Exposé IX. Groups of multiplicative type: homomorphisms into a group scheme

by A. Grothendieck

Version 1.0 of 8 November 2009: additions in proof of 3.6 bis, 4.4–7, 5.0–6, 6.1, 7.1, 8.2. 8.1 and 4.5 to be revised.¹

1. Definitions

Definition 1.1. Let $S$ be a prescheme and $G$ an $S$-prescheme in groups. One says that $G$ is a group of multiplicative type if $G$ is locally diagonalizable in the sense of the faithfully flat quasi-compact topology (cf. IV 6.3), i.e. if for every $s\in S$, there exist an open neighborhood $U$ of $s$ and a faithfully flat quasi-compact morphism $S^{\prime }\to U$ such that $G^{\prime }=G{\times }_S{S}^{\prime }$ is a diagonalizable $S^{\prime }$-group (I 4.4 and VIII 1.1).

— One says that $G$ is of quasi-isotrivial multiplicative type if it is even locally diagonalizable in the sense of the étale topology (IV 6.3), i.e. if in the preceding definition one can even take $S^{\prime }\to U$ étale surjective, or again (which amounts to the same, as one sees by taking the disjoint-sum scheme of the various $S^{\prime }$ attached to the various $s\in S$) if there exists $S^{\prime }\to S$ étale and surjective such that $G^{\prime }=G{\times }_S{S}^{\prime }$ is a locally diagonalizable $S^{\prime }$-group.

— If one can even choose $S^{\prime }\to S$ étale surjective and finite, one says that $G$ is of isotrivial multiplicative type.

— Finally, one says that $G$ is a group of locally trivial (resp. locally isotrivial*) multiplicative type if every $s\in S$ admits an open neighborhood $U$ such that $G{\times }_{S}U$ is a diagonalizable $U$-group (resp. a group of isotrivial multiplicative type, i.e. there exists an étale surjective finite morphism $S^{\prime }\to U$ such that $G{\times }_S{S}^{\prime }$ is a diagonalizable $S^{\prime }$-group).*

Comments 1.2. One will note that the five preceding notions all derive from the notion of diagonalizable group by the process of localization, in the sense of five different "topologies" associated with (Sch).

We agree, generally, that when the word "locally" is not made explicit by specifying the topology envisaged, the Zariski topology is meant, in accordance with received terminology. In the terminology introduced here, "of locally trivial multiplicative type" is equivalent to "locally diagonalizable" of VIII 1.1, and likewise "trivial" translates as "diagonalizable". Among the five notions introduced, one has the following diagram of implications, which results from the corresponding relations between the topologies in play:

                 multiplicative type (general)
                            ⇑
                      quasi-isotrivial
                            ⇑
            locally isotrivial ⇐ isotrivial
                  ⇑                   ⇑
            locally trivial ⇐    trivial

From the practical point of view, let us point out at once that all the groups of multiplicative type we shall encounter will be quasi-isotrivial; thus, we shall see in the following Exposé (X 4.5) that when $G$ is of finite type over $S$, then $G$ is automatically quasi-isotrivial, but we shall give examples where it is not locally isotrivial. We shall likewise see there that $G$ may be locally trivial without being isotrivial nor, a fortiori, trivial (which easily implies that the inclusions of the preceding diagram are strict).

On the other hand, we shall also see there (X 5.16) that when $S$ is locally noetherian and normal (more generally geometrically unibranch), every group of multiplicative type of finite type over $S$ is necessarily isotrivial, and moreover trivial as soon as it is locally trivial (or even merely trivial on a dense open). This explains that most groups of multiplicative type one will encounter in practice will doubtless be isotrivial, all the more so since we shall see later that the maximal tori of semisimple group schemes are automatically isotrivial.

Definition 1.3. Let $S$, $G$ be as in 1.1. One says that $G$ is a torus if it is locally isomorphic, in the sense of the faithfully flat quasi-compact topology, to a group of the form $G_m^r$ (where $r$ is an integer $⩾0$).

