We have (L, a)

« «

x1 00a

(x1 , x2 , x3 ) ’ and z ’ 1 0 0 .

x2

x3 010

From this example, it readily follows that (L, a) is a central simple F -algebra

for all Galois C3 -algebras L and all a ∈ F — , since (L, a) —F Fsep (L —F Fsep , a)

and L —F Fsep Fsep — Fsep — Fsep . Of course, this is also easy to prove without

extending scalars: see for instance Draxl [?, p. 49].

The main result of this subsection is the following:

§19. CENTRAL SIMPLE ALGEBRAS OF DEGREE THREE 303

(19.2) Theorem (Wedderburn). Every central simple F -algebra of degree 3 is

cyclic.

The proof below is due to Haile [?]; it is free from restrictions on the charac-

teristic of F . However, the proof can be somewhat simpli¬ed if char F = 3: see

Draxl [?, p. 63] or Jacobson [?, p. 80].

(19.3) Lemma. Let A be a central simple F -algebra of degree 3 and let x ∈ A — .

If Trd(x) = Trd(x’1 ) = 0, then x3 = Nrd(x).

Proof : If the reduced characteristic polynomial of x is

X 3 ’ Trd(x)X 2 + Srd(x)X ’ Nrd(x),

then the reduced characteristic polynomial of x’1 is

Srd(x) 2 Trd(x) 1

X3 ’ X+ X’ .

Nrd(x) Nrd(x) Nrd(x)

Therefore, Trd(x’1 ) = Srd(x) Nrd(x’1 ), and it follows that the reduced character-

istic polynomial of x takes the form X 3 ’ Nrd(x) if Trd(x) = Trd(x’1 ) = 0.

Proof of Theorem (??): In view of Example (??), it su¬ces to prove that central

division F -algebras of degree 3 are cyclic. Let D be such a division algebra. We

claim that it su¬ces to ¬nd elements y, z ∈ D such that z ∈ F , y ∈ F (z) and

Trd(x) = Trd(x’1 ) = 0 for x = z, yz, yz 2 . Indeed, the lemma then shows

z 3 = Nrd(z), (yz)3 = Nrd(yz), (yz 2 )3 = Nrd(yz 2 );

since Nrd(yz 2 ) = Nrd(yz) Nrd(z) it follows that

(yz 2 )3 = (yz)3 z 3

hence, after cancellation,

zyz 2y = yzyz 2.

By dividing each side by z 3 = Nrd(z), we obtain

(zyz ’1)y = y(zyz ’1),

which shows that zyz ’1 ∈ F (y). We have zyz ’1 = y since y ∈ F (z), hence we may

de¬ne a Galois C3 -algebra structure on F (y) by letting ρ(y) = zyz ’1; then,

D F (y), Nrd(z) .

We now proceed to construct elements y, z satisfying the required conditions.

Let L ‚ D be an arbitrary maximal sub¬eld. Considering D as a bilinear

space for the nonsingular bilinear form induced by the reduced trace, pick a nonzero

element u1 ∈ L⊥ . Since dim (u’1 F )⊥ ©L ≥ 2, we may ¬nd u2 ∈ (u’1 F )⊥ ©L such

1 1

that u2 ∈ u1 F . Set z = u1 u2 . We have Trd(z) = 0 because u1 ∈ L and u’1 ∈ L,

’1 ⊥

2

and Trd(z ’1 ) = 0 because u2 ∈ (u’1 F )⊥ . Moreover, z ∈ F since u2 ∈ u1 F .

1

Next, pick a nonzero element v1 ∈ F (z)⊥ F (z). Since dim(zF + z ’1 F )⊥ = 7,

we have

v1 (zF + z ’1 F )⊥ © F (z) = {0};

’1

we may thus ¬nd a nonzero element v2 in this intersection. Set y = v2 v1 . Since

v1 ∈ F (z), we have y ∈ F (z). On the other hand, since v1 ∈ F (z)⊥ and v2 ∈ F (z),

we have

’1 ’1

Trd(yz 2 ) = Trd(v1 z 2 v2 ) = 0.

Trd(yz) = Trd(v1 zv2 ) = 0 and

304 V. ALGEBRAS OF DEGREE THREE

’1

Since v1 v2 ∈ (zF + z ’1 F )⊥ and z 3 = Nrd(z), we also have

’1

Trd(z ’1 y ’1 ) = Trd(z ’1 v1 v2 ) = 0,

’1

Trd(z ’2 y ’1 ) = Nrd(z) Trd(zv1 v2 ) = 0.

The elements y and z thus meet our requirements.

19.B. Classi¬cation of involutions of the second kind. Let K be a quad-

ratic ´tale extension of F , and let B be a central simple28 K-algebra of degree 3

e

such that NK/F (B) is split. By Theorem (??), this condition is necessary and su¬-

cient for the existence of involutions of the second kind on B which ¬x the elements

of F .

We aim to classify those involutions up to conjugation, by means of the as-

sociated trace form on Sym(B, „ ) (see §??). We therefore assume char F = 2

throughout this subsection.

(19.4) De¬nition. Let „ be an involution of the second kind on B which is the

identity on F . Recall from §?? the quadratic form Q„ on Sym(B, „ ) de¬ned by

Q„ (x) = TrdB (x2 ) for x ∈ Sym(B, „ ).

Let ± ∈ F be such that K F [X]/(X 2 ’ ±). Proposition (??) shows that Q„ has

a diagonalization of the form:

Q„ = 1, 1, 1 ⊥ 2 · ± · q„

where q„ is a 3-dimensional quadratic form of determinant 1. The form

π(„ ) = ± · q„ ⊥ ±

is uniquely determined by „ up to isometry and is a 3-fold P¬ster form since det q„ =

∼

1. We call it the P¬ster form of „ . Every F -isomorphism (B, „ ) ’ (B , „ ) of

’

∼

algebras with involution induces an isometry of trace forms Sym(B, „ ), Q„ ’ ’

∼

Sym(B , „ ), Q„ , hence also an isometry π(„ ) ’ π(„ ). The P¬ster form π(„ ) is

’

therefore an invariant of the conjugacy class of „ . Our main result (Theorem (??))

is that it determines this conjugacy class uniquely.

(19.5) Example. Let V be a 3-dimensional vector space over K with a nonsingular

hermitian form h. Let h = δ1 , δ2 , δ3 K be a diagonalization of this form (so that

δ1 , δ2 , δ3 ∈ F — ) and let „ = „h be the adjoint involution on B = EndK (V ) with

respect to h.

Propositions (??) and (??) show that

Q„ 1, 1, 1 ⊥ 2 · ± · δ1 δ2 , δ1 δ3 , δ2 δ3 .

Therefore,

π(„ ) = ± · ’δ1 δ2 , ’δ1 δ3 .

(19.6) Theorem. Let B be a central simple K-algebra of degree 3 and let „ , „

be involutions of the second kind on B ¬xing the elements of F . The following

conditions are equivalent:

(1) „ and „ are conjugate, i.e., there exists u ∈ B — such that

„ = Int(u) —¦ „ —¦ Int(u)’1 ;

28 We A — Aop for some central simple F -algebra A of

allow K F — F , in which case B

degree 3.

§19. CENTRAL SIMPLE ALGEBRAS OF DEGREE THREE 305

(2) Q„ Q„ ;

(3) π(„ ) π(„ ).

Proof : (??) ’ (??) Conjugation by u de¬nes an isometry from Sym(B, „ ), Q„

to Sym(B, „ ), Q„ .

(??) ” (??) This follows by Witt cancellation, in view of the relation between

Q„ and π(„ ).

(??) ’ (??) If K F — F , then all the involutions on B are conjugate to

the exchange involution. We may therefore assume K is a ¬eld. If B is split, let

B = EndK (V ) for some 3-dimensional vector space V . The involutions „ and „

are adjoint to nonsingular hermitian forms h, h on V . Let

h = δ 1 , δ2 , δ3 and h = δ1 , δ2 , δ3

K K

(with δ1 , . . . , δ3 ∈ F — ) be diagonalizations of these forms. As we observed in (??),

we have

Q„ 1, 1, 1 ⊥ 2 · ± · δ1 δ2 , δ1 δ3 , δ2 δ3

and

Q„ 1, 1, 1 ⊥ 2 · ± · δ1 δ2 , δ1 δ3 , δ2 δ3 .

Therefore, condition (??) implies

± · δ 1 δ2 , δ1 δ3 , δ2 δ3 ± · δ 1 δ2 , δ1 δ3 , δ2 δ3 .

It follows from a theorem of Jacobson (see Scharlau [?, Theorem 10.1.1]) that the

hermitian forms δ1 δ2 , δ1 δ3 , δ2 δ3 K and δ1 δ2 , δ1 δ3 , δ2 δ3 K are isometric; then

δ 1 δ2 δ3 · h δ 1 δ2 , δ1 δ3 , δ2 δ3 δ 1 δ2 , δ1 δ3 , δ2 δ3 δ 1 δ2 δ3 · h ,

K K

hence the hermitian forms h, h are similar. The involutions „ , „ are therefore

conjugate. This completes the proof in the case where B is split.

The general case is reduced to the split case by an odd-degree scalar extension.

Suppose B is a division algebra and let „ = Int(v) —¦ „ for some v ∈ B — , which may

be assumed symmetric under „ . By substituting Nrd(v)v for v, we may assume

Nrd(v) = µ2 for some µ ∈ F — . Let L/F be a cubic ¬eld extension contained in B.

(For example, we may take for L the sub¬eld of B generated by any noncentral

„ -symmetric element.) The algebra BL = B —F L is split, hence the argument

given above shows that „L and „L are conjugate:

—

„L = Int(u) —¦ „L —¦ Int(u)’1 = Int u„L (u) —¦ „L for some u ∈ BL ,

hence

v = »u„L (u) for some » ∈ KL = K —F L.

Since „ (v) = v, we have in fact » ∈ L— . Let ι be the nontrivial automorphism

of KL /L. Since Nrd(u) = µ2 , by taking the reduced norm on each side of the

preceding equality we obtain:

µ2 = »3 Nrd(u) · ι Nrd(u) ,

—

hence » = µ»’1 Nrd(u) · ι µ»’1 Nrd(u) and, letting w = µ»’1 Nrd(u)u ∈ BL ,

v = w„L (w).

By (??), there exists w ∈ B — such that v = w „ (w ). Therefore,

„ = Int(w ) —¦ „ —¦ Int(w )’1 .

