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 .

1 2

(20.13) Examples. (1) GL1 (A) is connected.

(2) For a central simple algebra A, G = SL1 (A) is connected since F [G] is is

the quotient of a polynomial ring modulo the ideal generated by the irreducible

polynomial Nrd(X) ’ 1.

(3) µn is an example of a non-connected group scheme.

(20.14) Lemma (Homogeneity property of Hopf algebras). Let A be a Hopf al-

gebra which is ¬nitely generated over F = Falg . Then for any pair of maximal

ideals M , M ‚ A there exists an F -algebra automorphism ρ : A ’ A such that

ρ(M ) = M .

Proof : We may assume that M is the augmentation ideal in A. Let f be the

canonical projection A ’ A/M = F , and set ρ = (Id A — f ) —¦ c. One checks that the

map IdA — (f —¦ i) —¦ c is inverse to ρ, i.e., ρ ∈ AutF (A). Since (u — IdA ) —¦ c = IdA ,

it follows that u —¦ ρ = (u — f ) —¦ c = f —¦ (u — IdA ) —¦ c = f and ρ(M ) = ρ(ker f ) =

ker u = M .

Let nil(A) be the set of all nilpotent elements of A; it is an ideal of A, and equals

the intersection of all the prime ideals of A. The algebra A/ nil(A) is denoted by

Ared .

(20.15) Proposition. Let G be an algebraic group scheme over F and let A =

F [G]. Then the following conditions are equivalent:

(1) G is connected.

(2) A is connected.

(3) Ared is connected.

(4) Ared is a domain.

Proof : The implications (??) ” (??) ” (??) ⇐ (??) are easy.

For (??) ’ (??) we may assume that F = Falg . Since G is an algebraic group

the scheme A is ¬nitely generated. Hence the intersection of all maximal ideals

in A containing a given prime ideal P is P (Bourbaki [?, Ch.V, §3, no. 4, Cor. to

Prop. 8 (ii)]) and there is a maximal ideal containing exactly one minimal prime

ideal. By the Lemma above, each maximal ideal contains exactly one minimal

prime ideal. Hence any two di¬erent minimal prime ideals P and P are coprime:

332 VI. ALGEBRAIC GROUPS

P + P = A. Let P1 , P2 , . . . , Pn be all minimal prime ideals. Since Pi = nil(A),

we have Ared = A/ nil(A) A/Pi by the Chinese Remainder Theorem. By

assumption Ared is connected, hence n = 1 and Ared = A/P1 is a domain.

Constant and ´tale group schemes. Let H be a ¬nite (abstract) group.

e