Exposé XII. Maximal tori, the Weyl group, Cartan subgroups, the reductive center of smooth and affine group schemes

by A. Grothendieck

From the present Exposé onwards, in contrast to the preceding Exposés, we make use of the known results on the structure of smooth algebraic groups over an algebraically closed field $k$, and above all of Borel's theory of affine algebraic groups, expounded in the Séminaire Chevalley 56/57: "Classification des groupes de Lie algébriques".¹ Following the usage in the theory of algebraic groups, we refer to this Séminaire by the sigil BIBLE. For the next Exposés, we shall need in particular the results of BIBLE 4, 5, 6, 7 (the number after BIBLE refers to the Exposé number). It seems, moreover, that the theory of schemes brings no notable simplification to Borel's theory as exposed in BIBLE. That is why it has not seemed useful to reproduce it in the present Séminaire, our aim here being to deduce, from results known over algebraically closed fields, analogous results valid over an arbitrary base prescheme. (Things will be different for Chevalley's structure theory of semisimple groups, which, it appears, is best treated ab ovo over an arbitrary base prescheme.)

In the oral Exposés (which we have followed in Nos. 1 to 4) we restricted ourselves to affine group preschemes over $S$, relying in an essential way on the representability theorems of Exp. XI N° 4. In the present notes (cf. Nos. 6 to 8) we show rapidly how the affine hypotheses can be eliminated by a simpler method that does not use the most delicate results of Exposé XI. For other generalizations of the results contained in the present Exposé, see also Exposés XV and XVI. (It is evident that the content of Exposés XI, XII, XV, XVI ought to be completely rewritten.)

1. Maximal tori

1.0. Let first $G$ be an algebraic group over an algebraically closed field $k$. By a maximal torus of $G$ one means an algebraic subgroup $T$ of $G$ which is a torus (meaning here, since $k$ is algebraically closed, that it is isomorphic to a group of the form $G_m^r$), and which is maximal for this property. Note that, $k$ being perfect, $G_{red}$ is a subgroup of $G$, smooth over $k$, so it is essentially an algebraic group in the sense of BIBLE. Moreover, every reduced subgroup of $G$ (such as a torus) is automatically a subgroup of $G_{red}$, so the maximal tori of $G$ coincide with the maximal tori of $G_{red}$. When $G$ is affine, hence $G_{red}$ is affine, a fundamental theorem of Borel tells us that two maximal tori of $G$ are conjugate by an element of $G(k)=G_{red}(k)$ (BIBLE 6, th. 4 c)); in particular they have the same dimension. We shall call the common dimension of the maximal tori of $G$ the reductive rank of $G$. Note moreover that the restriction that $G$ be affine is unnecessary, as follows from a known theorem of Chevalley, asserting that every connected smooth algebraic group over $k$ is an extension of an abelian variety by an affine group; cf. N° 6. In Nos. 1 to 4 we shall mostly restrict ourselves to affine group preschemes over the base.

Let $G$ be a smooth algebraic group over $k$, and $T$ a maximal torus of $G$; the centralizer $C(T)$ of $T$ in $G$ will be called the Cartan subgroup of $G$ associated with $T$. For us this is a group subpreschema defined thanks to (VIII, 6.7), but one will note that, $G$ being smooth over $k$, the same holds for the Cartan subgroup $C$ by (XI, 5.3); hence in this case $C$ is the unique group subpreschema of $G$ smooth over $k$ (i.e. an algebraic subgroup of $G$ in the sense of BIBLE) such that $C(k)$ is the subgroup of $G(k)$ centralizing $T(k)$, i.e. it is essentially the centralizer in the sense of BIBLE. By the conjugation theorem already cited, the Cartan subgroups associated with the various maximal tori are conjugate to each other, hence have the same dimension. We shall call their common dimension the nilpotent rank of $G$; it equals that of $G_{red}$. Let ${\rho }_r(G)$ and ${\rho }_n(G)$ denote the reductive and nilpotent ranks of $G$; then one has

$${\rho }_r(G)⩽{\rho }_n(G),$$

and the difference

ρ_u(G) = ρ_n(G) − ρ_r(G) = dim C/T

might be called, when $G$ is affine, the unipotent rank of $G$. When $G$ is smooth, affine, and connected, then $C$ is a nilpotent connected algebraic group (BIBLE 6, th. 6 a) and c)), hence (BIBLE 6, th. 2) isomorphic to the product $C_s\times C_u$, where $C_s=T$ is the maximal torus we started with (which is a fortiori a maximal torus in $C$), and where $C_u$ is a smooth unipotent subgroup, i.e. a successive extension of groups isomorphic to $G_a$ (BIBLE 6 th. 1 cor. 1 and 7, th. 4). One then has also, in this case:

$${\rho }_u(G)=\dim C_u.$$

Remark 1.1. Besides the three notions of rank we have just made precise for an affine algebraic group, there are two others that are useful, namely the semisimple rank ${\rho }_s(G)$, which is by definition the reductive rank of the quotient $G/R$, where $R$ is the radical of $G$, and finally the infinitesimal rank ${\rho }_i(G)$, which is by definition the nilpotent rank of the Lie algebra of $G$ (which will be defined and studied in the following Exposé). We shall not use them in the present Exposé. Let us only signal the inequalities:

ρ_s ⩽ ρ_r ⩽ ρ_n ⩽ ρ_i.

Lemma 1.2. Let $G$ be an algebraic group over the algebraically closed field $k$, $T$ an algebraic subgroup of $G$, $k^{\prime }$ an algebraically closed extension of $k$, $G^{\prime }$ and $T^{\prime }$ the groups obtained from $G$, $T$ by extension of the base. For $T$ to be a maximal torus of $G$, it is necessary and sufficient that $T^{\prime }$ be a maximal torus of $G^{\prime }$.

This is an immediate consequence of the "principle of finite extension" (EGA IV, 9.1.1).

Definition 1.3. Let $S$ be a prescheme, $G$ an $S$-prescheme in groups of finite type, $T$ a group subpreschema of $G$. One says that $T$ is a maximal torus of $G$ if

a) $T$ is a torus (IX, 1.3) and

b) for every $s\in S$, denoting by $\bar{s}$ the spectrum of an algebraic closure of $\kappa (s)$, $T_{\bar{s}}$ is a maximal torus in $G_{\bar{s}}$.

Remarks 1.4. It follows from 1.2 that when $S$ is the spectrum of an algebraically closed field one recovers the usual definition, and that property 1.3 is stable under any base change. Note that, by (X, 8.8), one may in Definition 1.3 replace condition a) by:

a′) $T$ is of finite presentation and flat over $S$.

One should beware that a maximal torus in the sense of Definition 1.3 is indeed maximal among the subtori of $G$ (as follows at once from (IX, 2.9)), but that the converse cannot be valid—the base $S$ being, say, connected—unless $G$ effectively admits a maximal torus in the sense of 1.3, which is not the case in general (even if $S=\mathrm{Spec}(\mathbb{Z})$ and $G$ is "semisimple"; see also 1.6). We shall see however in XIV that the converse holds when $S$ is artinian, or when $S$ is a local scheme and $G$ is "reductive": in this case, every torus of $G$ is contained in a maximal torus.

Definition 1.5. Let $G$ be an algebraic group over a field $k$. The reductive rank (resp. nilpotent rank*, resp.* unipotent rank*, etc.) of $G$ is the reductive rank (resp. …) of $G_{\bar{k}}$, where $\bar{k}$ is an algebraic closure of $k$.*

One will note that, in view of 1.2 and the commutation of the formation of $Cent{r}_G(T)$ with base extension, the notions of rank introduced in 1.5 are invariant under extension of the base field; on the other hand, for $k$ algebraically closed, they coincide with those introduced at the beginning of the present number.

Remark 1.6. It is not difficult to construct an affine smooth group scheme $G$ over the spectrum $S$ of a discrete valuation ring, whose generic fiber is isomorphic to $G_m$, and whose special fiber is isomorphic to $G_a$. Such a $G$ contains no torus except the trivial torus $T$ (reduced to the unit subgroup), which is evidently not a maximal torus. More precisely, in the special fiber $G_0=G_{a,k}$, T_0 is indeed a maximal torus, but in the generic fiber $G_1=G_{m,K}$, T_1 is no longer a maximal torus ($k$ is the residue field, $K$ the fraction field of the valuation ring under consideration). One sees also from this example that the reductive rank of $G_s$ ($s\in S$) is not a continuous function of $s$. One has however the following results:

Theorem 1.7. Let $S$ be a prescheme, $G$ an affine smooth $S$-prescheme in groups over $S$. For every $s\in S$, consider ${\rho }_r(s)={\rho }_r(G_s)$ and ${\rho }_n(s)={\rho }_n(G_s)$, the reductive and nilpotent ranks of $G_s$ (1.5). With these notations, one has the following:

a) The function ${\rho }_r$ on $S$ is lower semicontinuous, the function ${\rho }_n$ on $S$ is upper semicontinuous, hence the function ${\rho }_u={\rho }_n-{\rho }_r$ is upper semicontinuous.

b) The following conditions (stable under arbitrary base change) are equivalent:

(i) The function ${\rho }_r$ on $S$ is locally constant.

(ii) There exists, locally for the étale topology, a maximal torus in $G$.

(ii bis) There exists, locally for the faithfully flat quasi-compact topology, a maximal torus in $G$.

c) Let T_1, T_2 be two maximal tori in $G$ (which implies that one is under the conditions of b)). Then T_1, T_2 are conjugate locally for the étale topology, i.e. there exists an étale surjective morphism $S^{\prime }\to S$ such that the subgroups $(T_1{)}_{S^{\prime }}$ and $(T_2{)}_{S^{\prime }}$ of $G_{S^{\prime }}$ are conjugate by a section of $G_{S^{\prime }}$ over $S^{\prime }$.

d) Under the conditions of b), i.e. when ${\rho }_r$ is locally constant, the same holds for ${\rho }_n$ (hence also for ${\rho }_u={\rho }_n-{\rho }_r$).

