all maximal ideals in A is zero since A is reduced, therefore f — is injective and f is

surjective.

(22.4) Proposition. Let f : G ’ H be a surjective homomorphism of algebraic

group schemes.

(1) If G is connected (resp. smooth), then H is connected (resp. smooth).

(2) Let H be a subgroup of H. Then the restriction of f to f ’1 (H ) is a surjective

homomorphism f ’1 (H ) ’ H .

Proof : (??) is clear. For (??), let J ‚ A = F [H] be the Hopf ideal corresponding

to H . Hence the ideal J = f — (J) · B ‚ B = F [G] corresponds to f ’1 (H ),

and the homomorphism F [H ] = A/J ’ B/J = F [f ’1 (H )] is injective since

(f — )’1 (J ) = J (see Bourbaki [?, Ch.I, §3, no. 5, Prop. 8 (ii)]).

The isomorphism criterion. Propositions (??) and (??) imply that

(22.5) Proposition. Let f : G ’ H be a homomorphism of algebraic group sche-

mes with H smooth. Then the following conditions are equivalent:

(1) f is an isomorphism.

(2) f is injective and surjective.

(3) falg : G(Falg ) ’ H(Falg ) is an isomorphism and df is injective.

(22.6) Example. Let f : Gm ’ Gm be the pth -power homomorphism where p =

char F . Clearly, falg is an isomorphism, but f is not since df = 0.

Factor group schemes.

(22.7) Proposition. Let f : G ’ H be a surjective homomorphism of group sche-

mes with kernel N . Then any group scheme homomorphism f : G ’ H vanishing

on N factors uniquely through f .

Proof : Let A = F [H] and B = F [G]. The two natural homomorphisms B

B —A B, being elements in G(B —A B), have the same image in H(B —A B) and

hence they are congruent modulo N (B —A B). Hence the two composite maps

—

f

F [H ] ’ ’ B

’ B —A B

coincide. By the faithful ¬‚atness of B over A the equalizer of B B —A B is A,

—

thus the image of f lies in A.

The proposition shows that a surjective homomorphism f : G ’ H is uniquely

determined (up to isomorphism) by G and the normal subgroup N . We write

H = G/N and call H the factor group scheme G modulo N .

(22.8) Proposition. Let G be a group scheme and let N be a normal subgroup

in G. Then there is a surjective homomorphism G ’ H with the kernel N , i.e.,

the factor group scheme G/N exists.

Reference: See Waterhouse [?, §16.3].

§22. FACTOR GROUPS 341

Exact sequences. A sequence of homomorphisms of group schemes

f g

(22.9) 1’N ’ G’ H ’1

’’

is called exact if f induces an isomorphism of N with ker(g) and g is surjective or,

equivalently, f is injective and H G/ im(f ). For any group scheme homomor-

phism g : G ’ H we have an exact sequence 1 ’ ker(g) ’ G ’ im(g) ’ 1, i.e.,

im(g) G/ ker(g).

(22.10) Proposition. A sequence as in (??) with H smooth is exact if and only

if

fR gR

(1) 1 ’ N (R) ’’ G(R) ’’ H(R) is exact for every R ∈ Alg F and

(2) galg : G(Falg ) ’ H(Falg ) is surjective.

Proof : It follows from (??) that N = ker(g) and from Proposition (??) that g is

surjective.

(22.11) Proposition. Suppose that (??) is exact. Then

dim G = dim N + dim H.

Proof : We may assume that F = Falg and that G (hence also H) is connected.

Put A = F [H], B = F [G], C = F [N ]. We have a bijection of represented functors

∼

G — N ’ G —H G,

’ (g, n) ’ (g, gn).

By Yoneda™s lemma there is an F -algebra isomorphism B —A B B —F C. We

compute the Krull dimension of both sides. Denote by Quot(S) the ¬eld of fractions

of a domain S; let K = Quot(Ared ) and L = Quot(Bred ); then

dim(B —A B) = dim(Bred —Ared Bred ) = tr.degF Quot(L —K L)red

= 2 · tr.degK (L) + tr.degF (K) = 2 · tr.degF (L) ’ tr.degF (K)

= 2 · dim G ’ dim H.

On the other hand

dim(B —F C) = dim(G — N ) = dim G + dim N

by Proposition (??).

(22.12) Corollary. Suppose that in (??) N and H are smooth. Then G is also

smooth.