With the notation of 1.1, this means therefore that one can choose $S^{\prime }$ such that $G^{\prime }$ be isomorphic to a group of the form $(G_{m,S^{\prime }}{)}^r$. One will note that the integer $r$ depends on $s\in S$: it is the dimension of the fiber $G_s=G{\otimes }_S\kappa (s)$. It is a locally constant function of $s$, as one verifies at once. There is occasion to generalize this remark:

Definition 1.4. Let $G$ be a diagonalizable group scheme over a field $k$, so that $G$ is isomorphic to a group $D_k(M)$, where $M$ is an ordinary commutative group, defined up to isomorphism by this condition — more precisely $M\simeq {\mathrm{Hom}}_{k-gr.}(G,G_{m,k})$ (VIII 1.3). The isomorphism class of $M$ is called the type of the diagonalizable group $G$; it is evidently invariant under extension of the base field.

If now $G$ is a group scheme of multiplicative type over an arbitrary prescheme $S$, then for every $s\in S$, there exists an extension $k$ of $\kappa (s)$ such that $G{\times }_S\mathrm{Spec}(k)$ is a diagonalizable $k$-group;² its type is then independent of the chosen extension by the preceding remark, and will be called the type of $G$ at $s$, or type of $G_s$. In particular, if $G$ itself is diagonalizable, hence isomorphic to a group of the form $D_S(M)$, then for every $s\in S$, the type of $G$ at $s$ equals the class of the group $M$.

In general, if $G$ is a group of multiplicative type over $S$ and $M$ an ordinary commutative group, one will say that $G$ is of type $M$ if it is of type $M$ at every point $s\in S$, in other words if $G$ is locally isomorphic to $D_S(M)$ in the sense of the faithfully flat quasi-compact topology.

Remark 1.4.1.³ a) One sees immediately that for a given group of multiplicative type $G$ over $S$, the function $s\mapsto typeof{G}_s$ is locally constant on $S$: indeed, with the notation of 1.1, if $G^{\prime }$ is of type $M$, then $G$ is of type $M$ on the neighborhood $U$ of $s$. It follows that one has a canonical partition of $S$ into a disjoint sum of preschemes $S_i$, such that for every $i$, $G_{S_i}$ is of type $M_i$, where the $M_i$ are pairwise non-isomorphic commutative groups.

b) In particular, if $S$ is connected, the type of the fibers of $G$ is constant, i.e. there exists a commutative group $M$ such that $G$ is of type $M$. Finally, if $G$ is a torus, the type of $G$ at $s$ is characterized by the integer $\dim G_s$ (indeed, $G_s$ is of type ${\mathbb{Z}}^n$, where $n=\dim G_s$).

Remark 1.5. a) It is trivial on definitions 1.1 and 1.3 that these are "compatible with base extension". Thus, if $G$ is a prescheme in groups over $S$, and $S^{\prime }\to S$ is a base-change morphism, then if $G$ is of multiplicative type (resp. of isotrivial multiplicative type, etc.), the same holds for $G^{\prime }$.

b) When moreover $S^{\prime }\to S$ is faithfully flat and quasi-compact, then if $G^{\prime }$ is of multiplicative type, resp. a torus, the same holds for $G$. If moreover $S^{\prime }\to S$ is étale (resp. finite étale), and $G^{\prime }$ quasi-isotrivial (resp. isotrivial), the same holds for $G$.

c) Finally, returning to an arbitrary base-change morphism $f:S^{\prime }\to S$, if $s^{\prime }\in S^{\prime }$ and $s=f(s^{\prime })$, then the type of $G^{\prime }$ at $s^{\prime }$ is equal to that of $G$ at $s$, since $G_{s^{\prime }}^{\prime }=G_s{\otimes }_{\kappa (s)}\kappa (s^{\prime })$.

2. Extension of certain properties of diagonalizable groups to groups of multiplicative type

The extensions in question are essentially trivial consequences of the results of the preceding Exposé, given Definitions 1.1 and the "local" nature (in the sense of the faithfully flat quasi-compact topology) of the results concerned.

