Exposé XIX. Reductive groups: generalities

by M. Demazure

¹ The remainder of this Séminaire is devoted to the study of reductive groups. Its principal aim is the generalization of Chevalley's classical results (Bible and Tôhoku)² to arbitrary base schemes, the two central results being the uniqueness theorem (Exp. XXIII, Th. 4.1 and Cor. 5.1 to 5.10) and the existence theorem (Exp. XXV, Th. 1.1) for "pinned" reductive group schemes corresponding to prescribed "root data". The proofs employed are inspired by Chevalley's, the technique of schemes permitting one to lend them increased efficiency.

The results of the first volume of the Bible (Exp. 1 to 13) will be used systematically. By contrast, we shall prove directly over an arbitrary scheme the results of the second volume (in particular the isomorphism theorem); a knowledge of the proofs over an algebraically closed field is therefore not absolutely indispensable.

In the proof of these two fundamental results, we shall use only the most elementary results from the theory of groups of multiplicative type, contained essentially in Exp. VIII and IX; we shall, on the other hand, make essential use of the results of Exposé XVIII.³ The reader specifically interested in the existence and uniqueness theorems may, on a first reading, skip Exposés X to XVII.

1. Reminders on groups over an algebraically closed field

1.1. In this number, $k$ will always denote an algebraically closed field. As announced above, the only results from Bible used in what follows are found in volume 1 (Exposés 1 to 13). All the results of Bible (loc. cit.) concerning semisimple groups are valid more generally for reductive groups (definition below) and their proof is identical, with the following innocuous modifications:

Exp. 9, § 4, Definition 3: see 1.6.1 below.
Exp. 12, § 4, Theorem 2, e): suppress "finite".
Exp. 13, § 3, Theorem 2: replace "rank" by "semisimple rank".
Exp. 13, § 4, Corollary 2 to Theorem 3: replace "rank" by "reductive rank".

1.2. Let $G$ be a smooth affine connected $k$ -group. The radical of $G$ (Bible, § 9.4, Prop. 2)⁴ is the reduced subgroup associated with the neutral component of the intersection of the Borel subgroups of $G$ ; it is also the largest smooth connected solvable invariant subgroup of $G$ ; we shall denote it $r a d (G)$ .

The unipotent radical of $G$ is the unipotent part of the radical of $G$ ; it is also the largest smooth connected unipotent invariant subgroup of $G$ ; we shall denote it $r a d_{u} (G)$ .

1.3. Let $G$ be a smooth affine connected $k$ -group, and $Q$ a torus of $G$ . Then the centralizer $C e n t r_{G} (Q)$ of $Q$ in $G$ is a closed subgroup of $G$ (Exp. VIII, 6.7), smooth (Exp. XI, 2.4), and connected (cf. Bible, § 6.6, Th. 6 a) or [Ch05], § 6.7, Th. 7 a)).

One has the fundamental relation:

rad_u(Centr_G(Q)) = rad_u(G) ∩ Centr_G(Q).

First,⁵ $U = r a d_{u} (G) \cap C e n t r_{G} (Q)$ is an invariant unipotent subgroup of $C e n t r_{G} (Q)$ ; let us show that it is smooth and connected. If we make $Q$ operate on $r a d_{u} (G)$ by inner automorphisms, $U$ is none other than the scheme of invariants of this operation. But this scheme is smooth and connected, by Lemma 1.4 below.

Consequently, $U$ is a closed subgroup of $r a d_{u} (C e n t r_{G} (Q))$ . On the other hand, by Bible, Exp. 12, § 3, Cor. to Th. 1, one has rad_u(Centr_G(Q))(k) ⊂ rad_u(G)(k). The equality (1.3.1) follows.

Let us signal a particular case of the foregoing result: if $G$ is a smooth affine connected $k$ -group and if $T$ is a maximal torus of $G$ , then

Centr_G(T) = T · (Centr_G(T) ∩ rad_u(G)).

Indeed (cf. Bible, § 6.6, Th. 6 c) or [Ch05], § 6.7, Th. 7 c)), $C e n t r_{G} (T)$ is a solvable group, hence the semi-direct product of its maximal torus $T$ and its unipotent radical.

By the density theorem (cf. Bible, § 6.5, Th. 5 a) or [Ch05], § 6.6, Th. 6 a)), the union of the $C e n t r_{G} (T)$ , as $T$ runs through the set of maximal tori of $G$ , contains a dense open subset of $G$ ; it therefore follows from (1.3.2):

Corollary 1.3.3. If $G$ is a smooth affine connected $k$ -group, then the union of the $T \cdot r a d_{u} (G)$ , where $T$ runs through the set of maximal tori of $G$ , contains a dense open subset of $G$ .

Notations 1.4.0. ⁶ We shall systematically use the following notation: if $S$ is a scheme and $s$ a point of $S$ , we denote by $\overset{s}{ˉ}$ the spectrum of an algebraic closure $κ \overset{s}{ˉ}$ of $κ (s)$ .

Lemma 1.4. ⁶ Let $S$ be a scheme, $Q$ an $S$ -torus operating on an $S$ -group scheme $H$ , separated and smooth over $S$ .

(i) The functor of invariants $H^{Q}$ is representable by a closed subscheme of $H$ , smooth over $S$ .

(ii) If moreover $H$ is affine over $S$ , and has connected fibers, then $H^{Q}$ does too.

Proof. First, by VIII 6.5 (d),⁷ since $H$ is separated over $S$ and $Q$ essentially free over $S$ , $H^{Q}$ is representable by a closed subscheme of $H$ . In particular, if $H$ is affine over $S$ , then so is $H^{Q}$ .

Consider now the semi-direct product $G = H \cdot Q$ ; it is smooth and separated over $S$ . Then, the centralizer $C e n t r_{G} (Q)$ is representable by a closed subscheme of $G$ , of finite presentation over $G$ , by Exp. XI 6.11 (a),⁷ and it is smooth over $S$ by Exp. XI 2.4.

Since $C e n t r_{G} (Q) = H^{Q} \cdot Q$ , it follows that $H^{Q}$ is smooth over $S$ . (Indeed, using the section $σ : H^{Q} \to C e n t r_{G} (Q)$ deduced from the unit section $ϵ_{Q}$ of $Q$ , one sees at once that $H^{Q}$ is formally smooth over $S$ ; and since moreover $ϵ_{Q}$ is of finite presentation, $H^{Q}$ is locally of finite presentation over $S$ .) This proves (i).

Finally, suppose $H$ affine over $S$ and with connected fibers. Then each geometric fiber $G_{\overset{s}{ˉ}}$ of $G$ is a smooth affine connected $κ \overset{s}{ˉ}$ -group, hence, by the first assertion of 1.3, so is

Centr_{G_s̄}(Q_s̄) = (Centr_G(Q))_s̄ = (H^Q)_s̄ · Q_s̄,

and this entails that $(H^{Q})_{\overset{s}{ˉ}}$ is connected.

1.5. We recall that the reductive rank of the smooth affine $k$ -group $G$ is the common dimension of the maximal tori of $G$ . We shall denote it $r g re d (G / k)$ or $r g re d (G)$ . In order that $r g re d (G / k) = 0$ , it is necessary and sufficient that $G$ be unipotent (i.e. that $G = r a d_{u} (G)$ ), by Bible, § 6.4, Cor. 1 to Th. 4 or [Ch05], § 6.5, Cor. 1 to Th. 5.

If $H$ is an invariant subgroup of the smooth affine $k$ -group $G$ , then the quotient $G / H$ is affine and smooth (Exp. VI_B, 11.17 and VI_A, 3.3.2 (iii)). Moreover (Bible, § 7.3, Th. 3, a) and c)), one has

rgred(G) = rgred(G/H) + rgred(H).

Definition 1.6.1. ⁸ One says that the $k$ -group $G$ is reductive if it is smooth, affine, and connected, and if $r a d (G)$ is a torus, i.e. if $r a d_{u} (G) = e$ .

Lemma 1.6.2. Let $G$ be a reductive $k$ -group.

(i) If $Q$ is a torus of $G$ , then $C e n t r_{G} (Q)$ is reductive.

(ii) In particular, if $T$ is a maximal torus of $G$ , then $C e n t r_{G} (T) = T$ .

(iii) The center of $G$ is contained in every maximal torus, hence is diagonalizable.

(iv) The radical of $G$ is the (unique) maximal torus of $C e n t r (G)$ .

Proof. Indeed, (i) follows from (1.3.1), (ii) from (1.3.2), and (iii) follows from (ii). Finally, the maximal torus of $C e n t r (G)$ (i.e. the neutral component $C e n t r (G)^{0}$ ) is a smooth connected solvable subgroup, invariant in $G$ , hence contained in $r a d (G)$ ; conversely, $r a d (G)$ is an invariant torus in $G$ , hence central (Bible, § 4.3, Cor. to Prop. 2), whence (iv).

Remark 1.6.3. If $G$ is reductive and if $dim (G) \neq = r g re d (G)$ , then $dim (G) - r g re d (G) ⩾ 2$ . Indeed, this difference is always even (cf. 1.10 below).

1.7. Let $G$ be a smooth affine connected $k$ -group and $H$ a smooth connected invariant subgroup. Then

rad(H) = (rad(G) ∩ H)⁰_red   and   rad_u(H) = (rad_u(G) ∩ H)⁰_red,

as one sees at once. In particular, if $G$ is reductive, so is $H$ .

Let $f : G \to G^{'}$ be a surjective morphism of smooth affine connected $k$ -groups. Then

$f (r a d_{u} (G)) = r a d_{u} (G^{'}) .$

In particular, if $G$ is reductive, so is $G^{'}$ .

Let us prove (1.7.2). First, $f$ sends $r a d_{u} (G)$ into $r a d_{u} (G^{'})$ . Introducing $H = (f^{- 1} (r a d_{u} (G^{'})))_{re d}^{0}$ , which contains $r a d_{u} (G)$ , one has $r a d_{u} (H) = r a d_{u} (G)$ and one is reduced to the case where $G = H$ , i.e. where $G^{'}$ is unipotent. Since the union of the $T \cdot r a d_{u} (G)$ (as $T$ runs through the set of maximal tori of $G$ ) is dense in $G$ , the union of the $f (T) f (r a d_{u} (G))$ is dense in $G^{'}$ ; but $f (T)$ consists of semisimple elements, so $f (T) = e$ , $G^{'}$ being unipotent; this shows that $f (r a d_{u} (G))$ is dense in $G^{'}$ . Therefore, by Bible, § 5.4, Lemma 4 or [Ch05], § 6.1, Lemma 1,⁹ $f (r a d_{u} (G))$ is an open subgroup of $G^{'}$ ; the latter being connected, it follows that $f (r a d_{u} (G)) = G^{'}$ . (N.B.: one can prove under the same hypotheses that $f (r a d (G)) = r a d (G^{'})$ .)