306 V. ALGEBRAS OF DEGREE THREE

(19.7) Remark. If (B, „ ) and (B , „ ) are central simple K-algebras of degree 3

with involutions of the second kind leaving F elementwise invariant, the condition

π(„ ) π(„ ) does not imply (B, „ ) (B , „ ). For example, if K F — F all the

forms π(„ ) are hyperbolic since they contain the factor ± , but (B, „ ) and (B , „ )

are not isomorphic if B B .

(19.8) De¬nition. An involution of the second kind „ on a central simple K-

algebra B of degree 3 is called distinguished if π(„ ) is hyperbolic. Theorem (??)

shows that the distinguished involutions form a conjugacy class. If K F — F , all

involutions are distinguished (see the preceding remark).

(19.9) Example. Consider again the split case B = EndK (V ), as in (??). If

the hermitian form h is isotropic, then we may ¬nd a diagonalization of the form

h = 1, ’1, » K for some » ∈ F — , hence the computations in (??) show that the

adjoint involution „h is distinguished. Conversely, if h is a nonsingular hermitian

form whose adjoint involution „h is distinguished, then „h and „h are conjugate,

hence h is similar to h. Therefore, in the split case the distinguished involutions

are those which are adjoint to isotropic hermitian forms.

We next characterize distinguished involutions by a condition on the Witt index

of the restriction of Q„ to elements of trace zero. This characterization will be used

to prove the existence of distinguished involutions on arbitrary central simple K-

algebras B of degree 3 such that NK/F (B) is split, at least when char F = 3.

Let

Sym(B, „ )0 = { x ∈ Sym(B, „ ) | TrdB (x) = 0 }

and let Q0 be the restriction of the bilinear form Q„ to Sym(B, „ )0 . We avoid

„

the case where char F = 3, since then Q0 is singular (with radical F ). Assuming

„

char F = 2, 3, let w(Q0 ) be the Witt index of Q0 .

„ „

(19.10) Proposition. Suppose char F = 2, 3. The following conditions are equiv-

alent:

(1) „ is distinguished;

(2) w(Q0 ) ≥ 2;

„

(3) w(Q0 ) ≥ 3.

„

√

Proof : Let K = F ( ±). The subspace Sym(B, „ )0 is the orthogonal complement

of F in Sym(B, „ ) for the form Q„ ; since Q„ (1) = 3, it follows that Q„ = 3 ⊥ Q0 .

„

Since 1, 1, 1 3, 2, 6 , it follows from (??) that

Q0 2· 1, 3 ⊥ ± · q„ .

„

Since π(„ ) = ± · q„ ⊥ ± , we have

Q0 = 2 · 3, ± + π(„ ) in W F .

„

By comparing dimensions on each side, we obtain

w Q0 = w 3, ± ⊥ π(„ ) ’ 1.

„

Condition (??) implies that w 3, ± ⊥π(„ ) ≥ 4, hence the equality above yields (??).

On the other hand, condition (??) shows that w 3, ± ⊥ π(„ ) ≥ 3, from which

it follows that π(„ ) is isotropic, hence hyperbolic. Therefore, (??) ’ (??). Since

(??) ’ (??) is clear, the proof is complete.

§19. CENTRAL SIMPLE ALGEBRAS OF DEGREE THREE 307

The relation between TrdB and SrdB (see (??)) shows that for x ∈ Sym(B, „ )0 ,

the condition that TrdB (x2 ) = 0 is equivalent to SrdB (x) = 0. Therefore, the totally

isotropic subspaces of Sym(B, „ )0 for the form Q0 can be described as the subspaces

„

3

consisting of elements x such that x = NrdB (x). We may therefore reformulate

the preceding proposition as follows:

(19.11) Corollary. Suppose char F = 2, 3. The following conditions are equiva-

lent:

(1) „ is distinguished;

there exists a subspace U ‚ Sym(B, „ )0 of dimension 2 such that u3 = NrdB (u)

(2)

all u ∈ U ;

for

there exists a subspace U ‚ Sym(B, „ )0 of dimension 3 such that u3 = NrdB (u)

(3)

all u ∈ U .

for

We now prove the existence of distinguished involutions:

(19.12) Proposition. Suppose29 char F = 2, 3. Every central simple K-algebra B

such that NK/F (B) is split carries a distinguished involution.

Proof : In view of (??), we may assume that B is a division algebra. Let „ be

an arbitrary involution of the second kind on B which is the identity on F . Let

L ‚ Sym(B, „ ) be a cubic ¬eld extension of F and let u be a nonzero element in

the orthogonal complement L⊥ of L for the quadratic form Q„ . We claim that the

involution „ = Int(u) —¦ „ is distinguished.

In order to prove this, consider the F -vector space

U = L © (u’1 F )⊥ · u’1 .

Since Q„ is nonsingular, we have dim U ≥ 2. Moreover, since L ‚ Sym(B, „ ), we

have U ‚ Sym(B, „ ). For x ∈ L©(u’1 F )⊥ , x = 0, we have Trd(xu’1 ) = 0 because

x ∈ (u’1 F )⊥ , and Trd(ux’1 ) = 0 because u ∈ L⊥ and x’1 ∈ L. Therefore, for all

nonzero y ∈ U we have

Trd(y) = Trd(y ’1 ) = 0,

hence also y 3 = Nrd(y), by (??). Therefore, Corollary (??) shows that „ is distin-

guished.

(19.13) Remark. If char F = 3, the proof still shows that for every central simple

K-algebra B such that NK/F (B) is split, there exists a unitary involution „ on B

and a 2-dimensional subspace U ‚ Sym(B, „ ) such that u3 ∈ F for all u ∈ U .

´

19.C. Etale subalgebras. As in the preceding subsection, we consider a cen-

tral simple algebra B of degree 3 over a quadratic ´tale extension K of F such that

e

NK/F (B) is split. We continue to assume char F = 2, 3, and let ι be the nontrivial

automorphism of K/F . Our aim is to obtain information on the cubic ´tale F -

e

algebras L contained in B and on the involutions of the second kind which are the

identity on L.

29 See (??) for a di¬erent proof, which works also if char F = 3.

308 V. ALGEBRAS OF DEGREE THREE

Albert™s theorem. The ¬rst main result is a theorem of Albert [?] which

asserts the existence of cubic ´tale F -subalgebras L ‚ B with discriminant ∆(L)

e

isomorphic to K. For such an algebra, we have LK L — ∆(L) Σ(L), by (??),

hence LK can be endowed with a Galois S3 -algebra structure. This structure may

be used to give an explicit description of (B, „ ) as a cyclic algebra with involution.

(19.14) Theorem (Albert). Suppose30 char F = 2, 3 and let K be a quadratic

´tale extension of F . Every central simple K-algebra B such that N K/F (B) is split

e

contains a cubic ´tale F -algebra L with discriminant ∆(L) isomorphic to K.

e

Proof : (after Haile-Knus [?]). We ¬rst consider the easy special cases where B is

not a division algebra. If K F — F , then B A — Aop for some central simple F -

algebra A of degree 3. Wedderburn™s theorem (??) shows that A contains a Galois

C3 -algebra L0 over F . By (??), we have ∆(L0 ) F — F , hence we may set

0

L = {( , ) | ∈ L0 }.

If K is a ¬eld and B is split, then we may ¬nd in B a subalgebra L isomorphic to

F — K. Identifying B with M3 (K), we may then choose

±«

f00

L = 0 k 0 f ∈ F , k ∈ K .

00k

Proposition (??) shows that ∆(L) K.

For the rest of the proof, we may thus assume B is a division algebra. Let „ be

a distinguished involution on B. By (??), there exists a subspace U ‚ Sym(B, „ )

of dimension 2 such that u3 = Nrd(u) for all u ∈ U . Pick a nonzero element u ∈ U .

Since dim U = 2, the linear map U ’ F which carries x ∈ U to Trd(u’1 x) has a

nonzero kernel; we may therefore ¬nd a nonzero v ∈ U such that Trd(u’1 v) = 0.

Consider the reduced characteristic polynomial of u’1 v:

Nrd(X ’ u’1 v) = X 3 + Srd(u’1 v)X ’ Nrd(u’1 v).

By substituting 1 and ’1 for X and multiplying by Nrd(u), we obtain

Nrd(u ’ v) = Nrd(u) + Nrd(u) Srd(u’1 v) ’ Nrd(v)

and

Nrd(u + v) = Nrd(u) + Nrd(u) Srd(u’1 v) + Nrd(v),

hence

Nrd(u + v) ’ Nrd(u ’ v) = 2 Nrd(v).

On the other hand, since u, v, u + v, u ’ v ∈ U and x3 = Nrd(x) for all x ∈ U , we

obtain

Nrd(u + v) = (u + v)3

= Nrd(u) + (u2 v + uvu + vu2 ) + (uv 2 + vuv + v 2 u) + Nrd(v)

and

Nrd(u ’ v) = (u ’ v)3

= Nrd(u) ’ (u2 v + uvu + vu2 ) + (uv 2 + vuv + v 2 u) ’ Nrd(v),

30 See Exercise ?? for the case where char F = 3.

§19. CENTRAL SIMPLE ALGEBRAS OF DEGREE THREE 309

hence

Nrd(u + v) ’ Nrd(u ’ v) = 2(u2 v + uvu + vu2 ) + 2 Nrd(v).

By comparing the expressions above for Nrd(u + v) ’ Nrd(u ’ v), it follows that

u2 v + uvu + vu2 = 0.

De¬ne

t1 = u’1 v = Nrd(u)’1 u2 v,

t2 = u’1 t1 u = Nrd(u)’1 uvu,

t3 = u’1 t2 u = Nrd(u)’1 vu2 ,

so that

t1 + t 2 + t 3 = 0

and conjugation by u permutes t1 , t2 , and t3 cyclically. Moreover, since u and v

are „ -symmetric, we have „ (t2 ) = t2 and „ (t1 ) = t3 .

Let w = t’1 t3 . Suppose ¬rst that w ∈ K. Since Nrd(t2 ) = Nrd(t3 ), we have

2

3

Nrd(w) = w = 1. If w = 1, then t2 = t3 , hence also t3 = t1 , a contradiction to

t1 + t2 + t3 = 0. Therefore, w is a primitive cube root of unity. Conjugating each

side of the relation t3 = wt2 by u, we ¬nd t2 = wt1 ; hence K(t1 ) = K(t2 ) = K(t3 )

and conjugation by u is an automorphism of order 3 of this sub¬eld. Cubing the

equations t2 = wt1 and t3 = wt2 , we obtain t3 = t3 = t3 , hence this element is

1 2 3

invariant under conjugation by u and therefore t3 = t3 = t3 ∈ K — . Since „ (t2 ) = t2 ,

1 2 3

we have in fact t3 ∈ F — , hence Proposition (??) shows that F (t2 ) is a sub¬eld of B

2

F [X]/(X 2 + X + 1). On the other hand, by applying

with discriminant ∆ F (t2 )

„ to each side of the equation t2 = wt1 , we ¬nd t2 = t3 „ (w). Since t3 = wt2 , it

follows that „ (w) = w ’1 , so that K = F (w) and therefore K F [X]/(X 2 +X +1).