Definition 2.0.⁴ We denote by $M$ the set of faithfully flat quasi-compact morphisms.

Proposition 2.1. Let $G$ be a group of multiplicative type over a prescheme $S$. One has the following:

a) $G$ is faithfully flat over $S$, and affine over $S$ (a fortiori quasi-compact over $S$).

b) $G$ of finite type over $S$ $\iff$ $G$ of finite presentation over $S$ $\iff$ for every $s\in S$, the type of $G$ at $s$ is given by a commutative group of finite type.

c) $G$ finite over $S$ $\iff$ for every $s\in S$, the type of $G$ at $s$ is given by a finite commutative group $\iff$ (if $S$ quasi-compact) $G$ is of finite type over $S$ and is annihilated by an integer $n>0$.

c′) $G$ integral over $S$ $\iff$ for every $s\in S$, the type of $G$ at $s$ is given by a torsion commutative group.

d) $G$ is the unit group $\iff$ for every $s\in S$, the type of $G$ at $s$ is given by the zero commutative group.

e) $G$ is smooth over $S$ $\iff$ for every $s\in S$, the type of $G$ at $s$ is given by a commutative group of finite type whose torsion subgroup is of order prime to the characteristic of $\kappa (s)$.

These statements follow from VIII 2.1, taking into account that the properties in question descend along faithfully flat quasi-compact morphisms (cf. SGA 1, VIII or EGA IV₂, § 2).

Using VIII 3.5, one obtains likewise:

Proposition 2.2. Let $G$ be a group of multiplicative type and of finite type over $S$; then for every integer $n\ne 0$, the kernel ${}_{nG}$ of $n\cdot i{d}_G$ is a group of multiplicative type, finite over $S$.

Proposition 2.3. Let $G$ be a group of multiplicative type over the prescheme $S$, operating freely on the right on the $S$-prescheme $X$ affine over $S$. Then:

a) The equivalence relation defined by $G$ in $X$ is $M$-effective (IV 3.4), and $Y=X/G$ is affine over $S$.

b) If moreover $X$ is of finite presentation (resp. of finite type) over $S$, the same holds for $Y$.

The first assertion follows from VIII 5.1, which treats the case where $G$ is diagonalizable, and from IV 3.5.2, which allows one to reduce to that case, given that faithfully flat quasi-compact morphisms $S^{\prime }\to S$ are morphisms of effective descent for the fibered category of affine schemes over others, i.e. for every $Y^{\prime }$ affine over $S^{\prime }$ endowed with a descent datum relative to $S^{\prime }\to S$, this descent datum is effective, i.e. $Y^{\prime }$ comes from a $Y$ affine over $S$ (cf. SGA 1, VIII 2.1).

For the second assertion one is likewise reduced to the diagonalizable case VIII 5.8, since the finiteness conditions envisaged descend along faithfully flat quasi-compact morphisms (SGA 1, VIII 3.3 and 3.6⁵). Proceeding as in VIII, Corollaries 5.5 to 5.7, one deduces from 2.3:

Corollary 2.4. Under the conditions of 2.3, the graph morphism

X ×_S G ⟶ X ×_S X

is a closed immersion. For every section $\sigma$ of $X$ over $S$, the corresponding morphism $G\to X$, $g\mapsto \sigma \cdot g$ is a closed immersion.

In particular:

Corollary 2.5. Let $u:G\to H$ be a monomorphism of $S$-preschemes in groups, with $G$ of multiplicative type and $H$ affine over $S$. Then $u$ is a closed immersion, $H/G=Y$ exists and is affine over $S$, and finally $H$ is a principal homogeneous bundle over $Y$ with group G_Y.

Remark 2.6. Let $u:G\to H$ be a monomorphism of $S$-preschemes in groups, with $G$ of multiplicative type and of finite type over $S$, $H$ of finite presentation over $S$ and separated over $S$. Then one can show that $u$ is a closed immersion, using VIII 7.12 (see remark VIII 7.13 b)).