Proof. a) Note that for any morphism $S^{\prime }\to S$, setting $G^{\prime }=G{\times }_S{S}^{\prime }$, the functions ${\rho }_r^{\prime }$ etc. on $S^{\prime }$ defined in terms of $G^{\prime }$ as ${\rho }_r$ etc. in terms of $G$ are obtained simply by composing the latter with $S^{\prime }\to S$. When $S^{\prime }\to S$ is faithfully flat quasi-compact, it follows that ${\rho }^{\prime }$ is continuous, resp. upper semicontinuous, resp. lower semicontinuous, if and only if $\rho$ is, the Zariski topology of $S$ being in effect a quotient of that of $S^{\prime }$ (SGA 1, VIII 4.3). Consequently, the assertions of a) are local for the faithfully flat quasi-compact topology. Let then $s\in S$; we wish to show that the set $U$ of $t\in S$ such that ${\rho }_r(t)⩾{\rho }_r(s)$ (resp. ${\rho }_n(t)⩽{\rho }_n(s)$) is a neighborhood of $s$. By the principle of finite extension, there exists a finite extension $k$ of $\kappa (s)$ such that $G_k$ admits a maximal torus. There then exist an open neighborhood $U$ of $s$, and a finite surjective locally free morphism $S^{\prime }\to U$ such that the fiber $S_s^{\prime }$ is $\kappa (s)$-isomorphic to $\mathrm{Spec}(k)$ (cf. EGA III, 10.3.2, where the noetherian hypothesis is manifestly unnecessary). Since $S^{\prime }\to S$ is an open morphism (SGA 1, IV 6.6), we are reduced to the case where $S^{\prime }=S$, i.e. to the case where there exists a maximal torus $T_s$ in $G_s$. Furthermore, by (XI, 5.8 a)), possibly replacing $S$ by an $S^{\prime }$ étale over $S$ equipped with a point $s^{\prime }$ above $s$, we may suppose that $T_s$ is the fiber at $s$ of a subtorus $T$ of $G$. Then for every $t\in S$, ${\rho }_r(t)={\rho }_r(G_t)⩾\dim T_t=\dim T_s={\rho }_r(G_s)={\rho }_r(s)$, which proves that ${\rho }_r$ is lower semicontinuous. On the other hand, by (XI, 5.3), the functor

$$C=Cent{r}_G(T)$$

is representable by $C=Cent{r}_G(T)$,² a closed group subpreschema of $G$ smooth over $S$. So there exists a neighborhood $U$ of $s$ such that $t\in U$ implies $\dim C_t=\dim C_s={\rho }_n(s)$. The upper semicontinuity of ${\rho }_n$ is then a consequence of the relation

ρ_n(t) ⩽ dim C_t   for every t ∈ S,

which is itself contained in the following purely geometric lemma:

Lemma 1.8. Let $k$ be a field, $G$ an affine smooth algebraic group over $k$, $T$ a torus in $G$, $C$ its centralizer; then one has

$${\rho }_n(G)⩽\dim C.$$

Indeed, one may suppose $k$ algebraically closed, so that $T$ is contained in a maximal torus T_0. Let C_0 be the centralizer of the latter; then $C_0\subset C$, hence $\dim C_0⩽\dim C$. QED.

b) If ${\rho }_r$ is locally constant, then for every torus $T$ in $G$, and every $s\in S$, if $T_s$ is a maximal torus in $G_s$, then there exists an open neighborhood $U$ of $s$ such that $T|U$ is a maximal torus in $G|U$. Using now the reasoning of a), one sees that (i) ⇒ (ii bis). On the other hand (ii bis) ⇒ (i), for when $G$ admits a maximal torus $T$, then it is evident that ${\rho }_r(s)=\dim T_s$ is a locally constant function of $s$; now we signaled at the beginning of the proof of a) that the question of continuity of ${\rho }_r$ was local for the faithfully flat topology. Hence (i) ⇔ (ii bis), and obviously (ii) ⇒ (ii bis); it remains to prove the converse implication (i) ⇒ (ii). For this, introduce the functor $F$ of (XI, 4.1),³ which is therefore a smooth separated prescheme over $S$, and consider the subfunctor $\mathcal{T}$ of $F$, whose value for an $S^{\prime }$ over $S$ is the set of maximal tori in $G_{S^{\prime }}$. Using the previous remark that, granted (i), a torus in $G_{S^{\prime }}$ which is maximal in the fiber of some point $s\in S^{\prime }$ is maximal above an open neighborhood of $s$, one sees that $\mathcal{T}$ is representable by an open subpreschema of $F$, and is therefore smooth and separated over $S$. Since the structural morphism $\mathcal{T}\to S$ is evidently surjective, it admits a section locally for the étale topology by (XI, 1.10), which proves (i) ⇒ (ii).

c) This is an immediate consequence of (XI, 5.4 bis), taking into account Borel's conjugation theorem recalled at the beginning of the number.

d) By the remark at the beginning of the proof in a), and taking into account b), we may suppose that there exists a maximal torus $T$ in $G$. If $C$ is its centralizer, then $C$ is representable and smooth over $S$ by (XI, 5.3), so the function $s\mapsto {\rho }_n(s)=\dim C_s$ is indeed locally constant.

The proof of 1.7 is complete. We shall refer to the conditions envisaged in 1.7 b) by saying that in this case $G$ is of locally constant reductive rank. Let us note:

Corollary 1.9. Let $G$ be as in 1.7, and let $s\in S$ be such that ${\rho }_u(s)=0$, i.e. ${\rho }_r(s)={\rho }_n(s)$ (i.e. the Cartan subgroups of $G_{\bar{k}}$ are tori, where $\bar{k}$ is an algebraic closure of $\kappa (s)$). Then there exists an open neighborhood $U$ of $s$ over which ${\rho }_r$ and ${\rho }_n$ are constant; in particular, for every $t\in U$, the unipotent rank ${\rho }_u(t)$ of $G_t$ is zero.

Indeed, this follows immediately from 1.7 a) and from the inequality ${\rho }_r(t)⩽{\rho }_n(t)$ for every $t\in S$.

Note also that we have proved, at the same time as b), the

Corollary 1.10. Let $G$ be as in 1.7, and suppose $G$ of locally constant reductive rank. Consider the functor

$$\mathcal{T}:(Sch{)}^{\circ }⟶(Ens),$$

such that for every $S^{\prime }$ over $S$, one has

𝒯(S′) = the set of maximal tori of G_{S′}.

Then $\mathcal{T}$ is representable by a smooth, separated prescheme of finite type over $S$.

It remains to verify that $\mathcal{T}$ is of finite type over $S$. The question being local for the faithfully flat quasi-compact topology, we may suppose that $G$ admits a maximal torus $T$. By (XI, 5.3 bis) and the bis version of 5.5, $Nor{m}_G(T)$ and $G/Nor{m}_G(T)$ are representable by preschemes $Nor{m}_G(T)$ and $G/Nor{m}_G(T)$, and $\mathcal{T}$ is isomorphic to $G/Nor{m}_G(T)$.⁴ The morphism $G\to \mathcal{T}$ defined by $g\mapsto int(g)(T)$ being surjective, and $G$ quasi-compact over $S$, the same holds for $\mathcal{T}$, which completes the proof.

Remarks 1.11. a) The prescheme $\mathcal{T}$ of 1.10 will rightly be called the prescheme of maximal tori of $G$. We shall see in N° 5 that it is in fact affine over $S$. One can verify this directly when $G$ is isomorphic to a closed group subpreschema of a prescheme of the form $GL(n)$, using (XI, 4.6) and remarking that, by the reasoning of the proof of 1.7 b), $\mathcal{T}$ identifies with a subpreschema both open and closed of the prescheme $F$ which represents the multiplicative-type subgroups of $G$.

b) It is possible to give a proof of 1.7—and hence of 1.9—not using the delicate results of XI, but only the easy results (XI, 3.12) and (XI, 6.2), working only with multiplicative-type subgroups finite over the base (morally, the groups ${}_{n}T$ where $T$ is a maximal torus); compare N° 7. The same remark applies to the proof of 1.10.

We end this number by giving examples in which there exists a unique maximal torus.

Proposition 1.12. Let $G$ be an $S$-prescheme in groups of multiplicative type and of finite type. Then $G$ admits a unique maximal torus, and every torus in $G$ is contained in this maximal torus.

Uniqueness obviously follows from the latter assertion, which characterizes the maximal torus as the largest subtorus of $G$. From uniqueness it follows that the existence question is local for the faithfully flat quasi-compact topology, which allows us to suppose $G$ diagonalizable, i.e. of the form $D_S(M)$, with $M$ a finitely generated commutative group. Let M_0 be the quotient of $M$ by its torsion subgroup; I claim that the torus $T=D_S(M_0)$ in $G$ is a maximal torus and a largest subtorus. Indeed, a subtorus $T^{\prime }$ of $G$ is locally diagonalizable for the fpqc topology, hence to prove $T^{\prime }\subset T$ we may suppose $T^{\prime }$ diagonalizable, hence of the form $D_S(N)$, where $N$ is a free quotient of $M$ (VIII 1.4 and 3.2 b)), hence $N$ is a quotient of M_0, hence $T^{\prime }\subset T$. Since the construction of $T$ as $D_S(M_0)$ is compatible with any base extension, this shows at the same time that $T$ is a maximal torus of $G$, and completes the proof. In the case where $G$ is smooth over $S$, 1.12 may be generalized:

Proposition 1.13. Let $G$ be an $S$-prescheme in groups of finite presentation over $S$. Suppose that $G$ admits, locally for the faithfully flat quasi-compact topology, a central maximal torus. Then $G$ admits (globally) a unique maximal torus,⁵ and it is the largest subtorus of $G$.

Here again, uniqueness is trivial from the last assertion, and renders existence a local question for fpqc, which allows us to suppose that $G$ admits a central maximal torus $T$. We show that every torus $R$ of $G$ is contained in $T$.

This follows from the

Lemma 1.14. Let $G$ be an $S$-prescheme in groups of finite presentation over $S$, $T$ a maximal torus of $G$, $R$ a subtorus of $G$, and suppose that $T$ and $R$ commute. Then $R\subset T$.