The theorem is thus proved if w ∈ K, since then ∆ F (t2 ) K.

Suppose next that w ∈ K, hence K(w) is a cubic extension of K. Since

t1 + t2 + t3 = 0, we have

Int(u’1 )(w) = t’1 t1 = ’t’1 (t2 + t3 ) = ’1 ’ w’1 ∈ K(w),

3 3

hence Int(u’1 ) restricts to a K-automorphism θ of K(w). If θ = Id, then w =

’1 ’ w’1 , hence w is a root of an equation of degree 2 with coe¬cients in K, a

contradiction. Therefore, θ is nontrivial; it is of order 3 since u3 ∈ F — .

Now consider the action of „ :

„ (w) = t1 t’1 = t2 (t’1 t1 )t’1 = ’t2 (1 + w)t’1 .

2 2 2 2

This shows that the involution „ = Int(t’1 ) —¦ „ satis¬es

2

„ (w) = ’1 ’ w ∈ K(w).

Therefore, „ de¬nes an automorphism of order 2 of K(w). We claim that θ and „

generate a group of automorphisms of K(w) isomorphic to the symmetric group S3 .

Indeed,

’w

„ —¦ θ(w) = ’1 + (1 + w)’1 =

1+w

and

θ2 —¦ „ (w) = ’(1 + w ’1 )’1 = „ —¦ θ(w).

310 V. ALGEBRAS OF DEGREE THREE

Therefore, K(w)/F is a Galois extension with Galois group S3 . Let L = K(w)„ ,

the sub¬eld of elements ¬xed by „ . This sub¬eld is a cubic ´tale extension of F .

e

Since L/F is not cyclic, we have ∆(L) F — F , by Corollary (??). However,

LK = L — K K(w) is a cyclic extension of K, hence ∆(LK ) K — K. Therefore,

∆(L) K.

Suppose L ‚ B is a cubic ´tale F -algebra with discriminant ∆(L) isomorphic

e

to K. Let LK = L—K L—∆(L). By (??), we have L—∆(L) Σ(L), hence L—K

can be given a Galois S3 -algebra structure over F . Under any of these S3 -algebra

structures, the automorphism Id — ι gives the action of some transposition, and K

is the algebra of invariant elements under the action of the alternating group A3 . It

follows that LK can be given a Galois C3 -algebra structure, since A3 C3 = Z/3Z.

We ¬x such a structure and set ρ = 1 + 3Z ∈ C3 , as in §??. Since conjugation by

a transposition yields the nontrivial automorphism of A3 , we have

(Id — ι) ρ(x) = ρ2 Id — ι(x) for x ∈ LK .

By (??), there exist involutions „ of the second kind on B ¬xing the elements of

L. We proceed to describe these involutions in terms of the cyclic algebra structure

of B. It will be shown below (see (??)) that these involutions are all distinguished.

(This property also follows from (??)).

(19.15) Proposition. Suppose „ is an involution of the second kind on B such

that L ‚ Sym(B, „ ). The algebra B is a cyclic algebra:

B = L K • LK z • L K z 2

where z is subject to the relations: „ (z) = z, zx = ρ(x)z for all x ∈ L K and

z3 ∈ F —.

Proof : We ¬rst consider the case where B is split. We may then assume B =

EndK (LK ) and identify x ∈ LK with the endomorphism of multiplication by x. The

involution „ is the adjoint involution with respect to some nonsingular hermitian

form

h : LK — LK ’ K.

Since HomK (LK , K) is a free module of rank 1 over LK , the linear form x ’ h(1, x)

is of the form x ’ TLK /K ( x) for some ∈ LK . For x, y ∈ LK , we then have

h(x, y) = h 1, „ (x)y = TLK /K „ (x)y .

If were not invertible, then we could ¬nd x = 0 in LK such that x = 0. It follows

that h „ (x), y = 0 for all y ∈ LK , a contradiction. So, ∈ L— . Moreover, since

K

h(y, x) = ι h(x, y) for all x, y ∈ LK , we have

TLK /K „ (y)x = „ TLK /K „ (x)y for x, y ∈ LK ,

hence „ ( ) = since the bilinear trace form on LK is nonsingular. Therefore,

∈ L— .

Note that the restriction of „ to LK is Id — ι, hence

„ —¦ ρ(x) = ρ2 —¦ „ (x) for x ∈ LK .

’1

Consider β = ρ ∈ EndK (LK ). For x, y ∈ LK we have

h β(x), y = TLK /K „ —¦ ρ(x)y = TLK /K ρ2 —¦ „ (x)y

§19. CENTRAL SIMPLE ALGEBRAS OF DEGREE THREE 311

and

h x, β(y) = TLK /K „ (x)ρ(y) .

Since TLK /K ρ2 (u) = TLK /K (u) for all u ∈ LK , we also have

h x, β(y) = TLK /K ρ2 —¦ „ (x)y ,

hence h(β(x), y) = h x, β(y) for all x, y ∈ LK and therefore „ (β) = β. Clearly,

β —¦ x = ρ(x) —¦ β for x ∈ L, and β 3 = ’1 ρ( ’1 )ρ2 ( ’1 ) ∈ F — . Therefore, we may

choose z = β. This proves the proposition in the case where B is split.

If B is not split, consider the F -vector space:

S = { z ∈ Sym(B, „ ) | zx = ρ(x)z for x ∈ LK }.

The invertible elements in S form a Zariski-open set. Extension of scalars to a

splitting ¬eld of B shows that this open set is not empty. Since B is not split, the

¬eld F is in¬nite, hence the rational points in S are dense. We may therefore ¬nd

an invertible element in S.

If z ∈ S, then z 3 centralizes LK , hence z 3 ∈ LK . Since z 3 commutes with z

and z ∈ Sym(B, „ ), we have z 3 ∈ F . Therefore, every invertible element z ∈ S

satis¬es the required conditions.

´

Etale subalgebras and the invariant π(„ ). We now ¬x an involution of

the second kind „ on the central simple K-algebra B of degree 3 and a cubic ´tale e

F -algebra L ‚ Sym(B, „ ). We assume throughout that char F = 2. We will give a

special expression for the quadratic form Q„ , hence also for the P¬ster form π(„ ),

taking into account the algebra L (see Theorem (??)). As an application, we prove

the following statements: if an involution is the identity on a cubic ´tale F -algebra

e

of discriminant isomorphic to K, then it is distinguished; moreover, every cubic

´tale F -subalgebra in B is stabilized by some distinguished involution.

e

The idea to obtain the special form of Q„ is to consider the orthogonal decom-

position Sym(B, „ ) = L ⊥ M where M = L⊥ is the orthogonal complement of L

for the quadratic form Q„ :

M = { x ∈ Sym(B, „ ) | TrdB (x ) = 0 for ∈ L }.

We show that the restriction of Q„ to M is the transfer of some hermitian form

HM on M . This hermitian form is actually de¬ned on the whole of B, with values

in LK —K LK , where LK = L — K ‚ B.

We ¬rst make B a right LK —LK -module as follows: for b ∈ B and 1 , 2 ∈ LK ,

we set:

b—( — 2) = 1 b 2.

1

(19.16) Lemma. The separability idempotent e ∈ LK — LK satis¬es the following

properties relative to —:

(1) — e = for all ∈ L, and Trd(x) = Trd(x — e) for all x ∈ B;

(2) (x — e) = (x ) — e = ( x) — e = (x — e) for all x ∈ B, ∈ LK ;

(3) B — e = LK ;

(4) x — e = 0 for all x ∈ MK = M — K.

From (??) and (??) it follows that multiplication by e is the orthogonal projection

B ’ LK for the trace bilinear form.

312 V. ALGEBRAS OF DEGREE THREE

3

Proof : (??) Let e = i=1 ui — vi . Proposition (??) shows that ui vi = 1; since

i

LK is commutative, it follows that

3 3

—e= ui v i = ( ui v i ) = for ∈ LK .

i=1 i=1

Moreover, for x ∈ B we have

3 3

Trd(x — e) = Trd( ui xvi ) = Trd ( vi ui )x = Trd(x).

i=1 i=1

(??) For x ∈ B and ∈ LK ,

(x — e) = (x — e) — (1 — ) = x — e(1 — ) .

By (??), we have e(1 — ) = (1 — )e = ( — 1)e, hence, by substituting this in the

preceding equality:

(x — e) = (x ) — e = ( x) — e.

Similarly, (x — e) = (x — e) — ( — 1) = x — e( — 1) and e( — 1) = ( — 1)e, hence

we also have

(x — e) = ( x) — e.

(??) Property (??) shows that B — e centralizes LK , hence B — e = LK .

(??) For x ∈ MK and ∈ LK we have by (??) and (??):

Trd (x — e) = Trd (x ) — e = Trd(x ) = 0.

From (??), we obtain (x — e) ∈ MK © LK = {0}.

(19.17) Example. The split case. Suppose B = EndK (V ) for some 3-dimensional

vector space V and „ = „h is the adjoint involution with respect to some hermitian

form h on V . Suppose also that L F — F — F and let e1 , e2 , e3 ∈ L be the

primitive idempotents. There is a corresponding direct sum decomposition of V

into K-subspaces of dimension 1:

V = V1 • V2 • V3

such that ei is the projection onto Vi with kernel Vj • Vk for {i, j, k} = {1, 2, 3}.

Since e1 , e2 , e3 are „ -symmetric, we have for x, y ∈ V and i, j = 1, 2, 3, i = j:

h ei (x), ej (y) = h x, ei —¦ ej (y) = 0.

Therefore, the subspaces V1 , V2 , V3 are pairwise orthogonal with respect to h. For

i = 1, 2, 3, pick a nonzero vector vi ∈ Vi and let h(vi , vi ) = δi ∈ F — . We may use

the basis (v1 , v2 , v3 ) to identify B with M3 (K); the involution „ is then given by

« «

’1 ’1

ι(x11 ) δ1 ι(x21 )δ2 δ1 ι(x31 )δ3

x11 x12 x13

„ x21 x22 x23 = δ2 ι(x12 )δ1 δ2 ι(x32 )δ3 ,

’1 ’1

ι(x22 )

’1 ’1

x31 x32 x33 δ3 ι(x13 )δ1 δ3 ι(x23 )δ2 ι(x33 )

so that

±«

x δ2 a δ3 b

δ1 ι(a) δ 3 c

y

Sym(B, „ ) = x, y, z ∈ F , a, b, c ∈ K .

