A = Map(H, F )

of all functions H ’ F . For h ∈ H, let eh be the characteristic function of {h};

this map is an idempotent in A, and we have A = h∈H F · eh . A Hopf algebra

structure on A is given by

1 if h = 1,

c(eh ) = ex — e y , i(eh ) = eh’1 , u(eh ) =

0 if h = 1.

xy=h

The group scheme over F represented by A is denoted Hconst and called the constant

group scheme associated to H. For any connected F -algebra R ∈ Alg F , Hconst (R) =

H.

A group scheme G over F is said to be ´tale if F [G] is an ´tale F -algebra. For

e

e

example, constant group schemes are ´tale and, for any algebraic group scheme G,

e

the group scheme π0 (G) is ´tale. If G is ´tale, then G(Fsep ) is a ¬nite (discrete)

e e

group with a continuous action of “ = Gal(Fsep /F ). Conversely, given a ¬nite

group H with a continuous “-action by group automorphisms, we have a “-action on

the Fsep -algebra A = Map(H, Fsep ). Let Het be the ´tale group scheme represented

e

“

by the (´tale) Hopf algebra A . Subgroups of Het are ´tale and correspond to

e e

“-stable subgroups of H.

(20.16) Proposition. The two functors

´ Finite groups with

Etale group schemes

←’

continuous “-action

over F

G ’ G(Fsep )

Het ← H

are mutually inverse equivalences of categories. In this equivalence constant group

schemes correspond to ¬nite groups with trivial “-action.

Proof : This follows from Theorem (??).

Diagonalizable group schemes and group schemes of multiplicative

type. Let H be an (abstract) abelian group, written multiplicatively. We have a

structure of a Hopf algebra on the group algebra F H over F given by c(h) = h—h,

i(h) = h’1 and u(h) = 1. The group scheme represented by F H is said to be

diagonalizable and is denoted Hdiag . Clearly,

Hdiag (R) = Hom(H, R— ), R ∈ Alg F .

The group-like elements in F H are of the form h — h for h ∈ H. Hence the

character group (Hdiag )— is naturally isomorphic to H. For example, we have

Zdiag = Gm , (Z/nZ)diag = µn .

A group scheme G over F is said to be of multiplicative type if Gsep (= GFsep ) is

diagonalizable. In particular, diagonalizable group schemes are of multiplicative

§20. HOPF ALGEBRAS AND GROUP SCHEMES 333

type. Let G be of multiplicative type. The character group (Gsep )— has a natural

continuous action of “ = Gal(Fsep /F ). To describe this action we observe that the

group of characters (Gsep )— is isomorphic to the group of group-like elements in

Fsep [Gsep ]. The action is induced from the natural action on action on Fsep [Gsep ].

Conversely, given an abelian group H with a continuous “-action, the Hopf alge-

bra of “-stable elements in Fsep [Hdiag ] = Fsep H represents a group scheme of

multiplicative type which we denote Hmult . Clearly,

Hmult (R) = Hom“ H, (R —F Fsep )— .

(20.17) Proposition. The two contravariant functors

Group schemes of

Abelian groups with

multiplicative type ←’

continuous “-action

over F

(Gsep )—

G ’

Hmult ← H

de¬ne an equivalence of categories. Under this equivalence diagonalizable group

schemes correspond to abelian groups with trivial “-action.

An algebraic torus is a group scheme of multiplicative type Hmult where H is a

free abelian group of ¬nite rank. A torus T is said to be split if it is a diagonalizable

group scheme, i.e., T = Hdiag (Zn )diag = Gm — · · ·—Gm (n factors) is isomorphic

to the group scheme of diagonal matrices in GLn (F ). Any torus T is split over Fsep .

Cartier Duality. Let H be a ¬nite abelian (abstract) group with a continuous

“-action and let “ = Gal(Fsep /F ). One can associate two group schemes to H:

Het and Hmult . We discuss the relation between these group schemes. A group

scheme G over F is called ¬nite if dimF F [G] < ∞. The order of G is dimF F [G].

For example an ´tale group scheme G is ¬nite. Its order is the order of G(Fsep ).

e

Let G be a ¬nite commutative group scheme over F ; then A = F [G] is of ¬nite

dimension. Consider the dual F -vector space A— = HomF (A, F ). The duals of

the ¬ve structure maps on A, namely the unit map e : F ’ A, the multiplication

m : A —F A ’ A and the maps c, i, u de¬ning the Hopf algebra structure on A,

yield ¬ve maps which de¬ne a Hopf algebra structure on A— . The associated group

scheme is denoted GD and is called Cartier dual of G. Thus, F [GD ] = F [G]— and

GDD = G.

