Exposé XVI. Groups of zero unipotent rank

by M. Raynaud¹

1. An immersion criterion

1.1. Examples of monomorphisms of group preschemes that are not immersions.

We are going to construct a prescheme $S$, two $S$-group preschemes $G$ and $H$, and an $S$-group monomorphism $u:G\to H$ that is not an immersion. The groups $G$ and $H$ will be of finite presentation over $S$, flat over $S$, and $G$ will even be smooth over $S$ (cf. Exp. VIII, paragraph and footnote (∗) preceding 7.1; see also XVII App. III, 4).

a) Take first for $S$ the spectrum of a discrete valuation ring $A$, of unequal characteristic and of residue characteristic equal to 2. Let $t$ be the generic point of $S$ and $s$ the closed point. Take for $G$ the open subgroup of the constant group $G_0=(\mathbb{Z}/2\mathbb{Z}{)}_S$ obtained by removing the point of the closed fiber $(\mathbb{Z}/2\mathbb{Z}{)}_s$ distinct from the identity element, and take for $H$ the group of multiplicative type (µ₂)_S of second roots of unity (Exp. I 4.4.4). In view of the choice of $S$, $H_t$ is isomorphic to $(\mathbb{Z}/2\mathbb{Z}{)}_t$ while $H_s$ is a radicial group. One has an evident morphism $G_0\to H$ defined by the section $(-1)$ of $H$, whence a morphism $u:G\to H$ which is a monomorphism but is not an immersion (otherwise it would necessarily be an isomorphism (cf. Exp. VIII 7)). Here, Exp. VIII 7.9 does not apply, since ${}_{2}G=G$ is not finite over $S$.

b) Denote by $K$ the $S$-group prescheme, étale and non-separated, obtained from the unit group by "doubling" the unique point of the closed fiber. Take for $H$ the product group (µ₂)_S ×_S K. Let $a$ (resp. $b$) be the unique section of µ₂ (resp. $K$) over $S$ distinct from the unit section, and let $h=(a,b)$ be the corresponding section of $H$. The datum of $h$ defines an $S$-group morphism

u : G = (ℤ/2ℤ)_S ⟶ H.

It is clear that $u$ is a monomorphism and is not an immersion. Here, Exp. VIII 7.9 does not apply, since ${}_{2}H=H$ is not separated over $S$.

c) More interesting is the following example, in which $G$ is a smooth group with connected fibers (and even $G=({\mathbb{G}}_a{)}_S$) (cf. Exp. VIII 7.10).

Take for $S$ the spectrum of a discrete valuation ring $A$ of equal characteristic 2, and let $\pi$ be a uniformizer of $A$ and $\mathfrak{m}$ the maximal ideal of $A$.

Consider the additive group $({\mathbb{G}}_a{)}_S$, the product group $({\mathbb{G}}_a\times {\mathbb{G}}_a{)}_S$ of ring A[X, Y], and let $G_1$ be the closed subgroup of equation

$$(1)X^2+X-\pi Y=0.$$

The group $G_1$ is smooth over $S$, its generic fiber is isomorphic to $({\mathbb{G}}_a{)}_t$ and its closed fiber is isomorphic to the product of $({\mathbb{G}}_a{)}_s$ by $(\mathbb{Z}/2\mathbb{Z}{)}_s$. One may take for the factor $(\mathbb{Z}/2\mathbb{Z}{)}_s$ the subgroup $N_s$ of $(G_1{)}_s$ of equation

$$(2)Y=X^2+X=0.$$

Let $N^{\prime }$ and $N^{\prime \prime }$ be two group subschemes of $G_1$, isomorphic to $(\mathbb{Z}/2\mathbb{Z}{)}_S$, whose closed fibers coincide with $N_s$ and whose generic fibers are distinct. The groups $N^{\prime }$ and $N^{\prime \prime }$ are therefore defined by the data of two sections of $G_1$ over $S$ with coordinates $(a^{\prime },b^{\prime })$ (resp. $(a^{\prime \prime },b^{\prime \prime })$) such that

(3)    a′² + a′ − πb′ = a″² + a″ − πb″ = 0,
       a′ and a″ ≡ 1 (mod 𝔪), a′ ≠ a″,
       b′ and b″ ≡ 0 (mod 𝔪).

(One may take, for example, (1, 0) and $(1+{\pi }^2,\pi +{\pi }^3)$.)

The quotient groups $G^{\prime }=G_1/N^{\prime }$ and $G^{\prime \prime }=G_1/N^{\prime \prime }$ are representable (Exp. V 4.1). Let $u$ be the canonical morphism

u : G₁ ⟶ G′ ×_S G″,

and let $H$ be the group subscheme of $G^{\prime }{\times }_S{G}^{\prime \prime }$ equal to the schematic closure in $G^{\prime }{\times }_S{G}^{\prime \prime }$ of $u_t((G_1{)}_t)$, so that $u$ factors through $H$. The kernel of $u$ is $K=N^{\prime }\cap N^{\prime \prime }$, whose generic fiber is the unit group and whose closed fiber is equal to $N_s$; in particular, $K$ is not flat over $S$. On the generic fiber, $u_t$ is therefore an isomorphism of $(G_1{)}_t$ onto $H_t$. On the other hand, I claim that $H_s$ is not smooth. Indeed, if $H_s$ were smooth, then since $u_s$ is a morphism with finite kernel and $(G_1{)}_s$ and $H_s$ have the same dimension (namely 1), $u_s$ would be a flat morphism; since $G_1$ and $H$ are flat, $u$ would be a flat morphism and consequently Ker u would be flat over $S$, which is not the case. It is clear that the restriction of $u$ to the connected component $G$ of $G_1$ is a monomorphism (the fibers of $K\cap G$ are unit groups, so $K\cap G$ is the unit group (Exp. VI_B 2.9)), but it is not an immersion (otherwise it would be an open immersion and $H_s$ would be smooth). Here, Exp. VIII 7.9 does not apply, since ${}_{2}G=G$ is not finite over $S$.

For lovers of equations, let us say that in the example above one may take for $H$ the closed subgroup of $({\mathbb{G}}_a{\times }_S{\mathbb{G}}_a{)}_S$ with ring $A[V,W]/(F)$, where

F = (a″b′ − b″a′)(a″²V − a′²W) − (a″V − a′W)²

(where $a^{\prime },a^{\prime \prime },b^{\prime },b^{\prime \prime }$ satisfy (3)). For $G$, one takes the group $({\mathbb{G}}_a{)}_S$ with ring A[T], and for morphism $u:G\to H$ the $S$-morphism defined by the maps

V ↦ a′(a′T + πT²),    W ↦ a″(a″T + πT²).

Remark 1.2. The preceding construction is inspired by the method of Koizumi–Shimura.² It does not apply when $S$ is the spectrum of a ring of unequal characteristic, the points of finite order of $G$ being then "too rigid".

1.3. Statement of the immersion criterion.

Theorem 1.3. Let $S$ be a locally noetherian prescheme, $G$ an $S$-group prescheme, smooth over $S$, of finite type, possessing locally for the fpqc topology Cartan subgroups (Exp. XV 6.1 and 7.3 (i)), $H$ an $S$-group prescheme locally of finite type, $u:G\to H$ an $S$-group monomorphism. Then:

a) If $G$ has connected fibers, in order that $u$ be an immersion, it is necessary and sufficient that for every $S$-scheme $S^{\prime }$ which is the spectrum of a complete discrete valuation ring with algebraically closed residue field, and for every Cartan subgroup $C$ of $G_{S^{\prime }}$, the restriction of $u_{S^{\prime }}$ to $C$ be an immersion.

b) In order that $u$ be an immersion (resp. a closed immersion), it is necessary and sufficient that for every $S^{\prime }$ as above and every Cartan subgroup $C$ of the connected component $(G_{S^{\prime }}{)}^0$ of $G_{S^{\prime }}$, the restriction of $u_{S^{\prime }}$ to the normalizer (Exp. XI 6.11) $N$ of $C$ in $G_{S^{\prime }}$ be an immersion (resp. a closed immersion).

Before proving 1.3, let us state some applications.

Corollary 1.4. Let $S$ be a prescheme, $G$ an $S$-group prescheme smooth over $S$, of finite presentation over $S$, of zero unipotent rank (Exp. XV 6.1 ter) and possessing locally for the fpqc topology maximal tori, $H$ an $S$-group prescheme, $u:G\to H$ an $S$-group monomorphism. Suppose further that either $H$ is of finite presentation over $S$, or $S$ is locally noetherian and $H$ locally of finite type. Then:

a) If $G$ has connected fibers, $u$ is an immersion.

b) If $H$ is separated over $S$ and if for every $S^{\prime }$ over $S$ and every maximal torus $T$ of $G_{S^{\prime }}$ the Weyl group

$$W=Nor{m}_{G_{S^{\prime }}}(T)/Cent{r}_{G_{S^{\prime }}}^0(T)$$

is representable (a condition always realized if $G$ has connected fibers (Exp. XV 7.1 (iv))) and is finite over $S$, then $u$ is a closed immersion.

Corollary 1.5. Let $S$ be a prescheme, $G$ an $S$-group prescheme smooth over $S$, of finite presentation over $S$, with connected fibers, $H$ an $S$-group prescheme, $u:G\to H$ an $S$-group monomorphism. Suppose either $H$ of finite presentation over $S$, or $S$ locally noetherian and $H$ locally of finite type. Then:

a) If $G$ is reductive, i.e. is affine over $S$ with reductive fibers (cf. Exp. XIX), $u$ is an immersion, and a closed immersion if $H$ is separated.

b) If $G$ has affine, solvable, connected fibers, of zero unipotent rank, $u$ is an immersion, and a closed immersion if $H$ is separated.

