TC over a ¬eld F of characteristic not 2 and let
Mn (C) = Mn (F ) — C.
For X = (cij ) ∈ Mn (C), let X = (cij ) where c ’ c, c ∈ C, is conjugation. Let
¯
± = diag(±1 , ±2 , . . . , ±n ) ∈ GLn (F ). Let
t
Hn (C, ±) = { X ∈ Mn (C) | ±’1 X ± = X }.
Let n ≥ 3. If C is associative, Hn (C, ±) and twisted forms of Hn (C, ±) are Jordan
1
algebras of classical type for the product X q Y = 2 (XY + Y X) where XY is the
usual matrix product. In particular they are special. If n = 3 and C = C is a Cayley
algebra, H3 (C, ±) (and twisted forms of H3 (C, ±)) are Jordan algebras for the same
multiplication (see for example Jacobson [?, Chap. III, Theorem 1, p. 127]). For
n = 2 we still get Jordan algebras since H2 (C, ±) can be viewed as a subalgebra
of H3 (C, ±) with respect to a Peirce decomposition ([?, Chap. III, Sect. 1]) relative
to the idempotent diag(1, 0, 0). In fact, H2 (C, ±) is, for any Hurwitz algebra C,
separable of degree 2 hence special (see Exercise 3). However the algebra H 3 (C, ±)
and twisted forms of H3 (C, ±) are exceptional Jordan algebras (Albert [?]). In fact
they are not even homomorphic images of special Jordan algebras (Albert-Paige
[?] or Jacobson [?, Chap. I, Sect. 11, Theorem 11]). Conversely, any central simple
exceptional Jordan algebra is a twisted form of H3 (C, ±) for some Cayley algebra
C (Albert [?, Theorem 17]).
The elements of J = H3 (C, ±) can be represented as matrices
« 
±’1 ±3 c2
ξ1 c3 ¯
1
a = ±’1 ±1 c3 c1  , ci ∈ C, ξi ∈ F
(37.8) ¯ ξ2
2
’1
c2 ±3 ±2 c 1
¯ ξ3
and the generic minimal polynomial is (Jacobson [?, p. 233]):
PJ,a (X) = X 3 ’ TJ (a)X 2 + SJ (a)X ’ NJ (a)1
where
TJ (a) = ξ1 + ξ2 + ξ3 ,
SJ (a) = ξ1 ξ2 + ξ2 ξ3 + ξ1 ξ3 ’ ±’1 ±2 NC (c1 ) ’ ±’1 ±3 NC (c2 ) ’ ±’1 ±1 NC (c3 ),
3 1 2

NJ (a) = ξ1 ξ2 ξ3 ’ ±’1 ±2 ξ1 NC (c1 ) ’ ±’1 ±3 ξ2 NC (c2 ) ’ ±’1 ±1 ξ3 NC (c3 )
3 1 2
+ TC (c3 c1 c2 ).


Let
« 
¯
±’1 ±3 d2
·1 d3 1
¯
b = ± 2 ± 1 d 3 d1  ,
’1
di ∈ C, ·i ∈ F.
·2
¯
’1
d2 ±3 ±2 d 1 ·3
Let bC be the polar of NC . The bilinear trace form T : (a, b) ’ TJ (a q b) is given by
T (a, b) =
ξ1 ·1 + ξ2 ·2 + ξ3 ·3 + ±’1 ±2 bC (c1 , d1 ) + ±’1 ±3 bC (c2 , d2 ) + ±’1 ±1 bC (c3 , d3 )
3 1 2
or
T = 1, 1, 1 ⊥ bC — ±’1 ±2 , ±’1 ±3 , ±’1 ±1 ,
(37.9) 3 1 2
§37. JORDAN ALGEBRAS 515