Elements of the group (GD )(F ) are represented by F -algebra homomorphisms

F [G]— ’ F which, as is easily seen, are given by group-like elements of F [G].

Hence, GD (F ) G— , the character group of G.

Cartier duality is an involutory contravariant functor D on the category of

¬nite commutative group schemes over F .

The restriction of D gives an equivalence of categories

´ Finite group schemes of

Etale commutative

←’

multiplicative type over F

group schemes over F

More precisely, if H is a ¬nite abelian (abstract) group with a continuous “-action,

then

(Het )D = Hmult , (Hmult )D = Het .

334 VI. ALGEBRAIC GROUPS

(20.18) Example.

(Z/nZ)D = µn , µD = Z/nZ.

n

(We write Z/nZ for (Z/nZ)const .)

§21. The Lie Algebra and Smoothness

Let M be an A-module. A derivation D of A into M is an F -linear map

D : A ’ M such that

D(ab) = a · D(b) + b · D(a).

We set Der(A, M ) for the A-module of all derivations of A into M .

21.A. The Lie algebra of a group scheme. Let G be an algebraic group

scheme over F and let A = F [G]. A derivation D ∈ Der(A, A) is said to be left-

invariant if c —¦ D = (id — D) —¦ c. The F -vector space of left-invariant derivations

is denoted Lie(G) and is called the Lie algebra of G. The Lie algebra structure

on Lie(G) is given by [D1 , D2 ] = D1 —¦ D2 ’ D2 —¦ D1 .

Denote by F [µ] the F -algebra of dual numbers, i.e., F [µ] = F ·1•F ·µ with multi-

plication given by µ2 = 0. There is a unique F -algebra homomorphism κ : F [µ] ’ F

G(κ)

with κ(µ) = 0. The kernel of G(F [µ]) ’ ’ G(F ) carries a natural F -vector space

’’

structure: addition is the multiplication in G(F [µ]) and scalar multiplication is de-

¬ned by the formula a · g = G( a )(g) for g ∈ G(F [µ]), a ∈ F , where a : F [µ] ’ F [µ]

is the F -algebra homomorphism de¬ned by a (µ) = aµ.

(21.1) Proposition. There exist natural isomorphisms between the following F -

vector spaces:

(1) Lie(G),

(2) Der(A, F ) where F is considered as an A-module via the co-unit map u : A ’ F ,

(3) (I/I 2 )— where I ‚ A is the augmentation ideal,

G(κ)

(4) ker G(F [µ]) ’ ’ G(F ) .

’’

Proof : (??) ” (??) If D ∈ Lie(G), then u —¦ D ∈ Der(A, F ). Conversely, for

d ∈ Der(A, F ) one has D = (Id — d) —¦ c ∈ Lie(G).

(??) ” (??) Any derivation d : A ’ F satis¬es d(I 2 ) = 0, hence the restriction

d|I induces an F -linear form on I/I 2 . Conversely, if f : I ’ F is an F -linear map

such that f (I 2 ) = 0, then d : A = F · 1 • I ’ F given by d(± + x) = f (x) is a

derivation.

(??) ” (??) An element f of ker G(κ) is an F -algebra homomorphism f : A ’

F [µ] of the form f (a) = u(a) + d(a) · µ where d ∈ Der(A, F ).

(21.2) Corollary. If G is an algebraic group scheme, then dimF Lie(G) < ∞.

Proof : Since A is noetherian, I is a ¬nitely generated ideal, hence I/I 2 is ¬nitely

generated over A/I = F .

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

Then Der(A, A) is a ¬nitely generated free A-module and

rankA Der(A, A) = dim Lie(G).

§21. THE LIE ALGEBRA AND SMOOTHNESS 335

Proof : Let G be an algebraic group scheme over F . The map

(I/I 2 )— ’ Der(A, A),

Der(A, F ) d ’ (id — d) —¦ c

used in the proof of Proposition (??) extends to an isomorphism of A-modules

(Waterhouse [?, 11.3.])

∼

A —F (I/I 2 )— ’ Der(A, A).

’

The Lie algebra structure on Lie(G) can be recovered as follows (see Waterhouse

2

[?, p. 94]). Consider the commutative F -algebra R = F [µ, µ ] with µ2 = 0 = µ .

From d, d ∈ Der(A, F ) we build two elements g = u+d·µ and g = u+d ·µ in G(R).

’1

A computation of the commutator of g and g in G(R) yields gg g ’1 g = u+d ·µµ

where d = [d, d ] in Lie(G).