Proof of 1.4 from 1.3.

Reduction to the noetherian case. If $S$ is locally noetherian, the reduction is immediate, the properties to prove being local on $S$. In the second case, $G$ and $H$ are of finite presentation over $S$. Replacing $S$ by a smaller open if necessary, we may assume $S$ affine. By Exp. VI_B § 10, there exist a noetherian scheme $S_0$, an $S_0$-group prescheme $G_0$ smooth over $S_0$ (with connected fibers in case a)), of finite type over $S_0$, an $S_0$-group prescheme $H_0$ of finite type (separated in case b)), and an $S_0$-group morphism

$$u_0:G_0⟶H_0,$$

such that $G$, $H$, $u$ are obtained from $G_0$, $H_0$, $u_0$ by a base extension $S\to S_0$. The fact that $u$ is a monomorphism translates to $Keru=$ unit group; we may therefore assume that $u_0$ is a monomorphism. Since $G$ has zero unipotent rank ${\rho }_u$ and possesses, locally for the fpqc topology, maximal tori, the abelian rank ${\rho }_{ab}$ of the fibers of $G$ is locally constant (Exp. XV 8.18). But ${\rho }_u$ and ${\rho }_{ab}$ are locally constructible functions (Exp. XV 6.3 bis). A standard argument (cf. EGA IV 8.3.4) shows that one may choose $S_0$ and $G_0$ so that the unipotent rank (resp. the abelian rank) of the fibers of $G_0$ is zero (resp. locally constant). But then $G_0$ possesses locally maximal tori for the fpqc topology (Exp. XV 8.18), and the functor ${\mathcal{T}\mathcal{M}}_{G_0}$ of maximal tori of $G_0$ is representable by an $S_0$-scheme $X_0$, of finite type over $S_0$ (Exp. XV 8.15). Let $T_0$ be the "universal" maximal torus of $(G_0{)}_{X_0}$, $X$ the $S$-prescheme $X_0{\times }_{S_0}S$, $T=T_0{\times }_{S_0}S$ the universal maximal torus for $G$. By hypothesis, in case b), the Weyl group relative to $T$ is representable and finite over $X$. These two properties are compatible with projective limits of preschemes (Exp. VI_B 10.1 iii) and EGA IV 8.10.5). One may therefore choose $S_0$ so that the Weyl group of $T_0$ is finite over $X_0$. Under these conditions it is clear that to prove 1.4 we may replace $S$, $G$, $u$, $H$ by $S_0$, $G_0$, $u_0$, $H_0$, hence assume $S$ noetherian.

Let us use the valuative criterion provided by 1.3. We are reduced to the case where $S$ is the spectrum of a discrete valuation ring and where $G_0$ possesses a Cartan subgroup $C$.

Proof of 1.4 a). We must show that the restriction of $u$ to $C$ is an immersion (1.3 a)), which reduces us to the case where $G=C$ has nilpotent connected fibers. Standard reductions (cf. Exp. VIII proof of 7.1) reduce us to the case where $H$ is flat over $S$, $u$ is an isomorphism on the generic fiber, and $u$ is an isomorphism of underlying spaces on the closed fiber. The group $H$ is then with connected fibers, hence separated over $S$ (Exp. VI_B 5.2). Granting for a moment the lemma:

Lemma 1.6. Let $S$ be a prescheme, $G$ an $S$-group prescheme smooth over $S$, with connected, nilpotent fibers, of zero unipotent rank. Then:

i) $G$ is commutative.

ii) For every integer $n>0$, ${}_{n}G$ is a group prescheme, flat, quasi-finite over $S$, finite over $S$ if and only if the abelian rank, or the reductive rank, of $G$ is locally constant on $S$.

iii) For every integer $q>0$ invertible on $S$, the family of subgroups ${}_{q^n}G$ is universally schematically dense in $G$ relative to $S$ (EGA IV 11.10.8).

Lemma 1.6 applies to the group $G$, and since $G$ possesses locally maximal tori, the reductive rank of the fibers of $G$ is locally constant; hence (1.6 ii)), for every integer $n>0$, ${}_{n}G$ is finite over $S$. The fact that $u$ is an immersion then follows from Exp. VIII 7.9.

Proof of 1.6. Let us first examine the case where $S$ is the spectrum of a field:

Lemma 1.7. Let $k$ be a field of characteristic $p$, $G$ an algebraic $k$-group, smooth, connected, nilpotent, of zero unipotent rank. Then $G$ is a commutative group, extension of an abelian variety $A$ by a torus $T$. For every integer $n>0$, ${}_{n}G$ is a finite group, étale if $(n,p)=1$, defined by a $k$-algebra of rank $n^{{\rho }_r+2{\rho }_{ab}}$. If $(q,p)=1$, the family of subgroups ${}_{q^n}G(n\in \mathbb{N})$ is schematically dense in $G$.

Proof of 1.7. Let $Z$ be the center of $G$. The group $G/Z$ is affine (Exp. XII 6.1), of zero unipotent rank, smooth and connected, hence is a torus (Bible 4 th. 4); but then $G$ is commutative (Exp. XII 6.4). If $T$ is the unique maximal torus of $G$ (Exp. XV 3.4), it follows at once from Chevalley's theorem (Sém. Bourbaki 1956/57, No. 145) that $G$ is an extension of an abelian variety $A$ by $T$. For every integer $n>0$, raising to the $n$th power is an epimorphism in $T$; one deduces an exact sequence

0 ⟶ _n T ⟶ _n G ⟶ _n A ⟶ 0.

A classical theorem of Weil (A. Weil: Variétés abéliennes et courbes algébriques, § IX th. 33 cor. 1) tells us that ${}_{n}A$ is a finite group defined by a $k$-algebra of rank $n^{2{\rho }_{ab}}$. Since ${}_{n}T$ is a finite group of rank $n^{{\rho }_r}$, one deduces the announced structure of ${}_{n}G$.

Now let $H$ be the smallest closed subscheme of $G$ majorizing ${}_{q^n}G$ for every $n$. It follows from the foregoing and from Exp. XV 4.6 that $H$ is a smooth, connected subgroup, hence of the same type as $G$. Raising to the $n$th power in $H$ is an epimorphism (since ${}_{q}H$ is finite), so that one has the exact sequence

0 ⟶ _q H ⟶ _q G ⟶ _q(G/H) ⟶ 0.

It follows that ${}_q(G/H)=0$, hence $G/H=0$. That is to say, the subgroups ${}_{q^n}G$ are schematically dense in $G$.

Continuation of the proof of 1.6 i). To show that $G$ is commutative, we reduce by the usual procedure to the case where $S$ is noetherian, then to the case where $S$ is the spectrum of a local ring with closed point $s$. The center of $G$ is representable by a closed group subscheme $Z$ of $G$ (Exp. XI 6.11). To show that $Z=G$, it suffices to show that $Z=G$ after reduction by every power of the maximal ideal of the ring of $S$ (for $Z$ will then be an open subgroup of $G$, hence equal to $G$ since $G$ has connected fibers). This reduces us to the case where $S$ is local artinian.

Let $q$ be an integer invertible on $S$. For every integer $n$, ${}_{q^n}{G}_s$ is a subgroup of multiplicative type of $G_s$, étale and central (1.7). Since $G$ is smooth, ${}_{q^n}(G_s)$ lifts to an $S$-group subscheme étale and central $M_n$ of $G$ (Exp. XV 1.2). The family of flat subgroups $M_n$, $n\in \mathbb{N}$, is then schematically dense in $G$ (1.7 and EGA IV 11.10.9). Since $Z$ is closed in $G$ and majorizes all the $M_n$, we have $Z=G$.

1.6 ii). To see that ${}_{n}G$ is flat and quasi-finite over $S$, it suffices to show that raising to the $n$th power in $G$ is a flat and quasi-finite morphism. Since $G$ is flat over $S$, we are reduced to the case where $S$ is the spectrum of a field (EGA IV 11.3.10), and we noted in the proof of 1.7 that raising to the $n$th power was an epimorphism. Moreover ${}_{n}G$ is separated over $S$, since $G$ is separated over $S$ (Exp. VI_B 5.2). For ${}_{n}G$ to be finite over $S$, it is necessary and sufficient that the fibers of ${}_{n}G$ be the spectra of finite algebras, of locally constant rank (one sees this immediately by reducing to the case where $S$ is the spectrum of a henselian local ring). But, by 1.7, this condition amounts to saying that the abelian rank, or the reductive rank, of the fibers of $G$ is locally constant.

1.6 iii). To see that the family of subgroups ${}_{q^n}G(n\in \mathbb{N})$ is universally schematically dense in $G$, we again reduce to the case $S$ noetherian. Taking 1.7 and 1.6 b) into account, it suffices to apply EGA IV 11.10.9.

Proof of 1.4 b). We have reduced to the case where $S$ is the spectrum of a discrete valuation ring and where $G_0$ possesses a Cartan subgroup $C$. Let $N=Nor{m}_{G}C=Nor{m}_{G}T$, where $T$ is the unique maximal torus of $C$ (Exp. XII 7.1 a) and b)). Since $H$ is separated over $S$, it follows from 1.6 ii) and from Exp. VIII 7.12 that the restriction of $u$ to $C$ is a closed immersion. On the other hand, to prove that $u$ is a closed immersion, it suffices to show that this is the case for $u|N$ (1.3 b)). Since by hypothesis $W=N/C$ is representable by a finite group scheme, this will follow from the following lemma, applied to the exact sequence

$$0⟶C⟶N⟶W⟶1$$

(note that a proper immersion is a closed immersion).