Proposition 2.7. Let $u:G\to H$ be a homomorphism of $S$-groups of multiplicative type, with $H$ of finite type over $S$. Then:

(i) $G_0=Keru$ is an $S$-group of multiplicative type, the equivalence relation defined by G_0 in $G$ is $M$-effective, hence (IV 3.4) $u$ factors as

G ⟶ G/G_0 = I ⟶ H,

where $I\to H$ is a monomorphism⁶ of $S$-groups; $I$ is of multiplicative type, the equivalence relation in $H$ defined by $I$ is $M$-effective, consequently $H_0=H/I$ exists, and moreover H_0 is of multiplicative type.

(ii) H_0 and $I$ are of finite type, and the same holds for G_0 if $G$ is.

Proof. Proceeding as for 2.3, one is reduced to the case where $G$ and $H$ are diagonalizable, and then 2.7 reduces to VIII 3.4.

Let us note the following corollaries:

Corollary 2.8. a) Let $S$ be a prescheme. Then the category of $S$-groups of multiplicative type and of finite type is an abelian category.

b) Let $u:G\to H$ be a homomorphism in this category; for it to be a monomorphism (resp. an epimorphism, resp. an isomorphism) in this category, it is necessary and sufficient that $u$ be a monomorphism of preschemes (resp. that $u$ be faithfully flat, resp. that $u$ be an isomorphism of preschemes).

It suffices to note that the product of two $S$-groups of multiplicative type is again of multiplicative type (which is immediate), and evidently of finite type over $S$ if both factors are. The rest of 2.8 follows immediately from 2.7; the detail is left to the reader.

Corollary 2.9. Let $u:G\to H$ be a homomorphism of $S$-groups of multiplicative type and of finite type. Let $U$ be the set of points $s\in S$ such that $u_s:G_s\to H_s$ is a monomorphism (resp. faithfully flat, resp. an isomorphism). Then $U$ is both open and closed, and the induced homomorphism $u_U:G_U\to H_U$ is a monomorphism (resp. faithfully flat, resp. an isomorphism).

Let $P$ (resp. $Q$) be the kernel (resp. cokernel) of $u$. By 2.7, $Q$ exists and $P$ and $Q$ are of multiplicative type; moreover the formation of $P$ and of $Q$ commutes with every base change $S^{\prime }\to S$, in particular with the formation of fibers. On the other hand, $u$ is a monomorphism (resp. faithfully flat, resp. an isomorphism) if and only if $P=0$ (resp. $Q=0$, resp. $P=Q=0$). Thanks to these remarks, one is therefore reduced to verifying the following: if $R$ is an $S$-group of multiplicative type, then the set $U$ of $s\in S$ such that $R_s=0$ is open and closed, and $R_U=0$. But this is contained in Remark 1.4.1 a).

Corollary 2.10. Let $G$ be an $S$-group of multiplicative type and of finite type, and $H$, $H^{\prime }$ two subgroups of multiplicative type.

a) $H^{\prime \prime }=H\cap H^{\prime }$ is a subgroup of multiplicative type of $G$.

b) Let $U$ be the set of $s\in S$ such that $H_s\subseteq H_s^{\prime }$ (resp. $H_s=H_s^{\prime }$). Then $U$ is open and closed, and $H_U\subseteq H_U^{\prime }$ (resp. $H_U=H_U^{\prime }$).

Of course the intersection sign in $H\cap H^{\prime }$ denotes the intersection in the functorial sense, i.e. $H{\times }_G{H}^{\prime }$, which is evidently an $S$-subgroup of $G$; likewise the inclusion sign (resp. equality sign) denotes the order relation (resp. the equality) between subfunctors of $H$ (and not inclusion (resp. equality) of underlying sets).

Applying first 2.7 to the inclusion morphism $H\to G$, one finds that $H$ is of finite type; likewise $H^{\prime }$ is of finite type. It then follows from 2.8 that $H\cap H^{\prime }$ is of multiplicative type and of finite type (one will note that the canonical functor from the category envisaged in 2.8 to the category $(Sch)/S$ commutes with finite projective limits).