Indeed, since $R$ and $T$ commute, the morphism $R{\times }_{S}T\to G$ defined by $(r,t)\mapsto r\cdot t$ is a group homomorphism, hence by (IX, 6.8) it admits an image subgroup $T^{\prime }$ in $G$, which is a multiplicative-type group quotient of $R{\times }_{S}T$, hence a torus, evidently containing $T$. Since $T$ is a maximal torus, one has $T^{\prime }=T$ (1.4), hence $R\subset T$, which proves 1.14 and hence 1.13. In particular, using 1.7 b):

Corollary 1.15. Let $G$ be a commutative, smooth, and affine $S$-prescheme in groups over $S$, of locally constant reductive rank. Then $G$ admits a unique maximal torus, and this latter contains every subtorus of $G$.

Corollary 1.16. Let $G$ be a smooth and affine $S$-prescheme in groups over $S$. Suppose that for every $s\in S$, denoting by $\bar{s}$ the spectrum of an algebraic closure $\bar{k}$ of $\kappa (s)$, the geometric fiber $G_{\bar{s}}$ is a connected nilpotent algebraic group (in the sense of BIBLE, i.e. the group $G(\bar{k})$ of its $\bar{k}$-valued points is nilpotent). Suppose moreover that the reductive rank of $G$ is locally constant. Then $G$ admits a unique maximal torus $T$; moreover, $T$ is central and is the largest subtorus of $G$.

By 1.7 b), $G$ admits a maximal torus locally for fpqc, and by 1.13 we are reduced to proving that a maximal torus of $G$ is central. By (IX, 5.6 b)), we are reduced to the case where $S$ is the spectrum of a field, which one may suppose algebraically closed. Then $T(k)$ lies in the center of $G(k)$ by BIBLE 6 th. 2, which implies that $T$ lies in the center of $G$, since $Cent{r}_G(T)$ is a closed subscheme of $G$ (VIII 6.6) which contains the points of $G(k)$, hence is identical to $G$ by the Nullstellensatz (since $G$ is reduced).

Finally, for later reference, let us signal the following trivial proposition (of which we have already made implicit use):

Proposition 1.17. Let $G\supset H\supset T$ be preschemes in groups of finite type over $S$. Equivalent conditions:

(i) $T$ is a maximal torus of $G$.

(ii) $T$ is a maximal torus of $H$, and for every $s\in S$, one has equality of the reductive ranks ${\rho }_r(G_s)={\rho }_r(H_s)$.

2. The Weyl group

Let first $k$ be an algebraically closed field, and let $G$ be a smooth affine algebraic group over $k$. If $T$ is a maximal torus, $C$ its centralizer and $N$ its normalizer, then by (XI, 5.9) these are smooth closed subgroups of $G$, and $C$ is an open subgroup of $N$, so that $W=N/C$ is a finite étale group over $k$, hence determined by the group $W(k)$ of its $k$-valued points, as $W=W(k{)}_k$ (constant group defined by the ordinary finite group $W(k)$). The finite group $W(k)$ will be called the geometric Weyl group (or simply the Weyl group if no confusion is to be feared) of $G$ relative to $T$. By Borel's conjugation theorem, the Weyl groups relative to the various maximal tori are isomorphic to each other; this is why one sometimes speaks of "the" Weyl group of $G$, without specifying a maximal torus. Since the formation of $C$, $N$, and $N/C$ commutes with any base extension, one sees that if $k^{\prime }$ is an algebraically closed extension of $k$, the geometric Weyl group of $G_{k^{\prime }}$ relative to $T_{k^{\prime }}$ is canonically isomorphic to that of $G$ relative to $T$; consequently, "the" geometric Weyl group of $G$ (which is properly speaking an isomorphism class of ordinary finite groups) coincides with that of $G^{\prime }$.

This allows one, when $G$ is a smooth affine algebraic group over an arbitrary field $k$, to speak of the geometric Weyl group of $G$ as the isomorphism class of the geometric Weyl group of $G_{k^{\prime }}$, where $k^{\prime }$ is any algebraically closed extension of $k$. When $G$ admits a maximal torus $T$, then one may evidently form as above $C=Cent{r}_G(T)$, $N=Nor{m}_G(T)$, $W=N/C$ which is a finite étale group over $k$, called the Weyl group of $G$ relative to $T$; the geometric Weyl group is then nothing other than the class of the group of points of $W$ with values in any algebraically closed extension $k^{\prime }$ of $k$. Here, knowledge of the geometric Weyl group $W(k^{\prime })$ evidently no longer suffices in general to reconstruct the algebraic group $W$: one must also know the operations on $W(k^{\prime })$ of the Galois group of the separable algebraic closure of $k$ in $k^{\prime }$.

When finally $G$ is an $S$-prescheme in groups over an arbitrary base $S$, $G$ smooth and affine over $S$, and if $T$ is a maximal torus of $G$, then (XI, 5.9) still allows us to form the group

W(T) = N(T)/C(T),

which is an étale, separated, quasi-finite $S$-prescheme in groups over $S$. Its geometric fibers (relative to the algebraic closures of the residue fields $\kappa (s)$, $s\in S$) are the geometric Weyl groups of the fibers $G_s$. Consequently, (XI, 5.10) gives information on the variation of these groups with $s\in S$. We can make this information more precise and generalize it as follows:

Theorem 2.1. Let $S$ be a prescheme, $G$ a smooth and affine $S$-prescheme in groups over $S$. For every $s\in S$, let $w(s)$ be the geometric Weyl group of $G_s$, which is therefore an isomorphism class of finite groups. In the set $E$ of such classes, introduce the following preorder relation: $w⩽w^{\prime }$ if and only if $w$ and $w^{\prime }$ are represented by finite groups $W$ and $W^{\prime }$ such that $W$ is isomorphic to a quotient of a subgroup of $W^{\prime }$. With this convention:

a) The function $s\mapsto w(s)$ from $S$ to $E$ is lower semicontinuous.

b) Suppose that the reductive rank of $G$ is locally constant. Then the following conditions are equivalent:

(i) The function $s\mapsto w(s)$ is locally constant.

(i bis) The function $s\mapsto cardw(s)$ is locally constant.

(ii) There exists, locally for the étale topology (or merely for the fpqc topology), a maximal torus $T$, such that $W(T)$ is finite over its base.

(ii bis) For every $S^{\prime }$ over $S$, and every maximal torus $T$ of $G_{S^{\prime }}$, the associated Weyl group $W(T)$ is finite over $S^{\prime }$.

Proof. a) Proceeding as in 1.7 a), one is reduced, in order to prove that for every $s\in S$ there exists an open neighborhood $U$ of $s$ such that $t\in U$ implies $w(t)⩾w(s)$, to the case where there exists a torus $R$ in $G$ such that $R_s$ is a maximal torus in $G_s$. Let

$$W(R)=Nor{m}_G(R)/Cent{r}_G(R)$$

as in (XI, 5.9); this is an étale, separated, quasi-finite group prescheme over $S$. For every $t\in S$, let $w^{\prime }(t)\in E$ be its geometric fiber at $t$. Since $R_s$ is a maximal torus in $G_s$, and the formation of Norm, Centr, $Norm/Centr$ is compatible with any base extension, in particular with passage to the fibers, one sees that one has

$$w(s)=w^{\prime }(s);$$

I claim moreover that, for $t$ near $s$, one has the inequalities

w(t) ⩾ w′(t)   and   w′(t) ⩾ w′(s),

which will suffice to establish a). These two inequalities are contained in the two following lemmas:

Lemma 2.2. Let $S$ be a prescheme, $W$ an étale, separated, quasi-finite $S$-prescheme in groups over $S$. For every $s\in S$, let $f(s)$ be the class of the geometric fiber of $W$ at $s$, which is an element of the ordered set $E$ of isomorphism classes of finite groups, introduced in 2.1. Then the function $f:S\to E$ is lower semicontinuous. For it to be constant in a neighborhood of $s\in S$, it is necessary and sufficient that the same hold for the function $s\mapsto cardf(s)$, and for this it is necessary and sufficient that $W$ be finite over $S$ above an open neighborhood $U$ of $s$.

This result is a refinement, in terms of groups, of the one invoked in the proof of (XI, 5.10); we content ourselves with a sketch of the proof (which is of the most standard type). As usual, one is reduced to the case $S$ affine noetherian. One sees at once that the function $f$ is constructible (EGA 0_III, 9.3.1 and 9.3.2, and the sundries of EGA IV, 9), and by (EGA 0_III, 9.3.4) one is reduced, for the semicontinuity, to proving that if $t$ is a generalization of $s$, one has $f(t)⩾f(s)$. This reduces us, thanks to (EGA II, 7.1.7), to the case where $S$ is the spectrum of a discrete valuation ring, which one may suppose complete with algebraically closed residue field. But then, $s$ denoting the closed point of $S$, since $G$ is étale and separated over $S$, it contains a subscheme $G^{\prime }$ both open and closed, finite over $S$, such that $G_s^{\prime }=G_s$ (EGA II, 6.2.6), and one sees at once that $G^{\prime }$ is here a subgroup of $G$. Moreover $G^{\prime }$, being étale finite over $S=\mathrm{Spec}(V)$, with $V$ complete with algebraically closed residue field, is a constant group, hence of the form A_S, where $A=G^{\prime }(\kappa (s))=G(\kappa (s))$ has class $f(s)$. If $B$ is the geometric fiber of $G$ at the generic point $t$ of $S$, one has therefore a canonical monomorphism $A\to B$, which proves $f(s)⩽f(t)$. (N.B. this proof in fact proves the semicontinuity for an order relation on $E$ finer than the one indicated in 2.1.) The fact that $f$ is continuous at $s$ if and only if $s\mapsto cardf(s)$ is, follows from the fact that for $w,w^{\prime }\in E$, the relations $w⩽w^{\prime }$ and $cardw=card{w}^{\prime }$ imply $w=w^{\prime }$. The fact that this condition is equivalent to the finiteness of $W$ on a neighborhood of $s$ is then independent of the group structure on $W$, and has been signaled after (XI, 5.10); its proof can moreover easily be carried out by the preceding arguments, using the valuative criterion of properness (EGA II, 7.3.8).

Lemma 2.3. Let $G$ be a smooth affine algebraic group over an algebraically closed field $k$, $R\subset T$ two subtori, $W(R)$ and $W(T)$ the two associated finite groups as in (XI, 5.9), quotients of the normalizer by the centralizer. Then $W(R)$ is isomorphic to a quotient of a subgroup of $W(T)$.