Lemma 1.8. Let $S$ be a prescheme, $G$ an $S$-group prescheme of finite presentation over $S$, extension of a group prescheme $G^{\prime \prime }$, proper and of finite presentation over $S$, by a group prescheme $G^{\prime }$, of finite presentation and flat over $S$:

$$1⟶G^{\prime }⟶G⟶G^{\prime \prime }⟶1.$$

Let on the other hand $H$ be an $S$-group prescheme of finite presentation over $S$ (or locally of finite presentation if $S$ is locally noetherian) and $u:G\to H$ an $S$-group morphism. Then if the restriction of $u$ to $G^{\prime }$ is proper, $u$ is proper.

Proof of 1.8. We reduce as usual to the case where $S$ is noetherian (Exp. VI_B § 10 and Exp. XV 6.2) and we may then apply the valuative criterion of properness (EGA II 7.3.8). We therefore suppose that $S$ is the spectrum of a discrete valuation ring $A$, with closed point $s$ and generic point $t$. Let $x\in G(t)$ and $h\in H(S)$ be such that $u_t(x)=h(t)$. We must show that $x$ comes from a unique element $x$ of $G(S)$. It even suffices to prove the existence and uniqueness of $x$ after a faithfully flat extension of the discrete valuation ring $A$. Now let $y$ be the projection of $x$ in $G^{\prime \prime }(t)$. Since $G^{\prime \prime }$ is proper over $S$, $y$ comes from a unique element $y$ of $G^{\prime \prime }(S)$. Let $X$ be the inverse image of $y$ in $G$. The prescheme $X$ is faithfully flat over $S$ (since $G^{\prime }$ is faithfully flat over $S$ as is the morphism $G\to G^{\prime \prime }$ (Exp. VI_B § 9)). Replacing if necessary the discrete valuation ring by a faithfully flat one, we may assume that $X$ has a section $\ell$ over $S$ (EGA IV 14.5.8). Replacing $x$ by $x{\ell }_t^{-1}$, we may assume that $x\in G^{\prime }(t)$. But then the existence and uniqueness of $x$ follow from the fact that $u|G^{\prime }$ is proper.

Proof of 1.5.

a) We shall see in Exp. XIX that if $G$ is reductive, $G$ possesses locally for the faithfully flat topology maximal tori, has a finite Weyl group, and has zero unipotent rank. Assertion a) follows, in view of 1.4.

b) If $G$ has affine fibers of zero unipotent rank, $G$ possesses maximal tori locally for the fpqc topology (Exp. XV 8.18). It then suffices to apply 1.4, taking into account the fact that $G$, having connected, smooth, affine, solvable fibers, has unit Weyl group (Bible 6 th. 1 cor. 3).

Proof of 1.3 a).

Since the morphism $u$ is already a monomorphism, in order to see that $u$ is an immersion, it suffices to show that $u$ is proper at the points of $u(G)$ (EGA IV 15.7.1); for that we may use the valuative criterion of local properness (EGA IV 15.7.5). We are thus reduced to the case where $S$ is the spectrum of a complete discrete valuation ring with algebraically closed residue field, with closed point $s$ and generic point $t$. Since $G$ is flat over $S$, standard reductions (cf. Exp. VIII proof of 7.1) allow us to reduce to the case where $H$ is flat over $S$, $u_t$ is an isomorphism, and $u_s$ is an isomorphism of underlying spaces. To say that $u$ is an immersion is then equivalent to saying that $u$ is an isomorphism.

By hypothesis, the group $G$ possesses locally Cartan subgroups for the fpqc topology, so we may speak of the open set $U$ of regular points of $G$ (Exp. XV 7.3 i) and ii)).

Lemma 1.9. With the preceding hypotheses, in order that $u$ be an immersion, it suffices that $u|U$ be an immersion.

Indeed, to say that $u|U$ is an immersion means that there exist an open $V$ of $H$ and a closed $F$ of $V$ such that $u|U$ factors through $F$ and induces an isomorphism of $U$ onto $F$. Since $H_t$, and hence also $V_t$, is irreducible, one necessarily has $V_t=F_t$. But then $F_t$ majorizes the schematic closure of $V_t$ in $V$, which is equal to $V$ since $H$ is flat over $S$. In short, $V=F$. It follows that $u|U$ is an open immersion, hence $u$ is an open immersion (VI_B 2.6).

It remains to show that $u|U$ is an immersion, and for that we apply the valuative criterion of local properness. Replacing $S$ by the spectrum of a faithfully flat discrete valuation ring, we must show that if $h$ is a section of $H$ over $S$ whose image $h(S)$ is contained in $u(U)$, then $h$ is the image of a section $g$ of $U$ over $S$. It suffices to show that $h$ is contained in the image of a Cartan subgroup $C$ of $G$. Indeed, by hypothesis $u|C$ is an immersion, so $h$ is the image of a section $g$ of $C$, which is necessarily a section of $U$.

Let $a\in U(s)$ (resp. $b\in U(t)$) be such that $u_s(a)=h(s)$ (resp. $u_t(b)=h(t)$). Since $u_t$ is an isomorphism, $h(t)$ is a regular point of $H_t$, so is contained in a unique Cartan subgroup $D_t$ of $H_t$ (Exp. XIII 3.2). Let $D$ be the schematic closure of $D_t$ in $H$.

Lemma 1.10. i) $D_s$ is a nilpotent algebraic group (Exp. VI_B § 8).

ii) $\dim D_s=\nu =$ nilpotent rank of $G=$ nilpotent rank of $(H_s{)}_{red}$.

iii) $h(s)\in D_s(\kappa (s))$.

i) Since $H$ is separated (Exp. VI_B 5.2), so is $D$. Moreover, $D$ is flat over $S$, and its generic fiber is nilpotent. The fact that $D_s$ is nilpotent then follows from Exp. VI_B 8.4.

ii) Since $(H_s{)}_{red}\simeq G_s$, these two groups have the same nilpotent rank. Since $G$ possesses, locally for the fpqc topology, Cartan subgroups, $G_s$ and $G_t$ have the same nilpotent rank $\nu$. Finally, since $D$ is flat over $S$, one has $\dim D_s=\dim D_t=\nu$ (Exp. VI_B § 4).

c) Since $h_t$ factors through $D_t$, $h$ evidently factors through $D$, whence iii).

This being said, the following lemma proves that $(D_s{)}_{red}$ is a Cartan subgroup of $(H_s{)}_{red}\cong G_s$:

Lemma 1.11. Let $G$ be a smooth and connected algebraic group defined over an algebraically closed field $k$, $D$ a smooth nilpotent algebraic subgroup containing a regular element $a$ of $G(k)$. Then $D^0$ is contained in a Cartan subgroup of $G$. If moreover $\dim D$ is equal to the nilpotent rank of $G$, then $D$ is a Cartan subgroup of $G$ (and hence is connected).

Let $Z$ be the center of $G$, $G^{\prime }=G/Z$ (which is affine, Exp. XII 6.1), $a^{\prime }$, $D^{\prime }$ the images of $a$, $D$ in $G^{\prime }$. It follows immediately from the correspondence between Cartan subgroups of $G$ and Cartan subgroups of $G^{\prime }$ (Exp. XII 6.6 e)) that $a^{\prime }$ is a regular element of $G^{\prime }$ and that it suffices to prove the lemma for the pair $D^{\prime },G^{\prime }$; this allows us to assume $G$ affine.

Let $s$ then be the semisimple component of $a$, which belongs to $D(k)$ (Bible 4 th. 3) and is regular (Bible 7 th. 2 cor. 1). By Bible 6 th. 2, $s$ centralizes $D^0$, hence $D^0$ is contained in the connected centralizer of $s$, which is a Cartan subgroup of $G$ (Bible 7 th. 2). If now $\dim D=$ nilpotent rank of $G$, $D^0$ is therefore a Cartan subgroup of $G$, equal to $Cent{r}_G(T)$, where $T$ is the unique maximal torus of $D^0$ (Exp. XII 6.6). But $D$ is nilpotent, hence centralizes $T$ (Bible 6 th. 2), hence $D=D^0$.

Let us now work with the group $G$. The group $C_t=u_t^{-1}(D_t)$ is the unique Cartan subgroup of $G_t$ containing $b$. Let $C$ be the schematic closure of $C_t$ in $G$.

By functoriality of the schematic closure, $u|C$ factors through $D$, so $u_s(C_s)\subset D_s$. Since $u_s$ is an isomorphism of $G_s$ onto $(H_s{)}_{red}$, since dim C_s = dim C_t = ν = dim D_s (1.10), and since $D_s$ is connected (1.11), $u_s$ furnishes an isomorphism of $(C_s{)}_{red}$ onto $(D_s{)}_{red}$, hence $(C_s{)}_{red}$ is a Cartan subgroup of $G_s$. The following lemma shows that $C$ is in fact a Cartan subgroup of $G$.

Lemma 1.12. Let $S$ be the spectrum of a discrete valuation ring, $G$ an $S$-group prescheme smooth, of finite type, $C$ a group sub-prescheme of $G$, flat over $S$, such that the generic fiber $C_t$ is a Cartan subgroup of $G_t$ and the geometric reduced closed fiber $(C_s{)}_{red}$ is a Cartan subgroup of $G_s$. Then $C$ is a Cartan subgroup of $G$.