The formation of $H\cap H^{\prime }$ commutes with base extension, in particular with the formation of fibers. On the other hand, $H\subseteq H^{\prime }$ (resp. $H=H^{\prime }$) is equivalent to $H^{\prime \prime }=H$ (resp. $H^{\prime \prime }=H$ and $H^{\prime \prime }=H^{\prime }$). Consider then the canonical homomorphisms $H^{\prime \prime }\to H$ and $H^{\prime \prime }\to H^{\prime }$; then $U$ is the set of $s\in S$ such that the induced homomorphism $H_s^{\prime \prime }\to H_s$ is an isomorphism (resp. such that $H_s^{\prime \prime }\to H_s$ and $H_s^{\prime \prime }\to H_s^{\prime }$ are isomorphisms). By 2.9, it follows that $U$ is open and closed, and that $H_U^{\prime \prime }\to H_U$ (resp. $H_U^{\prime \prime }\to H_U$ and $H_U^{\prime \prime }\to H_U^{\prime }$) are isomorphisms. Hence the desired conclusion.

Proposition 2.11. Let $S$ be a prescheme, $G$ an $S$-group of multiplicative type and of finite type over $S$, $H$ a subgroup of multiplicative type of $G$, and $K=G/H$ (which is a group of multiplicative type, quotient of $G$).

(i) Suppose $G$ is trivial, i.e. diagonalizable. Then there exists a partition of $S$ into open-and-closed parts $S_i$, such that for every $i$, $H_{S_i}$ and $K_{S_i}$ are diagonalizable. In particular, if $S$ is connected, $H$ and $K$ are diagonalizable.

(ii) Same statement as in (i), replacing "diagonalizable" by "isotrivial", provided that $S$ is connected, or locally connected, or quasi-compact.

(iii) Suppose $G$ is locally trivial (resp. locally isotrivial, resp. quasi-isotrivial); then the same holds for $K$ and $H$.

Proof. (i) By hypothesis $G=D_S(M)$, where $M$ is a commutative group of finite type. For every quotient group $M_i$ of $M$, let $H_i=D_S(M_i)$ be the corresponding diagonalizable subgroup of $G$ (VIII 3.1). Let $S_i$ be the set of $s\in S$ such that $H_s=(H_i{)}_s$; by 2.10, $S_i$ is open and closed, and one has $H_{S_i}=(H_i{)}_{S_i}$, hence $H_{S_i}$ is diagonalizable, hence so is $K_{S_i}$ (VIII 3.1). Evidently the $S_i$ are pairwise disjoint; we claim that they cover $S$. This follows from the fact that for every $s$, $H_s$ is diagonalizable, as a subgroup of the diagonalizable group $G_s$ (cf. 8.1 below⁷); hence $H_s$ is of the form $D_{\kappa (s)}(M_i)$, by VIII, 1.5 and 3.2 b). Restricting to the family of non-empty $S_i$, the conclusion (i) appears.

(ii) By hypothesis, there exists $S^{\prime }\to S$ étale finite surjective such that $G_{S^{\prime }}$ is diagonalizable. Hence every point of $S^{\prime }$ has an open-and-closed neighborhood $U^{\prime }$ such that $H_{U^{\prime }}$ and $K_{U^{\prime }}$ are diagonalizable. Then the image $U$ of $U^{\prime }$ in $S$ is open and closed, and $S^{\prime }\to S$ still induces an étale finite surjective morphism $U^{\prime }\to U$; hence one sees that every point $s$ of $S$ has an open-and-closed neighborhood $U$ such that H_U and K_U are isotrivial. The conclusion (ii) follows at once.

(iii) Follows at once from (i) and (ii) and the definitions. Note moreover that the "quasi-isotrivial" case will also follow from the more general fact that "finite type ⇒ quasi-isotrivial", announced in 1.2 (cf. X 4.5).