Indeed, consider the diagram

T ↪ C(T) ↪ C(R)
         ↓     ↓
   N(T) ∩ N(R) ↪ N(R)
       ↓
     N(T)

then $(N(T)\cap N(R))/C(T)$ is a subgroup of $W(T)=N(T)/C(T)$, and one has an obvious homomorphism:

(N(T) ∩ N(R))/C(T) ⟶ W(R) = N(R)/C(R),

and everything reduces to proving that this last is surjective, i.e. that for every $k$-valued point $g$ of $N(R)$, there exists a $k$-valued point $c$ of $C(R)$ such that cg normalizes $T$, i.e. such that

int(c)(int(g)T) = T.

Now for this it suffices to note that int(g)T is a torus of $N(R)$ hence of $C(R)$ (which is an open subgroup thereof). Then $T$ and int(g)T are maximal tori of $C(R)$, since they are maximal in $G$, and one concludes by Borel's conjugation theorem.

This proves 2.3 and thereby 2.1 a).

b) We have already signaled that (i) and (i bis) are trivially equivalent; they imply (ii bis) by the converse to 2.2 or to (XI, 5.10) at choice; (ii bis) ⇒ (ii) thanks to 1.7 b); finally (ii) ⇒ (i bis), for one sees as in 1.7 a) that the condition (i) is local for fpqc, which allows us to suppose that $G$ admits a maximal torus $T$ such that $W(T)$ is finite over $S$, and one concludes again by (XI, 5.10). This completes the proof of 2.1.

3. Cartan subgroups

Definition 3.1. Let $G$ be a smooth $S$-prescheme in groups of finite type over the prescheme $S$. By a Cartan subgroup of $G$ one means a group subpreschema $C$ of $G$, smooth over $S$, such that for every $s\in S$, denoting by $\bar{s}$ the spectrum of an algebraic closure of $\kappa (s)$, $C_{\bar{s}}$ is a Cartan subgroup (cf. N° 1) of $G_{\bar{s}}$.

It is immediate that if $C$ is a Cartan subgroup of $G$, then for every $S^{\prime }$ over $S$, $C_{S^{\prime }}$ is a Cartan subgroup of $G_{S^{\prime }}$. If $S$ is the spectrum of an algebraically closed field, one recovers the notion recalled in N° 1. Finally, one verifies at once that the property for a group subpreschema $C$ of $G$ to be a Cartan subgroup is local in nature for the faithfully flat quasi-compact topology.

Theorem 3.2. Let $G$ over $S$ be as in 3.1, and suppose that $G$ is affine over $S$ and of locally constant reductive rank. Then the map

T ⟼ Centr_G(T)

induces a bijection from the set of maximal tori of $G$ onto the set of Cartan subgroups⁶ of $G$. If $C$ corresponds to $T$, then $T$ is the unique maximal torus

of $C$.

If $T$ is a maximal torus of $G$, the functor $Cent{r}_G(T)$ is representable, by (XI, 5.3) or (XI, 6.2), by a closed smooth subpreschema of $G$, $C=Cent{r}_G(T)$, and it follows from the definitions that $C$ is a Cartan subgroup of $G$. Moreover, $T$ is obviously a maximal torus of $C$, and being central, it is the unique maximal torus of $C$ (1.13). Hence the map $T\mapsto Cent{r}_G(T)$ from the set of maximal tori into the set of Cartan subgroups is injective; it remains to prove that it is surjective. Let then $C$ be a Cartan subgroup of $G$; we prove that it is of the form $Cent{r}_G(T)$, for $T$ a maximal torus of $G$. For this it suffices to find a maximal torus $T$ of $C$, for then $T$ will be a maximal torus of $G$ (since for every $s\in S$, $G_s$ and $C_s$ have the same reductive rank). By (IX, 5.6 b)) $T$ lies in the center of $C$, hence $C\subset C^{\prime }=Cent{r}_G(T)$, and then $C$ is a smooth subgroup of the smooth group $C^{\prime }$ over $S$, coinciding with $C^{\prime }$ fiber by fiber, whence $C=C^{\prime }$. Now since $G$ is of locally constant reductive rank, the same holds for $C$, hence $C$ admits a maximal torus locally for the étale topology by 1.7 b), and since this torus is central by the preceding reasoning, it follows by 1.13 that $C$ does admit a maximal torus, which completes the proof.

Corollary 3.3. Let $G$ be a smooth and affine $S$-prescheme in groups over $S$ of locally constant reductive rank, and let $\mathcal{C}:(Sch{)}^{\circ }/S\to (Ens)$ be the functor defined by

𝒞(S′) = the set of Cartan subgroups of G_{S′}.

Then the functor $\mathcal{C}$ is isomorphic to the functor $\mathcal{T}$ of 1.10, hence is representable by the same smooth, separated prescheme of finite type over $S$.

Corollary 3.4. Under the conditions of 3.2, if $C=Cent{r}_G(T)$, one has

$$Nor{m}_G(C)=Nor{m}_G(T).$$

Remark 3.5. When $G$ is not of locally constant reductive rank, it is nevertheless possible that $G$ admit Cartan subgroups (for example if $G$ has connected nilpotent fibers, $G$ is a Cartan subgroup of itself, but is not necessarily of locally constant reductive rank, cf. 1.6). In XV, we develop the theory of Cartan subgroups without supposing $G$ affine over $S$ or of locally constant reductive rank, using the theory of regular elements of the next Exposé. See also Nos. 6 and 7 for the elimination of the affine hypothesis.

4. The reductive center

Definition 4.1. Let $S$ be a prescheme, $G$ an $S$-group of finite presentation over $S$, with affine fibers, $Z$ a group subpreschema of $G$. One says that $Z$ is a reductive center of $G$ if

(i) $Z$ is central, and of multiplicative type, and

(ii) for every base change $S^{\prime }\to S$ and every central homomorphism $u:H\to G_{S^{\prime }}$, where $H$ is a group of multiplicative type and of finite type over $S^{\prime }$, $u$ factors through $Z_{S^{\prime }}$.

(For a variant of this notion when the fibers of $G$ are not supposed affine, cf. 8.6.)

Note that such a $Z$ is necessarily of finite type over $S$ (since its fibers are), hence is uniquely determined as the largest central multiplicative-type subgroup of $G$. It is easy to give examples where $G$ (smooth and affine over $S$) admits a largest central multiplicative-type subgroup $Z$, but where $Z$ is not a reductive center (i.e. $G$ does not admit a reductive center), cf. for example 1.6; it follows however easily from (IX, 6.8) that a subgroup $Z$ of $G$ is a reductive center of $G$ if and only if it is a largest central multiplicative-type subgroup, and retains this property under any base change; see also 4.3 below.

It is evident from 4.1 that if $Z$ is the reductive center of $G$, then for every base change $S^{\prime }\to S$, $Z_{S^{\prime }}$ is the reductive center of $G_{S^{\prime }}$. From this, and from the uniqueness of the reductive center, follows, thanks to the theory of faithfully flat quasi-compact descent (SGA 1, VIII):

Proposition 4.2. Let $G$ be an $S$-prescheme in groups of finite presentation over $S$ and with affine fibers. If $G$ admits a reductive center locally for the faithfully flat quasi-compact topology, then it admits a reductive center. For $Z$ to be a reductive center of $G$, it is necessary and sufficient that it be so locally for the fpqc topology.

Let us also note:

Proposition 4.3. Let $G$ be an $S$-prescheme in groups of finite presentation and with affine fibers, $Z$ a group subpreschema. For $Z$ to be a reductive center of $G$, it is necessary and sufficient that it be of multiplicative type, and that for every $s\in S$, $Z_s$ be a reductive center of $G_s$.

Indeed, it follows first from (IX, 5.6 b)) that $Z$ is then central. Since the conditions envisaged are stable under base change, it remains to prove that any central homomorphism $u:H\to G$, with $H$ of multiplicative type and of finite type, factors through $Z$. Now, $Z$ being central, $u$ and the canonical immersion $v:Z\to G$ define a group homomorphism

w : H ×_S Z ⟶ G.

By (IX, 6.8) this last admits an image group $K$, which is a multiplicative-type subgroup of $G$, and everything reduces to proving that $K=Z$. Now this is true fiber by fiber by the hypothesis on $Z$, and it now suffices to apply (IX, 5.1 bis), which completes the proof of 4.3.

One will note that in criterion 4.3, the hypothesis that for every $s\in S$, $Z_s$ is the reductive center of $G_s$ is in fact purely geometric, i.e. it suffices to verify it on the algebraic closure of $\kappa (s)$, as follows from the second assertion of 4.2.

Theorem 4.4. Let $G$ be an affine algebraic group over a field $k$. Then $G$

admits a reductive center.

Since the center of $G$ is representable by a closed subgroup of $G$ (VIII, 6.7), one is immediately reduced to the case where $G$ is commutative. Furthermore, by 4.2 one may suppose $k$ algebraically closed. We shall see in XVII that in this case $G$ is written as a product $Z\times U$, where $Z$ is of multiplicative type and $U$ is "unipotent", from which it will follow at once that $Z$ is indeed a reductive center of $G$. We shall give here a proof independent of the structure theorem for commutative algebraic groups in the general form just indicated.

Note that it follows from (IX, 6.8) that the set of multiplicative-type subgroups $H$ of $G$ is filtered upward. We show that it admits a largest element.

When $G$ is smooth over $k$, one applies the structure theorem BIBLE 4 th. 4,

$$G=G_s\times G_u,$$

with $G_s$ of multiplicative type and $G_u$ "unipotent", which means here that $G_u$ admits a composition series whose factors are subgroups (smooth if one insists) of $G_a$. (Indeed, $G_u/G_u^0$ is a unipotent group by BIBLE 4 cor. to th. 3, hence a $p$-group where $p$ is the characteristic exponent, thanks to BIBLE 4 prop. 4; on the other hand, $G_u^0$ admits a composition series with connected smooth quotients of dimension 1 by BIBLE 6 th. 1 cor. 1, and these are isomorphic to $G_a$ by BIBLE 7 th. 4.) Now one sees easily (cf. the lemma below) that every homomorphism from a multiplicative-type group to $G_a$, and hence also to $G_u$, is trivial, which proves indeed that every multiplicative-type subgroup of $G$ is contained in $G_s$.