We must show that $C$ is smooth over $S$, and it suffices to establish this point after a faithfully flat extension of the base. The hypotheses on $C$ imply that the nilpotent rank of the fibers of $G$ is constant, and we may therefore speak of the open set $U$ of regular points of $G$ (Exp. XV 7.3). Since $(C_s{)}_{red}$ is a Cartan subgroup of $G_s$, $C_s$ contains a regular point $a_s$ of $G_s$. But $C$ is flat over $S$, so (EGA IV 14.5.8), replacing the base by a faithfully flat extension if necessary, we may assume that $a_s$ is an element of $G(s)$ lifting to a section $a$ of $C(S)$. Since $U$ is open and contains $a(s)$, $U$ contains $a$. Let $C^{\prime }$ be the unique Cartan subgroup of $G$ containing $a$ (Exp. XV 7.3). One necessarily has $C_t=C_t^{\prime }$, and consequently $C^{\prime }$ and $C$ coincide with the connected component of the schematic closure of $C_t$ in $G$ (note that $C^{\prime }$ and $C$ are flat over $S$).

End of the proof of 1.3 a). By hypothesis, the restriction of $u$ to the Cartan subgroup $C$ of $G$ is an immersion. It is clear that $h(S)$ is contained in $u(C)=D$, hence $h$ is the image of a section $g$ of $C(S)$. QED.

Proof of 1.3 b).

We shall again use the valuative criterion of local properness (EGA IV 15.5) in the case of an immersion (resp. the valuative criterion of properness (EGA II 7.3.6) in the case of a closed immersion). This reduces us to the case where $S$ is the spectrum of a complete discrete valuation ring with algebraically closed residue field. Let $s$ be the closed point and $t$ the generic point of $S$. Replacing $H$ by the schematic closure in $H$ of $u_t(G_t)$, we may assume that $H$ is flat over $S$ and that $u_t$ is an isomorphism.

Let $G^0$ be the connected component of $G$. By 1.2 a) ³, $u|G^0$ is an immersion; denote by $H^0$ the group sub-prescheme of $H$ image of $G^0$. One has therefore $H_s^0=(H_s{)}_{red}^0$.

To verify the valuative criterion, we must show that every section $h$ of $H$ over $S$ such that $h(S)$ is contained in $u(G)$ in the case of an immersion (resp. every section $h$ of $H$ over $S$ in the case of a closed immersion) is the image of a section $g$ of $G$ over $S$.

Let $C$ be a Cartan subgroup of $G$, $D=u(C)$ the Cartan subgroup of $H^0$ image of $C$. The $S$-group $D^{\prime }=int(h)D$ has connected fibers, hence its underlying space is contained in that of $H^0$; moreover, being smooth over $S$, it is reduced, and consequently $D^{\prime }$ is a smooth group sub-prescheme of $H^0$. The fibers of $D^{\prime }$ are Cartan subgroups of the fibers of $H^0$, so $D^{\prime }$ is a Cartan subgroup of $H^0$, and $C^{\prime }=u^{-1}(D^{\prime })$ is a Cartan subgroup of $G$. But $Trans{p}_G^{str}(C,C^{\prime })$ is smooth and surjective over $S$ (Exp. XII 7.1 b)). Since $S$ is henselian with algebraically closed residue field, there exists $g\in G(S)$ such that $int(g)C=C^{\prime }$. Replacing $h$ by $u(g{)}^{-1}h$, we may assume that $h(t)$ normalizes $D_t$ (and that $h(s)$ is the image of an element of the normalizer of $C_s$ in $G_s$ in the case of an immersion). Let $N=Nor{m}_G(C)$. By hypothesis $u|N$ is an immersion (resp. a closed immersion), so $h$ is the image of a section of $N$ over $S$, which completes the proof.

2. A representability theorem for quotients

Let us "recall" the following result:

Theorem 2.1. Let $S$ be a prescheme, $X$ and $Y$ two $S$-preschemes, $f:X\to Y$ an $S$-morphism. Suppose we are in one of the following two cases:

a) The morphism $f$ is locally of finite presentation.

b) The prescheme $S$ is locally noetherian and $X$ is locally of finite type over $S$.

Then the following conditions are equivalent:

i) There exist an $S$-prescheme $X^{\prime }$ and a factorization of $f$:

f : X ──f′──→ X′ ──f″──→ Y,

where $f^{\prime }$ is a faithfully flat $S$-morphism, locally of finite presentation, and $f^{\prime \prime }$ is a monomorphism.

ii) The (first) projection

p₁ : X ×_Y X ⟶ X

is a flat morphism.

Moreover, if the preceding conditions are satisfied, $(X^{\prime },f^{\prime })$ is a quotient of $X$ by the equivalence relation defined by $f$ (for the fpqc topology), so that the factorization $f=f^{\prime \prime }\circ f^{\prime }$ of i) is unique up to isomorphism.

The proof of this delicate theorem will be found in EGA V; one may also consult the lecture by J.-P. Murre, Séminaire Bourbaki, May 1965, No. 294 th. 2 cor. 2, where the case $Y$ locally noetherian, $X$ of finite type over $Y$ is treated. We shall see that one may reduce to this case.

Let us first make some remarks:

a) i) ⇒ ii) is trivial. Indeed, the first projection

p′₁ : X ×_{X′} X ⟶ X

factors through $X{\times }_{Y}X$:

p′₁ : X ×_{X′} X ──u──→ X ×_Y X ──p₁──→ X.

The morphism $u$ is an isomorphism, since $f^{\prime \prime }$ is a monomorphism, and $p_1^{\prime }$ is flat, since $f^{\prime }$ is flat, so $p_1$ is flat.

b) The assertions of 2.1 are local on $Y$ (and so are local on $S$); they are also local on $X$, as follows easily from the fact that a flat morphism locally of finite presentation is open (EGA IV 11.3.1).

c) Under the hypotheses of 2.1 a), in view of the foregoing, we are reduced to the case where $X$ and $Y$ are affine and $f$ of finite presentation. Replacing $S$ by $Y$, we may assume $X$ and $Y$ of finite presentation over $S$. We then reduce to the case $S$ noetherian thanks to EGA IV 11.2.6.

d) Under the hypotheses of 2.1 b), we may assume $S$, $X$, $Y$ affine, $S$ noetherian and $X$ of finite type over $S$. Consider $Y$ as a filtered projective limit of affine schemes $Y_i$ of finite type over $S$. The schemes $X{\times }_{Y_i}X$ form a decreasing filtered family of closed subschemes of $X{\times }_{S}X$, whose projective limit is $X{\times }_{Y}X$. Since $X{\times }_{S}X$ is noetherian, one has $X{\times }_{Y_i}X=X{\times }_{Y}X$ for $i$ large enough, so that

f_i : X ──f──→ Y → Y_i

satisfies the hypotheses of 2.1 ii) if $f$ does. Since the equivalence relation defined by $f$ on $X$ coincides with that defined by $f_i$, it is clear that it suffices to prove ii) ⇒ i) for $f_i$, which reduces us to the case where $Y$ is of finite type over $S$.

Application to group preschemes

Theorem 2.2. Let $S$ be a prescheme, $G$ an $S$-group prescheme locally of finite presentation over $S$, acting on an $S$-prescheme $X$. Let $\xi \in X(S)$ be a section of $X$ over $S$ such that the stabilizer $H$ of $\xi$ in $G$ is an $S$-group sub-prescheme of $G$, flat over $S$. Then, if $X$ is locally of finite type over $S$, or if $S$ is locally noetherian, the homogeneous space quotient $G/H$ is representable by an $S$-prescheme, locally of finite presentation over $S$, and the $S$-morphism

f : G ⟶ X,    g ↦ g · ξ

factors as

       G
      ╱ ╲
     p   f
    ╱     ╲
   ↓       ↘
  G/H ──i──→ X,

where $p$ is the canonical projection, which is a faithfully flat morphism locally of finite presentation, and $i$ is a monomorphism. (For definiteness, we have assumed that $G$ acts on the left on $X$.)

Proof. The morphism $f$ makes $G$ into an $X$-prescheme. By definition of the stabilizer of $\xi$, the morphism

G ×_S H ⟶ G ×_X G,    (g, h) ↦ (g, gh)

is an isomorphism. Since $H$ is flat over $S$, $G{\times }_{S}H$ is flat over $G$, so the first projection $p_1:G{\times }_{S}G\to G$ is a flat morphism. Furthermore, if $X$ is locally of finite type over $S$, $f$ is locally of finite presentation (EGA IV 1.4.3 v)); otherwise $S$ is assumed locally noetherian. It then suffices to apply 2.1 to the morphism $f$.

It remains to see that $G/H$ is locally of finite presentation over $S$, but this follows immediately from Exp. V 9.1.

Corollary 2.3. Let $S$ be a prescheme, $G$ and $H$ two $S$-group preschemes, $u:G\to H$ an $S$-homomorphism. Suppose $G$ locally of finite presentation over $S$ and that either $H$ is locally of finite type over $S$, or $S$ is locally noetherian. Then, if $K=Ker(u)$ is flat over $S$, the quotient group $G/K$ is representable by an $S$-group prescheme locally of finite presentation over $S$, and $u$ factors as

       G ──u──→ H
       ╲       ↗
        p     i
         ↘   ↗
          G/K

where $p$ is the canonical projection and $i$ a monomorphism.

Proof. One applies 2.2 with $X=H$ and $\xi$ the unit section of $H$.

Corollary 2.4. Let $S$ be a prescheme, $G$ an $S$-group prescheme of finite presentation over $S$, $H$ an $S$-group prescheme smooth over $S$ with connected fibers (hence of finite presentation over $S$, by VI_B 5.5), $i:H\to G$ a monomorphism of $S$-groups, so that $H$ is a subgroup of $G$.