Thus T is nonsingular. The quadratic form SJ is the quadratic trace which is a
regular quadratic form. Furthermore one can check that
(37.10) bSJ (a, b) = TJ (a)TJ (b) ’ TJ (a q b).
We have T (1, 1) = 3; hence there exists an orthogonal decomposition
H3 (C, ±) = F · 1 ⊥ H3 (C, ±)0 , H3 (C, ±)0 = { x ∈ H3 (C, ±) | TJ (x) = 0 }
if char F = 3.
We call Jordan algebras isomorphic to algebras H3 (C, ±), for some Hurwitz
algebra C, reduced Freudenthal algebras and we call twisted forms of H 3 (C, ±)
Freudenthal algebras. If we allow C to be 0 in H3 (C, ±), the split cubic ´tale algebra
e
F — F — F can also be viewed as a special case of a reduced Freudenthal algebra.
Hence cubic ´tale algebras are Freudenthal algebras of dimension 3. Furthermore, if
e
char F = 3, it is convenient to view F as a Freudenthal algebra with norm NF (x) =
x3 . Freudenthal algebras H3 (C, s), with C = 0 or C a split Hurwitz algebra and
s = diag(1, ’1, 1) are called split. A Freudenthal algebra can have dimension 1, 3,
6, 9, 15, or 27. In dimension 3 Freudenthal algebras are commutative cubic ´tale F -
e
algebras and in dimension greater than 3 central simple Jordan algebras of degree 3
over F . The group scheme G of F -automorphisms of the split Freudenthal algebra
of dimension 27 is simple exceptional split of type F4 (see Theorem (??)). Since
the ¬eld extension functor j : F4 (F ) ’ F4 (Fsep ) is a “-embedding (see the proof of
Theorem (??)) Freudenthal algebras of dimension 27 (which are also called Albert
algebras) are classi¬ed by H 1 (F, G):
(37.11) Proposition. Let G be a simple split group of type F4 . Albert algebras (=
simple exceptional Jordan algebras of dimension 27) are classi¬ed by H 1 (F, G).
It is convenient to distinguish between Freudenthal algebras with zero divisors
and Freudenthal algebras without zero divisors (“division algebras”).
(37.12) Theorem. Let J be a Freudenthal algebra.
(1) If J has zero divisors, then J F — K, K a quadratic ´tale F -algebra, if
e
dimF J = 3, and J H3 (C, ±) for some Hurwitz algebra C, i.e., J is reduced if
dimF J > 3. Moreover, a Freudenthal algebra J of degree > 3 is reduced if and
only if J contains a split ´tale algebra L = F — F — F . More precisely, if e i ,
e
i = 1, 2, 3, are primitive idempotents generating L, then there exist a Hurwitz
algebra C, a diagonal matrix ± = diag(±1 , ±2 , ±3 ) ∈ GL3 (F ) and an isomorphism

φ : J ’ H3 (C, ±) such that φ(ei ) = Eii .

(2) If J does not have zero divisors, then either J = F + (if char F = 3), J = L+ for
a cubic (separable) ¬eld extension L of F , J = D + for a central division algebra D,
J = H(B, „ ) for a central division algebra B of degree 3 over a quadratic ¬eld
extension K of F and „ a unitary involution or J is an exceptional Jordan division
algebra of dimension 27 over F .
Reference: The ¬rst part of (??) and the last claim of (??) follow from the clas-
si¬cation theorem (??) and the fact, due to Schafer [?], that Albert algebras with
zero divisors are of the form H3 (C, ±). The last claim in (??) is a special case of
the coordinatization theorem of Jacobson [?, Theorem 5.4.2].
In view of a deep result of Springer [?, Theorem 1, p. 421], the bilinear trace
form is an important invariant for reduced Freudenthal algebras. The result was
generalized by Serre [?, Th´or`me 10] and Rost as follows:
ee
516 IX. CUBIC JORDAN ALGEBRAS


(37.13) Theorem. Let F be a ¬eld of characteristic not 2. Let J, J be reduced
Freudenthal algebras. Let T , resp. T , be the corresponding bilinear trace forms.
The following conditions are equivalent:
(1) J and J are isomorphic.
(2) T and T are isometric.
Furthermore, if (??) (or (??)) holds, J H3 (C, ±) and J H3 (C , ± ), then C
and C are isomorphic.
Proof : We may assume that J = H3 (C, ±) and J = H3 (C , ± ) with C, C = 0.
(??) implies (??) by uniqueness of the generic minimal polynomial. Assume now
that T and T are isometric. The bilinear trace of H3 (C, ±) is of the form
T = 1, 1, 1 ⊥ bC — ±’1 ±2 , ±’1 ±3 , ±’1 ±1
3 1 2
and a similar formula holds for T . Thus
’1 ’1 ’1
bC — ±’1 ±2 , ±’1 ±3 , ±’1 ±1
(37.14) bC — ±3 ±2 , ±1 ±3 , ±2 ±1 .
3 1 2