In the general case, consider the subgroup

$$G_0=G_{red}$$

of $G$, which is smooth over $k$, hence by the foregoing admits a largest multiplicative-type subgroup Z_0. The multiplicative-type subgroups of $G$ containing Z_0 correspond to multiplicative-type subgroups of $G^0=G/Z_0$. Now one has an exact sequence

$$0⟶G_0/Z_0⟶G/Z_0⟶G/G_0⟶0,$$

where by the foregoing, $G_0/Z_0$ has no multiplicative-type subgroup except the unit group. Since a subgroup of a multiplicative-type group is of multiplicative type (IX, 8), it follows that for every multiplicative-type subgroup $H$ of $G/Z_0$, one has $H\cap (G_0/Z_0)=0$, hence $H\to G/G_0$ is injective. Since $G/G_0$ is a radicial algebraic group, hence finite over $k$, this implies that $H$ is itself radicial and of rank bounded by that of $G/G_0$. This implies that the family of multiplicative-type subgroups of $G$ admits a largest element, say $Z$.

I claim that $G/Z$ has no multiplicative-type subgroup other than the unit subgroup. This follows from the fact (proved in 7.1.1) that a commutative extension of two multiplicative-type algebraic groups is of multiplicative type: if then $Z^{\prime }$ is the inverse image in $G$ of a multiplicative-type subgroup of $G/Z$, then $Z^{\prime }$ is of multiplicative type, hence $Z^{\prime }=Z$ by the maximal character of $Z$, hence $Z^{\prime }/Z=0$.

It now easily follows from the "principle of finite extension" that for any extension $K$ of $k$, $(G/Z{)}_K=G_K/Z_K$ likewise has no multiplicative-type subgroup except the unit group.

We can now prove that $Z$ is a reductive center of $G$. Indeed, let $u:H\to G_S$ be a homomorphism of $S$-groups, where $S$ is a prescheme over $k$ and $H$ a multiplicative-type group of finite type over $S$; we prove that $u$ factors through Z_S. This amounts to saying that the composite homomorphism $H\to G_S\to (G/Z{)}_S=G_S/Z_S$ is zero. Now setting $U=G/Z$, I claim that every homomorphism $u:H\to U_S$ is zero. Indeed, by (IX, 5.2) one is reduced to the case where $S$ is the spectrum of a field, and by (IX, 6.8) this then follows from the fact that U_K has no multiplicative-type subgroup other than 0. This completes the proof of 4.4. It only remains to give the proof of the

Lemma 4.4.1. Let $H$ be a multiplicative-type $S$-prescheme in groups; then every homomorphism of $S$-groups $u:H\to G_a$ is trivial.

Indeed, consider the module $O_S^2=E$ as an extension of $O_S=E^{\prime }$ by $O_S=E^{\prime \prime }$; then $G_a$ identifies with the prescheme of automorphisms of this extension, hence a homomorphism $u:H\to G_a$ identifies with an $H$-module structure on $E$ respecting the extension structure, i.e. such that $E^{\prime }$ is stable under $H$ and the operations induced by $H$ on $E^{\prime }$ and $E^{\prime \prime }$ are trivial. By (I, 4.7.3) it follows that $H$ operates trivially on $E$, hence $u$ is trivial. QED.

Remark 4.4.2. If $G$ is a nonzero abelian variety over $k$, then $G$ does not admit a reductive center in the sense of 4.1 with the restriction "$G$ affine" omitted, because for $n$ coprime to the characteristic, ${}_{n}G$ is étale over $k$ of order prime to the characteristic, hence of multiplicative type; now the family of ${}_{n}G$ is schematically dense in $G$, so that if there were a reductive center, it would be identical to $G$, which is absurd, since $G$ is not of multiplicative type. This is the reason why it is appropriate in 4.1 to impose the restriction that $G$ have affine fibers.

Lemma 4.5. Let $S$ be a prescheme, $G$ a smooth $S$-prescheme in groups over $S$, affine over $S$, with connected fibers, $T$ a maximal torus of $G$, $u:H\to G$ a central homomorphism, with $H$ a multiplicative-type $S$-prescheme in groups of finite type. Then $u$ factors through $T$.

Indeed, let $C$ be the centralizer of $T$, which is a closed group subpreschema of $G$ smooth over $S$ (XI, 5.3), hence affine over $S$. Since $T$ is in the center of $C$, it is invariant, and one may consider the quotient group $C/T=U$, which is representable (VIII, 5.1). Since $u$ is central, it factors through $C$, and everything reduces to proving that the composite homomorphism $H\to C\to U=C/T$ is trivial. By (IX, 5.2) one is reduced to the case where $S$ is the spectrum of a field, which one may suppose algebraically closed. But then by BIBLE 6, th. 2, $U$ is a connected algebraic group (smooth over $k$) which is "unipotent", meaning, as we have already observed, that it admits a composition series with quotients isomorphic to $G_a$. Hence every homomorphism from a multiplicative-type group of finite type $H$ to $U$ is trivial. This proves 4.5.

Corollary 4.6. Let $G$ be as in 4.5. If $G$ admits a reductive center, the latter is contained in every maximal torus of $G$.

Theorem 4.7. Let $S$ be a prescheme, $G$ an $S$-prescheme in groups, smooth over $S$, affine over $S$, with connected fibers.

a) For every $s\in S$, let $z(s)$ be the type of the reductive center of $G_s$ (which is defined by 4.4). Order the set $E$ of isomorphism classes of finitely generated $\mathbb{Z}$-modules, by declaring that the class of $M$ is greater than that of $M^{\prime }$ if $M^{\prime }$ is

isomorphic to a quotient of $M$. Then the map $S\to E$, $s\mapsto z(s)$, is lower semicontinuous.

b) For $G$ to admit a reductive center $Z$, it is necessary and sufficient that the preceding function $z:S\to E$ be locally constant. If this is so, $G/Z$ is representable (cf. VIII, 5.1), and $G/Z$ admits the unit subgroup as reductive center.

c) Suppose that the reductive rank of $G$ is locally constant (cf. 1.7 b)). Then $G$ admits a reductive center $Z$. If $G/Z$ is representable (for example, $G$ affine over $S$), then moreover the maximal tori $T$ of $G$ (resp. the Cartan subgroups $C$ of $G$) and $T^{\prime }$ (resp. $C^{\prime }$) of $G^{\prime }=G/Z$ are in bijective correspondence, to $T$ (resp. $C$) corresponding $T^{\prime }=T/Z$ (resp. $C^{\prime }=C/Z$), and to $T^{\prime }$ (resp. $C^{\prime }$) corresponding $T={\phi }^{-1}(T^{\prime })$ (resp. $C={\phi }^{-1}(C^{\prime })$), where $\phi :G\to G^{\prime }$ is the canonical homomorphism.

d) Let $T$ be a maximal torus of $G$, let $g$ denote the Lie algebra of $G$, and consider the homomorphism

$$\theta :T⟶GL(g),$$

induced by the adjoint representation of $G$ (II, 4). Then the kernel of $\theta$ is a reductive center of $G$.

Proof. a) and b). The proof of a) and of the first assertion of b) is essentially identical to that of 1.7 a) and b), and we refrain from reproducing the reasoning here. We signal only that one must in a) appeal to (IX, 5.6) (using the fact that $G$ has connected fibers). Let us prove the second assertion of b), i.e. that if $Z$ is a reductive center of $G$, then $G^{\prime }=G/Z$ admits the unit subgroup as reductive center. Note immediately that by (VIII, 5.1) the quotient group $G/Z$ is indeed representable; it is affine over $S$, of finite presentation over $S$ (VIII, 5.8) and even smooth over $S$ (for it suffices to see this on the fibers, $G^{\prime }$ being flat of finite presentation over $S$; now on the fibers we signaled in (VI_B, 9.2.(xii)) that a quotient of a smooth algebraic group over a field $k$ is smooth); finally, $G$ having connected fibers, the same holds for $G^{\prime }$. Thus $G^{\prime }$ satisfies the same starting hypotheses as $G$. To see that the unit subgroup of $G^{\prime }$ is a reductive center, we may restrict if we wish to the case where $S$ is the spectrum of a field (4.3). Let $Z^{\prime }$ be a central multiplicative-type subgroup of $G^{\prime }$; everything reduces to proving that $Z^{\prime }$ is reduced to the unit group. Let Z_1 be the inverse image of $Z^{\prime }$ in $G$, and consider the operations of Z_1 on $G$ induced by inner automorphisms of $G$. Since $Z$ is central, $Z$ operates trivially, so Z_1 operates by way of operations of $Z^{\prime }$ on $G$. Moreover $Z$ being in the center of $G$, Z_1 and hence $Z^{\prime }$ operate trivially on $Z$; in addition $Z^{\prime }$ being in the center of $G^{\prime }$, the operations of $Z^{\prime }$ on the quotient $G/Z=G^{\prime }$ are also trivial. Since $Z$ is central, it follows at once that the operations of $Z^{\prime }$ on $G$ come from a group homomorphism

u : Z′ ⟶ Hom_{S-gr}(G′, Z),

by

r(z′) · g = g · u(z′)(ϕ(g)),

where $r:Z^{\prime }\to {\mathrm{Aut}}_{S-gr}(G)$ is the envisaged representation of $Z^{\prime }$ by automorphisms of $G$, and $\phi :G\to G^{\prime }$ the canonical homomorphism. Now the datum of a group homomorphism $u$ as above amounts to the datum of a group homomorphism

v : G′ ⟶ Hom_{S-gr}(Z′, Z),

on the other hand by (X, 5.8) the right-hand side is representable and is an étale group over $S$, hence its unit section is an open immersion, hence Ker v is an open subgroup of $G^{\prime }$. Since $G^{\prime }$ has connected fibers, it is equal to $G^{\prime }$, hence $v$ is zero, hence $u$ is zero, hence $Z^{\prime }$ operates trivially on $G$, hence the same holds for Z_1, which is therefore central in $G$. Thus Z_1 is a commutative extension of a multiplicative- type group $Z^{\prime }$ by a multiplicative-type group $Z$, hence (since we are over a field) is a multiplicative-type group (7.1.1). Being central in $G$, it is contained in the reductive center $Z$, i.e. $Z_1=Z$, whence $Z^{\prime }=$ unit group, which completes the proof of b).