Any homomorphism of group schemes f : G ’ H induces a commutative dia-

gram

fF [µ]

G(F [µ]) ’ ’ ’ H(F [µ])

’’

¦ ¦

¦ ¦H(κ)

G(κ)

fF

G(F ) ’’’

’’ H(F )

and hence de¬nes an F -linear map df : Lie(G) ’ Lie(H), which is a Lie algebra

homomorphism, called the di¬erential of f . If f is a closed embedding (i.e., G is

a subgroup of H) then df is injective and identi¬es Lie(G) with a Lie subalgebra

of Lie(H).

In the next proposition we collect some properties of the Lie algebra; we assume

that all group schemes appearing here are algebraic.

(21.4) Proposition. (1) For any ¬eld extension L/F , Lie(GL ) Lie(G) —F L.

(2) Let fi : Gi ’ H be group scheme homomorphisms, i = 1, 2. Then

Lie(G1 —H G2 ) = Lie(G1 ) —Lie(H) Lie(G2 ).

In particular :

(a) For a homomorphism f : G ’ H and a subgroup H ‚ H,

Lie f ’1 (H ) (df )’1 Lie(H ) .

(b) Lie(ker f ) = ker(df ).

(c) Lie(G1 — G2 ) = Lie(G1 ) — Lie(G2 ).

(3) For any ¬nite separable ¬eld extension L/F and any algebraic group scheme G

over L, Lie RL/F (G) = Lie(G) as F -algebras.

(4) Lie(G) = Lie(G0 ).

Reference: See Waterhouse [?, Chap. 12].

(21.5) Examples. (1) Let G = V, V a vector space over F . The elements of

ker G(κ) have the form v · µ with v ∈ V arbitrary. Hence Lie(G) = V with the

trivial Lie product. In particular, Lie(Ga ) = F .

(2) Let G = GL1 (A) where A is a ¬nite dimensional associative F -algebra. El-

ements of ker G(κ) are of the form 1 + a · µ, a ∈ A. Hence Lie GL1 (A) = A.

One can compute the Lie algebra structure using R = F [µ, µ ]: the commutator of

336 VI. ALGEBRAIC GROUPS

1 + a · µ and 1 + a · µ in G(R) is 1 + (aa ’ a a) · µµ , hence the Lie algebra structure

on A is given by [a, a ] = aa ’ a a. In particular,

Lie GL(V ) = End(V ), Lie(Gm ) = F.

(3) For a central simple algebra A over F , the di¬erential of the reduced norm

homomorphism Nrd : GL1 (A) ’ Gm is the reduced trace Trd : A ’ F since

Nrd(1 + a · µ) = 1 + Trd(a) · µ. Hence,

Lie SL1 (A) = { a ∈ A | Trd(a) = 0 } ‚ A.

(4) The di¬erential of the nth power homomorphism Gm ’ Gm is multiplication

by n : F ’ F since (1 + a · µ)n = 1 + na · µ. Hence

0 if char F does not divide n,

Lie(µn ) =

F otherwise.

(5) If G is an ´tale group scheme, then Lie(G) = 0 since Der(A, A) = 0 for any

e

´tale F -algebra A.

e

(6) Let H be an (abstract) abelian group with a continuous “-action and let G =

Hmult . An element in ker G(κ) has the form 1 + f · µ where f ∈ Hom“ (H, Fsep ).

Hence,

Lie(G) = Hom“ (H, Fsep ) = Hom“ (G— , Fsep ).

sep

(7) Let Sv ‚ GL(V ) be the stabilizer of 0 = v ∈ V . An element of ker Sv (κ) has

the form 1 + ± · µ where ± ∈ End(V ) and (1 + ± · µ)(v) = v, i.e., ±(v) = 0. Thus,

Lie(Sv ) = { ± ∈ End(V ) | ±(v) = 0 }.

(8) Let ρ : G ’ GL(V ) be a homomorphism, 0 = v ∈ V . Then

Lie AutG (v) = { x ∈ Lie(G) | (df )(x)(v) = 0 }.

(9) Let G = Autalg (A) where A is a ¬nite dimensional F -algebra and let

ρ : GL(A) ’ GL Hom(A —F A, A)

be as in Example (??), (??). By computing over F [µ], one ¬nds that the di¬erential

dρ : End(A) ’ End Hom(A —F A, A)

is given by the formula

(dρ)(±)(φ)(a — a ) = ± φ(a — a ) ’ φ ±(a) — a ’ φ a — ±(a ) ,

hence the condition (dρ)(±)(v) = 0, where v is the multiplication map, is equivalent

to ± ∈ Der(A, A), i.e.,

Lie Autalg (A) = Der(A, A).