We show in the following Lemma (??) that (??) implies NC NC , hence C
C holds by Proposition (??), and we may identify C and C . Assume next
that C is associative. By Jacobson [?], (??) implies that the C-hermitian forms
’1 ’1 ’1
±’1 ±2 , ±’1 ±3 , ±’1 ±1 C and ±3 ±2 , ±1 ±3 , ±2 ±1 C are isometric. They are
3 1 2
similar to ±1 , ±2 , ±3 C , resp. ±1 , ±2 , ±3 C . Thus ±, ± de¬ne isomorphic unitary
involutions on M3 (C) and the Jordan algebras H3 (C, ±) and H3 (C, ± ) are isomor-
phic. If C is a Cayley algebra, the claim is much deeper and we need Springer™s
result, which says that H3 (C, ±) and H3 (C, ± ) are isomorphic if their trace forms
are isometric (Springer [?, Theorem 1, p. 421]), to ¬nish the proof.
For any P¬ster form •, let • = 1⊥ .
(37.15) Lemma. Let φn , ψn be n-P¬ster bilinear forms and χp , •p p-P¬ster bi-
linear forms for p ≥ 2. If φn — χp ψn — •p , then φn ψn and φn — χp ψn — •p .
Proof : We make computations in the Witt ring W F and use the same notation
for a quadratic form and its class in W F . Let q = φn — χp = ψn — •p . Adding
φn , resp. ψn on both sides , we get that q + φn and q + ψn lie in I n+p F , so that
ψn ’ φn ∈ I n+p F . Since ψn ’ φn can be represented by a form of rank 2n+1 ’ 2,
it follows from the Arason-P¬ster Hauptsatz (Lam [?, Theorem 3.1, p. 289]), that
ψn ’ φn = 0.
(37.16) Corollary. Let T = 1, 1, 1 ⊥ bNC — ’b, ’c, bc be the trace form of
J = H3 (C, ±) and let q be the bilinear P¬ster form bNC — b, c . The isometry
class of T determines the isometry classes of NC and q. Conversely, the classes of
NC and q determine the class of T .
Proof : The claim is a special case of Lemma (??).
(37.17) Remark. Theorem (??) holds more generally for separable Jordan alge-
bras of degree 3: In view of the structure theorem (??) the only cases left are
algebras of the type F — J(V, q), where the claim follows from Proposition (??),
and ´tale algebras of dimension 3 with zero divisors. Here the claim follows from
e
the fact that quadratic ´tale algebras are isomorphic if and only if their norms are
e
isomorphic (see Proposition (??)).
An immediate consequence of Theorem (??) is:
§38. CUBIC JORDAN ALGEBRAS 517


(37.18) Corollary. H3 (C, ±) is split for any ± if C is split.
(37.19) Remark. Conditions on ±, ± so that H3 (C, ±) and H3 (C, ± ) are isomor-
phic for a Cayley division algebra C are given in Albert-Jacobson [?, Theorem 5].
We conclude this section with a useful “Skolem-Noether” theorem for Albert
algebras:
(37.20) Proposition. Let I, I be reduced simple Freudenthal subalgebras of de-

gree 3 of a reduced Albert algebra J. Any isomorphism φ : I ’ I can be extended

to an automorphism of J.
Reference: See Jacobson [?, Theorem 3, p. 370].

However, for example, split cubic ´tale subalgebras of a reduced Albert al-
e
gebra J are not necessarily conjugate by an automorphism of J. Necessary and
su¬cient conditions are given in Albert-Jacobson [?, Theorem 9]. It would be in-
teresting to have a corresponding result for a pair of arbitrary isomorphic cubic
´tale subalgebras.
e
Another Skolem-Noether type of theorem for Albert algebras is given in (??).