δ1 ι(b) δ2 ι(c) z

Under this identi¬cation,

« « «

100 000 000

e1 = 0 0 0 , e2 = 0 1 0 , e3 = 0 0 0 ,

000 000 001

§19. CENTRAL SIMPLE ALGEBRAS OF DEGREE THREE 313

and L is the F -algebra of diagonal matrices in Sym(B, „ ). The separability idem-

potent is e = e1 — e1 + e2 — e2 + e3 — e3 . A computation shows that

±«

0 δ2 a δ3 b

M = δ1 ι(a) δ3 c a, b, c ∈ K .

0

δ1 ι(b) δ2 ι(c) 0

The K-algebra LK is the algebra of diagonal matrices in B and MK is the space of

matrices whose diagonal entries are all 0. Let

«

111

m = 1 1 1 ∈ B.

111

For x = (xij )1¤i,j¤3 ∈ B, we have

3

m—( xij ei — ej ) = x,

i,j=1

hence B is a free LK — LK -module of rank 1.

We now return to the general case, and let θ be the K-automorphism of LK —LK

which switches the factors:

θ( — 2) = — for 1, ∈ LK .

1 2 1 2

As in (??), we call e ∈ LK — LK the separability idempotent of LK . The charac-

terization of e in (??) shows that e is invariant under θ.

(19.18) Proposition. Consider B as a right LK —LK -module through the —-multi-

plication. The module B is free of rank 1. Moreover, there is a unique hermitian

form

H : B — B ’ L K — LK

with respect to θ such that for all x ∈ B,

(19.19)

2 2 2

1

H(x, x) = e(x — x ) + (1 ’ e) (x — e) — 1 + 1 — (x — e) ’ Trd(x ) ,

2

2

where x = x — e and x = x — (1 ’ e). (Note that x and x — e lie in LK , by (??).)

Proof : Since LK — LK is an ´tale F -algebra, it decomposes into a direct product

e

of ¬elds by (??). Let LK — LK L1 — · · · — Ln for some ¬elds L1 , . . . , Ln . Then

B B1 — · · · — Bn where Bi is a vector space over Li for i = 1, . . . , n. To see that

B is a free LK — LK -module of rank 1, it su¬ces to prove that dimLi Bi = 1 for

i = 1, . . . , n. Since dimF B = dimF (LK — LK ), it actually su¬ces to show that

dimLi Bi = 0 for i = 1, . . . , n, which means that B is a faithful LK — LK -module.

This property may be checked over a scalar extension of F . Since it holds in the

split case, as was observed in (??), it also holds in the general case. (For a slightly

di¬erent proof, see Jacobson [?, p. 44].)

Now, let b ∈ B be a basis of B (as a free LK — LK -module). We de¬ne a

hermitian form H on B by

H(b — »1 , b — »2 ) = θ(»1 )H(b, b)»2 for »1 , »2 ∈ LK — LK ,

where H(b, b) is given by formula (??). This is obviously the unique hermitian

form on B for which H(b, b) takes the required value. In order to show that the

hermitian form thus de¬ned satis¬es formula (??) for all x ∈ B, we may extend

314 V. ALGEBRAS OF DEGREE THREE

scalars to a splitting ¬eld of B and L, and assume we are in the split case discussed

in (??). With the same notation as in (??), de¬ne:

3

(19.20) H (x, y) = xji yij ei — ej ∈ LK — LK

i,j=1

for x = (xij )1¤i,j¤3 , y = (yij )1¤i,j¤3 ∈ B. Straightforward computations show that

H is hermitian and satis¬es formula (??) for all x ∈ B. In particular, H (b, b) =

H(b, b), hence H = H. This proves the existence and uniqueness of the hermitian

form H.

The hermitian form H restricts to hermitian forms on Sym(B, „ ) and on MK

which we discuss next.

Let ω be the K-semilinear automorphism of LK — LK de¬ned by

ω( — k1 ) — ( — k2 ) = — ι(k2 ) — — ι(k1 )

1 2 2 1

for 1, ∈ L and k1 , k2 ∈ K. The following property is clear from the de¬nition:

2

„ (x — ») = „ (x) — ω(») for x ∈ B and » ∈ LK — LK .

This shows that Sym(B, „ ) is a right module over the algebra (LK — LK )ω of ω-

invariant elements in LK — LK . Moreover, by extending scalars to a splitting ¬eld

of B and using the explicit description of H = H in (??), one can check that

H „ (x), „ (y) = ω H(x, y) for x, y ∈ B.

Therefore, the hermitian form H restricts to a hermitian form

HS : Sym(B, „ ) — Sym(B, „ ) ’ (LK — LK )ω .

Now, consider the restriction of H to MK . By (??), we have x — e = 0 for all

x ∈ MK , hence x — (1 ’ e) = x and the LK — LK -module action on B restricts to

an action of (1 ’ e) · (LK — LK ) on MK . Moreover, for x, y ∈ MK ,

H(x, y) = H x — (1 ’ e), y = (1 ’ e)H(x, y) ∈ (1 ’ e) · (LK — LK ),

hence H restricts to a hermitian form

HMK : MK — MK ’ (1 ’ e) · (LK — LK ).

Recall from (??) the embedding 3 : LK ’ (1 ’ e) · (LK — LK ). By (??) and (??),

3 induces a canonical isomorphism

∼

3: LK — ∆(LK ) ’ (1 ’ e) · (LK — LK )

’

which we use to identify (1’e)·(LK —LK ) with LK —∆(LK ). Since the image of 3

is the subalgebra of (1 ’ e) · (LK — LK ) of elements ¬xed by θ, the automorphism

of LK — ∆(LK ) corresponding to θ via 3 is the identity on LK and the unique

nontrivial K-automorphism on ∆(LK ). We call this automorphism also θ. Thus,

we may consider the restriction of H to MK as a hermitian form with respect to θ:

HMK : MK — MK ’ LK — ∆(LK ).

In particular, HMK (x, x) ∈ LK for all x ∈ MK .

(19.21) Lemma. For all x ∈ MK ,

2TLK /K HMK (x, x) = TrdB (x2 )

and

2

H x2 — (1 ’ e), x2 — (1 ’ e) .

NLK /K HMK (x, x) =N 3 (LK )/K

§19. CENTRAL SIMPLE ALGEBRAS OF DEGREE THREE 315

For all x ∈ MK and ∈ LK ,

#

HMK ( x, x) = HMK (x, x),

#

: LK ’ LK is the quadratic map de¬ned in (??).

where

Proof : It su¬ces to verify these formulas when B and L are split. We may thus

assume B and L are as in (??). For

«

0 x12 x13

x = x21 0 x23 ∈ MK ,

x31 x32 0

we have, using (??) and (??),

H(x, x) = x12 x21 (e1 — e2 + e2 — e1 ) + x13 x31 (e1 — e3 + e3 — e1 )

+ x23 x32 (e2 — e3 + e3 — e2 )

= 3 (x23 x32 e1 + x13 x31 e2 + x12 x21 e3 ),

hence HMK (x, x) = x23 x32 e1 + x13 x31 e2 + x12 x21 e3 ∈ LK . It follows that

Trd(x2 )

1

TLK /K HMK (x, x) = x23 x32 + x13 x31 + x12 x21 = 2

and

NLK /K H(x, x) = x23 x32 x13 x31 x12 x21 .

On the other hand,

«

0 x13 x32 x12 x23

x — (1 ’ e) = x23 x31 x21 x13 ,

2

0

x32 x21 x31 x12 0

hence

H x2 — (1 ’ e), x2 — (1 ’ e) = (x23 x32 x13 x31 x12 x21 )2 .

N 3 (LK )/K

For = 1 e1 + 2 e2 + 3 e3 ∈ LK , we have

«

0 1 x12 1 x13

2 x21 2 x23

0

x=

3 x31 3 x32 0

and

H( x, x) = 3 ( 2 3 e1 + 1 3 e2 + 1 2 e3 )H(x, x),

proving the last formula of the lemma, since # = 2 3 e1 + 1 3 e2 + 1 2 e3 . Alter-

nately, the last formula follows from the fact that x = x—( —1), hence H( x, x) =

( — ) · H(x, x), together with the observation that ( — )(1 ’ e) = 3 ( # ).

The ¬rst formula also follows from the de¬nition of H(x, x) and of 3 , since

(??) shows that TL/F (x2 — e) = Trd(x2 ).

Finally, we combine the restrictions HS and HMK of H to Sym(B, „ ) and to

MK to describe the restriction of H to Sym(B, „ ) © MK = M . The automorphism

of LK — ∆(LK ) = L — ∆(L) — K corresponding to ω under the isomorphism 3

is the identity on L and restricts to the unique nontrivial automorphism of ∆(L)

and of K. The F -subalgebra of elements ¬xed by ω therefore has the form L — E,

where E is the quadratic ´tale F -subalgebra of ω-invariant elements in ∆(L) — K.

e

—

F [X]/(X 2 ’ ±) and ∆(L) F [X]/(X 2 ’ δ),

If ±, δ ∈ F are such that K

F [X]/(X 2 ’ ±δ). The LK — ∆(LK )-module structure on MK restricts

then E

316 V. ALGEBRAS OF DEGREE THREE

to an L — E-module structure on M , and the hermitian form HMK restricts to a

hermitian form

HM : M — M ’ L — E

with respect to (the restriction of) θ. Note that θ on L — E is the identity on L and

restricts to the nontrivial automorphism of E, hence HM (x, x) ∈ L for all x ∈ M .

(19.22) Proposition. The hermitian form HM satis¬es:

TL—E/F HM (x, x) = Q„ (x) for x ∈ M .

The L — E-module M is free of rank 1; it contains a basis vector m such that

NL/F HM (m, m) ∈ F —2 .

Proof : The ¬rst formula readily follows from (??), since

TL—E/F HM (x, x) = 2TL/F HM (x, x) .

We claim that every element x ∈ M such that HM (x, x) ∈ L— is a basis of the

L —F E-module M . Indeed, if » ∈ L — E satis¬es x — » = 0, then HM (x, x — ») =

HM (x, x)» = 0, hence » = 0. Therefore, x — (L — E) is a submodule of M which

has the same dimension over F as M . It follows that M = x — (L — E), and that x

is a basis of M over L — E.