1.8. One says that the $k$ -group $G$ is semisimple if it is smooth, affine, and connected and if $r a d (G) = e$ , i.e. if $G$ is reductive and $C e n t r (G)$ is finite. If $G$ is an arbitrary smooth affine connected $k$ -group, then $G / r a d (G)$ is semisimple (Bible, § 9.4, Prop. 2), and $G / r a d_{u} (G)$ is reductive. One calls the semisimple rank of $G$ and denotes by $r g ss (G / k)$ or $r g ss (G)$ the reductive rank of $G / r a d (G)$ .

If $G$ is reductive, one therefore has

rgred(G) = rgss(G) + dim(rad(G)).

If $G$ is a smooth affine connected $k$ -group and if $Q$ is a central subtorus of $G$ , then $G / Q$ is semisimple if and only if $G$ is reductive and $Q = r a d (G)$ . Indeed, if $G / Q$ is semisimple, $Q \supset r a d (G)$ , so $r a d (G)$ is a torus, so $G$ is reductive, so $r a d (G)$ is the maximal torus of $C e n t r (G)$ , so $r a d (G) = Q$ . If $G$ is reductive and if $Q$ is a central subgroup then ( $G / Q$ is reductive and) $r g ss (G) = r g ss (G / Q)$ .

1.9. If $K$ is an algebraically closed extension of $k$ and if $G$ is a smooth affine connected $k$ -group, then $G$ is reductive (resp. semisimple) if and only if G_K is, and one has

$r g re d (G / k) = r g re d (G_{K} / K), r g ss (G / k) = r g ss (G_{K} / K) .$

1.10. Let $G$ be a smooth connected $k$ -group and let $T$ be a torus of $G$ .¹⁰ Denote by $g$ the $k$ -Lie algebra of $G$ , i.e. $g = L i e (G) = L i e (G / k) (k)$ ; likewise denote $t = L i e (T)$ . Then $g$ decomposes under the action of $T$ (via $T \to G$ and the adjoint operation of $G$ ) as

g = g₀ ⊕ ∐_{α ∈ R} g_α,

where the $α \in R$ are non-trivial characters of $T$ and the $g_{α}$ are $\neq = 0$ . The characters $α \in R$ are called the roots of $G$ with respect to $T$ . By Exp. II, 5.2.3 (i), one has

$g_{0} = L i e (C e n t r_{G} (T)) .$

In particular,¹¹ since $C e n t r_{G} (T)$ is connected (1.3 in the case where $G$ is affine, and Exp. XII 6.6 b) in the general case), $T$ is its own centralizer if and only if $g_{0} = L i e (T)$ .

This condition entails that $T$ is maximal and that $C e n t r (G) \subset T$ , hence by Exp. XII 8.8 d) one has:

Centr(G) = Ker(T → GL(g)) = ⋂_{α ∈ R} Ker(α);

moreover $C e n t r (G)$ is then affine, hence $G \to G / C e n t r (G)$ is affine; since the latter group is affine (Exp. XII 6.1),¹² so is $G$ .

When $G$ is reductive and $T$ maximal, the roots in the preceding sense coincide with the roots in the sense of Bible, § 12.2, Def. 1; the latter are indeed roots in the sense of this number (Bible, § 13.2, Th. 1, c)) and there are $dim (G) - dim (T)$ of them (Bible, § 13.4, Cor. 2 to Th. 3). Moreover, if $α$ is a root, so is $- α$ (Bible, § 12.2, Cor. to Prop. 1). (As usual, one writes the group structure of $Hom_{k - g r .} (T, G_{m, k})$ indifferently additively or multiplicatively.) It follows that, for $G$ reductive,

dim(G) − rgred(G) = Card(R)

is always even.

The semisimple rank of the reductive group $G$ is the rank of the subset $R$ of the $Q$ -vector space $Hom_{k - g r .} (T, G_{m, k}) \otimes Q$ (Bible, § 13.3, Th. 2).

Lemma 1.11. Let $k$ be an algebraically closed field, $G$ a smooth affine connected $k$ -group, $T$ a torus of $G$ , $W (T) = N or m_{G} (T) / C e n t r_{G} (T)$ the Weyl group of $G$ with respect to $T$ . The following conditions are equivalent:

(i) $G$ is reductive, $T$ is maximal, $r g ss (G) = 1$ .

(ii) $G$ is reductive, $T$ is maximal, $G \neq = T$ ; there exists a subtorus $Q$ of $T$ , of codimension 1 in $T$ , central in $G$ .

(iii) $G$ is not solvable and $dim (G) - dim (T) ⩽ 2$ .

(iv) $W (T) \neq = e$ and $dim (G) - dim (T) ⩽ 2$ .

(v) $W (T) = (Z /2 Z)_{k}$ and $dim (G) - dim (T) = 2$ .

Moreover, under these conditions, there are exactly two roots of $G$ with respect to $T$ ; they are opposite. Under the conditions of (ii), $Q = r a d (G)$ .

Proof. One has obviously (v) ⇒ (iv). One has (iv) ⇒ (iii) by Bible, § 6.1, Cor. 3 to Th. 1 or [Ch05], § 6.2, Cor. 3 to Th. 2. Let us prove (iii) ⇒ (ii): let $U$ be the unipotent radical of $G$ ; one knows that $G / U$ is reductive and is not a torus (otherwise $G$ would be solvable); one therefore has, by (1.10.4),

dim(G/U) − rgred(G/U) = Card(R) ⩾ 2;

but

rgred(G) = rgred(G/U) + rgred(U) = rgred(G/U),

whence

dim(G) − dim(U) − rgred(G) = dim(G/U) − rgred(G/U)
                           ⩾ 2 ⩾ dim(G) − dim(T) ⩾ dim(G) − rgred(G).

This gives $dim (U) = 0$ , hence $G$ is reductive, $r g re d (G) = dim (T)$ , hence $T$ is maximal, $dim (G) - dim (T) = 2$ . By Bible, § 10.4, Prop. 8, there exists a singular torus $Q$ of codimension 1 in $T$ ; then $C e n t r_{G} (Q)$ is reductive (1.6.2 (i)), non-solvable (definition of a singular torus), hence dim(Centr_G(Q)) − rgred(G) ⩾ 2, which proves that $G = C e n t r_{G} (Q)$ and $Q$ is central in $G$ .

Let us prove (ii) ⇒ (i). One knows that $G / Q$ is reductive (1.7) and that $r g re d (G / Q) = 1$ , hence $r g ss (G / Q) = 0$ or 1. The first case is impossible, for it would entail $r g ss (G) = 0$ , hence $G = T$ ;

one therefore has $r g re d (G / Q) = r g ss (G / Q) = 1$ , which proves that $G / Q$ is semisimple; hence $Q$ is the radical of $G$ and $r g ss (G) = 1$ .

Let us finally prove (i) ⇒ (v). If $Q$ is the radical of $G$ , one has $dim (T) - dim (Q) = 1$ and $Q$ is central in $G$ , hence $G = C e n t r_{G} (Q)$ , which proves that $Q$ is a singular torus; by Bible, § 11.3, Th. 2, one has $W_{G} (T) = (Z /2 Z)_{k}$ ; by Bible, § 12.1, Lemma 1, one has $dim (G) - dim (T) = 2$ . There are therefore at most two roots of $G$ with respect to $T$ ; but there are at least two, opposite to each other (1.10).

Proposition 1.12. Let $k$ be an algebraically closed field, $G$ a smooth connected $k$ -group, $T$ a torus of $G$ , $R$ the set of roots of $G$ with respect to $T$ , and

g = g₀ ⊕ ∐_{α ∈ R} g_α,    with g_α ≠ 0,

the decomposition of $g$ under $A d (T)$ . For each $α \in R$ , let $T_{α}$ be the maximal torus of $Ker (α)$ ¹³ and $Z_{α} = C e n t r_{G} (T_{α})$ . The following conditions are equivalent:

(i) $G$ is affine, reductive; $T$ is maximal.

(ii) $g_{0} = t$ , each $Z_{α}$ ( $α \in R$ ) is reductive.

(iii) $g_{0} = t$ , each $g_{α}$ ( $α \in R$ ) is of dimension 1; and if $α, β \in R$ and $q \in Q$ are such that $β = q α$ , then $q = \pm 1$ ; for each $α \in R$ , there exists $w_{α} \in G (k)$ which normalizes $T$ , centralizes $T_{α}$ , but does not centralize $T$ .

Moreover, under these conditions, each $Z_{α}$ is of semisimple rank 1 and one has $L i e (Z_{α}) = t \oplus g_{α} \oplus g_{- α}$ .

Proof. If $G$ is affine and reductive and if $T$ is maximal, each $Z_{α}$ is affine and reductive (1.6.2 (i)), with maximal torus $T$ ; moreover, $T$ is its own centralizer (1.6.2 (ii)), hence $g_{0} = t$ , which proves (i) ⇒ (ii).

On the other hand $g_{0} = t$ entails that $T$ is maximal and $G$ affine (cf. 1.10), hence each $Z_{α}$ is affine, smooth, and connected, by 1.3. In any case, one has, by Exp. II, 5.2.2,

Lie(Z_α) = g^{T_α} = g₀ ⊕ ∐_{β ∈ R ∩ Qα} g_β.

One therefore has

Lie(Z_α) ⊃ t ⊕ g_α ⊕ g_{−α},

which entails $Z_{α} \neq = T$ . Since $T_{α}$ is a subtorus of codimension 1 in $T$ , central in $Z_{α}$ , one obtains by 1.11, applied to $Z_{α}$ , the equivalence (ii) ⇔ (iii) and the supplements.