3. Infinitesimal properties: lifting and conjugation theorems

The fundamental infinitesimal properties of groups of multiplicative type follow from the following theorem:

Theorem 3.1. Let $S$ be an affine scheme, $H$ an $S$-group of multiplicative type, $F$ a quasi-coherent sheaf on $S$ on which $H$ operates (I 4.7). Then one has

H^i(H, F) = 0    for    i > 0,

where $H^i$ is the Hochschild cohomology studied in Exp. I, § 5.

Indeed, by loc. cit., 5.3, the Hochschild cohomology is described, in terms of the affine rings $A$ of $S$ and $B$ of $G$, and the module $M$ over $A$ defining $F$, as the cohomology of a complex of $A$-modules $C^{\bullet }(H,M)$, the formation of which commutes with every base change $A\to A^{\prime }$. Consequently, for a base change $A\to A^{\prime }$ with $A^{\prime }$ flat over $A$, one has

H^i(H′, F′) = H^i(H, F) ⊗_A A′,

and consequently, if one supposes even $A^{\prime }$ faithfully flat over $A$, in order to prove that $H^i(H,F)$ is zero, it suffices to prove that the same holds for $H^i(H^{\prime },F^{\prime })$. This remark reduces us to verifying 3.1 in the case where $G$ is diagonalizable; in that case this was proved in I 5.3.3.

Using the results of Exposé III, we shall deduce from 3.1 various consequences of geometric nature:

Theorem 3.2. Let $S$ be an affine scheme, S_0 an affine subscheme defined by an ideal $J$, $H$ a group of multiplicative type over $S$, $G$ an arbitrary prescheme in groups over $S$, $u,v:H\to G$ two homomorphisms of groups, $u_0,v_0:H_0\to G_0$ the homomorphisms deduced from u, v by the base change $S_0\to S$. If $J^2=0$ and $u_0=v_0$, then there exists a $g\in G(S)$ such that $v=int(g)\circ u$ and $g_0=e$.

This follows from III 2.1 (ii), and from 3.1 (vanishing of $H^1$).

Corollary 3.3. Under the preliminary conditions of 3.2, suppose moreover $G$ smooth over $S$, but assume only $J$ nilpotent (not necessarily of square zero). If there exists $g_0\in G_0(S_0)$ such that $v_0=int(g_0)\circ u_0$, then $g_0$ lifts to a $g\in G(S)$ such that $v=int(g)\circ u$.

By induction on the integer $n$ such that $J^n=0$, one is reduced to the case where $J$ is of square zero. Moreover, $G$ being smooth over $S$, one can lift $g_0$ to an element $g^{\prime }$ of $G(S)$. Replacing $v$ by $v^{\prime }=int(g^{\prime })\circ u$, one is then reduced to the situation of 3.2.

Corollary 3.4. Under the preliminary conditions of 3.2, with $J$ nilpotent, suppose moreover $u$ is central (for example $G$ commutative). Then $u_0=v_0$ implies $u=v$.

Indeed, the reduction to the case where $J$ has square zero is again immediate, and then it suffices to apply 3.2. In particular:

Corollary 3.5. Let $S$, S_0, $H$, $G$ be as in 3.2, with $J$ nilpotent. Let $u:H\to G$ be a homomorphism of $S$-groups such that $u_0:H_0\to G_0$ is the unit homomorphism. Then $u$ is the unit homomorphism.

Theorem 3.6. Let $S$ be an affine scheme, S_0 a subscheme defined by a nilpotent ideal $J$, $H$ a group of multiplicative type over $S$, $G$ a prescheme in groups smooth over $S$, $u_0:H_0\to G_0$ a homomorphism of the S_0-groups deduced by the base change $S_0\to S$.

Then there exists a homomorphism $u:H\to G$ of $S$-groups which lifts $u_0$ (and by 3.3 any two such liftings $u,u^{\prime }$ are conjugate by an element of $G(S)$ reducing along the unit element of $G(S_0)$).