The existence of elements x such that HM (x, x) ∈ L— is clear if F is in¬nite,

since the proof of (??) shows that NL/F HM (x, x) is a nonzero polynomial function

of x. It is also easy to establish when F is ¬nite. (Note that in that case the

algebra B is split).

To ¬nd a basis element m ∈ M such that NL/F HM (m, m) ∈ F —2 , pick any

x ∈ M such that HM (x, x) ∈ L— and set m = x2 — (1 ’ e). By (??), we have

2

∈ F —2 .

NL/F HM (m, m) = NL/F HM (x, x)

Let m ∈ M be a basis of M over L — E such that NL/F HM (m, m) ∈ F —2

and let = HM (m, m) ∈ L— . We then have a diagonalization HM = L—E ,

and Proposition (??) shows that the restriction Q„ |M of Q„ to M is the Scharlau

transfer of the hermitian form L—E :

Q„ |M = (TL—H/F )— ( L—H ).

We may use transitivity of the trace to represent the right-hand expression as the

F [X]/(X 2 ’ ±) and

transfer of a 2-dimensional quadratic space over L: if K

∆(L) F [X]/(X 2 ’ δ), so that E F [X]/(X 2 ’ ±δ), we have

(TL—E/L)— 2 , ’2±δ 2 · ±δ · L,

L—E L

hence

(19.23) Q „ |M 2 · ±δ · (TL/F )— .

L

This formula readily yields an expression for the form Q„ , in view of the orthog-

onal decomposition Sym(B, „ ) = L ⊥ M . In order to get another special expression,

we prove a technical result:

∈ L— such that NL/F ( ) ∈ F —2 , the quadratic form

(19.24) Lemma. For all

δ · (TL/F )— ⊥ ’1

L

is hyperbolic.

§19. CENTRAL SIMPLE ALGEBRAS OF DEGREE THREE 317

Proof : By Springer™s theorem on odd-degree extensions, it su¬ces to prove that

the quadratic form above is hyperbolic after extending scalars from F to L. We

may thus assume L F — ∆(L). Let = ( 0 , 1 ) with 0 ∈ F and 1 ∈ ∆(L); then

(TL/F )— = ⊥ (T∆(L)/F )— .

L 0 1 ∆(L)

By Scharlau [?, p. 50], the image of the transfer map from the Witt ring W ∆(L)

to W F is killed by δ , hence

δ · (TL/F )— =δ· in W F .

L 0

∈ F —2 , hence

On the other hand, NL/F ( ) = 0 N∆(L)/F ( 1 ) is a norm from ∆(L)

0

and therefore

δ· =δ in W F .

0

Here, ¬nally, is the main result of this subsection:

(19.25) Theorem. Let (B, „ ) be a central simple K-algebra of degree 3 with in-

volution of the second kind which is the identity on F and let L ‚ Sym(B, „ ) be

a cubic ´tale F -algebra. Let ±, δ ∈ F — be such that K F [X]/(X 2 ’ ±) and

e

F [X]/(X 2 ’ δ). Then, the quadratic form Q„ and the invariant π(„ )

∆(L)

satisfy:

Q„ 1, 2, 2δ ⊥ 2 · ±δ · (TL/F )— L

(19.26)

1, 1, 1 ⊥ 2δ · ± · (TL/F )— L

and

(19.27) π(„ ) ±· 1 ⊥ δ · (TL/F )— L

for some ∈ L— such that NL/F ( ) ∈ F —2 .

In particular, π(„ ) has a factorization: π(„ ) ± · • where • is a 2-fold

P¬ster form such that

•· δ =0 in W F .

Proof : Lemma (??) shows that the restriction of Q„ to L has a diagonalization:

Q „ |L 1, 2, 2δ .

Since Sym(B, „ ) = L ⊥ M , the ¬rst formula for Q„ follows from (??).

In W F , we have ±δ = ± · δ + δ . By substituting this in the ¬rst

formula for Q„ , we obtain:

Q„ = 1, 2, 2δ + 2δ · ± · (TL/F )— + 2 · δ · (TL/F )— in W F .

L L

Lemma (??) shows that the last term on the right equals 2 · δ . Since

2δ + 2 · δ = 2 and 1, 2, 2 = 1, 1, 1 ,

we ¬nd

Q„ = 1, 1, 1 + 2δ · ± · (TL/F )— in W F .

L

Since these two quadratic forms have the same dimension, they are isometric, prov-

ing the second formula for Q„ .

The formula for π(„ ) readily follows, by the de¬nition of π(„ ) in (??).

318 V. ALGEBRAS OF DEGREE THREE

According to Scharlau [?, p. 51], we have det(TL/F )— L = δNL/F ( ), hence

the form • = 1 ⊥ δ · (TL/F )— L is a 2-fold P¬ster form. Finally, Lemma (??)

shows that

δ · • = δ + δ · δ = 0 in W F .

(19.28) Corollary. Every unitary involution „ such that Sym(B, „ ) contains a

cubic ´tale F -algebra L with discriminant ∆(L) isomorphic to K is distinguished.

e

Proof : Theorem (??) yields a factorization π(„ ) = ± · • with • · δ = 0 in W F .

Therefore, π(„ ) = 0 if ± = δ.

So far, the involution „ has been ¬xed, as has been the ´tale subalgebra

e

L ‚ Sym(B, „ ). In the next proposition, we compare the quadratic forms Q„

and Q„ associated to two involutions of the second kind which are the identity

on L. By (??), we then have „ = Int(u) —¦ „ for some u ∈ L— .

(19.29) Proposition. Let δ ∈ F — be such that ∆(L) F [X]/(X 2 ’ δ). Let

u ∈ L— and let „u = Int(u) —¦ „ . For any ∈ L— such that

Q„ 1, 2, 2δ ⊥ 2 · ±δ · (TL/F )— ,

L

we have

1, 2, 2δ ⊥ 2 · ±δ · (TL/F )— u#

Q „u .

L

Proof : Left multiplication by u gives a linear bijection Sym(B, „ ) ’ Sym(B, „u )

which maps L to L and the orthogonal complement M of L in Sym(B, „ ) for the

form Q„ to the orthogonal complement Mu of L in Sym(B, „u ) for the form Q„u .

Lemma (??) shows that

HMK (ux, ux) = u# HMK (x, x) for x ∈ MK ,

∼

hence multiplication by u de¬nes a similitude (M, HM ) ’ (Mu , HMu ) with multi-

’

plier u# .

(19.30) Corollary. Let L be an arbitrary cubic ´tale F -algebra in B with ∆(L)

e

2 —

F [X]/(X ’ δ) for δ ∈ F .

(1) For every ∈ L— such that NL/F ( ) ∈ F —2 , there exists an involution „ which

is the identity on L and such that Q„ and π(„ ) satisfy (??) and (??).

(2) There exists a distinguished involution which is the identity on L.

Proof : (??) By (??), there is an involution of the second kind „0 such that L ‚

Sym(B, „0 ). Theorem (??) yields

Q „0 1, 1, 1 ⊥ 2δ · ± · (TL/F )— 0L

for some 0 ∈ L— with NL/F ( 0 ) ∈ F —2 . If ∈ L— satis¬es NL/F ( ) ∈ F —2 ,

then NL/F ( ’1 ) ∈ F —2 , hence Proposition (??) shows that there exists u ∈ L—

0

satisfying u# ≡ ’1 mod L—2 . We then have L u# 0 L , hence, by (??), the

0

involution „ = Int(u) —¦ „0 satis¬es the speci¬ed conditions.

(??) Choose 1 ∈ L— satisfying TL/F ( 1 ) = 0 and let = 1 NL/F ( 1 )’1 ; then

NL/F ( ) = NL/F ( 1 )’2 ∈ F —2 and TL/F ( ) = 0. Part (??) shows that there exists

an involution „ which is the identity on L and satis¬es

π(„ ) ±· 1 ⊥ δ · (TL/F )— .

L

EXERCISES 319

Since TL/F ( ) = 0, the form (TL/F )— L is isotropic. Therefore, π(„ ) is isotropic,

hence hyperbolic since it is a P¬ster form, and it follows that „ is distinguished.

As another consequence of (??), we obtain some information on the conju-

gacy classes of involutions which leave a given cubic ´tale F -algebra L elementwise

e

invariant:

(19.31) Corollary. Let L ‚ B be an arbitrary cubic ´tale F -subalgebra and let „

e

be an arbitrary involution which is the identity on L. For u, v ∈ L— , the involutions

„u = Int(u)—¦„ and „v = Int(v)—¦„ are conjugate if uv ∈ NLK /L (L— )·F — . Therefore,

K

the map u ∈ L— ’ „u induces a surjection of pointed sets from L— /NLK /L (L— )·F —

K

to the set of conjugacy classes of involutions which are the identity on L, where the

distinguished involution is „ .

Proof : We use the same notation as in (??) and (??); thus

1, 2, 2δ ⊥ 2 · ±δ · (TL/F )— u#

Q „u L

1, 1, 1 ⊥ 2δ · ± · (TL/F )— u# L

∈ L— such that NL/F ( ) ∈ F —2 and, similarly,

for some

1, 1, 1 ⊥ 2δ · ± · (TL/F )— v #

Q „v .

L

According to (??), the involutions „u and „v are conjugate if and only if Q„u Q„v .

In view of the expressions above for Q„u and Q„v , this condition is equivalent to:

± · (TL/F )— u# , ’v # =0 in W F,

L

or, using Frobenius reciprocity, to:

±, (uv)# · u#

(TL/F )— =0 in W F.

L L

If uv = NLK /L (»)µ for some » ∈ L— and some µ ∈ F — , then

K

(uv)# = NLK /L µNLK /K (»)»’1 ,

hence ±, (uv)# is hyperbolic.

Exercises

1. Let L be a ¬nite dimensional commutative algebra over a ¬eld F . Let µ : L —F

L ’ L be the multiplication map. Suppose L —F L contains an element e such

that e(x — 1) = e(1 — x) for all x ∈ L and µ(e) = 1. Show that L is ´tale. e

n

Hint: Let (ui )1¤i¤n be a basis of L and e = i=1 ui — vi . Show that

(vi )1¤i¤n is a basis of L and that T (ui , vj ) = δij for all i, j = 1, . . . , n, hence

T is nonsingular.

From this exercise and Proposition (??), it follows that L is ´tale if and

e

only if L —F L contains a separability idempotent of L.