It remains to prove (ii) ⇒ (i); one already knows that (ii) entails that $T$ is maximal and $G$ affine. Let $U$ be its unipotent radical; it is invariant in $G$ , its Lie algebra is invariant under $A d (T)$ . One therefore has

Lie(U) = (Lie(U) ∩ g₀) ⊕ ∐_{α ∈ R} (Lie(U) ∩ g_α).

By (1.3.1), one has

U ∩ T = U ∩ Centr_G(T) = rad_u(Centr_G(T)) = rad_u(T) = {e},
U ∩ Z_α = U ∩ Centr_G(T_α) = rad_u(Centr_G(T_α)) = rad_u(Z_α) = {e}.

One therefore has¹⁴

Lie(U) ∩ g₀ = Lie(U) ∩ t = Lie(U ∩ T) = 0,
Lie(U) ∩ g_α ⊂ Lie(U) ∩ Lie(Z_α) = Lie(U ∩ Z_α) = 0;

whence $L i e (U) = 0$ and $U = e$ , i.e. $G$ is reductive.

Corollary 1.13. Let $k$ be an algebraically closed field, $G$ a smooth connected $k$ -group, $T$ a torus of $G$ , $R$ the set of roots of $G$ with respect to $T$ , and

g = g₀ ⊕ ∐_{α ∈ R} g_α

as above. For each $α \in R$ , let $T_{α}$ be the maximal torus of $Ker (α)$ and $Z_{α} = C e n t r_{G} (T_{α})$ . The following conditions are equivalent:

(i) $G$ is affine, semisimple; $T$ is maximal.

(ii) $g_{0} = t$ , each $Z_{α}$ is reductive, and $⋂_{α \in R} Ker (α)$ is finite.

¹⁵ This follows from the equalities $r a d (G) = C e n t r (G)^{0}$ and $C e n t r (G) = ⋂_{α \in R} Ker (α)$ , established in 1.6.2 (iv) and (1.10.3).

2. Reductive group schemes. Definitions and first properties

Scholie 2.1. If $G$ is a group scheme over $S$ , the following properties are equivalent:

(i) $G$ is affine and smooth over $S$ , with connected fibers.

(ii) $G$ is affine and flat over $S$ , of finite presentation over $S$ , with geometrically integral fibers.

These properties are stable under base change and local for (fpqc).

¹⁶ Indeed, suppose (i) is satisfied. Since $G$ is affine and smooth over $S$ , it is of finite presentation over $S$ ; and since its fibers are smooth and connected, they are geometrically integral, by VI_A, 2.4.

Conversely, if (ii) is satisfied, the fibers of $G$ are connected and geometrically reduced, hence smooth (VI_A, 1.3.1); then $G$ is smooth over $S$ , by EGA IV₄, 17.5.2.

Of course, these properties are stable under base change: cf. EGA II, 1.5.1 for "affine", IV₁, 1.6.2 (iii) for "of finite presentation", IV₂, 2.1.4 for "flat", and IV₂, 4.6.5 (i) for "with geometrically integral fibers".

On the other hand, these properties are clearly local for the Zariski topology, so it suffices to verify that if $S^{'} \to S$ is a faithfully flat quasi-compact morphism and if $G_{S^{'}} \to S^{'}$ has the indicated properties, then so does $G \to S$ . This follows from EGA IV₂, 2.5.1 for "flat", 2.7.1 (vi) and (xiii) for "of finite presentation" and "affine", and 4.6.5 (i) for "with geometrically integral fibers" (and also EGA IV₄, 17.7.3 (ii) for "smooth").

2.2. Let $G$ be an $S$ -group scheme satisfying the preceding conditions, and $Q$ a torus (cf. IX, Def. 1.3) of $G$ .¹⁷ Then, by XI, 6.11 a) and XI, 2.4, $C e n t r_{G} (Q)$ is representable by a closed subgroup scheme of $G$ (hence affine over $S$ ), of finite presentation and smooth over $S$ ; moreover, since each geometric fiber $G_{\overset{s}{ˉ}}$ of $G$ is a smooth affine connected $κ \overset{s}{ˉ}$ -group, then, by the first assertion of 1.3, so is

$C e n t r_{G_{\overset{s}{ˉ}}} (Q_{\overset{s}{ˉ}}) = (C e n t r_{G} (Q))_{\overset{s}{ˉ}} .$

Lemma 2.3. Let $S$ be a scheme, $G$ an $S$ -group scheme smooth and affine over $S$ , with connected fibers, $T$ a torus of $G$ . The set of $s \in S$ such that $G_{s}$ is a reductive $s$ -group, of semisimple rank 1 and with maximal torus $T_{s}$ , is an open subset $U$ of $S$ .

Proof. Since $G$ and $T$ are flat over $S$ , the function

$s \mapsto dim (G_{s} / s) - dim (T_{s} / s)$

is locally constant; let $U_{1}$ be the open subset of points of $S$ where it equals 2.

¹⁸ By 6.3, the Weyl group

$W_{G} (T) = N or m_{G} (T) / C e n t r_{G} (T)$

is representable by an $S$ -group scheme étale and separated over $S$ , and the function

s ↦ number of points of W_G(T)_s

is lower semicontinuous. Let $U_{2}$ be the set of points of $S$ where this function is > 1; it is open. By 1.11, the set of $s$ such that $G_{s}$ is reductive, of semisimple rank 1, with maximal torus $T_{s}$ , is $U = U_{1} \cap U_{2}$ ; moreover, for every $s \in U$ , $W_{G} (T)_{s}$ has exactly two points.

Consequently (cf. SGA 1, I 10.9 and EGA IV₃, 15.5.1 and IV₄, 18.12.4), $W_{G} (T)_{U}$ is étale and finite over $U$ .

Remark 2.4. The group $W_{G} (T)_{U}$ is isomorphic to $(Z /2 Z)_{U}$ .

Indeed, by what precedes, it is étale and finite over $U$ ; since the functor of automorphisms of $(Z /2 Z)_{U}$ is the trivial group, it suffices to verify the assertion locally; but the assertion is immediate if the algebra $A$ defining $W_{G} (T)_{U}$ is a free O_U-module.¹⁹

Notations and reminders 2.5.0. ²⁰ Let $S$ be a scheme, $G$ an $S$ -group scheme, $ϵ : S \to G$ the unit section of $G$ . One has seen in II, § 4.11 that the functor $L i e (G / S)$ is representable by the vector fibration $V (ω_{G / S}^{1})$ (where $ω_{G / S}^{1} = ϵ * (Ω_{G / S}^{1})$ ), and one denotes

$g = L i e (G / S) = (ω_{G / S}^{1})^{V}$

the sheaf of sections of this vector fibration. Suppose moreover that $G$ is smooth over $S$ ; then $ω_{G / S}^{1}$ and hence $L i e (G / S)$ are locally free O_S-modules of finite type, and one has (cf. I 4.6.5.1):

Lie(G/S) = W(Lie(G/S)),

i.e. for every $S$ -scheme $S^{'}$ ,

Lie(G/S)(S′) = Γ(S′, Lie(G/S) ⊗ O_{S′}).

By II 4.1.1.1, the adjoint action of $G$ endows $L i e (G / S) = W (L i e (G / S))$ with a $G$ -O_S-module structure, hence $L i e (G / S)$ is a $G$ -O_S-module (cf. I 4.7.1). If moreover $G$ is affine over $S$ , this amounts to saying, by I 4.7.2, that $L i e (G / S)$ is an $A (G)$ -comodule.

If $T$ is a torus over $S$ (IX Def. 1.3), one says that $T$ is split ("trivial" in the original) if it is isomorphic to $G_{m, S}^{r}$ for some integer $r ⩾ 0$ , and one says that $T$ is trivialized if one has fixed such an isomorphism or, more generally, an isomorphism $T ≃ D_{S} (M)$ , where $M$ is a free abelian group of rank $r$ .

Let us finally recall (XII Def. 1.3) that a torus $T$ of $G$ is called maximal if, for every $s \in S$ , $T_{\overset{s}{ˉ}}$ is a maximal torus of $G_{\overset{s}{ˉ}}$ .

Theorem 2.5. Let $S$ be a scheme, $G$ an $S$ -group scheme affine and of finite presentation over $S$ , with connected fibers, and $s_{0}$ a point of $S$ . Suppose $G$ flat over $S$ at $ϵ (s_{0})$ and the geometric fiber $G_{\overset{s}{ˉ}_{0}}$ (reduced and) reductive (resp. semisimple). Then there exists an open subset $U$ of $S$ containing $s_{0}$ and an étale surjective morphism $S^{'} \to U$ such that:

(i) G_U is smooth over $S$ , with geometrically reductive (resp. semisimple) fibers, of constant reductive rank and semisimple rank.

(ii) $G_{S^{'}}$ possesses a split²¹ maximal torus $T$ , and the Weyl group (cf. 6.3)

W_{G_{S′}}(T) = Norm_{G_{S′}}(T) / Centr_{G_{S′}}(T) = Norm_{G_{S′}}(T) / T

is finite over $S^{'}$ .

Proof.²² Denote by $π : G \to S$ the structure morphism and $ϵ : S \to G$ the unit section. Since $G$ is flat over $S$ at $ϵ (s_{0})$ and $G_{\overset{s}{ˉ}_{0}}$ reduced (hence smooth over $\overset{s}{ˉ}_{0}$ , cf. VI_A 1.3.1), $G$ is smooth over $S$ at the point $ϵ (s_{0})$ (EGA IV₄, 17.5.1), i.e. there exists an open neighborhood $V$ of $ϵ (s_{0})$ such that $π ∣ V$ is smooth. Then, $S^{'} = ϵ^{- 1} (V)$ is an open subset of $S$ , and $G_{S^{'}}$ is smooth over $S^{'}$ at the points of $ϵ (S^{'})$ . Since $G$ has connected fibers, $G_{S^{'}}$ is smooth over $S^{'}$ , by VI_B, 3.10. So, replacing $S$ by $S^{'}$ , one may assume $G$ smooth over $S$ .