§38. Cubic Jordan Algebras
A separable Jordan algebra of degree 3 is either a Freudenthal algebra or is of
the form F + — J(V, q) where J(V, q) is the Jordan algebra of a quadratic space of
dimension ≥ 2 (see the structure theorem (??)); if J is a Freudenthal algebra, then
J is of the form F + (assuming char F = 3), L+ for L cubic ´tale, classical of type
e
A2 , B1 , C3 or exceptional of dimension 27. Let
PJ,a (X) = X 3 ’ TJ (a)X 2 + SJ (a)X ’ NJ (a)1
be the generic minimal polynomial of a separable Jordan algebra J of degree 3.
The element
x# = x2 ’ TJ (x)x + SJ (x)1 ∈ J
obviously satis¬es x q x# = NJ (x)1. It is the (Freudenthal ) adjoint of x and the
linearization of the quadratic map x ’ x#
x — y = (x + y)# ’ x# ’ y #
= 2x q y ’ TJ (x)y ’ TJ (y)x + bSJ (x, y)1
is the Freudenthal “—”-product.34 Let T (x, y) = TJ (x q y) be the bilinear trace
form. The datum (J, NJ , #, T, 1) has the following properties (see McCrimmon [?,
Section 1]):
(a) the form NJ : J ’ F is cubic, the adjoint # : J ’ J, x ’ x# , is a quadratic
map such that x## = N (x)x and 1 ∈ J is a base point such that 1# = 1;
(b) the nonsingular bilinear trace form T is such that
NJ (x + »y) = »3 NJ (y) + »2 T (x# , y) + »T (x, y # ) + NJ (x)
and T (x, 1)1 = 1 — x + x for x, y ∈ J and » ∈ F .

1
34 The (x + y)# ’ x# ’ y # .
—-product is sometimes de¬ned as 2
518 IX. CUBIC JORDAN ALGEBRAS


These properties are characteristic-free. Following McCrimmon [?] and Petersson-
Racine [?] (see also Jacobson [?, 2.4]), we de¬ne a cubic norm structure over any
¬eld F (even if char F = 2) as a datum (J, N, #, T, 1) with properties (??) and (??).
An isomorphism

φ : (J, N, #, T, 1) ’ (J , N , #, T , 1 )

∼ ∼
is an F -isomorphism J ’ J of vector spaces which is an isometry (J, N, T ) ’
’ ’
(J , N , T ), such that φ(1) = 1 and φ(x# ) = φ(x)# for all x ∈ J. We write Cubjord
for the groupoid of cubic norm structures with isomorphisms as morphisms.
(38.1) Examples. Forgetting the Jordan multiplication and just considering ge-
neric minimal polynomials, we get cubic norm structures on J = H3 (C, ±) and on
twisted forms of these. If J = L is cubic ´tale over F , NL = NL/F , TL = TL/F ,
e
and T is the trace form. If J is of classical type A, B, or D, then NJ is the
reduced norm and TJ is the reduced trace. If J is of classical type C , then NJ
is the reduced pfa¬an and TJ is the reduced pfa¬an trace. We also have cu-
bic structures associated to quadratic forms, as in the case of the Jordan algebra
J = F + — J(V, q). More generally, let J = (V , q , 1 ) be a pointed quadratic
space, i.e., 1 ∈ V is such that q (1 ) = 1, and let b be the polar of q . On
J = F • J we de¬ne NJ (x, v) = xq (v), 1 = (1, 1 ), TJ (x, v) = x + b (1 , v),
T (x, v), (y, w) = xy + b (v, w) where w = b (1 , w)1 ’ w and (x, v)# = q (v), xv .
Conversely, any cubic norm structure is of one of the types described above, see
for example Petersson-Racine [?, Theorem 1.1]. We refer to cubic norm structures
associated with Freudenthal algebras as Freudenthal algebras (even if they do not
necessarily admit a multiplication!). Cubic norm structures of the form H3 (C, ±)
for arbitrary C and ± are called reduced Freudenthal algebras.
(38.2) Lemma. Let (J, N, #, T, 1) be a cubic norm structure and set
x2 = T (x, 1)x ’ x# — 1 x3 = T (x, x)x ’ x# — x.
and
(1) Any element x ∈ J satis¬es the cubic equation
P (x) = x3 ’ TJ (x)x2 + SJ (x)x ’ NJ (x)1 = 0
where TJ (x) = T (x, 1) and SJ (x) = TJ (x# ). Furthermore we have
x# = x2 ’ TJ (x)x + SJ (x)1.
In particular any element x ∈ J generates a commutative associative cubic unital
algebra F [x] ‚ J.
(2) There is a Zariski-open, non-empty subset U of J such that F [x] is ´tale for

<< . .

. 68
( : 75)



. . >>