Proof : By Proposition (??)(b), ker(dg) = Lie(N ). Hence

dim Lie(G) = dim ker(dg) + dim im(dg) ¤ dim Lie(N ) + dim Lie(H),

= dim N + dim H = dim G,

and therefore, G is smooth.

A surjective homomorphism f : G ’ H is said to be separable if the di¬erential

df : Lie(G) ’ Lie(H) is surjective.

(22.13) Proposition. A surjective homomorphism f : G ’ H of algebraic groups

is separable if and only if ker(f ) is smooth.

342 VI. ALGEBRAIC GROUPS

Proof : Let N = ker(f ). By Propositions (??) and (??),

dim Lie(N ) = dim ker(df ) = dim Lie(G) ’ dim im(df ),

= dim G ’ dim im(df ) = dim N + dim H ’ dim im(df ),

= dim N + dim Lie(H) ’ dim im(df ).

Hence, N is smooth if and only if dim N = dim Lie(N ) if and only if dim Lie(H) ’

dim im(df ) = 0 if and only if df is surjective.

(22.14) Example. The natural surjection GL1 (A) ’ GL1 (A)/ Gm is separable.

f

(22.15) Proposition. Let 1 ’ N ’ G ’ H ’ 1 be an exact sequence of alge-

’

braic group schemes with N smooth. Then the sequence of groups

1 ’ N (Fsep ) ’ G(Fsep ) ’ H(Fsep ) ’ 1

is exact.

Proof : Since N = ker(f ), it su¬ces to prove only exactness on the right. We may

assume that F = Fsep . Let A = F [H], B = F [G], (so A ‚ B) and C = F [N ]. Take

any h ∈ H(F ) and consider the F -algebra D = B —A F where F is made into an

A-algebra via h. For any R ∈ Alg F with structure homomorphism ν : F ’ R, we

have

’1

HomAlg F (D, R) = { g ∈ HomAlg F (B, R) | g|A = ν —¦ h } = fR (ν —¦ h),

i.e., the F -algebra D represents the ¬ber functor

’1

R ’ P (R) := fR (ν —¦ h) ‚ G(R).

If there exists g ∈ G(F ) such that fF (g) = h, i.e. g ∈ P (F ), then there is a bijection

of functors „ : N ’ P , given by „ (R)(n) = n · (ν —¦ g) ∈ P (R). By Yoneda™s lemma

the F -algebras C and D representing the functors N and P are then isomorphic.

We do not know yet if such an element g ∈ P (F ) exists, but it certainly exists

over E = Falg since HomAlg E (DE , E) = … (a form of Hilbert Nullstellensatz).

Hence the E-algebras CE and DE are isomorphic. In particular, DE is reduced.

Then HomAlg F (D, F ) = … (see Borel [?, AG 13.3]), i.e., P (F ) = …, so h belongs to

the image of fF , and the described g exists.

Isogenies. A surjective homomorphism f : G ’ H of group schemes is called

an isogeny if N = ker(f ) is ¬nite, and is called a central isogeny if N (R) is central

in G(R) for every R ∈ Alg F .

(22.16) Example. The nth -power homomorphism Gm ’ Gm is a central isogeny.

Representations. Let G be a group scheme over F , with A = F [G]. A

representation of G is a group scheme homomorphism ρ : G ’ GL(V ) where V is

a ¬nite dimensional vector space over F . For any R ∈ Alg F the group G(R) then

acts on VR = V —F R by R-linear automorphisms; we write

g · v=ρR (g)(v), g ∈ G(R), v ∈ VR .

By taking R = A, we obtain an F -linear map

ρ : V ’ V —F A, ρ(v) = IdA · v

§22. FACTOR GROUPS 343

(where IdA ∈ G(A) is the “generic” element), such that the following diagrams

commute (see Waterhouse [?, §3.2])

ρ

V ’’’

’’ V —F A

¦ ¦

¦ ¦

(22.17) ρ Id—c

ρ—Id

V —F A ’ ’ ’ V —F A —F A,

’’

ρ

V ’ ’ ’ V —F A

’’

¦

¦

(22.18) Id—u

∼

V ’ ’ ’ V —F F.

’’

Conversely, a map ρ for some F -vector space V , such that the diagrams (??)

and (??) commute, yields a representation ρ : G ’ GL(V ) as follows: given g ∈

G(R), ρ(g) is the R-linear extension of the composite map

ρ Id—g

V ’ V —F A ’ ’ V —F R.