(10) Let NU be the normalizer of a subspace U ‚ V (see Example (??.??)). Since

the condition (1 + ± · µ)(u + u · µ) ∈ U + U · µ, for ± ∈ End(V ), u, u ∈ U , is

equivalent to ±(u) ∈ U , we have

Lie(NU ) = { ± ∈ End(V ) | ±(U ) ‚ U }.

§21. THE LIE ALGEBRA AND SMOOTHNESS 337

The dimension. Let G be an algebraic group scheme over F . If G is con-

nected, then F [G]red is a domain (Proposition (??)). The dimension dim G of G

is the transcendence degree over F of the ¬eld of fractions of F [G]red . If G is not

connected, we de¬ne dim G = dim G0 .

(21.6) Examples. (1) dim V = dimF V .

(2) dim GL1 (A) = dimF A.

(3) dim SL1 (A) = dimF A ’ 1.

(4) dim Gm = dim Ga = 1.

(5) dim µn = 0.

The main properties of the dimension are collected in the following

(21.7) Proposition. (1) dim G = dim F [G] (Krull dimension).

(2) G is ¬nite if and only if dim G = 0.

(3) For any ¬eld extension L/F , dim(GL ) = dim G.

(4) dim(G1 — G2 ) = dim G1 + dim G2 .

(5) For any separable ¬eld extension L/F and any algebraic group scheme G over L,

dim RL/F (G) = [L : F ] · dim G.

(6) Let G be a connected algebraic group scheme with F [G] reduced (i.e., F [G] has

no nilpotent elements) and let H be a proper subgroup of G. Then dim H < dim G.

Proof : (??) follows from Matsumura [?, Th. 5.6]; (??) and (??) are immediate

consequences of the de¬nition.

(??) Set A = F [G]. Since A is noetherian and dim A = 0, A is artinian. But

A is also ¬nitely generated, hence dimF A < ∞.

(??) We may assume that the Gi are connected and F = Falg . Let Li be the

¬eld of fractions of F [Gi ]red . Since F = Falg , the ring L1 —F L2 is an integral domain

and the ¬eld of fractions of F [G1 — G2 ]red is the ¬eld of fractions E of L1 —F L2 .

Thus, dim(G1 —G2 ) = tr.degF (E) = tr.degF (L1 )+tr.degF (L2 ) = dim G1 +dim G2 .

(??) With the notation of (??) we have

dim RL/F (G) = dim RL/F (G)sep

dim G„ = [L : F ] · dim G.

= dim( G„ ) =

„ ∈X „ ∈X

Smoothness. Let S be a commutative noetherian local ring with maximal

ideal M and residue ¬eld K = S/M . It is known (see Matsumura [?, p. 78]) that

dimK (M/M 2 ) ≥ dim S.

The ring S is said to be regular if equality holds. Recall that regular local rings are

integral domains (Matsumura [?, Th. 19.4]).

(21.8) Lemma. For any algebraic group scheme G over F we have dim F Lie(G) ≥

dim G. Equality holds if and only if the local ring F [G]I is regular where I is the

augmentation ideal of F [G].

Proof : Let A = F [G]. The augmentation ideal I ‚ A is maximal with A/I = F .

Hence, for the localization S = AI with respect to the maximal ideal M = IS we

have S/M = F and

dimF Lie(G) = dimF (I/I 2 ) = dimF (M/M 2 ) ≥ dim S = dim A = dim G,

proving the lemma.

338 VI. ALGEBRAIC GROUPS

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

Then the following conditions are equivalent:

(1) AL is reduced for any ¬eld extension L/F .

(2) AFalg is reduced.

(3) dimF Lie(G) = dim G.

If F is perfect, these conditions are also equivalent to

(4) A is reduced.

Proof : (??) ’ (??) is trivial.

(??) ’ (??) We may assume that F = Falg and that G is connected (since

F [G0 ] is a direct factor of A and hence is reduced). By (??) A is an integral domain.

Let K be its ¬eld of fractions. The K-space of derivations Der(K, K) is isomorphic

to Der(A, A) —A K, hence by (??)

dimF Lie(G) = rankA Der(A, A) = dimK Der(K, K).

But the latter is known to equal tr.degF (K) = dim G.

(??) ’ (??) We may assume that L = F = Falg . By (??) the ring AI is

regular hence is an integral domain and is therefore reduced. By the homogeneity

property (see (??)) AM is reduced for every maximal ideal M ‚ A. Hence, A is

reduced.