Suppose that $N=Nor{m}_G(H)$ (which is representable by a closed group sub-prescheme of $G$, of finite presentation over $S$ (Exp. XI 6.11)) is flat over $S$. Then $G/N$ is representable by an $S$-prescheme of finite presentation over $S$ and quasi-projective over $S$.

All the assertions to be proved, except the last, are local on $S$. To establish them, we may therefore assume $S$ quasi-compact and the relative dimension of $H$ over $S$ constant and equal to $r$. Let us proceed as in XV § 5. For every integer $n>0$, let $G_{(n)}$ (resp. $H_{(n)}$) be the $n$th normal invariant of the unit section of $G$ (resp. of $H$) (EGA IV 16). The sheaf of ${\mathcal{O}}_S$-modules $H_{(n)}$ is a quotient of $G_{(n)}$, and $H$ being smooth over $S$ of dimension $r$, $H_{(n)}$ canonically defines an element ${\xi }_n$ of $X_n=Gras{s}_{\phi (n,r)}{G}_{(n)}$ (EGA I 2nd ed. § 9) for a suitable integer $\phi (n,r)$. On the other hand, $G$ acts naturally on $G_{(n)}$ (and hence also on $X_n$) via the representation

G ⟶ Aut_{S-gr}(G),    g ↦ int(g).

Since $S$ is quasi-compact, there exists an integer $m$ such that for $n⩾m$ one has

N = Norm_G(H) = Norm_G H_{(n)}    (Exp. XI 6.11 b)).

That is to say, $N$ is the stabilizer of ${\xi }_n$ for $n$ large. The representability of $G/N$ therefore follows from 2.2. The fact that $G/N$ is of finite presentation over $S$ is a consequence of Exp. V 9.1. It remains to see that $G/N$ is quasi-projective over $S$, and for that we shall exhibit a canonical invertible sheaf on $G/N$, $S$-ample. Consider the functor $F:(Sch/S{)}^{\circ }\to Ens$ such that for every $S$-prescheme $T$ one has

F(T) =  set of T-subgroups H of G_T, representable, smooth over T,
        with connected fibers.

Such a group $H$ is of finite presentation over $S$ (Exp. VI_B 5.5) and $H\to G_T$ is a quasi-affine morphism (EGA IV 8.11.2). By effective descent of quasi-affine morphisms (SGA 1, VIII 7.9), one deduces that $F$ is a sheaf for the fpqc topology. Since $G/N$ is the sheaf associated to a subfunctor of $F$, one sees that there is a canonical monomorphism $G/N\to F$. There exists therefore a subgroup $H_0$ of $G{\times }_S(G/N)$, representable, smooth over $G/N$, with connected fibers, "universal" for the functor $G/N$. I claim that the invertible sheaf $L=(det(Lie{H}_0){)}^{-1}$ is $S$-ample. In this form, the assertion becomes local on $S$, and the proof is analogous to the one given in (Exp. XV 5.8).

The following corollary was announced in Exp. XIV 4.8 bis.

Corollary 2.5. Let $S$ be a prescheme, $G$ an $S$-group prescheme smooth and of finite presentation over $S$, with connected fibers, $P$ a parabolic subgroup of $G$ (Exp. XIV 4.8 bis). Then $G/P$ is representable by an $S$-prescheme smooth and projective over $S$.

It only remains to show (loc. cit.) that $G/P$ is representable by an $S$-prescheme quasi-projective, of finite presentation, which follows from 2.4, given that $P=Nor{m}_G(P)$ (loc. cit.) is therefore flat over $S$.

3. Groups with flat center

Proposition 3.1. Let $S$ be a prescheme, $G$ an $S$-group prescheme smooth and of finite presentation over $S$, with connected fibers. Suppose that the center $Z$ of $G$ (which is representable by Exp. XI 6.11) is flat over $S$. For every integer $n>0$, let $G_{(n)}$ denote the locally free ${\mathcal{O}}_S$-module equal to the $n$th normal invariant of $G$ along the unit section, and let ${\rho }_n$ be the natural "adjoint representation" of $G$ in $G{L}_S(G_{(n)})$. Then:

a) The quotient $G^{\prime }=G/Z$ is representable by an $S$-group prescheme, smooth, of finite presentation over $S$, quasi-affine, with affine connected fibers.

b) The representation ${\rho }_n$ factors through $G^{\prime }$:

        G ──ρ_n──→ GL_S(G_{(n)})
        ╲          ↗
         ↘        ↗ i_n
          G/Z ──

If $S$ is quasi-compact, $i_n$ is a monomorphism for $n$ large enough.

c) The functor ${\mathcal{T}}_{G^{\prime }}$ of subtori of $G^{\prime }$ is representable by an $S$-prescheme, smooth over $S$, which is a sum of a family of preschemes affine over $S$ if $S$ is quasi-compact.

Proof. The assertions of a) are local on $S$, which allows us to assume $S$ quasi-compact. By Exp. XI 6.11 b), $Z$ is equal to $Ker{\rho }_n$ for $n$ large enough, so this kernel is flat over $S$. The fact that $G/Z$ is representable and that $i_n$ is a monomorphism therefore follows from 2.3. Since $G$ is smooth and of finite presentation over $S$, the same is true of $G^{\prime }$ (Exp. VI_B § 9). The group $G{L}_S(G_{(n)})$ is affine over $S$, and a monomorphism of finite presentation is quasi-affine (EGA IV 8.11.2), hence $G^{\prime }$ is quasi-affine over $S$ and has affine fibers. Since $G^{\prime }$ is smooth over $S$, the functor ${\mathcal{T}}_{G^{\prime }}$ is formally smooth (Exp. XI 2.1 bis). Taking into account Exp. XI 4.6 and 4.3, the assertions of 3.1 c) will therefore follow from the next lemma:

Lemma 3.2. Let $S$ be a prescheme, $G$ and $H$ two $S$-group preschemes of finite presentation over $S$, $u:G\to H$ an $S$-group monomorphism, ${\mathcal{T}}_G$ (resp. ${\mathcal{T}}_H$) the functors of subtori of $G$ (resp. of $H$) (cf. Exp. XV § 8). Then the map $T\mapsto u(T)$ defines a monomorphism (XV 8.3 c))

$$\tilde{u}:{\mathcal{T}}_G⟶{\mathcal{T}}_H$$

which is representable by a closed immersion of finite presentation.

By the usual procedure, we are reduced to the following problem: let $T$ be a subtorus of $H$, $T^{\prime }=G{\times }_{T}H$ its inverse image in $G$, $u_T:T^{\prime }\to T$ the $S$-group monomorphism deduced from $u$. One must show that the $S$-functor ${\prod }_{T/S}(T^{\prime }/T)$ is representable by a closed sub-prescheme of $S$, of finite presentation. But $T$, being a torus, is smooth over $S$ with connected fibers, and it suffices to apply Exp. XI 6.10.

Theorem 3.3. Let $S$ be a prescheme, $G$ an $S$-group prescheme smooth and of finite presentation over $S$, with connected fibers, $Z$ the center of $G$. Suppose that $Z$ is flat over $S$ and that the unipotent rank (Exp. XV 6.1 ter) of $G$ is equal to that of $Z$. Then:

a) The group $G^{\prime }=G/Z$ is representable, and if $S$ is quasi-compact, the canonical morphism $i_n:G^{\prime }\to G{L}_S(G_{(n)})$ (cf. 3.1 b)) is an immersion for $n$ large.

b) The group $G^{\prime }$ is quasi-affine over $S$, with affine fibers, the center of $G^{\prime }$ is the unit group, and $G^{\prime }$ has zero unipotent rank.

c) The group $G^{\prime }$ possesses locally for the étale topology maximal tori, and these are also Cartan subgroups of $G^{\prime }$. The functor ${\mathcal{T}\mathcal{M}}_{G^{\prime }}$ of maximal tori of $G^{\prime }$ (Exp. XV § 8) is representable by an $S$-prescheme smooth and affine over $S$.

d) The group $G$ possesses locally for the étale topology Cartan subgroups, and the functor ${\mathcal{C}}_G$ of Cartan subgroups of $G$ is representable by an $S$-prescheme, smooth and affine over $S$.

Proof. By 3.1, the group $G^{\prime }$ is representable by an $S$-prescheme, smooth and quasi-affine over $S$, with affine fibers. In view of the correspondence between Cartan subgroups of the fibers of $G$ and of the fibers of $G^{\prime }$ (Exp. XII 6.6 e)), the hypothesis on the unipotent rank of $Z$ implies that $G^{\prime }$ has zero unipotent rank. Using now Exp. XV 8.18, one sees that $G^{\prime }$ possesses locally for the étale topology maximal tori. The fact that $i_n$ is an immersion for $n$ large then follows from 3.1 b) and 1.4 a). This completes the proof of a).

Let us now show c). Since $G^{\prime }$ has zero unipotent rank, it is clear that every maximal torus of $G^{\prime }$ is also a Cartan subgroup of $G^{\prime }$. The functor of maximal tori of $G^{\prime }$ is representable by a sub-prescheme both open and closed of the functor of subtori of $G^{\prime }$; moreover it is of finite type over $S$ (Exp. XV 8.15); it follows from 3.1 c) that this functor is representable by an $S$-prescheme, smooth and affine over $S$.