’ ’’

A ¬nite dimensional F -vector space V together with a map ρ as above is called an

A-comodule. There is an obvious notion of subcomodules.

A vector v ∈ V is said to be G-invariant if ρ(v) = v — 1. Denote by V G the

F -subspace of all G-invariant elements. Clearly, G(R) acts trivially on (V G ) —F R

for any R ∈ Alg F . For a ¬eld extension L/F one has (VL )GL (V G )L .

A representation ρ : G ’ GL(V ) is called irreducible if the A-comodule V has

no nontrivial subcomodules.

(22.19) Examples. (1) If dim V = 1, then GL(V ) = Gm . Hence a 1-dimensional

representation is simply a character.

(2) Let G be an algebraic group scheme over F . For any R ∈ Alg F the group G(R)

acts by conjugation on

ker G(R[µ]) ’ G(R) = Lie(G) —F R.

Hence we get a representation

Ad = AdG : G ’ GL Lie(G)

called the adjoint representation. When G = GL(V ) the adjoint representation

Ad : GL(V ) ’ GL End(V )

is given by conjugation: Ad(±)(β) = ±β±’1 .

Representations of diagonalizable groups. Let G = Hdiag be a diagonal-

izable group scheme, A = F [G] = F H . Let ρ : V ’ V —F A be the A-comodule

structure on a ¬nite dimensional vector space V corresponding to some represen-

tation ρ : G ’ GL(V ).

Write ρ(v) = fh (v) — h for uniquely determined F -linear maps fh : V ’ V .

h∈H

The commutativity of diagram (??) is equivalent to the conditions

fh if h = h ,

fh —¦ f h =

0 if h = h ,

344 VI. ALGEBRAIC GROUPS

and the commutativity of (??) gives fh (v) = v for all v ∈ V . Hence the maps fh

h∈H

induce a decomposition

(22.20) V= Vh , where Vh = im(fh ).

h∈H

A character h ∈ H = G— is called a weight of ρ if Vh = 0. A representation ρ of

a diagonalizable group is uniquely determined (up to isomorphism) by its weights

and their multiplicities mh = dim Vh .

§23. Automorphism Groups of Algebras

In this section we consider various algebraic group schemes related to algebras

and algebras with involution.

Let A be a separable associative unital F -algebra (i.e., A is a ¬nite product

of algebras which are central simple over ¬nite separable ¬eld extensions of F , or

equivalently, AF = A — F is semisimple for every ¬eld extension F of F ). Let L

be the center of A (which is an ´tale F -algebra). The kernel of the restriction ho-

e

momorphism Autalg (A) ’ Autalg (L) is denoted AutL (A). Since all L-derivations

of A are inner (see for example Knus-Ojanguren [?, p. 73-74]), it follows from

Example (??.??) that

Lie AutL (A) = DerL (A, A) = A/L.

We use the notation ad(a)(x) = [a, x] = ax ’ xa for the inner derivation ad(a)

associated to a ∈ A. Consider the group scheme homomorphism

Int : GL1 (A) ’ AutL (A), a ’ Int(a)

with kernel GL1 (L) = RL/F (Gm,L ). By Proposition (??) we have:

dim AutL (A) ≥ dim im(Int) = dim GL1 (A) ’ dim GL1 (L)

= dimF A ’ dimF L = dimF Lie AutL (A) .

The group scheme AutL (A) is smooth. This follows from Lemma (??) and

Proposition (??). By the Skolem-Noether theorem the homomorphism IntE is

surjective for any ¬eld extension E/F , hence Int is surjective by Proposition (??),

and AutL (A) is connected by Proposition (??). Thus we have an exact sequence

of connected algebraic groups

(23.1) 1 ’ GL1 (L) ’ GL1 (A) ’ AutL (A) ’ 1.

Assume now that A is a central simple algebra over F , i.e., L = F . We write

PGL1 (A) for the group Autalg (A), so that

PGL1 (A) GL1 (A)/ Gm , Lie PGL1 (A) = A/F,

and

PGL1 (A)(R) = AutR (AR ), R ∈ Alg F .

We say that an F -algebra R satis¬es the SN -condition if for any central simple

algebra A over F all R-algebra automorphisms of AR are inner. Fields and local

rings satisfy the SN -condition (see for example Knus-Ojanguren [?, p. 107]).

If R satis¬es the SN -condition then

PGL1 (A)(R) = (AR )— /R— .