Finally, assume F is perfect. Since the tensor product of reduced algebras over

a perfect ¬eld is reduced (see Bourbaki [?, Ch.V, §15, no. 5, Th´or`me 3]), it follows

ee

that AFalg is reduced if A is reduced. The converse is clear.

An algebraic group scheme G is said to be smooth if G satis¬es the equivalent

conditions of Proposition (??). Smooth algebraic group schemes are also called

algebraic groups.

(21.10) Proposition. (1) Let G be an algebraic group scheme over F and let L/F

be a ¬eld extension. Then GL is smooth if and only if G is smooth.

(2) If G1 , G2 are smooth then G1 — G2 is smooth.

(3) If char F = 0, all algebraic group schemes are smooth.

(4) An algebraic group scheme is smooth if and only if its connected component G 0

is smooth.

Proof : (??) and (??) follow from the de¬nition of smoothness and (??) (which is a

result due to Cartier) is given in Waterhouse [?, §11.4]. (??) follows from the proof

of (??).

(21.11) Examples. (1) GL1 (A), SL1 (A) are smooth for any central simple F -

algebra A.

´

(2) Etale group schemes are smooth.

(3) Hmult is smooth if and only if H has no p-torsion where p = char F .

Let F be a perfect ¬eld (for example F = Falg ), let G be an algebraic group

scheme over F and let A = F [G]. Since the ring Ared —F Ared is reduced, the

comultiplication c factors through

cred : Ared ’ Ared —F Ared ,

making Ared a Hopf algebra. The corresponding smooth algebraic group scheme

Gred is called the smooth algebraic group associated to G. Clearly Gred is a subgroup

of G and Gred (R) = G(R) for any reduced algebra R ∈ Alg F .

§22. FACTOR GROUPS 339

(21.12) Remark. The classical notion of an (a¬ne) algebraic group over an al-

gebraically closed ¬eld, as an a¬ne variety Spec A endowed with a group structure

corresponds to reduced ¬nitely generated Hopf algebras A, i.e., coincides with the

notion of a smooth algebraic group scheme. This is why we call such group schemes

algebraic groups. Therefore, for any algebraic group scheme G, one associates a

(classical) algebraic group (Galg )red over Falg . The notions of dimension, connected-

ness, Lie algebra, . . . given here then coincide with the classical ones (see Borel [?],

Humphreys [?]).

§22. Factor Groups

22.A. Group scheme homomorphisms.

The injectivity criterion. We will use the following

(22.1) Proposition. Let A ‚ B be Hopf algebras. Then B is faithfully ¬‚at over A.

Reference: See Waterhouse [?, §14.1].

A group scheme homomorphism f : G ’ H is said to be injective if ker f = 1,

or equivalently, if fR : G(R) ’ H(R) is injective for all R ∈ Alg F .

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

mes. The following conditions are equivalent:

(1) f is injective.

(2) f is a closed embedding (i.e., f — is surjective).

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

Proof : (??) ’ (??) By replacing H by the image of f we may assume that

f — : A = F [H] ’ F [G] = B

is injective. The elements in G(B—A B) given by the two natural maps B B—A B

have the same image in H(B —A B), hence they are equal. But B is faithfully ¬‚at

over A, hence the equalizer of B B —A B is A. Thus, A = B.

The implication (??) ’ (??) is clear.

(??) ’ (??) Let N = ker f . We have Lie(N ) = ker(df ) = 0, hence by Lemma

(??) dim N ¤ dim Lie(N ) = 0 and N is ¬nite (Proposition (??)). Then it follows

from Proposition (??) that N is smooth and hence ´tale, N = Het where H =

e

N (Fsep ) (see ??). But N (Fsep ) ‚ N (Falg ) = ker(falg ) = 1, hence N = 1 and f is

injective.

The surjectivity criterion.

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

mes. If H is smooth, the following conditions are equivalent:

(1) f is surjective (i.e., f — is injective).

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

Proof : (??) ’ (??) We may assume that F = Falg . Since B = F [G] is faithfully

¬‚at over A = F [H], any maximal ideal of A is the intersection with A of a maximal

ideal of B (Bourbaki [?, Ch.1, §3, no. 5, Prop. 8 (iv)]), or equivalently, any F -

algebra homomorphism A ’ F can be extended to B. (Note that we are not using

the smoothness assumption here.)

(??) ’ (??) Assume F = Falg . Any F -algebra homomorphism A ’ F factors

through f — , hence all maximal ideals in A contain ker f . But the intersection of

340 VI. ALGEBRAIC GROUPS