d) Let $Z=Ker\theta$, which is a multiplicative-type subgroup of $T$ (e.g. by (IX, 6.8)). By 4.5, every central homomorphism $u:H\to G$, with $H$ of multiplicative type and of finite type, factors through $T$, hence through $Z$. Since the formation of $Z$ is compatible with any base change, it remains to prove that $Z$ is central, i.e. that the centralizer $C$ of $Z$ equals $G$. Now by (XI, 5.3), $C$ is a closed smooth subgroup of $G$; on the other hand, since $Z$ operates trivially on $g$, one sees that $Lie(C)=g$. One easily concludes that $C=G$: indeed, the immersion $C\to G$ is étale, since it is so fiber by fiber (being an unramified homomorphism of smooth algebraic groups of the same dimension, VI_B, 1.3), and one may apply (X, 3.5). Thus $C\to G$ is an étale closed immersion, hence an open immersion (SGA 1, I 5.1), and since it is also a closed immersion and $G$ has connected fibers, it is an isomorphism.

c) By 1.7 b), $G$ admits a maximal torus locally for the fpqc topology, hence by 4.2 and by d) which we have just proved, $G$ admits a reductive center $Z$. We saw in d) that every maximal torus $T$ of $G$ contains $Z$. One sees at once that $T^{\prime }=T/Z$ is a torus; to prove that it is a maximal torus in $G^{\prime }$, one is reduced by Definition 1.3 to the case where $S$ is the spectrum of an algebraically closed field, and then the assertion is contained in BIBLE 7 th. 3 a). Conversely, let $T^{\prime }$ be a maximal torus of $G^{\prime }$, and let $T={\phi }^{-1}(T^{\prime })$; we prove that $T$ is a maximal torus of $G$. The question being local for the fpqc topology, we may suppose that $G$ already admits a maximal torus, say T_0, so that by the foregoing, $T_0^{\prime }=T_0/Z_0$ is a maximal torus of $G^{\prime }$. By the conjugation theorem 1.7 c), $T^{\prime }$ and $T_0^{\prime }$ are locally conjugate for the fpqc topology, hence (since $\phi :G\to G^{\prime }$ is covering for this topology, so every section of $G^{\prime }$ lifts locally to a section of $G$) $T$ and T_0 are also locally conjugate. Since T_0 is a maximal torus, the same therefore holds for $T$. One proceeds analogously for the Cartan subgroups. This completes the proof.

Let us give a useful translation of d), in the case where $T$ is diagonalizable, hence of the form $D_S(M)$, where $M$ is a finitely generated free $\mathbb{Z}$-module. Then (I, 4.7.3) the $T$-module $g$ decomposes as a direct sum of $T$-submodules $g_m$ ($m\in M$):

g = ⨁_{m ∈ M} g_m,

which are necessarily locally free. Suppose that for every $m\in M$, the rank of $g_m$ is constant (which is the case in particular if $S$ is connected). Then the set $R$ of $m\in M$ such that $g_m\ne 0$ ("roots") is finite. With this stated:

Corollary 4.8. Under the preceding conditions and with the preceding notations, the reductive center of $G$ is the intersection of the kernels of the root characters $m\in R$ on $T$. One thus has an isomorphism

$$Z\simeq D_S(N),$$

where $N$ is the quotient of $M$ by the subgroup generated by $R$.

Corollary 4.9. Let $G$ be an algebraic group over an algebraically closed field $k$. Suppose $G$ smooth over $k$, connected affine, with reductive center reduced to the unit group, and that the Lie algebra $g$ of $G$ is nilpotent. Then $G$ is "unipotent", i.e. admits a composition series with quotients isomorphic to $G_a$.

By BIBLE 6, th. 4, cor. 3 it suffices to prove that a maximal torus $T$ of $G$ is reduced to the unit group, or equivalently that the Lie algebra $t$ of $T$ is reduced to 0. Now it follows from the fact that the $T$-module $g$ decomposes according to the characters $\alpha$ of $T$ (I, 4.7.3) that for every $t\in t$, the operation $a{d}_g(t)$ on $g$ is semisimple. If then $g$ is nilpotent, $a{d}_g(t)$ is zero. Now by 4.7 d), the reductive center of $G$ being reduced to the identity element, the homomorphism $T\to GL(g)$ is a monomorphism, hence induces an injective application on the Lie algebras, which means (II, 4.5) that for $t\in t$, the relation $a{d}_g(t)=0$ implies $t=0$. This proves that $t=0$ and completes the proof.

Proposition 4.10. Let $G$ be an affine, smooth, connected algebraic group over an algebraically closed field $k$. Then the reductive center of $G$ is the intersection of the maximal tori of $G$.

Of course, this is the intersection in the schematic sense (or equivalently, in the sense of subfunctors of $G$), i.e. the largest closed subpreschema of $G$ majorized by the maximal tori of $G$. It follows from the noetherian character of $G$ that this is also the intersection of a suitable finite set of maximal tori of $G$.

Let $Z$ be the intersection in question; $Z$ is a closed subgroup of a torus, hence of multiplicative type; on the other hand by 4.5 it contains the reductive center of $G$. To prove that it is equal, it remains to prove that it is central. Since $G$ is connected, it suffices by (IX, 5.5) to prove that $Z$ is invariant. Now by construction $Z$ is invariant under the $int(g)$, with $g\in G(k)$, hence the normalizer $N$ of $Z$ (cf. VIII, 6.7) is a closed subgroup of $G$ containing the rational points of $G$. Since $G$ is reduced, one has $N=G$, which completes the proof.

Proposition 4.11. Let $S$ be a prescheme, $G$ a smooth $S$-prescheme in groups over $S$, affine over $S$, with connected fibers, and of zero unipotent rank, which implies (1.9) that

$G$ is of locally constant reductive rank, hence (1.10) that the "prescheme of maximal tori of $G$" is defined, say $\mathcal{T}$, and is smooth, separated, of finite type over $S$. Let $G$ operate on $\mathcal{T}$ via inner automorphisms, whence a homomorphism of group functors

$$u:G⟶{\mathrm{Aut}}_S(\mathcal{T}).$$

Under these conditions, the following three subfunctors of $G$ are identical:

(i) The reductive center $Z$ of $G$.

(ii) The center $Z^{\prime }$ of $G$.

(iii) The kernel $Z^{\prime \prime }=Keru$ of the preceding homomorphism.

In particular, the center of $G$ is representable by a multiplicative-type subgroup of $G$.

Proof. One has trivially $Z\subset Z^{\prime }\subset Z^{\prime \prime }$; it remains to prove that $Z^{\prime \prime }\subset Z$, which amounts to proving (the hypotheses being stable under base change) that every section $g$ of $G$ over $S$ which operates trivially on $\mathcal{T}$ is a section of $Z$. Introducing $G^{\prime }=G/Z$ and using 4.7 b) and c), which imply in particular that the prescheme ${\mathcal{T}}^{\prime }$ of maximal tori of $G^{\prime }$ is canonically isomorphic to $\mathcal{T}$, one is reduced to the case where $G=G^{\prime }$, i.e. to the case where the reductive center $Z$ of $G$ is zero. (N.B. note that by 4.7 c), the unipotent rank of $G^{\prime }$ equals that of $G$, hence is zero since that of $G$ is zero.) It thus remains to prove in this case that $g$ is the unit section of $G$. The usual reduction procedure brings us to the case where $S$ is noetherian, and even to the case where $S$ is local artinian (since to verify that $g$ is the unit section it suffices to verify when $S$ is locally noetherian that this is so after any base change $S^{\prime }\to S$ with $S^{\prime }$ local artinian). Now in this case the kernel $Z^{\prime \prime }$ of $u$ is representable (VIII, 6.2 a) and 6.5 c)) (Note that $Z^{\prime \prime }$ is representable by (XI, 6.8)). To prove that $Z^{\prime \prime }$ is reduced to the unit subgroup, it suffices by Nakayama to prove that the same holds for its fiber $Z_0^{\prime \prime }$. This thus reduces us to the case where $S$ is the spectrum of a field $k$, which one may evidently suppose algebraically closed. Now $Z^{\prime \prime }$ is contained in the stabilizer of every point of $\mathcal{T}(k)$, i.e. in the normalizer $N(T)$ of every maximal torus $T$ of $G$. Since the reductive rank of $G$ is zero, it follows by (XI, 5.9) that $T$ is an open subgroup of $N(T)$, hence the Lie algebra of $N(T)$ is identical to that of $T$. Hence the Lie algebra of $Z^{\prime \prime }$ is contained in that of $T$. On the other hand, it follows from 4.10 that the intersection of the Lie algebras of the maximal tori $T$ of $G$ is none other than the Lie algebra of the reductive center $Z$, hence here zero, since we have assumed $Z=0$. Consequently, the Lie algebra of $Z^{\prime \prime }$ is zero, i.e. $Z^{\prime \prime }$ is étale over $k$. Moreover $Z^{\prime \prime }$ is evidently invariant in $G$, and since $G$ is connected it follows easily that $Z^{\prime \prime }$ is in the center of $G$. It is therefore in $T=Cent{r}_G(T)$ for every maximal torus $T$, hence in the intersection of the maximal tori, i.e. in $Z=0$ by 4.10, which completes the proof.

5. Application to the scheme of multiplicative-type subgroups7

Theorem 5.1. Let $S$ be a prescheme, $G$ a smooth and affine $S$-prescheme in groups over $S$, $M$ the "prescheme of multiplicative-type subgroups of $G$", representing the functor explicit in (XI, 4.1). For every integer $n>0$, let $T_n$ be the subfunctor of $M$ such that $T_n(S^{\prime })=$ the set of multiplicative-type group subpreschemata $H$ of $G_{S^{\prime }}$ such that $n\cdot i{d}_H=0$, so that $T_n$ is representable, and is affine over $S$ (XI, 3.12 a)). Let $u_n:M\to T_n$ be the morphism defined by $u_n(H)={}_{n}H$ (where ${}_{n}H=Ker(n\cdot i{d}_H)$). With these notations, one has the following:

a) Every subpreschema $U$ of $M$ of finite type over $S$ is contained in a closed subpreschema of finite type over $S$, and every closed subpreschema $X$ of $M$ of finite type over $S$ is affine over $S$.

b) Suppose $S$ quasi-compact, and let $X$ be a closed subpreschema of $M$ of finite type over $S$. Then there exists an integer $n>0$ such that for every multiple $m$ of $n$, the induced morphism

$$u_m|X:X⟶T_m$$

is a closed immersion.