(23.2)

§23. AUTOMORPHISM GROUPS OF ALGEBRAS 345

We set PGL(V ) = PGL1 End(V ) = GL(V )/ Gm and call PGL(V ) the projective

general linear group; we write PGL(V ) = PGL n if V = F n .

23.A. Involutions. In this part we rediscuss most of the groups introduced

in Chapter ?? from the point of view of group schemes. Let A be a separable F -

algebra with center K and F -involution σ. We de¬ne various group schemes over

F related to A. Consider the representation

ρ : GL1 (A) ’ GL(A), a ’ x ’ a · x · σ(a) .

The subgroup AutGL1 (A) (1) in GL1 (A) is denoted Iso(A, σ) and is called the group

scheme of isometries of (A, σ):

Iso(A, σ)(R) = { a ∈ A— | a · σR (a) = 1 }.

R

An element 1 + a · µ, a ∈ A lies in ker Iso(A, σ)(κ) if and only if

(1 + a · µ) 1 + σ(a) · µ = 1,

or equivalently, a + σ(a) = 0. Hence,

Lie Iso(A, σ) = Skew(A, σ) ‚ A.

Consider the adjoint representation

± ’ (β ’ ±β±’1 )

ρ : GL(A) ’ GL EndF (A) ,

and denote the intersection of the subgroups Autalg (A) and AutGL(A) (σ) of GL(A)

by Aut(A, σ):

Aut(A, σ)(R) = { ± ∈ AutR (AR ) | ± —¦ σR = σR —¦ ± }.

A derivation x = ad(a) ∈ Der(A, A) = Lie Autalg (A) lies in Lie Aut(A, σ) if

and only if (1 + x · µ) —¦ σ = σ —¦ (1 + x · µ) if and only if x —¦ σ = σ —¦ x if and only if

a + σ(a) ∈ K. Hence

Lie Aut(A, σ) = { a ∈ A | a + σ(a) ∈ K }/K.

Denote the intersection of Aut(A, σ) and AutK (A) by AutK (A, σ). If an F -

algebra R satis¬es the SN -condition, then

AutK (A, σ)(R) = { a ∈ A— | a · σR (a) ∈ KR }/KR .

— —

R

The inverse image of AutK (A, σ) with respect to the surjection

Int : GL1 (A) ’ AutK (A)

(see ??) is denoted Sim(A, σ) and called the group scheme of similitudes of (A, σ).

Clearly,

Sim(A, σ)(R) = { a ∈ A— | a · σ(a) ∈ KR }

—

R

Lie Sim(A, σ) = { a ∈ A | a + σ(a) ∈ K }.

By Proposition (??) we have an exact sequence of group schemes

(23.3) 1 ’ GL1 (K) ’ Sim(A, σ) ’ AutK (A, σ) ’ 1.

Let E be the F -subalgebra of K consisting of all σ-invariant elements. We have a

group scheme homomorphism

µ : Sim(A, σ) ’ GL1 (E), a ’ a · σ(a).

346 VI. ALGEBRAIC GROUPS

The map µalg is clearly surjective. Hence, by Proposition (??), we have an exact

sequence

µ

(23.4) 1 ’ Iso(A, σ) ’ Sim(A, σ) ’ GL1 (E) ’ 1.

’

Unitary involutions. Let K/F be an ´tale quadratic extension, B be a cen-

e

tral simple algebra over K of degree n with a unitary F -involution „ . We use the

following notation (and de¬nitions) for group schemes over F :

U(B, „ ) = Iso(B, „ ) Unitary group

GU(B, „ ) = Sim(B, „ ) Group of unitary similitudes

PGU(B, „ ) = AutK (B, „ ) Projective unitary group

Assume ¬rst that K is split, K F — F . Then B A — Aop and „ is the exchange

involution. Let b = (a1 , aop ) ∈ B. The condition b·„ b = 1 is equivalent to a1 a2 = 1.

2

Hence we have an isomorphism

∼

a ’ a, (a’1 )op .

GL1 (A) ’ U(B, „ ),

’

The homomorphism

φ ’ (φ, φop )

Autalg (A) ’ PGU(B, „ ),

is clearly an isomorphism. Hence,

PGU(B, „ ) PGL1 (A).

Thus the group schemes U(B, „ ) and PGU(B, „ ) are smooth and connected. This

also holds when K is not split, as one sees by scalar extension. Furthermore the sur-

jection Aut(B, „ ) ’ Autalg (K) Z/2Z induces an isomorphism π0 Aut(B, „ )

Z/2Z. Hence PGU(B, „ ) is the connected component of Aut(B, „ ) and is as a sub-

group of index 2.