Since $G^{\prime }$ possesses, locally for the étale topology, Cartan subgroups, the same is true of $G$, and the functor of Cartan subgroups of $G$ is canonically isomorphic to that of $G^{\prime }$ (Exp. XV 7.3 iv)), so c) ⇒ d).

It remains to show b), and more precisely, it remains to prove that the center $Z^{\prime }$ of $G^{\prime }$ is the unit group. Since $Z^{\prime }$ is representable (Exp. XI 6.11) and of finite presentation over $S$, it suffices to show that the fibers of $Z^{\prime }$ are reduced to the unit group (Exp. VI_B 2.9), which reduces us to the case where $S$ is the spectrum of an algebraically closed field. The center $Z^{\prime }$ is evidently contained in every Cartan subgroup of $G^{\prime }$, hence in every maximal torus of $G^{\prime }$ by c), and is therefore of multiplicative type. Moreover, we shall see in Exp. XVII that if $G$ is a connected algebraic group with center $Z$, the center $Z^{\prime }$ of $G/Z$ is unipotent. In the present case, $Z^{\prime }$ being both of multiplicative type and unipotent, is reduced to the unit group (cf. Exp. XVII).

Examples of groups whose center is flat

Proposition 3.4. Let $S$ be a prescheme, $G$ an $S$-group prescheme of finite presentation over $S$, smooth, with connected fibers, $Z$ the center of $G$. Then $Z$ is flat over $S$ in the following two cases:

a) The unipotent rank ${\rho }_u$ (Exp. XV 6.1 ter) of the fibers of $G$ is zero.

b) i) $S$ is reduced.

ii) The dimension of the fibers of $Z$ is a locally constant function on $S$.

iii) The unipotent rank of $G$ is equal to the unipotent rank of $Z$.

Proof of 3.4 a). We shall prove at the same time:

Proposition 3.5. Under the hypotheses of 3.4 a), suppose moreover that $G$ possesses a maximal torus $T$ and let $C=Cent{r}_G(T)$ be the Cartan subgroup of $G$ associated to $T$ (Exp. XII 7.1). Then:

(i) $Z\cap T$ is a subgroup of multiplicative type of $G$.

(ii) $C$ is commutative and equal to $T\cdot Z$.

(iii) If the quotients $Z/Z\cap T$ and $C/T$ are representable, they are representable by abelian preschemes (i.e. $S$-group preschemes, smooth over $S$, whose fibers are abelian varieties) and the canonical monomorphism $Z/Z\cap T\to C/T$ is an isomorphism.

Using the general properties of passage to the limit proved in Exp. VI_B § 10 and Exp. XV 6.2, 6.3, and 6.3 bis, we reduce as usual to the case where $S$ is noetherian (note that the assertions of 3.4 and 3.5 are local on $S$). We noted in 3.1 that the hypotheses on $G$ imply that $Z$ is representable. To show that $Z$ is flat over $S$, we reduce by EGA 0_III 10.2.6 to the case where $S$ is local artinian. But then, $G$ being smooth, $G$ possesses locally for the étale topology maximal tori (Exp. XV 8.17). Replacing $S$ by a finite flat extension if necessary (which is allowable for proving that $Z$ is flat), we may assume that $G$ possesses a maximal torus $T$.

Let us show, in the same way, that to prove 3.5 we may reduce to the case $S$ artinian.

i) Since $Z$ is closed in $G$ (Exp. XI 6.11), $Z\cap T$ is a closed group subscheme of $T$. It follows from Exp. X 4.8 b) that $Z\cap T$ is of multiplicative type if and only if it is flat over $S$. As before, it suffices to establish that $Z\cap T$ is flat when $S$ is artinian.

ii) Since $G$ has zero unipotent rank, every Cartan subgroup $C$ of $G$ satisfies the hypotheses of Lemma 1.6, hence is commutative. The fact that $C=T\cdot Z$ will follow from iii).

iii) If $C/T$ is representable, $C/T$ is smooth over $S$ (Exp. VI_B 9) and its fibers are abelian varieties (1.7), so $C/T$ is an abelian prescheme. To show that the canonical monomorphism

i : Z/Z ∩ T ⟶ C/T

is an isomorphism, it suffices to verify this when $S$ is local artinian. Indeed, one will then deduce in succession that $Z/Z\cap T$ is flat over $S$ (EGA 0_III 10.2.6), then that $i$ is flat (EGA IV 11.3.10), then that $i$ is an open immersion (EGA IV 17.9.1), then that $i$ is an isomorphism (since $C/T$ has connected fibers).

We henceforth assume that $S$ is local artinian with closed point $s$ and that $G$ possesses a maximal torus $T$. Let $C=Cent{r}_G(T)$. The group $C$ is a Cartan subgroup of $G$ (Exp. XII 7.1) and majorizes $Z$.

The algebraic group $M_s=Z_s\cap T_s$ is a subgroup of multiplicative type of $G_s$ which is central. Since $T$ is smooth over $S$, $M_s$ lifts to a group subscheme $M$ of $T$, of multiplicative type (Exp. IX 3.6 bis) and contained in the center of $G$ (Exp. IX 3.9), hence contained in $Z\cap T$. Since $S$ is artinian and $M$ and $T$ are flat over $S$, the quotient groups $Z\cap T/M$, $Z^{\prime }=Z/M$, and $A=C/T$ are representable (Exp. VI_A §§ 4 and 5). By construction, $(Z\cap T/M{)}_s$ is the unit group, so $Z\cap T=M$ (Exp. VI_B 2.9); a fortiori it is flat over $S$, which proves 3.5 i).

By passage to the quotient, one has a canonical monomorphism $i:Z^{\prime }\to A$, which is therefore a closed immersion (Exp. VI_B 1.4.2). We have already noted that $A$ is an abelian scheme. It remains to show that $i$ is an isomorphism. This will prove 3.5 iii) and will imply that $Z^{\prime }$ is flat over $S$, hence that $Z$ is flat over $S$ (as an extension of $Z^{\prime }$ by the flat group $M$ (Exp. VI_B § 9)). Since $Z_s$ has the same abelian rank as $G_s$ (Exp. XII 6.1), $i_s$ is an epimorphism, hence an isomorphism. Let $q$ be an integer > 0, invertible on $S$. By the density theorem of 1.6 iii), to see that $i(Z^{\prime })=A$, it suffices to show that for every integer $n$ equal to a power of $q$, $i(Z^{\prime })$ majorizes ${}_{n}A$. Now let $M_s^0$ be the connected component of $M_s$. It is immediate by duality that raising to the $q$-th power in $M_s^0$ is an epimorphism. One deduces immediately that if $m_0$ is the exponent of $q$ in the factorization into prime factors of $card(M_s/M_s^0)$, the image of ${}_{n{m}_0}{Z}_s$ in $A_s\cong Z_s/M_s$ majorizes ${}_n{A}_s$. There exists therefore a subgroup of multiplicative type $M_s(n)$ of $Z_s$ whose image in $A_s$ is ${}_n{A}_s$. As before, one sees that $M_s(n)$ lifts to a subgroup of $G$, central and of multiplicative type, $M(n)$. The image of $M(n)$ in $A$ is a subgroup of multiplicative type (Exp. IX 6.8), hence necessarily equal to ${}_{n}A$, since it is so on the reduced fiber (Exp. IX 5.1 bis). This completes the proof of 3.4 a) and of 3.5.

Proof of 3.4 b). The assertion to be proved is local on $S$; we may therefore assume $S$ affine with ring $A$. Considering $A$ as an inductive limit of its sub-$\mathbb{Z}$-algebras of finite type, we reduce as above to the case where $S$ is noetherian reduced.

To show that $Z$ is flat, we have at our disposal a valuative criterion of flatness (EGA IV 11.8.1), which allows us to reduce to the case where $S$ is the spectrum of a complete discrete valuation ring with algebraically closed residue field, with generic point $t$ and closed point $s$. Let $Z^{\prime }$ be the schematic closure in $Z$ of $Z_t$. We must show that $Z^{\prime }=Z$, and it suffices even to show that $Z_s=Z_s^{\prime }$ (Exp. VI_B 2.6). By (Exp. VI_B § 4), one has the inequalities

dim Z_t = dim Z′_t = dim Z′_s ⩽ dim Z_s.

But by hypothesis $\dim Z_t=\dim Z_s$, hence $\dim Z_s^{\prime }=\dim Z_s$ and consequently $Z_s^{\prime }$ majorizes $(Z_s{)}_{red}^0$. It follows that $G_s^{\prime }=G_s/Z_s^{\prime }$ is affine (Exp. XII 6.1). In view of the correspondence between Cartan subgroups of $G$ and of $G^{\prime }$ (Exp. XII 6.6 e)), hypothesis 3.4 iii) implies that the unipotent rank of $G_s^{\prime }$ is zero, and consequently its Cartan subgroups are also its maximal tori. The image $Z_s^{\prime \prime }$ of $Z_s$ in $G_s^{\prime }$ is a central algebraic subgroup of $G_s^{\prime }$, hence contained in every Cartan subgroup of $G_s^{\prime }$, that is, in every maximal torus; $Z_s^{\prime \prime }$ is therefore a subgroup of multiplicative type of $G_s^{\prime }$. Under these conditions, we shall show in (Exp. XVII § 7) that there exists a finite subgroup of multiplicative type $M_s$ of $Z_s$ whose image in $Z_s/Z_s^{\prime }$ is $Z_s^{\prime \prime }$ (we used this fact in the proof of 3.4 a) when $Z_s^{\prime \prime }$ is étale). Since $G$ is smooth over $S$ and $S$ is the spectrum of a complete local ring, $M_s$ lifts to an $S$-subgroup of multiplicative type $M$ of $G$ (Exp. IX 3.6 bis and Exp. XV 1.6 b)) which is central (Exp. IX 5.6 a)). So $M_t$ is contained in $Z_t^{\prime }=Z_t$, and since $M$ is flat, $M$ is contained in $Z^{\prime }$. The group $M_s$ is therefore contained in $Z_s^{\prime }$, but this implies that $Z_s^{\prime \prime }$ is the unit group, i.e. $Z_s=Z_s^{\prime }$. This completes the proof of 3.4 b).