This follows from III 2.1 and 2.3, and from 3.1 (vanishing of $H^2$ for the existence of the lift of $u_0$, vanishing of $H^1$ for the uniqueness up to conjugation).

One can prove in the same way the following variants of 3.2 to 3.6 (which in fact entail the preceding statements, by applying them to the graph subgroups of the homomorphisms envisaged in 3.2 to 3.6):

Theorem 3.2 bis. Let $S$ be an affine scheme, S_0 a subscheme defined by an ideal $J$, $G$ an $S$-prescheme in groups, $H,H^{\prime }$ subpreschemes in groups, of multiplicative type, $G_0,H_0,H_0^{\prime }$ the groups over S_0 deduced by the base change $S_0\to S$. If $J^2=0$ and $H_0=H_0^{\prime }$, then there exists $g\in G(S)$ such that $H^{\prime }=int(g)(H)$ and $g_0=e$.

Corollary 3.3 bis. Under the preliminary conditions of 3.2 bis, suppose moreover $G$ smooth over $S$, and $J$ nilpotent (not necessarily of square zero). Let $g_0\in G_0(S_0)$ be such that $H_0^{\prime }=int(g_0)(H_0)$; then $g_0$ lifts to $g\in G(S)$ such that $H^{\prime }=int(g)(H)$.

Corollary 3.4 bis. Under the preliminary conditions of 3.2 bis, suppose $J$ nilpotent, $H$ central (for example $G$ commutative). Then $H_0=H_0^{\prime }$ implies $H=H^{\prime }$.

Theorem 3.6 bis. Let $S$ be an affine scheme, S_0 a subscheme defined by a nilpotent ideal $J$, $G$ an $S$-prescheme in groups smooth over $S$, H_0 a subprescheme in groups of $G_0=G{\times }_S{S}_0$, of multiplicative type. Then:

(a) There exists a subprescheme in groups $H$ of $G$, flat over $S$, such that $H{\times }_S{S}_0=H_0$.

(b) Such an $H$ is necessarily of multiplicative type.

(c) Finally, by 3.2 bis, any two such liftings $H,H^{\prime }$ of H_0 are conjugate by an element $g\in G(S)$ reducing along the unit element of $G(S_0)$.

Let us show how one may prove 3.2 bis (of which 3.3 bis and 3.4 bis are an immediate consequence) and assertion (a) of 3.6 bis. The latter follows from 3.1 (vanishing of $H^2$) and from III 4.37 (ii). Similarly 3.2 bis follows from 3.1 (vanishing of $H^1$) and from III 4.37 (i), at least when $G$ and $H$ are smooth over $S$.

But using the results of the following Exposé, one can very simply deduce 3.2 bis and 3.6 bis, in the general form stated here, from 3.2 resp. 3.6. For 3.2 bis, it suffices indeed to note that by X 2.1, $H$ and $H^{\prime }$ are isomorphic, which reduces us to 3.2.

For 3.6 bis, one notes that H_0 is necessarily of finite type over S_0:⁸ indeed, as a subprescheme of G_0, which is smooth over S_0, H_0 is locally of finite type over S_0; but, by 2.1 a), H_0 is affine over S_0, hence is of finite type over S_0. Then, by X 4.5, H_0 is quasi-isotrivial, hence comes, by X 2.1, from a group of multiplicative type $H$ over $S$.⁹ Then, by 3.6, there exists a homomorphism of $S$-groups $u:H\to G$ which lifts the immersion $H_0\hookrightarrow G_0$; since $H$ and H_0 (resp. $G$ and G_0) have the same underlying topological space, and since, for every $h\in H$, the morphism $O_{G,u(h)}\to O_{H,h}$ is surjective (since it is so after reduction modulo $J$, which is nilpotent), $u$ is also an immersion (cf. EGA I, 4.2.2).

Finally, for every lifting $H$ of H_0 to a flat subgroup of $G$, $H$ is necessarily of multiplicative type by X 2.3,¹⁰ which also proves assertion (b) of 3.6 bis. (The reader will verify that the results 3.2 bis to 3.6 bis are not used in Exposé X, hence that there is no vicious circle!)