By Exp. XI, Th. 4.1, the functor of subgroups of multiplicative type of $G$ is representable by an $S$ -scheme $M$ , smooth and separated over $S$ . Denote by $r_{0}$ the rank of $G_{\overset{s}{ˉ}_{0}}$ and consider the open subscheme $M_{r_{0}}$ of $M$ , which represents the functor of subtori of rank $r_{0}$ of $G$ (IX 1.4). Smoothness entails that $M_{r_{0}}$ admits a rational point over a finite separable extension of $κ (s_{0})$ (cf. EGA IV₄, 17.15.10 (iii)). Thus there exists $S^{'} \to S$ étale equipped with a point $s_{0}^{'}$ mapping to $s_{0}$ such that $G_{s_{0}^{'}}$ admits a torus of rank $r_{0}$ . Therefore, replacing $S$ by $S^{'}$ , one may assume that $G_{s_{0}}$ admits a torus of rank $r_{0}$ .

By "Hensel's lemma" (cf. XI, 1.10), the smoothness of $M_{r_{0}}$ permits one to lift this torus to an $S^{'}$ -torus $T$ of $G$ , where $S^{'} \to S$ is étale equipped with a point $s_{0}^{'}$ mapping to $s_{0}$ . By Exp. X, 4.5 (see also 6.1²³), there exists an étale morphism $f : S^{''} \to S^{'}$ splitting $T$ and such that $f^{- 1} (s_{0}^{'}) \neq = \emptyset$ . Since an étale morphism is open and the assertions of (i) are local for the étale topology, one may therefore assume that $G$ possesses a split torus $T$ ,²⁴ maximal at $s_{0}$ . Write therefore $T = D_{S} (M)$ and let

g = ∐_{m ∈ M} g_m

be the decomposition of $g = L i e (G / S)$ under $A d (T)$ (Exp. I, 4.7.3).

Put $t = L i e (T)$ and, for every $m \in M$ , denote $g_{m} (s_{0}) = g_{m} \otimes_{O_{S}} κ (s_{0})$ .²⁵ Let $R$ be the set of $m \in M$ , $m \neq = 0$ , such that $g_{m} (s_{0}) \neq = 0$ .²⁶ Since $G_{\overset{s}{ˉ}_{0}}$ is reductive, one has $g_{0} (s_{0}) = t (s_{0})$ , hence

g(s₀) = t(s₀) ⊕ ∐_{α ∈ R} g^α(s₀).

Since the modules in question are locally free, one may, by shrinking $S$ if necessary, assume the $g_{α}$ free and

g = t ⊕ ∐_{α ∈ R} g_α.

One recalls, cf. Exp. XII, 1.12, that a group of multiplicative type possesses a unique maximal torus (this is moreover essentially trivial by descent, the diagonalizable case being evident). Let then $T_{α}$ be the maximal torus of the kernel of $α$ and $Z_{α} = C e n t r_{G} (T_{α})$ .²⁷ By 2.2, $Z_{α}$ is affine and smooth over $S$ , with connected fibers, and by 1.12, its fiber $(Z_{α})_{\overset{s}{ˉ}_{0}}$ is reductive, of semisimple rank 1, with maximal torus $T_{\overset{s}{ˉ}_{0}}$ . By 2.3, there therefore exists an open subset $U_{α}$ of $S$ , containing $s_{0}$ , such that $Z_{α} ∣ U_{α}$ has reductive fibers.

Set $U = ⋂_{α \in R} U_{α}$ . By 1.12, for every $s \in U$ , $G_{\overset{s}{ˉ}}$ is reductive, with maximal torus $T_{\overset{s}{ˉ}}$ and the set of roots of $G_{\overset{s}{ˉ}}$ with respect to $T_{\overset{s}{ˉ}}$ is identified with $R$ , so that

rgred(G_s̄) = dim(T) = rg(M),     rgss(G_s̄) = rg(R)   (cf. 1.10).

One has therefore proved (i) and the first assertion of (ii); it remains to prove that the Weyl group $W_{G_{U}} (T_{U})$ is finite over $U$ , i.e. "that it has the same number of points in each geometric fiber" (cf. SGA 1, I 10.9 and EGA IV₃, 15.5.1 and IV₄, 8.12.4).

For this, it suffices to remark that the geometric fiber of this group at $s \in U$ is determined by the situation $R \subset M$ , as the constant group associated with the "abstract Weyl group of this root system", and in particular is independent of the point $s$ , cf. Bible, § 11.3, Th. 2 (see also Exp. XXII, no. 3).

Corollary 2.6. Let $G$ be an $S$ -group affine and smooth over $S$ , with connected fibers. The set of points $s \in S$ such that $G_{\overset{s}{ˉ}}$ is reductive (resp. semisimple) is an open subset $U$ of $S$ , and the functions

s ↦ rgred(G_s̄/s̄),     s ↦ rgss(G_s̄/s̄)

are locally constant on $U$ .

Definition 2.7. An $S$ -group (= $S$ -group scheme) $G$ is called reductive (resp. semisimple*) if it is affine and smooth over $S$ , with geometrically connected fibers, and reductive (resp. semisimple).*

The property of being reductive (resp. semisimple) for an $S$ -group $G$ is stable under base change and local for the (fpqc) topology.

2.8. Let $G$ be a reductive $S$ -group. For every torus (resp. maximal torus) $Q$ of $G$ , $Z (Q) = C e n t r_{G} (Q)$ is reductive (resp. equals $Q$ ). This follows at once from 2.2 and 1.6. Applying 2.5 to $C e n t r_{G} (Q)$ , one deduces from it that $Q$ is contained (locally for the étale topology) in a maximal torus.

Remark 2.9. Using as usual the technique of EGA IV₃, § 8, the finite-presentation hypotheses, and Theorem 2.5, one sees that if $G$ is a reductive group over $S$ , there exists an open covering of $S$ , say $U_{i}$ , such that each $G_{U_{i}}$ comes by base change from a reductive group over a noetherian ring (in fact, a finitely generated algebra over $Z$ ). Likewise, if $G$ possesses a split maximal torus $T$ , one may further assume that $T_{U_{i}}$ comes from a split maximal torus of the preceding group, ….

3. Roots and root systems of reductive group schemes

3.1. Let $S$ be a scheme, $T$ an $S$ -torus operating linearly on a locally free O_S-module of finite type $F$ (cf. I, § 4.7). For every character $α$ of $T$ (i.e. $α \in Hom_{S - g r .} (T, G_{m, S})$ ), one defines a subfunctor of $W (F)$ by

W(F)^α(S′) = { x ∈ W(F)(S′) | t·x = α(t) x for all t ∈ T(S″), S″ → S′ }.

Lemma. ²⁸ Then $W (F)^{α} = W (F^{α})$ , where $F^{α}$ is a submodule of $F$ , locally a direct summand in $F$ , hence also locally free.

Indeed, the assertion is local for the (fpqc) topology, and one may assume $T = D_{S} (M)$ , where $M$ is a (free) abelian group of finite type. Then $α$ is identified with a locally constant function from $S$ to $M$ (Exp. VIII 1.3), and shrinking $S$ if necessary, one may assume that this function is constant. One is then reduced to Exp. I, 4.7.3.

Definition 3.2. Let $S$ be a scheme, $G$ an $S$ -group scheme smooth and with connected fibers,²⁹ $T$ a torus of $G$ . Denote $g = L i e (G / S)$ and let $T$ operate on $g$ through the adjoint representation of $G$ .

One says that the character $α$ of $T$ is a root of $G$ with respect to $T$ if the following equivalent conditions are satisfied:

(i) For every $s \in S$ , $α_{\overset{s}{ˉ}}$ is a root of $G_{\overset{s}{ˉ}}$ with respect to $T_{\overset{s}{ˉ}}$ (1.10).

(ii) $α$ is non-trivial on each fiber and $g^{α} (s) \neq = 0$ for every $s \in S$ .

Lemma 3.3. Let $S$ be a scheme, $T$ an $S$ -torus, $α$ a character of $T$ . The following conditions are equivalent:

(i) $α$ is non-trivial on each fiber, that is to say: for every $s \in S$ , $α_{\overset{s}{ˉ}}$ is distinct from the unit character of $T_{\overset{s}{ˉ}}$ .

(ii) For every $S^{'} \to S$ , $S^{'} \neq = \emptyset$ , $α_{S^{'}}$ is distinct from the unit character of $T_{S^{'}}$ .

(iii) The morphism $α$ is faithfully flat.

³⁰ It is clear that (ii) ⇒ (i), and one sees easily that (iii) ⇒ (i). One has (i) ⇒ (ii), for if $s^{'} \in S^{'}$ lies over $s$ and if $α_{s^{'}}$ is the unit character, then so is $α_{s}$ . Finally, to prove (i) ⇒ (iii), one reduces to the case where $T$ is diagonalizable and concludes by Exp. VIII 3.2 a).

3.4. ³¹ Let $G$ be a reductive $S$ -group scheme, $T$ a maximal torus of $G$ . Let $α$ be a root of $G$ with respect to $T$ . Then, by 2.5.0 and 1.12, $g^{α}$ is a locally free O_S-module of rank one. Moreover, by 1.10, $- α$ is also a root of $G$ with respect to $T$ . In particular, if $G$ is of semisimple rank 1, one has by 1.11 and 1.12:

Lemma 3.5. Let $S$ be a scheme, $G$ a reductive $S$ -group of semisimple rank 1, $T$ a maximal torus of $G$ . If $α$ is a root of $G$ with respect to $T$ , then $- α$ is also one, and one has

g = t ⊕ g^α ⊕ g^{−α},

where $g^{α}$ and $g^{- α}$ are locally free of rank 1.

Definition 3.6. Let $S$ be a scheme, $G$ a reductive $S$ -group, $T$ a maximal torus of $G$ , $R$ a set of roots of $G$ with respect to $T$ . One says that $R$ is a root system of $G$ with respect to $T$ if the following equivalent conditions are satisfied:

(i) For every $s \in S$ , $α \mapsto α_{\overset{s}{ˉ}}$ is a bijection of $R$ onto the set of roots of $G_{\overset{s}{ˉ}}$ with respect to $T_{\overset{s}{ˉ}}$ .

(ii) The elements of $R$ are distinct on each fiber (i.e. if $α, α^{'} \in R$ , $α \neq = α^{'}$ , then $α - α^{'}$ (= $α α^{' - 1}$ ) is non-trivial on each fiber), and for every $s \in S$ , one has

dim(G_s/κ(s)) − dim(T_s/κ(s)) = Card(R).

(iii) One has $g = t \oplus ∐_{α \in R} g^{α}$ .

The equivalence of these various conditions is trivial.