2. Let G = {g1 , . . . , gn } be a ¬nite group of order n, and let L be a commutative

algebra of dimension n over a ¬eld F , endowed with an action of G by F -algebra

automorphisms. Show that the following conditions are equivalent:

(a) L is a Galois G-algebra;

(b) the map Ψ : L —F L = LL ’ Map(G, L) de¬ned by Ψ( 1 — 2 )(g) = g( 1 ) 2

is an isomorphism of L-algebras;

320 V. ALGEBRAS OF DEGREE THREE

(c) for some basis (ei )1¤i¤n of L, the matrix gi (ej ) 1¤i,j¤n ∈ Mn (L) is in-

vertible;

(d) for every basis (ei )1¤i¤n of L, the matrix gi (ej ) 1¤i,j¤n ∈ Mn (L) is

invertible.

3. Suppose L is a Galois G-algebra over a ¬eld F . Show that for all ¬eld exten-

sions K/F , the algebra LK is a Galois G-algebra over K.

4. Show that every ´tale algebra of dimension 2 is a Galois (Z/2Z)-algebra.

e

5. (Saltman) Suppose L is an ´tale F -algebra of dimension n. For i = 1, . . . , n, let

e

—n

πi : L ’ L denote the map which carries x ∈ L to 1 — · · · — 1 — x — 1 — · · ·— 1

(where x is in the i-th position). For i < j, let πij : L —F L ’ L—n be de¬ned

by πij (x — y) = πi (x)πj (y). Let s = 1¤i<j¤n πij (1 ’ e) where e is the

separability idempotent of L. Show that s is invariant under the action of the

symmetric group Sn on L—n by permutation of the factors, and that there is

an isomorphism of Sn -algebras over F :

s · L—n .

Σ(L)

Hint: If L = F — · · · — F and (ei )1¤i¤n is the canonical basis of L, show

that s = σ∈Sn eσ(1) — · · · — eσ(n) .

6. (Barnard [?]) Let L = F [X]/(f ) where

f = X n ’ a1 X n’1 + a2 X n’2 ’ · · · + (’1)n an ∈ F [X]

is a polynomial with no repeated roots in an algebraic closure of F . For

k = 1, . . . , n, let sk ∈ F [X1 , . . . , Xn ] be the k-th symmetric polynomial:

sk = i1 <···<ik Xi1 · · · Xik . Show that the action of Sn by permutation of the

indeterminates X1 , . . . , Xn induces an action of Sn on the quotient algebra

R = F [X1 , . . . , Xn ]/(s1 ’ a1 , s2 ’ a2 , . . . , sn ’ an ).

Establish an isomorphism of Sn -algebras: Σ(L) R.

7. (Berg´-Martinet [?]) Suppose L is an ´tale algebra of odd dimension over a

e e

¬eld F of characteristic 2. Let L = L — F and S = SL /F . Show that

F [t]/(t2 + t + a)

∆(L)

where a ∈ F is a representative of the determinant of S .

8. Let B be a central division algebra of degree 3 over a ¬eld K of characteristic 3,

and let u ∈ B F be such that u3 ∈ F — . Show that there exists x ∈ B — such

that ux = (x + 1)u and x3 ’ x ∈ F .

Hint: (Jacobson [?, p. 80]) Let ‚u : B ’ B map x to ux ’ xu. Show that

3 2

‚u = 0. Show that if y ∈ B satis¬es ‚u (y) = 0 and ‚u (y) = 0, then one may

take x = u(‚u y)’1 y.

9. (Albert™s theorem (??) in characteristic 3) Let B be a central division algebra of

degree 3 over a ¬eld K of characteristic 3. Suppose K is a quadratic extension

of some ¬eld F and NK/F (B) splits. Show that B contains a cubic extension

of K which is Galois over F with Galois group isomorphic to S3 .

Hint: (Villa [?]) Let „ be a unitary involution on B as in (??). Pick

u ∈ Sym(B, „ ) such that u3 ∈ F , u ∈ F , and use Exercise ?? to ¬nd x ∈ B such

3

that ux = (x+1)u. Show that x+„ (x) ∈ K(u), hence x+„ (x) ∈ F . Use this

information to show TrdB x„ (x) + „ (x)x = ’1, hence SrdB x ’ „ (x) = ’1.

Conclude by proving that K x ’ „ (x) is cyclic over K and Galois over F with

Galois group isomorphic to S3 .

NOTES 321

Notes

§??. The notion of a separable algebraic ¬eld extension ¬rst occurs, under the

name of algebraic extension of the ¬rst kind, in the fundamental paper of Steinitz [?]

on the algebraic theory of ¬elds. It was B. L. van der Waerden who proposed the

term separable in his Moderne Algebra, Vol. I, [?]. The extension of this notion to

associative (not necessarily commutative) algebras (as algebras which remain semi-

simple over any ¬eld extension) is already in Albert™s “Structure of Algebras” [?],

¬rst edition in 1939. The cohomological interpretation (A has dimension 0 or,

equivalently, A is projective as an A — Aop -module) is due to Hochschild [?]. A

systematic study of separable algebras based on this property is given in Auslander-

Goldman [?]. Commutative separable algebras over rings occur in Serre [?] as un-

´

rami¬ed coverings, and are called ´tales by Grothendieck in [?]. Etale algebras over

e

¬elds were consecrated as a standard tool by Bourbaki [?].

Galois algebras are considered in Grothendieck (loc. ref.) and Serre (loc. ref.).

A systematic study is given in Auslander-Goldman (loc. ref.). Further developments

may be found in the Memoir of Chase, Harrison and Rosenberg [?] and in the notes

of DeMeyer-Ingraham [?].

The notion of the discriminant of an ´tale F -algebra, and its relation to the

e

trace form, are classical in characteristic di¬erent from 2. (In this case, the dis-

criminant is usually de¬ned in terms of the trace form, and the relation with per-

mutations of the roots of the minimal polynomial of a primitive element is proved

subsequently.) In characteristic 2, however, this notion is fairly recent. A formula

for the discriminant of polynomials, satisfying the expected relation with the per-

mutation of the roots (see (??)), was ¬rst proposed by Berlekamp [?]. For an ´tale e

F -algebra L, Revoy [?] suggested a de¬nition based on the quadratic forms SL/F or

S 0 , and conjectured the relation, demonstrated in (??), between his de¬nition and

Berlekamp™s. Revoy™s conjecture was independently proved by Berg´-Martinet [?]

e

and by Wadsworth [?]. Their proofs involve lifting the ´tale algebra to a discrete

e

valuation ring of characteristic zero. A di¬erent approach, by descent theory, is due

to Waterhouse [?]; this approach also yields a de¬nition of discriminant for ´tale e

algebras over commutative rings. The proof of (??) in characteristic 2 given here

is new.

Reduced equations for cubic ´tale algebras (see (??)) (as well as for some higher-

e

dimensional algebras) can be found in Serre [?, p. 657] (in characteristic di¬erent

from 2 and 3) and in Berg´-Martinet [?, §4] (in characteristic 2).

e

§??. The fact that central simple algebras of degree 3 are cyclic is another

fundamental contribution of Wedderburn [?] to the theory of associative algebras.

Albert™s di¬cult paper [?] seems to be the ¬rst signi¬cant contribution in the

literature to the theory of algebras of degree 3 with unitary involutions. The classi-

¬cation of unitary involutions on such an algebra, as well as the related description

of distinguished involutions, comes from Haile-Knus-Rost-Tignol [?]. See (??) and

(??) for the cohomological version of this classi¬cation.

322 V. ALGEBRAS OF DEGREE THREE

CHAPTER VI

Algebraic Groups

It turns out that most of the groups which have occurred thus far in the book

are groups of points of certain algebraic group schemes. Moreover, many construc-

tions described previously are related to algebraic groups. For instance, the Cli¬ord

algebra and the discriminant algebra are nothing but Tits algebras for certain semi-

simple algebraic groups; the equivalences of categories considered in Chapter ??,

for example of central simple algebras of degree 6 with a quadratic pair and cen-

tral simple algebras of degree 4 with a unitary involution over an ´tale quadratic

e

extension (see §??), re¬‚ect the fact that certain semisimple groups have the same

Dynkin diagram (D3 A3 in this example).

The aim of this chapter is to give the classi¬cation of semisimple algebraic

groups of classical type without any ¬eld characteristic assumption, and also to

study the Tits algebras of semisimple groups.

In the study of linear algebraic groups (more generally, a¬ne group schemes) we

use a functorial approach equivalent to the study of Hopf algebras. The advantage

of such an approach is that nilpotents in algebras of functions are allowed (and they

really do appear when considering centers of simply connected groups over ¬elds of

positive characteristic); moreover many constructions like kernels, intersections of

subgroups, are very natural. A basic reference for this approach is Waterhouse [?].

The classical view of an algebraic group as a variety with a regular group structure

is equivalent to what we call a smooth algebraic group scheme.

The classical theory (mostly over an algebraically closed ¬eld) can be found

in Borel [?], Humphreys [?], or Springer [?]. We also refer to Springer™s survey

article [?]. (The new (1998) edition of [?] will contain the theory of algebraic groups

over non algebraically closed ¬elds.) We use some results in commutative algebra

which can be found in Bourbaki [?], [?], [?], and in the book of Matsumura [?].

The ¬rst three sections of the chapter are devoted to the general theory of group

schemes. In §?? we de¬ne the families of algebraic groups related to an algebra with

involution, a quadratic form, and an algebra with a quadratic pair. After a short

interlude (root systems, in §??) we come to the classi¬cation of split semisimple

groups over an arbitrary ¬eld. In fact, this classi¬cation does not depend on the

ground ¬eld F , and is essentially equivalent to the classi¬cation over the algebraic

closure Falg (see Tits [?], Borel-Tits [?]).

The central section of this chapter, §??, gives the classi¬cation of adjoint semi-

simple groups over arbitrary ¬elds. It is based on the observation of Weil [?] that

(in characteristic di¬erent from 2) a classical adjoint semisimple group is the con-

nected component of the automorphism group of some algebra with involution. In

arbitrary characteristic the notion of orthogonal involution has to be replaced by

the notion of a quadratic pair which has its origin in the fundamental paper [?] of

323

324 VI. ALGEBRAIC GROUPS

Tits. Groups of type G2 and F4 which are related to Cayley algebras (Chapter ??)

and exceptional Jordan algebras (Chapter ??), are also brie¬‚y discussed.

In the last section we de¬ne and study Tits algebras of semisimple groups. It

turns out that for the classical groups the nontrivial Tits algebras are the »-powers

of a central simple algebra, the discriminant algebra of a simple algebra with a

unitary involution, and the Cli¬ord algebra of a central simple algebra with an

orthogonal pair”exactly those algebras which have been studied in the book (and

nothing more!).