Proof. a) To prove the first assertion of a), one takes for $X$ the schematic closure of $U$ (EGA I, 9.5.1 and 9.5.3), which is defined since the immersion $i:U\to M$ is quasi-compact (because $U$ is of finite type over $S$ and $M$ is separated over $S$ (4.1)). It therefore remains to prove that such an $X$ is affine over $S$, which will prove at the same time the second assertion of a). In the foregoing form, one sees that the question is local on $S$, which one may therefore suppose affine. Then $U$, being of finite type over $S$, is contained in a quasi-compact open of $M$, and this brings us to the case where $U$ itself is an open, of finite type over $S$, i.e. quasi-compact. (N.B. A closed subscheme of a scheme affine over $S$ is affine over $S$.)

It suffices to prove that such a $U$ is majorized by a closed subpreschema of $M$ which is affine. In this form, the usual reduction procedure brings us at once to the case where $S$ is noetherian. One proceeds similarly for b), which reduces to the case where $S$ is affine noetherian.

Let us take up again the inverse limit $T$ of the $T_n$ used in the proof of (XI, 4.1), which is an affine prescheme (but not of finite type) over $S$, and whose local rings are noetherian, as was seen at the beginning of the proof of (IX, 3.7). (N.B. For $t\in T$, $t=(t_n)$, the transition morphisms $T_m\to T_n$ being smooth and the dimension of the $T_m$ being bounded, there exists an $n_0$ such that the $T_n\to T_{n_0}$ are étale at $t_n$ for every $n$ multiple of $n_0$.) The proof of loc. cit., or (XI, 3.11), show that the canonical morphism

$$u:M⟶T$$

is an immersion, and induces isomorphisms on the local rings (but one will note that $u$ is not in general an open immersion nor a quasi-compact morphism). Let $U$ be a quasi-compact open of $M$; we shall prove that its closure in $T$ is contained in $M$, which proves (if $X$ denotes the schematic closure of $U$ in $M$) that $u|X:X\to T$ is a closed immersion, hence that $X$ is affine, and that will prove a). Since $U$ is noetherian, it has only finitely many irreducible components, and every element $t\in T$ in the closure of $U$ is a specialization of an element $x$ of $U$. Since the local ring of $t$ in $T$ is noetherian, the same holds for the local ring $A$ of $t$ in $\bar{x}$ endowed with the induced reduced structure.

The canonical morphism of the noetherian integral local scheme $S^{\prime }=\mathrm{Spec}(A)$ into $T$ then sends the closed point of $S^{\prime }$ to $t$, the generic point to $x\in M$, and one must show under these conditions that $t\in M$. Possibly replacing $A$ by the quotient of a suitable complete local $A$-algebra flat over $A$ (EGA 0_III, 10.3.1) by a minimal prime ideal, we may suppose that $A$ is complete with algebraically closed residue field, (and we could even reduce to the case where it is moreover a discrete valuation ring thanks to EGA II, 7.1.7). One is thus reduced (making the change of notation: $S^{\prime }$ denoted by $S$) to the

Lemma 5.2. Let $S$ be the spectrum of a complete noetherian integral local ring with algebraically closed residue field, $G$ a smooth and affine $S$-prescheme in groups over $S$, $(H(n){)}_{n>0}$ a system of multiplicative-type subgroups of $G$, indexed by the integers $n>0$, $\eta$ the generic point of $S$. Suppose:

a) If $m$ is a multiple of $n$, one has $H(n)={}_{n}H(m)$.

b) There exists a multiplicative-type subgroup $H_{\eta }$ of $G_{\eta }$ such that one has $H(n{)}_{\eta }={}_n(H_{\eta })$ for every $n>0$.

Under these conditions, there exists a multiplicative-type subgroup $H$ of $G$ such that for every $n>0$, one has $H(n)={}_{n}H$.

For each $n$, let $C_n=Cent{r}_G(H(n))$, which is a closed group subpreschema of $G$, smooth over $S$ (XI, 5.3). The set of integers $n>0$ being ordered by divisibility, the $C_n$ form a decreasing family of closed subpreschemata of $G$, and since $G$ is noetherian, this family is stationary. Let $C$ be the common value of the $C_n$ for $n$ large. Then $C$ satisfies the same conditions as $G$; moreover the $H(n)$ are in fact subgroups of $C$. Hence, replacing $G$ by $C$, we may suppose that the $H(n)$ are central subgroups of $G$.

Let then $s$ be the closed point of $S$, and $Z_s$ the reductive center of $G_s$ (defined thanks to 4.4). By the definition (4.1) $Z_s$ contains the $H(n{)}_s$. Since $A$ is complete, $Z_s$ comes from a multiplicative-type group subpreschema $Z$ of $G$ (XI, 5.8). I claim that $Z$ contains the $H(n)$. Indeed, since $H(n)$ is central, it commutes with $Z$ in $G$, hence one deduces a group homomorphism $Z\times H(n)\to G$, which by (IX, 6.8) admits an image group which is a multiplicative-type subgroup $K$ of $G$ containing $Z$, and everything reduces to proving that $K=Z$, which follows from $K_s=Z_s$ and (IX, 5.1 bis).

One is thus reduced to proving the analogue of 5.2, but with $G$ replaced by a group $Z$ of multiplicative type and of finite type over $S$ (not necessarily smooth over $S$). Since $A$ is complete with algebraically closed residue field, $Z$ is diagonalizable (X, 3.3 and 1.4), hence of the form $D_S(M)$, with $M$ a finitely generated commutative group. Consequently every multiplicative-type subgroup $H$ of $Z$ is diagonalizable (IX, 2.11 (i)), hence of the form $D_S(N)$, where $N$ is a quotient group of $M$ (VIII, 3.2). Thus the $H(n)$ correspond to quotient groups $M(n)$ of $M$, and the existence of a multiplicative-type subgroup $H$ of $Z$ such that $H(n)={}_{n}H$ for every $n$ amounts to that of a quotient group $N$ of $M$ such that $M(n)=N/nN$ for every $n$. Now this follows at once from the fact that there exists a subgroup $H_{\eta }$ of $G_{\eta }$ such that $H(n{)}_{\eta }={}_n(H_{\eta })$ for every $n$. This completes the proof of 5.2 and hence of a).

b) We already know that $u|X:X\to T$ is a closed immersion. It easily follows that for $m$ large, the composite $u_m|X:X\to T_m$ is also a closed immersion. Since we will not need this fact in the sequel, we omit the details of the proof.

Corollary 5.3. With the notations of 5.1, let $U$ be a part of $M$ both open and closed, of finite type over $S$. Then $U$ is affine over $S$ for the structure induced by $M$, and if $S$ is quasi-compact, there exists an integer $n>0$ such that for every multiple $m$ of $n$, the induced morphism $u_m|U:U\to T_m$ is an open and closed immersion (i.e. an isomorphism onto an open and closed part of $T_m$, equipped with the induced structure).

The first assertion follows from 5.1 a), the second from 5.1 b), taking into account that $u_m:M\to T_m$ is smooth (XI, 2.2 bis) and that a smooth immersion, i.e. étale, is an open immersion.

Corollary 5.4. Let $S$ be a prescheme, $G$ a smooth and affine $S$-prescheme in groups over $S$, of locally constant reductive rank (1.7 b)), $\mathcal{T}$ the "prescheme of maximal tori of $G$" (1.10). Then $\mathcal{T}$ is smooth and affine over $S$. If $T$ is a maximal torus of $G$, $N(T)$ its normalizer, then $G/N(T)$ (XI, 5.3 bis) is affine over $S$. The same holds for $G/C(T)$ (where $C(T)$ is the centralizer of $T$) provided that $W(T)=N(T)/C(T)$ is finite over $S$ (cf. 2.1 b)).

The second assertion is contained in the first, since by the conjugation theorem, $G/N(T)$ is isomorphic to $\mathcal{T}$ (XI, 5.5 bis). For the first assertion, one notes that by construction, $\mathcal{T}$ is isomorphic to an open and closed subpreschema of $M$ (for one may suppose the reductive rank of $G$ constant and equal to $r$, and then $\mathcal{T}$ is the subpreschema of $M$ corresponding to the subtori of relative dimension $r$, i.e. the largest subpreschema of $M$ over which the "universal multiplicative-type subgroup" $H\subset G_M$ is a torus of relative dimension $r$). One may therefore apply 5.3. Finally, for the last assertion, one notes that $G/C(T)$ is finite over $G/N(T)\simeq (G/C(T))/W(T)$, hence is affine since $G/N(T)$ is.

Corollary 5.5. Let $G$ be a smooth and affine algebraic group over a field $k$. Then the scheme $M$ of multiplicative-type subgroups of $G$ is a direct sum of affine schemes over $S$. For every multiplicative-type subgroup $H$ of $G$, if $C$ and $N$ denote respectively its centralizer and its normalizer, the quotients $G/C$ and $G/N$ are affine.

Using (XI, 5.1 bis), one sees that the saturation under $G$ operating on $M$ of any finite closed part of $M$ is open: indeed, one is reduced to the case where $k$ is algebraically closed, hence to the case of the trajectory of a $k$-rational point $x$, but then by loc. cit. the morphism $g\mapsto g\cdot x$ from $G$ to $M$ is smooth hence open, hence its image is open. Let $U$ be the union of the classes of the closed points of $M$ for the equivalence relation defined by the operations of $G$. Then $U$ is open and contains every closed point of $M$, hence by the Nullstellensatz is identical to $M$. Thus $M$ is a disjoint union of opens, which are therefore necessarily closed, hence $M$ is the sum prescheme of preschemata $M_i$, each of which is a $G$-trajectory of a closed point, hence is quasi-compact, hence of finite type. By 5.3 the $M_i$ are therefore affine. If $H$ is a multiplicative-type subgroup of $G$, it corresponds to a $k$-rational point $x$ of $M$, and $G/N$ identifies with the trajectory of $x$ under $G$ (XI, 5.5 bis). It is therefore affine by the foregoing. Since $C$ is an open subgroup of $N$ (XI, 5.9), $G/C$ is finite over $G/N$ (since the latter is isomorphic to $(G/C)/(N/C)$), hence affine since $G/C$ is.