Lemma 3.7. Let $S$ be a scheme, $G$ a reductive $S$ -group, $T$ a maximal torus of $G$ , $R$ a root system of $G$ with respect to $T$ . Every root of $G$ with respect to $T$ is locally on $S$ equal to an element of $R$ .

Proof. Visible on condition (iii).

Put $M = Hom_{S - g r .} (T, G_{m, S})$ ; it is a twisted constant $S$ -group scheme (Exp. X 5.6). If $G$ admits a root system $R$ with respect to $T$ , then the inclusion $R ↪ M (S)$ defines a morphism $R_{S} \to M$ , where R_S is the constant $S$ -scheme defined by $R$ ; thanks to condition (ii), one sees easily that this morphism is an open and closed immersion whose image is none other than $⋃_{α \in R} α (S)$ (each $α \in R$ being considered as a section $S \to M$ ).

Let $R$ be the functor of roots of $G$ with respect to $T$ : by definition, $R (S^{'})$ is the set of roots of $G_{S^{'}}$ with respect to $T_{S^{'}}$ for every $S^{'} \to S$ ; if $S^{'} = \emptyset$ , one sets $R (\emptyset) = e$ , and if $S^{'} \neq = \emptyset$ then the inclusion $R ↪ M (S^{'})$ identifies $R$ with a root system of $G_{S^{'}}$ with respect to $T_{S^{'}}$ , and therefore, by 3.7, one has

𝓡(S′) = Hom_{loc. const.}(S′, R),

which shows that $R$ is representable by R_S.

If one now no longer supposes necessarily that $G$ possesses a root system relative to $T$ , $R$ is in any case a subsheaf of $M$ for (fpqc). Locally for this topology, $G$ possesses a root system with respect to $T$ (take for example $T$ split and use the proof of Th. 2.5). By Exp. IV 4.6.8 and the theory of descent of open (resp. closed) subschemes,³² one obtains:

Proposition 3.8. Let $S$ be a scheme, $G$ an $S$ -group, $T$ a maximal torus of $G$ . The functor $R$ of roots of $G$ with respect to $T$ is representable by a twisted constant finite $S$ -scheme (Exp. X 5.1) which is an open and closed subscheme of $Hom_{S - g r .} (T, G_{m, S})$ .

For $R \subset Hom_{S - g r .} (T, G_{m, S})$ to be a root system of $G$ with respect to $T$ , it is necessary and sufficient that the morphism $R_{S} \to Hom_{S - g r .} (T, G_{m, S})$ defined by the preceding inclusion induce an isomorphism $R_{S} \sim R$ .

3.9. Let $S$ be a scheme, $G$ a reductive $S$ -group, $T$ a maximal torus of $G$ , $α$ a root of $G$ with respect to $T$ (i.e. a section of $R$ ). Consider the kernel $Ker (α)$ of $α$ , its unique maximal torus $T_{α}$ , and the centralizer of the latter, $Z_{α} = C e n t r_{G} (T_{α})$ . It is an $S$ -group closed in $G$ , reductive (2.8) of semisimple rank 1 (1.12). Moreover,

Lie(Z_α/S) = t ⊕ g^α ⊕ g^{−α},

so $α, - α$ is a root system of $Z_{α}$ with respect to $T$ .

3.10. Let $S$ be a scheme, $G$ a reductive $S$ -group, $T$ a maximal torus of $G$ , $α$ a root of $G$ with respect to $T$ . If $q \in Q$ and if $q α$ is a root of $G$ with respect to $T$ , then $q = 1$ or $q = - 1$ . This follows at once from 1.12.

4. Roots and vector group schemes

4.1. Let $S$ be a scheme, $F$ a locally free O_S-module of finite type. The $S$ -scheme $W (F)$ is smooth over $S$ . Its Lie algebra is canonically isomorphic to $F$ . Indeed, one has a canonical isomorphism

$W (F) \sim L i e (W (F) / S) = W (L i e (W (F) / S))$

(Exp. II, 4.4.1 and 4.4.2). We shall always identify $F$ and $L i e (W (F) / S)$ .

Lemma 4.2. Let $S$ be a scheme, $V$ a vector fibration over $S$ , smooth over $S$ . Then there exists a unique isomorphism of O_S-modules

$exp : W (L i e (V / S)) \sim V$

inducing the identity on the Lie algebras. ³³

Proof. Indeed, $V = V (F)$ for some quasi-coherent O_S-module $F$ . Since $V$ is smooth over $S$ , then $F ≃ ω_{V / S}^{1}$ is locally free of finite type, and therefore

$L i e (V / S) = V (ω_{V / S}^{1}) ≃ W (L i e (V / S)) .$

Moreover, by Exp. II loc. cit., one has a canonical isomorphism

$V \sim L i e (V / S) ≃ W (L i e (V / S))$

and one has at once the uniqueness of exp, since $W$ is fully faithful.

4.3. Notations. If $V$ is a vector bundle over $S$ , one will denote by $V \times$ the open subset of $V$ obtained by removing the zero section. Write the group law of $V$ in multiplicative notation. The action of O_S on $V$ defining the module structure will then be written exponentially

O_S × V → V,    (x, v) ↦ v^x.
       S

One has $(v v^{'})^{x} = v^{x} v^{' x}$ , $v^{x + x^{'}} = v^{x} v^{x^{'}}$ , $v^{x x^{'}} = (v^{x})^{x^{'}}$ . In particular, if one restricts the operator law to $G_{m, S}$ , then $V \times$ is stable and is therefore endowed with a structure of object with group of operators $G_{m, S}$ . We shall again write this law as $(z, v) \mapsto v^{z}$ .

Definition 4.4.1. ³⁴ Let $L$ be an invertible module on $S$ and $W (L)$ the associated vector bundle. Then $W (L) \times$ is a principal homogeneous bundle (locally trivial) under $G_{m, S}$ . One denotes $Γ (S, L) \times = W (L) \times (S)$ .

Scholie 4.4.2. There is a bijective correspondence between the isomorphisms of O_S-modules $O_{S} ≃ L$ , the isomorphisms of O_S-modules $O_{S} ≃ W (L)$ ,³⁵ and the sections $S \to W (L) \times$ .

This correspondence is effected by $f \mapsto W (f) \mapsto W (f) (1)$ . It is compatible with extension of the base. One may therefore consider $W (L) \times$ as the "scheme of trivializations of $W (L)$ ".

4.5. Let $S$ be a scheme, $T$ a torus over $S$ , $P$ an $S$ -group with group of operators $G_{m, S}$ (for example a vector bundle), $α$ a character of $T$ . One denotes $T \cdot_{α} P$ the semi-direct product of $P$ by $T$ , where $T$ operates on $P$ by the composite morphism

T --α--> G_{m,S} ----> Aut_{S-gr.}(P).

Definition 4.6. Let $S$ be a scheme, $G$ an $S$ -group scheme, $T$ a subgroup of $G$ , $α$ a character of $T$ , $L$ an O_S-module. Let

p : W(L) → G          [^N.D.E-XIX-35]

be a morphism of $S$ -group functors. One says that $p$ is normalized by $T$ with multiplier $α$ if it satisfies the following equivalent conditions:

(i) $p$ is a morphism of objects with group of operators $T$ , if one makes $T$ operate on $W (L)$ by $α$ and on $G$ by inner automorphisms. In other words, for every $S^{'} \to S$ and all $t \in T (S^{'})$ and $x \in W (L) (S^{'}) = Γ (S^{'}, L \otimes O_{S^{'}})$ , one has

int(t) p(x) = p(α(t) x).

(ii) The morphism $T \cdot_{α} W (L) \to G$ defined by the product in $G$ (i.e. by $(t, x) \mapsto t \cdot p (x)$ ) is a morphism of groups.

Lemma 4.7. Under the conditions of 4.6, suppose moreover that $G$ is smooth and has connected fibers,³⁶ $T$ is a maximal torus of $G$ , $L$ is invertible. If $p$ is a monomorphism and $α$ non-zero on each fiber, then $α$ is a root of $G$ with respect to $T$ .

Proof. Indeed $L i e (p) : L \to g$ is a monomorphism which factors through $g^{α}$ .

Proposition 4.8. Under the conditions of 4.7, suppose that $G$ is reductive, and that $p$ is a monomorphism. Then $α$ is a root of $G$ with respect to $T$ and $L i e (p)$ induces an isomorphism

$L i e (p) : L \sim g^{α} .$

Proof. Indeed, by virtue of 4.7 and the fact that $g^{α}$ is invertible, it suffices to prove that $α$ is non-zero on each fiber. Let then $s \in S$ be such that $α_{s} = 0$ (= 1 in multiplicative notation). If $X$ is a non-zero section of $L \otimes_{S} κ (s)$ , then $p (X)$ is a unipotent element $\neq = e$ of $G (\overset{s}{ˉ})$ which centralizes $T_{\overset{s}{ˉ}}$ , which is impossible, since the latter is its own centralizer.

Corollary 4.9. Under the conditions of 4.8, there exists a monomorphism of groups with operators $T$ (i.e. normalized by $T$ with multiplier $α$ )

$W (g^{α}) \to G$

which induces on the Lie algebras the canonical morphism $g^{α} \to g$ .

We shall see shortly that 4.9 is verified in fact whenever $G$ is a reductive group and $α$ a root of $G$ with respect to $T$ , and that such a morphism is unique.

Reminder 4.10. Let $k$ be an algebraically closed field, $G$ a reductive $k$ -group, $T$ a maximal torus of $G$ , $α$ a root of $G$ with respect to $T$ . There exists a monomorphism

$p : G_{a, k} \to G$

normalized by $T$ with multiplier $α$ .

See Bible, § 13.1, Th. 1.

4.11. Let us conclude this section with a technical result that will be useful to us. Let $S$ be a scheme, and $L$ an invertible O_S-module. Let $q$ be an integer > 0 such that $x \mapsto x^{q}$ is an endomorphism of the $S$ -group $G_{a, S}$ . (If $S \neq = \emptyset$ , one has $q = 1$ , or $q = p^{n}$ , $p$ being a prime number that is zero on $S$ ; this follows at once from the elementary fact: the gcd of the binomial coefficients (q choose i), for $i \neq = 0, q$ , is $p$ if $q = p^{n}$ , $p$ prime, and 1 in the contrary case.) The morphism defined by the $q$ -th power

$L \to L^{\otimes q}$

is a morphism of sheaves of abelian groups. It defines by base change a morphism of $S$ -group schemes:

$W (L) \to W (L^{\otimes q}) .$

In particular, if $L^{'}$ is another invertible module and if one has a morphism of O_S-modules

$h : L^{\otimes q} \to L^{'},$

one deduces from it a morphism of $S$ -group schemes:

W(L) → W(L′),    x ↦ h(x^q).