The kernel of the reduced norm homomorphism Nrd : U(B, „ ) ’ GL 1 (K) is

denoted SU(B, „ ) and called the special unitary group. Clearly,

SU(B, „ )(R) = { b ∈ (B —F R)— | b · „R (b) = 1, NrdR (b) = 1 },

Lie SU(B, „ ) = { x ∈ Skew(B, „ ) | Trd(x) = 0 }.

The group scheme SU(B, „ ) is smooth and connected since, when K is split,

SU(B, „ ) = SL1 (A) (as the description given above shows). The kernel N of

the composition

f : SU(B, „ ) ’ U(B, „ ) ’ PGU(B, „ )

satis¬es

N (R) = { B ∈ (K —F R)— | b · „R (b) = 1, bn = 1 }.

In other words,

NK/F

N = ker RK/F (µn,K ) ’ ’ ’ µn,F ,

’’

hence N is a ¬nite group scheme of multiplicative type and is Cartier dual to Z/nZ

where the Galois group “ acts through Gal(K/F ) as x ’ ’x. Subgroups of N

correspond to (cyclic) subgroups of Z/nZ, which are automatically “-invariant.

Since falg is surjective, f is surjective by Proposition (??). Clearly, f is a

central isogeny and

PGU(B, „ ) SU(B, „ )/N.

§23. AUTOMORPHISM GROUPS OF ALGEBRAS 347

Symplectic involutions. Let A be a central simple algebra of degree n =

2m over F with a symplectic involution σ. We use the following notation (and

de¬nitions):

Sp(A, σ) = Iso(A, σ) Symplectic group

GSp(A, σ) = Sim(A, σ) Group of symplectic similitudes

PGSp(A, σ) = Aut(A, σ) Projective symplectic group

Assume ¬rst that A is split, A = EndF (V ), hence σ = σh where h is a nonsingular

alternating bilinear form on V . Then Sp(A, σ) = Sp(V, h), the symplectic group

of (V, h),

Sp(V, h)(R) = { ± ∈ GL(VR ) | hR ±(v), ±(v ) = hR (v, v ) for v, v ∈ VR }.

The associated classical algebraic group is connected of dimension m(2m + 1)

(Borel [?, 23.3]).

Coming back to the general case, we have

dim Lie Sp(A, σ) = dim Skew(A, σ) = m(2m + 1) = dim Sp(A, σ),

hence Sp(A, σ) is a smooth and connected group. It follows from the exactness of

µ

1 ’ Sp(A, σ) ’ GSp(A, σ) ’ Gm ’ 1

’

(see ??) and Corollary (??) that GSp(A, σ) is smooth.

The exactness of

1 ’ Gm ’ GSp(A, σ) ’ PGSp(A, σ) ’ 1

(see ??) implies that PGSp(A, σ) is smooth. Consider the composition

f : Sp(A, σ) ’ GSp(A, σ) ’ PGSp(A, σ)

whose kernel is µ2 . Clearly, falg is surjective, hence f is surjective and PGSp(A, σ)

is connected. Therefore, f is a central isogeny and PGSp(A, σ) Sp(A, σ)/µ2 .

In the split case the group PGSp(V, h) = PGSp(A, σ) is called the projective

symplectic group of (V, h).

Orthogonal involutions. Let A be a central simple algebra of degree n over F

with an orthogonal involution σ. We use the following notation

O(A, σ) = Iso(A, σ)

GO(A, σ) = Sim(A, σ)

PGO(A, σ) = Aut(A, σ)

Consider the split case A = EndF (V ), σ = σb , where b is a nonsingular symmetric

(non-alternating, if char F = 2) bilinear form. Then

O(A, σ)(R) = { ± ∈ GL(VR ) | b(±v, ±v ) = b(v, v ) for v, v ∈ VR }.

n(n’1)

The associated classical algebraic group has dimension (Borel [?]). On the

2

other hand,

n(n’1)

if char F = 2,

2

dim Lie O(A, σ) = dim Skew(A, σ) = n(n+1)

if char F = 2.

2

Hence O(A, σ) (and the other groups) are not smooth if char F = 2. To get smooth

groups also in characteristic 2 we use a di¬erent context, described in the next two

subsections.

348 VI. ALGEBRAIC GROUPS