This proof shows at the same time:

Corollary 5.6. Under the conditions of 5.5 for $G$ and $H$, the subscheme $U$ of $M$ "of multiplicative-type subgroups of $G$ that are locally conjugate to $H$" is an open and closed subpreschema of $G$.

In picturesque language, every multiplicative-type subgroup $H^{\prime }$ of $G$ that is a limit of subgroups of $G$ conjugate to $H$, is itself conjugate to $H$.

Remarks 5.7. Let $S$ be a prescheme, $G$ a smooth and affine $S$-prescheme over $S$, $H$ an $S$-prescheme of multiplicative type and of finite type, and set $M={\mathrm{Hom}}_{S-gr}(H,G)$, which by (XI, 4.2) is representable and is smooth over $S$ and separated over $S$. One can then prove for $M$ a result entirely analogous to 5.1, either by reducing to 5.1 by an argument analogous to that of (XI, 4.2), or by proceeding directly by an argument modeled on that of 5.1. From this one deduces corresponding variants for 5.3, 5.5, 5.6, which the reader will formulate.

6. Maximal tori and Cartan subgroups of not-necessarily-affine algebraic groups (algebraically closed base field)

Lemma 6.1. Let $G$ be a connected algebraic group over a field $k$, $Z$ an algebraic subgroup of finite index in its center; then $G/Z$ is affine.

One may suppose that $Z$ is the center of $G$, since a scheme finite over an affine scheme is affine. Consider the vector spaces

P_n = P_n(G) = O_{G,e}/m_{G,e}^{n+1}   (n an integer ⩾ 0),

where $m_{G,e}$ is the maximal ideal; then $G$ operates on the $P_n(G)$ by the adjoint representation, and if $Z_n$ is the kernel of the corresponding homomorphism

$$a{d}_n:G⟶GL(P_n),$$

one verifies easily (using the fact that $G$ is connected) that $Z$ is the intersection of the $Z_n$, hence ($G$ being noetherian) equal to one of the $Z_n$. But $a{d}_n$ defines, on passage to the quotient, a monomorphism $G/Z_n\to GL(P_n)$, which is therefore a closed immersion, hence $G/Z_n$ is affine, and consequently $G/Z$ is.

Lemma 6.2. Let $G$ be a smooth algebraic group over the field $k$, $Z$ a central algebraic subgroup, $G^{\prime }=G/Z$, $u:G\to G^{\prime }$ the canonical homomorphism, $T$ a multiplicative-type subgroup in $G$, $T^{\prime }=u(T)$ the image group, $C$ the centralizer of $T$ in $G$, $C^{\prime }$ that of $T^{\prime }$ in $G^{\prime }$. Then one has

$$C^{\prime 0}\subset u(C).$$

One may suppose $k$ algebraically closed. Let

$$C_1=(u^{-1}(C^{\prime }{)}^0{)}_{red},$$

it suffices to prove that $C_1\subset C$, since $u(C_1)$ is connected of finite index in $C^{\prime }$ hence equal set-theoretically to $C^{\prime 0}$, hence equal to $C^{\prime 0}$ since $C^{\prime }$ and consequently $C^{\prime 0}$ are smooth.

Consider the morphism

$$(g,t)\mapsto gt{g}^{-1}{t}^{-1}$$

from $G\times G$ to $G$; it induces a morphism

ϕ : C_1 × T ⟶ Z_1,   where Z_1 = Z_red,

since the left-hand side being reduced, it suffices to see that for $g\in C_1(k)$, $t\in T(k)$, one has $gt{g}^{-1}{t}^{-1}\in Z(k)$, which comes from the fact that $g$ centralizes $T$ modulo $Z$. One sees easily (by computing on $k$-rational points) that $\phi (g,t)$ is additive in $g$ and additive in $t$, hence is "bilinear"; I claim that (with Z_1 smooth and C_1 connected) this homomorphism is identically zero, which will prove indeed that $C_1\subset C$. Using the density theorem for $T$, we are reduced to the case where $T$ is finite, i.e. where there exists an integer $n>0$ such that $n\cdot i{d}_T=0$. Note that $\phi$ is defined by a group homomorphism

$$C_1⟶{\mathrm{Hom}}_{S-gr}(T,Z_1),$$

now Z_1 being commutative and smooth, the right-hand side is representable by an étale algebraic group over $k$, and C_1 being connected, every group homomorphism from C_1 to the latter is zero. QED.

Corollary 6.3. Under the preceding conditions, suppose $C^{\prime }$ connected; then one has

C′ = u(C),   C = u⁻¹(C′).

Indeed, then $C^{\prime }=C^{\prime 0}$; on the other hand $C$ obviously contains $Z$, hence is equal to $u^{-1}(u(C))$.

Lemma 6.4. Let $G$ be an algebraic group over the field $k$, $Z$ a central algebraic subgroup such that $G/Z=G^{\prime }$ is a torus. Then $G$ is commutative, and if $k$ is algebraically closed, there exists a torus $T$ in $G$ such that $u(T)=G^{\prime }$, where $u$ is the canonical homomorphism $G\to G/Z=G^{\prime }$.

One may suppose $k$ algebraically closed. Consider again the morphism $G\times G\to G$ defined by $(g,h)\mapsto gh{g}^{-1}{h}^{-1}$; then (since $Z$ is central) this morphism factors through a morphism $G^{\prime }\times G^{\prime }\to G$, and since $G^{\prime }$ is commutative, this latter takes its values in $Z$, and even in $Z_1=Z_{red}$, since $G^{\prime }\times G^{\prime }$ is reduced. One sees as above that the morphism $\phi :G^{\prime }\times G^{\prime }\to Z_{red}$ thus obtained is bilinear, hence zero, which proves that $G$ is commutative. To find a torus $T$ of $G$ such that $u(T)=G^{\prime }$, one may (replacing $G$ by $G_{red}^0$ if necessary) suppose $G$ smooth and connected, and moreover (replacing $Z$ by $Z_{red}^0$ if necessary) that $Z$ is smooth and connected. By a well-known theorem of Chevalley,⁸ $Z$ is an extension of an abelian variety by an affine group, which reduces us at once to proving our assertion in each of the two following cases: 1°) $Z$ is affine, 2°) $Z$ is an abelian variety. In case 1°), $G$ is affine and the result is well known (BIBLE 7 th. 3 a)). Suppose then that $Z$ is an abelian variety. Since every homomorphism from the additive group $G_a$ to the torus $G^{\prime }=R$ or to the abelian variety $Z$ is trivial, it follows that the same holds for every homomorphism from $G_a$ to $G$, hence $G$ does not contain a subgroup isomorphic to $G_a$. By Chevalley's structure theorem already invoked, one has an exact sequence

$$(\ast )0⟶T⟶G⟶A⟶0,$$

where $A$ is an abelian variety, and $T$ an affine smooth connected group. Since the latter is commutative and contains no additive subgroup, it follows that $T$ is a torus (and it is evidently the unique maximal torus of $G$). Everything reduces to proving that every epimorphism

$$u:G⟶R,$$

where $R$ is a torus, satisfies $u(T)=R$. Set

Hom_gr(T, G_m) = M,   Hom_gr(R, G_m) = P,

(these are finitely generated free $\mathbb{Z}$-modules which recover $T$, $R$ up to isomorphism by $T=D_k(M)$, $R=D_k(P)$), and let

$$B=Ex{t}_{k-gr}^1(A,G_m),$$

($B$ is also the set of $k$-rational points of the abelian variety dual to $A$). One evidently has

(××)  Ext¹(A, T) = Hom_gr(M, B),   Ext¹(A, R) = Hom_gr(P, B),

in particular the extension $G$ of $A$ by $T$ is given by a homomorphism

$$\theta :M⟶B.$$

On the other hand the exact sequence (∗) gives the exact sequence

0 ⟶ Hom(A, R) ⟶ Hom(G, R) ⟶ Hom(T, R) ⟶ Ext¹(A, R),

moreover $\mathrm{Hom}(A,R)=0$, and $\mathrm{Hom}(T,R)\to Ex{t}^1(A,R)$ identifies with the homomorphism

Hom(P, M) ⟶ Hom(P, B)

deduced from $\theta :M\to B$. Setting

$$N=Ker(\theta ),$$

one therefore finds a canonical bijection

Hom(G, R) ≃ Hom(P, N) ≃ Hom(S, R),   where S = D_k(N),

which can be described geometrically at once as follows:

Lemma 6.5. Let $G$ be an extension of an abelian variety $A$ by a torus $T=D_k(M)$, defined by a homomorphism $\theta :M\to B=Ex{t}^1(A,G_m)$ (base field $k$ algebraically closed). Let $N=Ker\theta$, $S=D_k(N)$ the corresponding torus, isomorphic to $T/U$ where $U=D_k(M/N)$; consider the extension $H=G/U$ of $A$ by $S$. This extension splits,⁹ hence one has a unique projection of $H$ onto $S$, whence a unique homomorphism

$$v:G⟶S$$

extending the canonical homomorphism $T\to S$. With this stated, for every torus $R$ and every homomorphism $u:G\to R$, there exists a unique homomorphism $u^{\prime }:S\to R$ such that $u=u^{\prime }v$. In particular, one has $Im(u)=Im(u^{\prime })=u(T)$.

This shows in particular that if $u$ is an epimorphism, the same holds for its restriction to $T$, which completes the proof of 6.4.

Theorem 6.6. Let $G$ be a smooth and connected algebraic group over an algebraically closed field $k$.

a) The maximal tori of $G$ are conjugate.

b) Let $T$ be a torus in $G$; then its centralizer is smooth and connected.

c) The map $T\mapsto Cent{r}_G(T)$ establishes a bijective correspondence between the set of maximal tori $T$ of $G$ and the set of Cartan subgroups (N° 1) of $G$. For an algebraic subgroup $C$ of $G$ to be a Cartan subgroup, it is necessary and sufficient that it be smooth, nilpotent, and set-theoretically equal to its connected normalizer (and then it is even equal to its connected normalizer); one then has $C=Cent{r}_G(T)$, where $T$ is the unique maximal torus of $C$, and $Nor{m}_G(C)=Nor{m}_G(T)$.