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Consider the algebra
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

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