These notations laid down, one has:

Proposition 4.12. Let $S$ be a scheme, $T$ (resp. $T^{'}$ ) an $S$ -torus, $L$ (resp. $L^{'}$ ) an invertible O_S-module, $α$ (resp. $α^{'}$ ) a character of $T$ (resp. $T^{'}$ ).³⁷ Let $f : T \to T^{'}$ be a morphism of groups and $g : W (L) \to W (L^{'})$ a morphism of $S$ -schemes (not necessarily a morphism of $S$ -groups) satisfying the following condition:

(⋆)    g(α(t) x) = α′(f(t)) g(x)

for all $x \in W (L) (S^{'})$ , $t \in T (S^{'})$ , $S^{'} \to S$ . Let $s_{0} \in S$ be such that $α_{s_{0}} \neq = 0$ .

a) Suppose that $g$ sends the zero section to the zero section and that for every integer $n > 0$ , one has $(α^{'} \circ f)_{s_{0}} \neq = n α_{s_{0}}$ . Then $g = 0$ in a neighborhood of $s_{0}$ .

b) Suppose that $g$ is a morphism of groups such that $g_{s_{0}} \neq = 0$ . Then there exist an open subset $U$ of $S$ containing $s_{0}$ and an integer $q > 0$ such that $x \mapsto x^{q}$ is an endomorphism of $G_{a, U}$ and $(α^{'} \circ f)_{U} = q α_{U}$ .

c) Suppose that $(α^{'} \circ f)_{s_{0}} = q α_{s_{0}}$ , where $q$ is an integer > 0 such that $x \mapsto x^{q}$ is an endomorphism of $G_{a, S}$ . Then there exist an open subset $U$ of $S$ containing $s_{0}$ and a unique morphism of O_S-modules

$h : L^{\otimes q} ∣ U \to L^{'} ∣ U$

such that $g_{U}$ is the composite morphism

W(L)_U --x↦x^q--> W(L^{⊗q})_U --W(h)--> (L′)_U.

Proof. Let us prove (a). Since the conclusion is local on $S$ , one may assume that

$W (L) = W (L^{'}) = G_{a, S}$ and therefore that $g$ is expressed as a polynomial

g(X) = ∑_{n ⩾ 0} a_n X^n,    a_n ∈ Γ(S, O_S).

The condition (⋆) linking $f$ and $g$ is written as an identity in $Γ (S^{'}, O_{S^{'}}) [X]$ :

∑_{n ⩾ 0} a_n α′(f(t)) X^n = ∑_{n ⩾ 0} a_n α(t)^n X^n,

that is, for every $n ⩾ 0$ , every $S^{'} \to S$ and every $t \in T (S^{'})$ ,

$a_{n} (α^{'} (f (t)) - α (t)^{n}) = 0.$

For each $n ⩾ 0$ , let $S_{n}$ be the set of $s \in S$ such that $(α^{'} \circ f)_{s} = n α_{s}$ . One knows (Exp. IX 5.3) that the $S_{n}$ are open and closed, and that $(α^{'} \circ f)_{S_{n}} = n α_{S_{n}}$ . Moreover, since $α_{s_{0}} \neq = 0$ , one may, shrinking $S$ if necessary, assume that $α$ is non-zero on each fiber (same reference), which entails that the $S_{n}$ are disjoint. Shrinking $S$ if necessary, one may therefore assume that one is in one of the two following cases: there exists an $n$ such that $S = S_{n}$ , or all the $S_{n}$ are empty.

Let $m ⩾ 0$ be such that $S_{m} = \emptyset$ ; I claim that then $a_{m} = 0$ ; indeed $α^{'} \circ f$ and $m α$ are distinct on each fiber of $S$ , and one has:

Lemma 4.13. Let $S$ be a scheme, $T$ an $S$ -torus, $α$ and $α^{'}$ two characters of $T$ distinct on each fiber; there exists a family $S_{i} \to S$ covering for (fpqc), and for each $i$ a $t_{i} \in T (S_{i})$ , such that $α (t_{i}) - α^{'} (t_{i}) = 1$ .

Proof. The lemma is trivial, by reduction to the diagonalizable case, then to the case $T = G_{m, S}$ .

Let us resume the proof of Proposition 4.12; we have just proved (a). In cases (b) and (c), there exists an $n$ such that $S = S_{n}$ (with $n = q$ in (c)). By the preceding result, one has therefore $a_{m} = 0$ for $m \neq = n$ , which proves that $g$ is written

g(X) = a_n X^n,    a_n ∈ Γ(S, O_S).

This proves (c) at once. In case (b), one knows that $a_{n} (s_{0}) \neq = 0$ , one may therefore assume $a_{n}$ invertible on $S$ , which entails that $x \mapsto x^{n}$ is an endomorphism of $G_{a, S}$ (by virtue of the hypothesis of (b)) and completes the proof.

5. An instructive example

5.1. Let $k$ be an algebraically closed field of characteristic 0. Put $A = k [t]$ , the polynomial ring in one variable over $k$ , and $S = Spec (A)$ . Consider the following Lie algebra $g$ over O_S: as a module, it is free of dimension 3, with basis {X, Y, H}; the multiplication table is

[X, Y] = 2t H,    [H, X] = X,   [H, Y] = −Y.

For $s \in S$ , $s \neq = s_{0}$ (the point defined by $t = 0$ ), the fiber $g (s) = g \otimes_{A} κ (s)$ is isomorphic to the Lie algebra of the group $PG L_{2, κ (s)}$ . For $s = s_{0}$ , it is a solvable Lie algebra.

5.2. Let $G_{1}$ be the group scheme of automorphisms of $g$ : for every $S^{'} \to S$ , $G_{1} (S^{'})$ is the group of automorphisms of the $O_{S^{'}}$ -Lie algebra $g \otimes_{O_{S}} O_{S^{'}}$ . It is a closed subscheme of the group $G L (g)$ of automorphisms of the O_S-module $g$ . Let $S^{'} \to S$ and $u \in M_{3} (Γ (S^{'}, O_{S^{'}}))$ considered as an endomorphism of the O_S-module $g \otimes_{O_{S}} O_{S^{'}}$ :

u(X) = a X + b Y + e H,
u(Y) = b′ X + a′ Y + e′ H,
u(H) = c X + c′ Y + d H.

One sees at once that $u$ is a section of $G_{1}$ if and only if $d e t (u)$ is invertible and one has the relations:³⁸

(1)  a(d − 1) = e c,            (1′)  a′(d − 1) = e′ c′,
(2)  b(d + 1) = e c′,           (2′)  b′(d + 1) = e′ c,
(3)  e = 2t(b c − a c′),        (3′)  e′ = 2t(b′ c′ − a′ c),
(4)  2t c = e b′ − a e′,        (4′)  2t c′ = b e′ − e a′,
                       (5)  2t(a a′ − b b′) = 2t d.

Lemma 5.3. The relations (1), (1′), (2), (2′) imply

det(u) = a a′ (2 − d) + b b′ (d + 2),
a a′ − b b′ = d · det(u).

Proof. Indeed, the first assertion is obtained at once by inserting the relations (1), (1′), (2), (2′) into the expansion of $d e t (u)$ :

det(u) = a a′ d + b e′ c + b′ c′ e − a′ e c − a e′ c′ − b b′ d
       = a a′ d + b b′ (d + 1) + b b′ (d + 1) − a a′ (d − 1) − b b′ d − a a′ (d − 1)
       = a a′ (d − d + 1 − d + 1) + b b′ (d + 1 + d + 1 − d)
       = a a′ (2 − d) + b b′ (d + 2).

Multiplying this relation by $d$ , one obtains

d · det(u) = a a′ (2d − d²) + b b′ (d² + 2d).

But the relation (1) × (1′) = (2) × (2′) gives at once

a a′ (d − 1)² = b b′ (d + 1)².

Combining the two preceding relations, one finds at once the second sought formula.

5.4. Consider then $G = G_{1} \cap S L (g)$ . It is the closed subgroup of $G_{1}$ defined by the equation $d e t (u) = 1$ . It is therefore an affine group over $S$ .

Proposition 5.5. The group $G$ is smooth over $S$ .

To prove the proposition, we shall need the following lemmas.

Lemma 5.6. Let $U$ be the open subset of $End_{A} (g) ≃ W (M_{3} (O_{S}))$ defined by the condition "the product a d is invertible", i.e. the open subset $End_{A} (g)_{f}$ , where $f$ is the function defined by $f (u) = a d$ . Then $U \cap G$ is the closed subscheme of $U$ defined by the six equations:

(1),  (2),  (2′),  (3),  (3′)   and   (D) : a a′ − b b′ = d.

Proof. It is first clear that these six relations are verified by every "point" of $G$ (Lemma 5.3), in particular by every "point" of $U \cap G$ . Conversely, it must be shown that if $u \in U (S^{'})$ (for every $S^{'} \to S$ ), and if $u$ verifies the six conditions of the statement, then $d e t (u) = 1$ and $u$ also verifies (1′), (4), (4′) and (5).

One has first (D) ⇒ (5). By (2) and (2′), one has

b b′ (d + 1) = b c e′ = b′ c′ e.

But by (3) and (3′), writing $2 t (b c - a c^{'}) (b^{'} c^{'} - a^{'} c)$ in two ways:

(b c − a c′) e′ = (b′ c′ − a′ c) e.

Combining with the preceding relation, this gives $a c^{'} e^{'} = a^{'} ce$ . But by (1), $a^{'} ce = a^{'} a (d - 1)$ , which proves $a (a^{'} (d - 1) - e^{'} c^{'}) = 0$ and entails (1′), since $a$ is supposed invertible.

Thus, (1), (2), (2′) and (1′) are verified, hence by Lemma 5.3 and (D) one has $d (d e t (u) - 1) = 0$ . Since $d$ is supposed invertible, this entails $d e t (u) = 1$ .

Let us prove (4) and (4′). Let us do it for example for (4′), the other calculation being deduced from it by symmetry.