§20. Hopf Algebras and Group Schemes

This section is mainly expository. We refer to Waterhouse [?] for proofs and

more details.

Hopf algebras. Let F be a ¬eld and let A be a commutative (unital, associa-

tive) F -algebra with multiplication m : A —F A ’ A. Assume we have F -algebra

homomorphisms

c : A ’ A —F A (comultiplication)

i: A ’ A (co-inverse)

u: A ’ F (co-unit)

which satisfy the following:

(a) The diagram

c

A ’’’

’’ A —F A

¦ ¦

¦ ¦

c c—Id

Id—c

A —F A ’ ’ ’ A — F A —F A

’’

commutes.

(b) The map

c u—Id

A ’ A —F A ’ ’ F — F A = A

’ ’’

equals the identity map Id : A ’ A.

(c) The two maps

c i—Id m

A ’ A —F A ’ ’ A — F A ’ A

’ ’’ ’

u ·1

A’ F ’ A

’ ’

coincide.

An F -algebra A together with maps c, i, and u as above is called a (commutative)

Hopf algebra over F . A Hopf algebra homomorphism f : A ’ B is an F -algebra

homomorphism preserving c, i, and u, i.e., (f — f ) —¦ cA = cB —¦ f , f —¦ iA = iB —¦ f , and

uA = uB —¦ f . Hopf algebras and homomorphisms of Hopf algebras form a category.

If A is a Hopf algebra over F and L/F is a ¬eld extension, then AL together

with cL , iL , uL is a Hopf algebra over L. If A ’ B and A ’ C are Hopf algebra

homomorphisms then there is a canonical induced Hopf algebra structure on the

F -algebra B —A C.

Let A be a Hopf algebra over F . An ideal J of A such that

c(J) ‚ J —F A + A —F J, i(J) ‚ J and u(J) = 0

§20. HOPF ALGEBRAS AND GROUP SCHEMES 325

is called a Hopf ideal. If J is a Hopf ideal, the algebra A/J admits the structure

of a Hopf algebra and there is a natural surjective Hopf algebra homomorphism

A ’ A/J. For example, J = ker(u) is a Hopf ideal and A/J = F is the trivial

Hopf F -algebra. The kernel of a Hopf algebra homomorphism f : A ’ B is a Hopf

ideal in A and the image of f is a Hopf subalgebra in B.

20.A. Group schemes. Recall that Alg F denotes the category of unital com-

mutative (associative) F -algebras with F -algebra homomorphisms as morphisms.

Let A be a Hopf algebra over F . For any unital commutative associative F -

algebra R one de¬nes a product on the set

GA (R) = HomAlg F (A, R)

by the formula f g = mR —¦ (f —F g) —¦ c where mR : R —F R ’ R is the multiplication

in R. The de¬ning properties of Hopf algebras imply that this product is associative,

u

with a left identity given by the composition A ’ F ’ R and left inverses given

’

by f ’1 = f —¦ i; thus GA (R) is a group.

For any F -algebra homomorphism f : R ’ S there is a group homomorphism

GA (f ) : GA (R) ’ GA (S), g ’ f —¦ g,

hence we obtain a functor

GA : Alg F ’ Groups.

Any Hopf algebra homomorphism A ’ B induces a natural transformation of

functors GB ’ GA .

(20.1) Remark. Let A be an F -algebra with a comultiplication c : A ’ A —F A.

Then c yields a binary operation on the set GA (R) for any F -algebra R. If for any R

the set GA (R) is a group with respect to this operation, then A is automatically

a Hopf algebra, that is, the comultiplication determines uniquely the co-inverse i

and the co-unit u.

An (a¬ne) group scheme G over F is a functor G : Alg F ’ Groups isomorphic

to GA for some Hopf algebra A over F . By Yoneda™s lemma (see for example

Waterhouse [?, p. 6]) the Hopf algebra A is uniquely determined by G (up to

an isomorphism) and is denoted A = F [G]. A group scheme G is said to be

commutative if G(R) is commutative for all R ∈ Alg F .

A group scheme homomorphism ρ : G ’ H is a natural transformation of

functors. For any R ∈ Alg F , let ρR be the corresponding group homomorphism

G(R) ’ H(R). By Yoneda™s lemma, ρ is completely determined by the unique

Hopf algebra homomorphism ρ— : F [H] ’ F [G] (called the comorphism of ρ) such

that ρR (g) = g —¦ ρ— .

Group schemes over F and group scheme homomorphisms form a category.

We denote the set of group scheme homomorphisms (over F ) ρ : G ’ H by

HomF (G, H) The functors

Group schemes Commutative Hopf

←’

over F algebras over F

G ’ F [G]

GA ← A

326 VI. ALGEBRAIC GROUPS

de¬ne an equivalence of categories. Thus, essentially, the theory of group schemes

is equivalent to the theory of (commutative) Hopf algebras.

For a group scheme G over F and for any R ∈ Alg F the group G(R) is called

the group of R-points of G. If f : R ’ S is an injective F -algebra homomorphism,

then the homomorphism G(f ) : G(R) ’ G(S) is also injective. If L/E is a Galois

extension of ¬elds containing F , with Galois group ∆ = Gal(L/E), then ∆ acts

naturally on G(L); Galois descent (Lemma (??)) applied to the algebra L[G] shows

that the natural homomorphism G(E) ’ G(L) identi¬es G(E) with the subgroup

G(L)∆ of Galois stable elements.

(20.2) Examples. (1) The trivial group 1(R) = 1 is represented by the trivial

Hopf algebra A = F .

(2) Let V be a ¬nite dimensional vector space over F . The functor

V : Alg F ’ Groups, R ’ V R = V —F R

(to additive groups) is represented by the symmetric algebra F [V] = S(V — ) of the

dual space V — . Namely one has

HomAlg F S(V — ), R = HomF (V — , R) = V —F R.

for any R ∈ Alg F The comultiplication c is given by c(f ) = f — 1 + 1 — f , the

co-inverse i by i(f ) = ’f , and the co-unit u by u(f ) = 0 for f ∈ V — .

In the particular case V = F we have the additive group, written Ga . One has

Ga (R) = R and F [Ga ] = F [t].

(3) Let A be a unital associative F -algebra of dimension n. The functor

R ’ (AR )—

GL1 (A) : Alg F ’ Groups,

1

is represented by the algebra B = S(A— )[ N ] where N : A ’ F is the norm map

considered as an element of S n (A— ). For,

HomAlg F (B, R) = { f ∈ HomAlg F S(A— ), R | f (N ) ∈ R— }

= { a ∈ AR | N (a) ∈ R— } = (AR )— .

The comultiplication c is induced by the map

A— ’ A — — A —

dual to the multiplication m. In the particular case A = EndF (V ) we set GL(V ) =

GL1 (A) (the general linear group), thus GL(V )(R) = GL(VR ).

1

If V = F n , we write GLn (F ) for GL(V ). Clearly, F [GLn (F )] = F [Xij , det X ]

where X = (Xij ).

If A = F we set Gm = Gm,F = GL1 (A) (the multiplicative group). Clearly,

Gm (R) = R— , F [Gm ] = F [t, t’1 ] with comultiplication c(t) = t — t, co-inverse

i(t) = t’1 , and co-unit u(t) = 1.

A group scheme G over F is said to be algebraic if the F -algebra F [G] is ¬nitely

generated. All the examples of group schemes given above are algebraic.

Let G be a group scheme over F and let L/F be a ¬eld extension. The functor

GL : Alg L ’ Groups, GL (R) = G(R)

is represented by F [G]L = F [G] —F L, since

HomAlg L (F [G]L , R) = HomAlg F (F [G], R) = G(R), R ∈ Alg L .

§20. HOPF ALGEBRAS AND GROUP SCHEMES 327

The group scheme GL is called the restriction of G to L. For example we have

GL1 (A)L = GL1 (AL ).

Subgroups. Let G be a group scheme over F , let A = F [G], and let J ‚ A

be a Hopf ideal. Consider the group scheme H represented by A/J and the group

scheme homomorphism ρ : H ’ G induced by the natural map A ’ A/J. Clearly,

for any R ∈ Alg F the homomorphism ρR : H(R) ’ G(R) is injective, hence we can

identify H(R) with a subgroup in G(R). H is called a (closed ) subgroup of G and

ρ a closed embedding. A subgroup H in G is said to be normal if H(R) is normal

in G(R) for all R ∈ Alg F .

(20.3) Examples. (1) For any group scheme G, the augmentation Hopf ideal I =

ker(u) ‚ F [G] corresponds to the trivial subgroup 1 since F [G]/I F .

(2) Let V be an F -vector space of ¬nite dimension. For v ∈ V , v = 0, consider the

functor

Sv (R) = { ± ∈ GL(VR ) | ±(v) = v } ‚ GL(V )(R).

To show that Sv is a subgroup of GL(V ) (called the stabilizer of v) consider an

1

F -basis (v1 , v2 , . . . , vn ) of V with v = v1 . Then F [GL(V )] = F [Xij , det X ] and Sv

corresponds to the Hopf ideal in this algebra generated by X11 ’ 1, X21 , . . . , Xn1 .

(3) Let U ‚ V be a subspace. Consider the functor

NU (R) = { ± ∈ GL(VR ) | ±(UR ) = UR } ‚ GL(V )(R).

To show that NU is a subgroup in GL(V ) (called the normalizer of U ) consider

an F -basis (v1 , v2 , . . . , vn ) of V such that (v1 , v2 , . . . , vk ) is a basis of U . Then NU

1

corresponds to the Hopf ideal in F [Xij , det X ] generated by the Xij for i = k + 1,

. . . , n; j = 1, 2, . . . , k.

Let f : G ’ H be a homomorphism of group schemes, with comorphism

f : F [H] ’ F [G]. The ideal J = ker(f — ) is a Hopf ideal in F [H]. It corre-

—

sponds to a subgroup in H called the image of f and denoted im(f ). Clearly, f

decomposes as

¯

f h

G ’ im(f ) ’ H

’ ’

where h is a closed embedding. A homomorphism f is said to be surjective if f —

¯

is injective. Thus the f above is surjective. Note that for a surjective homomor-

phism, the induced homomorphism of groups of points G(R) ’ H(R) need not be

surjective. For example, the nth power homomorphism f : Gm ’ Gm is surjective

since its comorphism f — : F [t] ’ F [t] given by f — (t) = tn is injective. However for

R ∈ Alg F the nth power homomorphism fR : R— ’ R— is not in general surjective.

A character of a group scheme G over F is a group scheme homomorphism

G ’ Gm . Characters of G form an abelian group denoted G— .