Example of a smooth group prescheme with connected fibers whose center is not flat

Let $S$ be the spectrum of a discrete valuation ring $A$, $\pi$ a uniformizer of $A$, $t$ the generic point of $S$, $s$ the closed point. Let $G$ be the smooth and affine $S$-group with ring $B=A[T,T^{-1},U]/F$ with $F=1-T+\pi U$, the composition law being defined by

((t, u), (t′, u′)) ↦ (tt′, πuu′ + u + u′).

The generic fiber $G_t$ is therefore isomorphic to the multiplicative group $({\mathbb{G}}_m{)}_t$, while the closed fiber $G_s$ is isomorphic to the additive group $({\mathbb{G}}_a{)}_s$. The function $T$ is invertible in $\Gamma (G,{\mathcal{O}}_G)$ and defines an $S$-group morphism $\phi :G\to ({\mathbb{G}}_m{)}_S$ which is an isomorphism on the generic fiber and the zero morphism on the closed fiber. The datum of $\phi$ allows one to construct the semidirect product group $H=G\cdot ({\mathbb{G}}_a{)}_S$ (note that $({\mathbb{G}}_m{)}_S$ acts on $({\mathbb{G}}_a{)}_S$). The center $Z$ of $H$ is the unit group on the generic fiber and is equal to $H_s$ on the closed fiber, hence is not flat over $S$.

4. Groups with affine fibers, of zero unipotent rank

Theorem 4.1. Let $S$ be a prescheme, $G$ an $S$-group prescheme smooth over $S$, of finite presentation, with affine, connected fibers, of zero unipotent rank (Exp. XV 6.1 ter). Then:

a) The center $Z$ of $G$ is a group of multiplicative type (and is therefore a reductive center of $G$ (Exp. XII 4.1)).

b) $G$ satisfies conditions a) to d) of 3.3.

c) $G$ possesses locally for the étale topology maximal tori, and these are also Cartan subgroups of $G$. The functor ${\mathcal{T}\mathcal{M}}_G$ of maximal tori of $G$ is representable by an $S$-prescheme smooth and affine over $S$.

d) $G$ is quasi-affine over $S$. Moreover, if $T$ is a maximal torus of $G$, $N=Nor{m}_{G}T$ its normalizer in $G$, $W=N/T$ the Weyl group relative to $T$, the following conditions are equivalent:

(i) $G$ is affine over $S$.

(ii) $G^{\prime }=G/Z$ is affine over $S$.

(iii) $N$ is affine over $S$.

(iv) $W$ is affine over $S$.

These conditions are always realized if $S$ is locally noetherian of dimension $⩽1$.

Proof. Since $G$ has affine fibers (and so abelian rank zero) and zero unipotent rank, $G$ possesses locally for the étale topology maximal tori (Exp. XV 8.18), and every maximal torus of $G$ is evidently a Cartan subgroup of $G$, which proves the first part of c). To see that $Z$ is of multiplicative type, we may assume, by what precedes, that $G$ possesses a maximal torus $T$. Since $T$ is also a Cartan subgroup, $T$ majorizes $Z$ (since $T=Cent{r}_G(T)$ by Exp. XII 7.1), and $Z=Z\cap T$ is of multiplicative type by 3.5 i). This proves a). Assertion b) is clear, given a). On the other hand the functor ${\mathcal{T}\mathcal{M}}_G$ is isomorphic to the functor of Cartan subgroups of $G$ (Exp. XII 7.1), hence is smooth and affine over $S$ (3.3 c)), which completes the proof of c). It remains to prove d).

Proof of d). Since $G^{\prime }$ is quasi-affine (3.5 b)) and $Z$ is of multiplicative type, hence affine over $S$, $G$ is quasi-affine (Exp. VI_B § 9).

i) ⇔ ii). If $G$ is affine, $G^{\prime }$ is affine by Exp. IX 2.3. If $G^{\prime }$ is affine, $G$ is affine as an extension of an affine group by an affine group (Exp. VI_B § 9).

i) ⇔ iii). If $G$ is affine, $N$ is affine since closed in $G$ (Exp. XI 6.11 a)). Moreover $G/N$ is isomorphic to the functor ${\mathcal{T}\mathcal{M}}_G$ (conjugacy of maximal tori, cf. Exp. XII 7.1 b)), so is affine over $S$ by c). Hence if $N$ is affine over $S$, $G$ is affine, as an "extension" of an affine homogeneous space by an affine group (Exp. VI_B § 9).

iii) ⇔ iv). If $N$ is affine, $W=N/T$ is affine (Exp. IX 2.3) and conversely by Exp. VI_B § 9.

Moreover, $N/T$ is representable by an $S$-group prescheme, étale, separated over $S$, of finite type (Exp. XV 7.1 iv)), hence quasi-finite over $S$. The last assertion of d) will therefore follow from the following lemma, applied with $X=W$ and $Y=S$:

Lemma 4.2. Let $S$ be a locally noetherian prescheme of dimension 1, $X$ and $Y$ two $S$-preschemes locally of finite type over $S$, $u:X\to Y$ an $S$-morphism quasi-finite and separated. Suppose that for every point $s$ of $S$ the morphism $u_s:X_s\to Y_s$ deduced from $u$ by the base change $\mathrm{Spec}\kappa (s)\to S$ is finite. Then $u$ is an affine morphism.

The assertion to be proved is local on $Y$, which allows us to assume $Y$ (and hence also $X$) of finite type over $S$. By EGA IV 8, one sees that it suffices to prove 4.2 when $S$ is the spectrum of a local ring $A$. By fpqc descent we may assume $A$ complete, then $A$ reduced (EGA II 1.6.4), then $A$ normal (Nagata's theorem (EGA 0_IV 22) and Chevalley's theorem (EGA II 6.7.1)). If $A$ is a field, $u$ is finite by hypothesis, hence affine. Otherwise $A$ is a discrete valuation ring; let $s$ be the closed point of $S$, $t$ the generic point, $\pi$ a uniformizer of $A$. Let $y$ be a point of $Y_s$. Applying EGA IV 8 once more, we may replace $Y$ by the spectrum $Y^{\prime }$ of ${\mathcal{O}}_{Y,y}$ and $X$ by $X^{\prime }=X{\times }_Y{Y}^{\prime }$; we may even assume ${\mathcal{O}}_{Y,y}$ complete. Since $u$ is quasi-finite and separated, by EGA II 6.2.6 $X^{\prime }$ is a sum of two schemes $X_1$ and $X_2$ with $X_1$ finite over $Y^{\prime }$ (hence affine) and $X_2$ such that $u(X_2)$ does not contain the closed point $y$ of $Y^{\prime }$. I claim that under these conditions $X_2$ does not meet the closed fiber $X_s^{\prime }$. Indeed, by hypothesis $X_s^{\prime }\to Y_s^{\prime }$ is finite, hence the restriction of this morphism to the closed subset $(X_2{)}_s$ is finite and its image in $(Y^{\prime }{)}_s$ is a closed subset. Since this image does not contain the closed point of the local scheme $(Y^{\prime }{)}_s$, $(X_2{)}_s$ is empty. Since $u_t$ is finite, the restriction to $(X_2{)}_t=X_2$ of the morphism $X_t^{\prime }\to Y_t^{\prime }$ is finite. Moreover, one has $Y_t^{\prime }=Y_{\pi }^{\prime }$, so the open immersion $Y_t^{\prime }\to Y^{\prime }$ is affine. In short, the composed morphism $X_2\to Y_t^{\prime }\to Y^{\prime }$ is affine, and it follows that the morphism $X^{\prime }=X_1\bigsqcup X_2\to Y^{\prime }$ is affine.

Corollary 4.3. Let $S$ be a prescheme, $G$ an $S$-group prescheme smooth and of finite presentation over $S$, with affine, solvable, connected fibers, of zero unipotent rank. Then $G$ is affine over $S$. If moreover the center of $G$ is the unit group and if $S$ is quasi-compact, the canonical morphism $i_n:G\to G{L}_S(G_{(n)})$ (cf. 3.1 b)) is a closed immersion for $n$ large enough.

To prove that $G$ is affine over $S$, we may assume that $G$ possesses a maximal torus $T$ (4.1 c)). Since $G$ has solvable fibers, the Weyl group relative to $T$ is the unit group (Bible 6 th. 1 cor. 3) and condition 4.1 d) iv) is satisfied. If the center of $G$ is the unit group, $i_n$ is a monomorphism for $n$ large enough (3.1), hence is a closed immersion (1.5 b)).

5. Application to reductive and semisimple groups

Definition 5.1. An $S$-group prescheme $G$ is said to be reductive (resp. semisimple) if $G$ is smooth and affine over $S$, with reductive (resp. semisimple) fibers.

