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 ??)