By (3), (3′) and (D), one has at once

a′ e + b e′ = −2t(a a′ − b b′) c′ = −2t d c′.

Combining (1′) and (2), one has also³⁹

a′ e + b e′ = −d(b e′ − e a′),

which completes the proof of (4′), $d$ being supposed invertible.

Lemma 5.7. $G$ is smooth over $S$ along the unit section.

Proof. By 5.6 and SGA 1, II 4.10, it suffices to prove that the differentials of the functions

a(d − 1) − e c,    b(d + 1) − e c′,    b′(d + 1) − e′ c,
e − 2t(b c − a c′),    e′ − 2t(b′ c′ − a′ c),    a a′ − b b′ − d,

at the points of the unit section of $G$ are linearly independent. But denoting by a capital letter the differential of the corresponding lower-case, these are⁴⁰

D,    2B,    2B′,    E + 2t C′,    E′ + 2t C,    A + A′ − D,

which are indeed linearly independent modulo every $(t - λ)$ , $λ \in k$ .⁴¹

Lemma 5.8. For $s \in S$ , $s \neq = s_{0}$ , the fiber $G_{s}$ is connected and semisimple.

Proof. ⁴² Indeed, since $s \neq = s_{0}$ , $g (s)$ is isomorphic to the Lie algebra of $PG L_{2, κ (s)}$ and, on the other hand, one has $G_{s} = (G_{1})_{s}$ ; but it is known that the group of automorphisms of the Lie algebra of $PG L_{2}$ over a field of characteristic 0 is $PG L_{2}$ itself, which is connected and semisimple.

Lemma 5.9. The fiber $G_{s_{0}}$ is solvable and has two connected components which are of the following form:

              ⎛ a   0   0 ⎞                          ⎛ 0    b   0 ⎞
G⁰_{s₀} =     ⎜ 0  a⁻¹  0 ⎟      and     G⁻_{s₀} =   ⎜ b⁻¹  0   0 ⎟ .
              ⎝ c   c′  1 ⎠                          ⎝ c    c′ −1 ⎠

Proof. Indeed, one has $e = e^{'} = 0$ , since $t = 0$ at $s_{0}$ . One then solves immediately the equations (1), (1′), … (5) and (D).

Lemma 5.10. The matrix

      ⎛ 0   1   0 ⎞
w =   ⎜ 1   0   0 ⎟
      ⎝ 0   0  −1 ⎠

is a section of $G$ over $S$ , such that $w (s_{0}) \in G_{s_{0}}^{-}$ .

Let us now prove 5.5.⁴³ Denote by $G^{0}$ the union of the neutral components of the fibers of $G$ (that is to say the complement of $G_{s_{0}}^{-}$ ); since $G$ is smooth over $S$ along the unit section (5.7), $G^{0}$ is an open subgroup of $G$ , smooth over $S$ , by VI_B, 3.10. Since, by translation, $G$ is evidently smooth at the points of $w (S)$ , $G$ is smooth over $S$ .

5.11. Consider the morphism $G_{m, S} \to G^{0}$ defined by

        ⎛ z   0   0 ⎞
z ↦     ⎜ 0  1/z  0 ⎟ .
        ⎝ 0   0   1 ⎠

It is a monomorphism that defines a torus $T$ of $G^{0}$ . I claim that one has

$T = C e n t r_{G} (T) = C e n t r_{G^{0}} (T) .$

It suffices indeed to verify the first equality. Since these are smooth subgroups of $G$ over $S$ , it suffices to verify that they have the same geometric points. For the fibers at points $s \neq = s_{0}$ , this follows from the fact that $PG L_{2, κ (s)}$ is reductive and that $T_{s}$ is a maximal torus of it for reasons of dimensions (cf. 1.11). On the fiber at $s_{0}$ , the computation is done immediately. It follows in particular that $T$ is a maximal torus of $G$ and of $G^{0}$ .

5.12. The section $w$ of $G$ defined in 5.9 normalizes $T$ . It follows at once (cf. 2.4) that the Weyl group of $G$ is isomorphic to $(Z /2 Z)_{S}$ , and in particular finite over $S$ .

W_G(T) = Norm_G(T) / T = (Z/2Z)_S.

On the other hand, $W_{G^{0}} (T)$ is not finite over $S$ : it "lacks a point" above $s_{0}$ .

5.13. The open immersion $G^{0} \to G$ is not a closed immersion (since $G^{0}$ is dense in $G$ ); it is however an affine morphism (and therefore $G^{0}$ is affine over $S$ ). Indeed, since $G_{s_{0}}^{0}$ is closed in $G_{s_{0}}$ , which is closed in $G$ , the complement $U$ of $G_{s_{0}}^{0}$ in $G$ is open; $G^{0}$ and $U$ form an open covering of $G$ and it suffices to verify that the immersions $G^{0} \to G^{0}$ and $G^{0} \cap U \to U$ are affine; for the first this is trivial, for the second, one notes that $U \cap G^{0}$ is defined in $U$ by the equation $b \neq = 0$ .

We have therefore constructed an affine smooth $S$ -group, with connected fibers, $G^{0}$ , possessing a maximal torus $T$ which is its own centralizer and whose Weyl group $W_{G^{0}} (T)$ is not finite (compare with Theorem 2.5).

6. Local existence of maximal tori. The Weyl group

In the course of the proof of 2.5, we used a result from Exp. XI on the local existence for the étale topology of maximal tori; the proof of Exp. XI uses a fine representability result (XI 4.1). In the particular case which occupies us, one may give another proof, based on the ideas of Exp. XII no. 7, and of a much more elementary nature.

Proposition 6.1. Let $S$ be a scheme, $G$ a smooth, affine $S$ -group scheme with connected fibers over $S$ , $s_{0}$ a point of $S$ such that the maximal tori of the geometric fiber $G_{\overset{s}{ˉ}_{0}}$ are their own centralizer. There exists an étale morphism $S^{'} \to S$ covering $s_{0}$ , and a split maximal torus $T$ of $G_{S^{'}}$ .

Proof. First, one may assume $S$ affine.⁴⁴ Since $G$ is of finite presentation over $S$ , one may assume $S$ noetherian, then local, then henselian with separably closed residue field (cf. EGA IV, 8.12, § 8.8, and § 18.8). Set therefore $S = Spec (A)$ , $A$ henselian with separably closed residue field $k = κ (s_{0})$ . Choose a maximal torus $T_{0}$ of $G_{0} (= G_{k})$ (one exists, for example because the scheme of maximal tori of $G_{0}$ is smooth over $k$ , Exp. XII, 7.1 c)); since $k$ is separably closed, $T_{0}$ is split (cf. X 1.4) and is therefore given by a monomorphism of groups

$f_{0} : G_{m, k}^{r} \to G_{0} .$

Let $m$ be an integer > 1 prime to the characteristic of $k$ . By Exp. VIII 6.7, for every $h > 0$ , $C e n t r_{G_{0}} (_{m^{h}} T_{0})$ is representable by a closed subscheme of $G_{0}$ . Since the $_{m^{h}} T_{0}$ are schematically dense in $T_{0}$ (cf. Exp. IX 4.10) and $G_{0}$ is noetherian, there exists an $h$ such that

$C e n t r_{G_{0}} (_{m^{h}} T_{0}) = C e n t r_{G_{0}} (T_{0}) = T_{0} .$

Put $n = m^{h}$ ; since $n$ is invertible on $S$ , $_{n} G_{m, S}$ is isomorphic to $(Z / n Z)_{S}$ ; $f_{0}$ therefore defines a monomorphism of groups

$u_{0} : (Z / n Z)_{k}^{r} \to G_{0}$

such that $C e n t r_{G} (u_{0}) = T_{0}$ . Now the $S$ -functor

$P = Hom_{S - g r .} ((Z / n Z)_{S}^{r}, G)$

is representable by an $S$ -scheme of finite type (as a closed subscheme of $r$ copies of $_{n} G = Hom_{S - g r .} ((Z / n Z)_{S}, G) = Ker (G - - n - - > G)$ ). But $P$ is smooth over $S$ (Exp. IX 3.6), hence $u_{0} \in P (k)$ lifts to a section $u \in P (S)$ (Hensel's lemma, Exp. XI 1.11):

$u : (Z / n Z)_{S}^{r} \to G .$

Consider $H = C e n t r_{G} (u)$ ; it is a closed subgroup scheme of $G$ , by Exp. VIII 6.5 e), and one has $H_{0} = T_{0}$ by hypothesis.⁴⁵ Moreover, $H$ is smooth over $S$ : indeed, let $S^{'} = Spec (A)$ be an affine scheme over $S$ , $u^{'} : (_{n} G_{m, S^{'}})^{r} \to G_{S^{'}}$ the morphism deduced from $u$ by base change, $S_{J}^{'} = Spec (A / J)$ , where $J$ is an ideal of square zero, and let $x \in H (S_{J}^{'})$ ; since $G$ is smooth, $x$ lifts to an element $g$ of $G (S^{'})$ ; then $v = in t (g) (u^{'})$ verifies $v_{J} = u_{J}^{'}$ and therefore, by IX 3.2, there exists an element $g^{'}$ of $G (S^{'})$ such that $g_{J}^{'} = e$ and $in t (g^{'}) (v) = u^{'}$ ; then $h = g^{'} g$ belongs to $H (S^{'})$ and verifies $h_{J} = x$ .

Let then $H^{0}$ be the neutral component of $H$ ; it is a subgroup scheme of $G$ , smooth and with connected fibers, whose special fiber is a torus. By Exp. X 8.1, it is a torus, necessarily split (Exp. X 4.6). Put $H^{0} = T$ and let $C = C e n t r_{G} (T)$ , which is a closed subgroup of $G$ (Exp. VIII 6.5 e)), smooth (Exp. XI 2.4). Consider $C^{0}$ (in fact $C^{0} = C$ , but we do not need to know it); then $C^{0} \supset T$ and these are two smooth groups with connected fibers. They coincide at $s_{0}$ , hence in a neighborhood. Shrinking $S$ if necessary, one may therefore assume $C^{0} = T$ , hence a fortiori $T$ maximal.

Remark 6.2. The proof shows in particular that the reductive rank of $G_{\overset{s}{ˉ}}$ is constant in a neighborhood of $s = s_{0}$ .