5.1.1. Reductive groups will be systematically studied from Exp. XIX onwards. In this section, we shall need the following properties, which will be proved in Exp. XIX (without using the developments of the present Exposé).

a) Let $S$ be a prescheme, $G$ an $S$-group prescheme smooth and affine over $S$, with connected fibers, $s$ a point of $S$ such that $G_s$ is reductive (resp. semisimple). Then there exists a neighborhood $U$ of $s$ such that $G|U$ is reductive (resp. semisimple).

b) If $G$ is reductive, $G$ possesses locally for the étale topology maximal tori, and if $T$ is a maximal torus of $G$, the Weyl group $W=Nor{m}_G(T)/T$ is finite over $S$.

c) A reductive group has zero unipotent rank.

This being admitted, we propose to improve assertion a) above.

Theorem 5.2. Let $S$ be a prescheme, $G$ an $S$-group prescheme smooth and of finite presentation over $S$, with connected fibers, $s$ a point of $S$. Then:

(i) If the fibers of $G$ are affine and $G_s$ is reductive, there exists an open neighborhood $U$ of $s$ such that $G|U$ is reductive.

(ii) If $G_s$ is semisimple, there exists an open neighborhood $U$ of $s$ such that $G|U$ is semisimple.

Using Exp. XV 6.2 i) and Exp. VI_B § 10, one reduces to the case where $S$ is noetherian.

In case (ii), consider the center $Z$ of $G$, which is representable (Exp. XI 6.11 a)). Since $G_s$ is semisimple, it is well known that $Z_s$ is finite. Consequently (Exp. VI_B § 4), there exists a neighborhood $U$ of $s$ such that $Z$ is quasi-finite over $U$. For every point $t$ of $U$, $G_t$ is then affine (Exp. XII 6.1). Replacing $S$ by a smaller open if necessary, we may therefore assume that the fibers of $G$ are affine, both in case (ii) and in case (i).

To prove 5.2 it suffices, by a), to prove that $G$ is affine over $S$. Since $G$ has affine fibers, we know that the functor of subtori of $G$ is representable by a smooth $S$-prescheme (Exp. XV 8.11 and 8.9). Replacing $S$ by an étale extension covering $s$ (which is allowable), we may therefore assume that there exists a subtorus $T$ of $G$ such that $T_s$ is a maximal torus of $G_s$. But then $C=Cent{r}_G(T)$ is a smooth subgroup of $G$, with connected fibers, majorizing $T$, such that $C_s=T_s$ (since $C_s$ is a Cartan subgroup of $G_s$ and $G_s$ has zero unipotent rank by c)). It follows that $C=T$, hence $T$ is a Cartan subgroup and a maximal torus of $G$; a fortiori, the unipotent rank of $G$ is zero and one may apply 4.1.

By 4.1 d), it suffices to show that the Weyl group $W$ relative to $T$ is affine over a neighborhood $U$ of $s$. In fact we shall see that $W$ is even finite over a neighborhood of $s$. Since $W$ is quasi-finite over $S$ (Exp. XV 7.1 iv)), to say that $W$ is finite over a neighborhood $U$ of $s$ is equivalent to saying that the morphism $W\to S$ is proper at $s$ (EGA IV 15.7 and EGA III 4.4.2); to establish this we have at our disposal the valuative criterion of local properness (EGA IV 15.7), which reduces us to the case where $S$ is the spectrum of a discrete valuation ring, with closed point $s$. But then, by the last assertion of 4.1 d), $G$ is affine over $S$, and we may apply property a) recalled above to conclude that $G$ is reductive (resp. semisimple). Using now property b), we conclude that $W$ is indeed finite over $S$, which completes the proof of 5.2.

6. Applications: extension of certain rigidity properties of tori to groups of zero unipotent rank

Proposition 6.1. Let $S$ be a prescheme, $G$ an $S$-group prescheme smooth over $S$, of finite presentation, with connected affine fibers, and whose center is of multiplicative type (for example $G$ of zero unipotent rank, cf. 4.1 a)). Then every closed normal subgroup $K$ of $G$, of finite presentation over $S$, quasi-finite over $S$, is finite over $S$.

Indeed, for every geometric point $s$ above $S$, $(K_s{)}_{red}$ is a finite étale subgroup of $G_s$, normal hence central (since $G_s$ is connected). Consequently $K_0=Z\cap K$ (where $Z$ denotes the center of $G$) has the same underlying space as $K$, and it suffices to show that $K_0$ is finite over $S$. Now $K_0$ is a closed, quasi-finite subgroup of the multiplicative-type group $Z$, hence is finite, as one sees immediately (locally on $S$, $K_0$ will be majorized by ${}_{n}Z$ for a suitable integer $n$, and ${}_{n}Z$ is finite over $S$).

Corollary 6.2. Let $S$ and $G$ be as above, $K$ an $S$-group sub-prescheme of $G$, of finite presentation over $S$, normal and closed in $G$, $s$ a point of $S$. If $K_s$ is finite (resp. is the unit group), there exists an open neighborhood $U$ of $s$ such that $K|U$ is finite over $S$ (resp. is the unit group).

In view of the upper semi-continuity of the dimension of the fibers of $K$ (Exp. VI_B § 4), we may already assume, replacing $S$ by a smaller open if necessary, that $K$ is quasi-finite over $S$, hence finite (6.1). If moreover $K_s$ is the unit group, one deduces easily by Nakayama's lemma that $K$ is the unit group over $\mathrm{Spec}{\mathcal{O}}_s$, hence over a neighborhood of $s$.

Proposition 6.3. Let $S$ be a prescheme, $G$ an $S$-group prescheme smooth and of finite presentation over $S$, with affine, connected fibers, of zero unipotent rank, $H$ an $S$-group prescheme, $s$ a point of $S$, $u:G\to H$ an $S$-group morphism such that $Ker{u}_s$ is central. Suppose moreover that $H$ is of finite presentation over $S$ or that $S$ is locally noetherian and $G$ is locally of finite type over $S$. Then there exists an open neighborhood $U$ of $s$ such that if $K=Keru$, $K|U$ is central, of multiplicative type. Moreover, $G^{\prime }=(G|U)/(K|U)$ is representable and the morphism $u^{\prime }:G^{\prime }\to H|U$ deduced from $u$ by passage to the quotient is an immersion.

Proof. Let $Z$ be the center of $G$, and $K_0=K\cap Z$.

a) $K_0$ is a subgroup of multiplicative type over a neighborhood $U$ of $s$. Indeed, replacing $S$ by an étale extension over a neighborhood $U$ of $s$ if necessary (which is legitimate), we may assume that $G$ possesses a maximal torus $T$ (4.1 c)). But then $T\cap K$ is a group of multiplicative type (Exp. XV 8.3) whose fiber at $s$ is central by hypothesis, so $T\cap K$ is central over a neighborhood of $s$ (Exp. IX 5.6 a)) and consequently coincides with $K_0$.

b) Let us show that $K_0=K$ over a neighborhood $U$ of $s$. By a) we may already assume that $K_0$ is of multiplicative type, hence flat over $S$. Since K₀_s = K_s, the natural immersion $K_0\to K$ is then open over a neighborhood $U$ of $s$ (Exp. VI_B 2.6). If $t\in U$, the image of $K_t$ in the group $G_t^{\prime }=G_t/Z_t$ is then an étale finite group, normal in $G_t^{\prime }$ (since $K_t$ is normal in $G_t$), hence central, hence reduced to the unit group (3.3 b)). This says that $K_t$ is contained in $Z_t$, hence is equal to K₀_t, whence $K=K_0$ over $U$.

c) The representability of $G^{\prime }$ then follows from 2.3; the fact that $u^{\prime }$ is an immersion is contained in 1.3 a), taking 4.1 c) applied to $G^{\prime }$ into account.

Proposition 6.4. Let $S$ be a prescheme, $G$ an $S$-group prescheme smooth, of finite presentation over $S$, with connected fibers, $s$ a point of $S$, such that $G_s$ is generated by its subtori (Exp. XII 8.2) (for example $G_s$ affine of zero unipotent rank), $H$ an $S$-group prescheme, $u$ and $v:G\to H$ two $S$-homomorphisms such that $u_s=v_s$ and such that $u_s$ is central. Suppose moreover that $H$ is of finite presentation over $S$, or that $S$ is locally noetherian. Then there exists a neighborhood $U$ of $s$ such that $u|U=v|U$.

We reduce as usual to the case where $S$ is noetherian (to study the condition "$G_s$ is generated by its subtori", one uses Exp. VI_B 7.4). Since $G$ has connected fibers, to show that $u=v$ over a neighborhood $U$ of $s$, it suffices

${\mathcal{O}}_{S,s}$, which reduces us to the case where $S$ is local artinian.

But then, the functor of maximal subtori of $G$ is representable by an $S$-scheme $X$, smooth of finite type over $S$ (Exp. XV 8.17). Let $T$ be the maximal subtorus of G_X, universal for the functor $X$. The hypothesis made on $G_s$ means that the algebraic subgroup of $G_s$ generated (Exp. VI_B § 7) by the $\kappa (s)$-morphism $f:T_s\to G_s$ (composite of the immersion $T_s\to G_{X_s}$ and the canonical projection $G_{X_s}\to G_s$) is equal to $G_s$ itself. But $X_s$ is geometrically connected (since if $T$ is a maximal torus of $G_s$, $N$ its normalizer in $G_s$, $X_s$ is isomorphic to $G_s/N$), and the image of $T_s$ under $f$ contains the unit section of $G_s$; it then follows from Exp. VI_B 7.4 that there exist an integer $N>0$ and an $S$-morphism $f^N:T^N\to G$ (where $T^N=T{\times }_S\cdots {\times }_{S}T$ ($N$ factors) and $f^N$ depends only on $f$ and on the multiplication law of $G$) such that $(f^N{)}_s$ is surjective.