A character χ : G ’ Gm is uniquely determined by the element f = χ— (t) ∈

F [G]— which satis¬es c(f ) = f — f . The elements f ∈ F [G]— satisfying this

condition are called group-like elements. The group-like elements form a subgroup

of G isomorphic to G— .

Let A be a central simple algebra over F . The reduced norm homomorphism

Nrd : GL1 (A) ’ Gm

is a character of GL1 (A).

328 VI. ALGEBRAIC GROUPS

Fiber products, inverse images, and kernels. Let fi : Gi ’ H, i = 1, 2,

be group scheme homomorphisms. The functor

(G1 —H G2 )(R) = G1 (R) —H(R) G2 (R)

= { (x, y) ∈ G1 (R) — G2 (R) | (f1 )R (x) = (f2 )R (y) }

is called the ¬ber product of G1 and G2 over H. It is represented by the Hopf

algebra F [G1 ] —F [H] F [G2 ].

(20.4) Examples. (1) For H = 1, we get the product G1 — G2 , represented by

F [G1 ] —F F [G2 ].

(2) Let f : G ’ H be a homomorphism of group schemes and let H be the sub-

group of H given by a Hopf ideal J ‚ F [H]. Then G —H H is a subgroup in G

given by the Hopf ideal f — (J) · F [G] in F [G], called the inverse image of H and

denoted f ’1 (H ). Clearly

f ’1 (H )(R) = { g ∈ G(R) | fR (g) ∈ H (R) }.

(3) The group f ’1 (1) in the preceding example is called the kernel of f , ker(f ),

ker(f )(R) = { g ∈ G(R) | fR (g) = 1 }.

The kernel of f is the subgroup in G corresponding to the Hopf ideal f — (I) · F [G]

where I is the augmentation ideal in F [H].

’1

(4) If fi : Hi ’ H are closed embeddings, i = 1, 2, then H1 —H H2 = f1 (H2 ) =

’1

f2 (H1 ) is a subgroup of H1 and of H2 , called the intersection H1 © H2 of H1

and H2 .

(5) The kernel of the nth power homomorphism Gm ’ Gm is denoted µn = µn,F

and called the group of nth roots of unity. Clearly,

µn (R) = { x ∈ R— | xn = 1 }

and F [µn ] = F [t]/(tn ’ 1) · F [t].

(6) Let A be a central simple algebra over F . The kernel of the reduced norm

character Nrd : GL1 (A) ’ Gm is denoted SL1 (A). If A = End(V ) we write SL(V )

for SL1 (A) and call the corresponding group scheme the special linear group.

(7) Let ρ : G ’ GL(V ) be a group scheme homomorphism and let 0 = v ∈ V . The

inverse image of the stabilizer ρ’1 (Sv ) is denoted AutG (v),

AutG (v)(R) = { g ∈ G(R) | ρR (g)(v) = v }.

(8) Let A be an F -algebra of ¬nite dimension (not necessarily unital, commutative,

associative). Let V = HomF (A —F A, A) and let v ∈ V be the multiplication map

in A. Consider the group scheme homomorphism

ρ : GL(A) ’ GL(V )

given by

ρR (±)(f )(a — a ) = ± f ±’1 (a) — ±’1 (a ) .

The group scheme AutGL(A) (v) for this v is denoted Autalg (A). The group of

R-points Autalg (A)(R) coincides with the group AutR (A) of R-automorphisms of

the R-algebra AR .

§20. HOPF ALGEBRAS AND GROUP SCHEMES 329

The corestriction. Let L/F be a ¬nite separable ¬eld extension and let G

be a group scheme over L with A = L[G]. Consider the functor

(20.5) RL/F (G) : Alg F ’ Groups, R ’ G(R —F L).

(20.6) Lemma. The functor RL/F (G) is a group scheme.

Proof : Let X = X(L) be the set of all F -algebra homomorphisms „ : L ’ Fsep .

The Galois group “ = Gal(Fsep /F ) acts on X by γ „ = γ —¦ „ . For any „ ∈ X let

A„ be the tensor product A —L Fsep where Fsep is made an L-algebra via „ , so that

a — x = a — „ ( )x for a ∈ A, ∈ L and x ∈ Fsep . For any γ ∈ “ and „ ∈ X the

map

γ„ : A„ ’ Aγ„ , a — x ’ a — γ(x)

is a ring isomorphism such that γ„ (xu) = γ(x) · γ„ (u) for x ∈ Fsep , u ∈ A„ .

Consider the tensor product B = —„ ∈X A„ over Fsep . The group “ acts contin-

uously on B by

γ(—a„ ) = —a„ where aγ„ = γ„ (a„ ).

The Fsep -algebra B has a natural Hopf algebra structure arising from the Hopf

algebra structure on A, and the structure on B, compatible with the action of “.

Hence the F -algebra B = B “ of “-stable elements is a Hopf algebra and by Lemma

(??) we get B —F Fsep B.

We show that the F -algebra B represents the functor RL/F (G). For any F -

algebra R we have a canonical isomorphism

HomAlg Fsep (B, R —F Fsep )“ .

HomAlg F (B, R)

A “-equivariant homomorphism B ’ R —F Fsep is determined by a collection of

Fsep -algebra homomorphisms {f„ : A„ ’ R —F Fsep }„ ∈X such that, for all γ ∈ “

and „ ∈ X, the diagram

f„

A„ ’ ’ ’ R —F Fsep

’’

¦ ¦

¦ ¦Id—γ

γ„

fγ„

Aγ„ ’ ’ ’ R —F Fsep

’’

commutes. For the restrictions g„ = f„ |A : A ’ R —F Fsep we have

(Id — γ) · g„ = gγ„ .

Hence the image of g„ is invariant under Gal(Fsep /„ L) ‚ “ and im g„ ‚ R —F („ L).

It is clear that the map

h = (Id — „ )’1 —¦ g„ : A ’ R —F L

is independent of the choice of „ and is an L-algebra homomorphism. Conversely,

any L-algebra homomorphism h : A ’ R —F L de¬nes a collection of maps f„ by

f„ (a — x) = [(Id — „ )h(a)]x.

Thus, HomAlg F (B, R) = HomAlg L (A, R —F L) = G(R —F L).

The group scheme RL/F (G) is called the corestriction of G from L to F .

330 VI. ALGEBRAIC GROUPS

(20.7) Proposition. The functors restriction and corestriction are adjoint to each

other, i.e., for any group schemes H over F and G over L, there is a natural

bijection

HomF H, RL/F (G) HomL (HL , G).

Furthermore we have

[RL/F (G)]Fsep G„ ,

„ ∈X

where G„ = GFsep , with Fsep made an L-algebra via „ .

Proof : Both statements follow from the proof of Lemma (??).

(20.8) Example. For a ¬nite dimensional L-vector space V , RL/F (V) = VF

where VF = V considered as an F -vector space.

(20.9) Remark. Sometimes it is convenient to consider group schemes over ar-

bitrary ´tale F -algebras (not necessarily ¬elds) as follows. An ´tale F -algebra L

e e

decomposes canonically into a product of separable ¬eld extensions,

L = L 1 — L2 — · · · — L n ,

(see Proposition (??)) and a group scheme G over L is a collection of group

schemes Gi over Li . One then de¬nes the corestriction RL/F (G) to be the product

of the corestrictions RLi /F (Gi ). For example we have

GL1 (L) = RL/F (Gm,L )

for an ´tale F -algebra L. Proposition (??) also holds in this setting.

e

The connected component. Let A be a ¬nitely generated commutative F -

algebra and let B ‚ A be an ´tale F -subalgebra. Since the Fsep -algebra B —F Fsep

e

is spanned by its idempotents (see Proposition (??)), dimF B is bounded by the

(¬nite) number of primitive idempotents of A —F Fsep . Furthermore, if B1 , B2 ‚ A

are ´tale F -subalgebras, then B1 B2 is also ´tale in A, being a quotient of the tensor

e e

product B1 —F B2 . Hence there exists a unique largest ´tale F -subalgebra in A,

e

which we denote π0 (A).

(20.10) Proposition. (1) The subalgebra π0 (A) contains all idempotents of A.

Hence A is connected (i.e., the a¬ne variety Spec A is connected, resp. A has no

non-trivial idempotents) if and only if π0 (A) is a ¬eld.

(2) For any ¬eld extension L/F , π0 (AL ) = π0 (A)L .

(3) π0 (A —F B) = π0 (A) —F π0 (B).

Reference: See Waterhouse [?, §6.5].

(20.11) Proposition. Let A be a ¬nitely generated Hopf algebra over F . Then A

is connected if and only if π0 (A) = F .

Proof : The “if” implication is part of (??) of Proposition (??). We show the

converse: the co-unit u : A ’ F takes the ¬eld π0 (A) to F , hence π0 (A) = F .

We call an algebraic group scheme G over F connected if F [G] is connected

(i.e., F [G] contains no non-trivial idempotents) or, equivalently, if π0 (F [G]) = F .

Let G be an algebraic group scheme over F and let A = F [G]. Then c π0 (A) ,

being an ´tale F -subalgebra in A—F A, is contained in π0 (A—F A) = π0 (A)—F π0 (A)

e

(see (??) of Proposition (??)). Similarly, we have i π0 (A) ‚ π0 (A). Thus, π0 (A) is

§20. HOPF ALGEBRAS AND GROUP SCHEMES 331

a Hopf subalgebra of A. The group scheme represented by π0 (A) is denoted π0 (G).

There is a natural surjection G ’ π0 (G). Clearly, G is connected if and only if

π0 (G) = 1. Propositions (??) and (??) then imply:

(20.12) Proposition. (1) Let L/F be a ¬eld extension and let G be an algebraic

group scheme over F . Then π0 (GL ) = π0 (G)L . In particular, GL is connected if

and only if G is connected.

(2) π0 (G1 — G2 ) = π0 (G1 ) — π0 (G2 ). In particular, the Gi are connected if and

only if G1 — G2 is connected.

Let G be an algebraic group scheme over F and let A = F [G]. The co-unit

homomorphism u maps all but one primitive idempotent of A to 0, so let e be

the primitive idempotent with u(e) = 1. Since π0 (A)e is a ¬eld, π0 (A)e = F and

I0 = π0 (A) · (1 ’ e) is the augmentation ideal in π0 (A). Denote the kernel of

G ’ π0 (G) by G0 . It is represented by the algebra A/A · I0 = A/A(1 ’ e) = Ae.

Since Ae is connected, G0 is connected; it is called the connected component of G.

We have (G1 — G2 )0 = G0 — G0 and for any ¬eld extension L/F , (GL )0 = (G0 )L .