Proposition 6.3. Let $S$ be a scheme, $G$ an $S$ -group scheme smooth and of finite presentation over $S$ , $Q$ a subtorus of $G$ .

(i) $C e n t r_{G} (Q)$ and $N or m_{G} (Q)$ are representable by closed subgroup schemes, smooth (and therefore of finite presentation) over $S$ .

(ii) $C e n t r_{G} (Q)$ is an open and closed subscheme of $N or m_{G} (Q)$ . The quotient $W_{G} (Q) = N or m_{G} (Q) / C e n t r_{G} (Q)$ is representable by an open subgroup scheme of $Aut_{S - g r .} (Q)$ ; it is therefore an $S$ -group scheme quasi-finite, étale and separated over $S$ .

(iii) For every $s \in S$ , put

$w (s) = N or m_{G (\overset{s}{ˉ})} (Q (\overset{s}{ˉ})) / C e n t r_{G (\overset{s}{ˉ})} (Q (\overset{s}{ˉ})) .$

Then $s \mapsto w (s)$ is lower semicontinuous, and is constant in a neighborhood of $s$ if and only if $W_{G} (Q)$ is finite over $S$ in a neighborhood of $s$ .

Proof. By Exp. XI 6.11, $C e n t r_{G} (Q)$ and $N or m_{G} (Q)$ are representable by closed subschemes of finite presentation of $G$ . These are smooth by Exp. XI 2.4 and 2.4 bis, which proves (i). Assertions (ii) and (iii) are then proved as in Exp. XI 5.9 and 5.10, whose proof uses in fact only (i) and not the fine theorems Exp. XI 4.1 and 4.2.

Bibliography

[Bible] C. Chevalley (with the collaboration of P. Cartier, A. Grothendieck, M. Lazard), Classification des groupes de Lie algébriques, 1956–58.

[Ch05] C. Chevalley, Classification des groupes algébriques semi-simples (with the collaboration of P. Cartier, A. Grothendieck, M. Lazard), Collected Works, vol. 3, Springer, 2005.

[Tô55] C. Chevalley, Sur certains groupes simples, Tôhoku Math. J. (2) 7 (1955), 14–66.

[TO70] J. Tate & F. Oort, Group schemes of prime order, Ann. Scient. Éc. Norm. Sup. (4), t. 3 (1970), 1–21.

Footnotes

N.D.E.: Version of 13/10/2024.

N.D.E.: See the bibliographical references at the end of this Exposé. In particular, a re-edition of the Séminaire Chevalley 1956–58, cited [Bible], revised by P. Cartier, was published in 2005, cf. [Ch05].

N.D.E.: More precisely, Proposition XVIII 2.3 (extension of a "generic homomorphism" between groups) is used in XXII 4.1.11 and then in Exp. XXIII (proof of the uniqueness theorem), and also in Exp. XXIV; Theorem XVIII 3.7 (construction of a group from a group germ) is used only in Exp. XXV.

⁴

N.D.E.: The reference 9-06 (= Exp. 9, p. 6) has been replaced by § 9.4 (= Exp. 9, § 4), which applies equally well to [Bible] and to [Ch05]. When the numbering of [Ch05] differs from [Bible], which is the case in Exp. 6, both references will be indicated explicitly.

⁵

N.D.E.: The following has been detailed.

⁶

N.D.E.: These notations, which figured in 2.3 of the original, have been inserted here; on the other hand, the statement and proof of 1.4 have been detailed.

⁷

N.D.E.: See also the additions made in VI_B, 6.2.4 (d) and 6.5.5 (a).

⁸

N.D.E.: 1.6 has been transformed into 1.6.1 to 1.6.3. Note that in this Séminaire every reductive (resp. semisimple, cf. 1.8) $k$ -group is, by definition, connected.

⁹

N.D.E.: The following has been detailed.

¹⁰

N.D.E.: The following sentence has been added.

¹¹

N.D.E.: The following has been detailed.

¹²

N.D.E.: See also the addition VI_B, 6.2.6.

¹³

N.D.E.: i.e., the connected component of $Ker (α)$ .

¹⁴

N.D.E.: Note that if $U$ , $L$ are two subgroup schemes of $G$ , one has Lie(U) ∩ Lie(L) = Lie(U ∩ L).

¹⁵

N.D.E.: The following sentence has been added.

¹⁶

N.D.E.: The following has been added.

¹⁷

N.D.E.: The following has been detailed.

¹⁸

N.D.E.: The following has been detailed. Note, on the other hand, that section 6 is independent of the rest of this Exposé.

¹⁹

N.D.E.: The hypothesis entails that $A$ is locally free of rank 2, and since the augmentation ideal $I$ is a direct factor of $A$ , then, replacing $U$ by a sufficiently small affine open subset $Spec (R)$ , one may assume that $I = Γ (U, I)$ is a free $R$ -module of rank 1. If $x$ is a generator of $I$ , one then has $x^{2} = a x$ for a certain $a \in R$ , and the hypothesis that $A = Γ (U, A)$ be étale entails that $a$ is invertible in $R$ , and one then sees easily that $A$ is the affine algebra of the constant $R$ -group $Z /2 Z$ (compare with the first lines of [TO70]).

²⁰

N.D.E.: This paragraph of notations and reminders has been added.

²¹

N.D.E.: "Split" has been added.

²²

N.D.E.: The proof has been detailed.

²³

N.D.E.: which is independent of the rest of this Exposé.

²⁴

N.D.E.: In all this volume, "trivial torus" has been replaced by "split torus".

²⁵

N.D.E.: The preceding sentence has been added.

²⁶

N.D.E.: Note that, $g$ being a finite locally free O_S-module, $R$ is a finite set.

²⁷

N.D.E.: The following sentence has been detailed.

²⁸

N.D.E.: This statement has been made into a lemma, in order to put it in evidence.

²⁹

N.D.E.: The original added the hypothesis that $G$ be of finite presentation over $S$ , which does not seem to be used in what follows. In any case, $G$ being smooth over $S$ and with connected fibers, it is quasi-compact and separated over $S$ (VI_B, 5.5), hence of finite presentation over $S$ .

³⁰

N.D.E.: The original has been simplified; it invoked Exp. IX, 5.2. Let us note, however, that loc. cit., 5.3 shows that if $S$ is connected, then the conditions of the lemma are verified as soon as (i) is verified at one point $s$ of $S$ .

³¹

N.D.E.: The following has been modified, in order to recall the hypotheses of 3.2 and to add that $T$ is a maximal torus.

³²

N.D.E.: see SGA 1, VIII 4.4 or EGA IV₂, 2.7.1.

³³

N.D.E.: We have kept the original proof; one can also detail it as follows. Let $F$ be a quasi-coherent O_S-module, $V = V (F)$ . Denote by $π$ the projection $V \to S$ and $ϵ$ the zero section $S \to V$ . Then $Ω_{V / S}^{1} = π * F$ , whence

(1)    ω¹_{V/S} = ε* Ω¹_{V/S} ≃ ε* π* F ≃ F,

and therefore $L i e (V / S) = (ω_{V / S}^{1})^{V} ≃ F^{V}$ . If one supposes $V$ smooth over $S$ then, by (1), $F$ is locally free of finite type, and therefore

(2)    V = V(F) ≃ W(Fⱽ) ≃ W(Lie(V/S)).

Now, if $U$ is an $S$ -scheme equipped with a section $τ : S \to U$ , and if one is given an isomorphism $ϕ : V \sim U$ of "pointed" $S$ -schemes, i.e. such that $ϕ \circ ϵ = τ$ (for example if $U$ is an $S$ -group), then $ϕ$ induces an isomorphism $d ϕ : F^{V} \sim L i e (U / S)$ , whence an isomorphism $ψ : W (F^{V}) \sim W (L i e (U / S))$ making the appropriate diagram commute, which permits one to identify $U$ with $W (L i e (U / S))$ . Since the functor $W$ is fully faithful, there exists a unique isomorphism $exp : W (L i e (U / S)) \sim U$ such that $L i e (ψ^{- 1} \circ exp) = i d$ . In fact, one can see directly that $L i e (ψ) = i d$ , so that $exp = ψ$ .

³⁴

N.D.E.: For later references, 4.4 has been transformed into 4.4.1 and 4.4.2.

³⁵

N.D.E.: Here, one identifies the vector bundle $W (L)$ with the functor in O_S-modules that it represents.

³⁶

N.D.E.: cf. N.D.E. (28).

³⁷

N.D.E.: "consider the semi-direct product $T \cdot_{α} W (L)$ , resp. $T^{'} \cdot_{α^{'}} W (L^{'})$ " has been removed. On the other hand, the hypothesis in (b) that $g$ be a morphism of groups, combined with (⋆), amounts to saying that the morphism $(t, x) \mapsto (f (t), g (x))$ is a morphism of groups from $T \cdot_{α} W (L)$ to $T^{'} \cdot_{α^{'}} W (L^{'})$ .

³⁸

N.D.E.: The equality $[u (X), u (Y)] = 2 t u (H)$ (resp. $[u (H), u (X)] = u (X)$ , resp. $[u (H), u (Y)] = - u (Y)$ ) gives the relations (4), (4′), and (5) (resp. (1)–(3), resp. (1′)–(3′)).

³⁹

N.D.E.: The sign has been corrected.

⁴⁰

N.D.E.: $A + A^{'} + B$ has been corrected to $A + A^{'} - D$ .

⁴¹

N.D.E.: " $(t - a)$ , $a \in A$ " has been corrected to " $(t - λ)$ , $λ \in k$ ".

⁴²

N.D.E.: The following sentence has been slightly modified.

⁴³

N.D.E.: The following sentence has been modified, by adding the reference to VI_B, 3.10.

⁴⁴

N.D.E.: The preceding sentence as well as the reference to EGA IV in what follows have been added.

⁴⁵

N.D.E.: The reference VIII 6.5 e) has been added in what precedes, and the following sentence has been detailed.

⁴⁶

N.D.E.: Here, one has written $W (L)$ (with $W$ in boldface) since, for an arbitrary O_S-module $L$ , $W (L)$ is not necessarily representable. But in what follows, $L$ will be assumed locally free of finite rank (and even invertible), in which case the functor is representable by the vector bundle $V (L^{V})$ , and one will denote it simply $W (L)$ .

Séminaire de Géométrie Algébrique du Bois-Marie — English