Proposition 3.8.¹¹ Let $S$ be a prescheme, S_0 a closed subprescheme defined by a locally nilpotent ideal $J$, $G$ an $S$-group of multiplicative type, $X$ an $S$-prescheme locally of finite presentation. Suppose that $G$ operates on $X$, in such a way that $G_0=G{\times }_S{S}_0$ operates trivially on $X_0=X{\times }_S{S}_0$. Under these conditions, $G$ operates trivially on $X$.

The proof is that of III 2.4 b); one can also reduce to loc. cit. by noting that 3.8 reduces at once to the case where $G$ is diagonalizable, which is the case envisaged in loc. cit.

Corollary 3.9. Let $S,S_0$ be as above, and $u:G\to H$ a homomorphism of $S$-groups, with $G$ of multiplicative type and $H$ locally of finite presentation over $S$. Suppose that the homomorphism $u_0:G_0\to H_0$ deduced by base change $S_0\to S$ is central. Then $u$ is central.

Indeed, it suffices to apply 3.8 by taking $X=H$ and making $G$ operate on $H$ by $(g,h)\mapsto int(u(g))\cdot h=u(g)hu(g{)}^{-1}$.

4. The density theorem

This theorem (cf. 4.7 below)¹², together with the theorem¹³ of N° 7, will be the essential tool in the present Exposé and the two following, for passing from the infinitesimal properties of groups of multiplicative type, which have just been developed, to the "finite" properties.

Definition 4.1. Let $X$ be a prescheme. A family $(Z_i{)}_{i\in I}$ of subpreschemes of $X$ is said to be schematically dense if for every open $U$ of $X$, and every closed subprescheme U_0 of $U$ which majorizes the $Z_i\cap U$, one has $U_0=U$.

One says that a subprescheme $Z$ of $X$ is schematically dense in $X$ if such is the case of the family reduced to $Z$.

One sees immediately (cf. EGA IV₃, 11.10.1) that the definition is equivalent to saying that for every open $U$ of $X$, every section $f$ of O_U which is zero on the $Z_i\cap U$ is zero, which also means that the intersection of the kernels of the canonical homomorphisms

$$u_i:O_X⟶(v_i{)}_{\ast }(O_{Z_i})$$

is zero, where $v_i:Z_i\to X$ is the canonical immersion.¹⁴ When $X$ lies over a prescheme $S$, this is again equivalent to saying that for every open $U$ of $X$ and every pair $(u,v)$ of morphisms from $U$ into an $S$-prescheme $Y$ separated over $S$ that coincide on the $Z_i\cap U$, one has $u=v$. (Indeed, the relation $u=v$ is equivalent to $U_0=U$, where U_0 is the inverse image of the diagonal of $Y{\times }_{S}Y$ by the $S$-morphism $X\to Y{\times }_{S}Y$ defined by $(u,v)$; this diagonal is a closed subprescheme of $Y{\times }_{S}Y$, hence U_0 is a closed subprescheme of $U$, majorizing the $Z_i\cap U$ by the hypothesis on $(u,v)$; hence if the family $(Z_i)$ is schematically dense, one will have $U_0=U$, hence $u=v$; the converse implication is seen by simply taking $Y=\mathrm{Spec}{O}_S[T]$.)

With the terminology introduced in EGA I, end of N° 9.5, to say that the subprescheme $Z$ of $X$ is schematically dense also means that $X$ is identical to the adherence subprescheme of $Z$ in $X$.

Lemma 4.2. Let $X$ be a flat $S$-prescheme, $(Z_i{)}_{i\in I}$ a family of subpreschemes of $X$ flat over $S$. Let S_0 be a subprescheme of $S$, defined by a nilpotent ideal $J$; suppose the modules $J^n/J^{n+1}$ locally free over S_0.

Let $X_0,Z_i^{\prime }$ be deduced from $X,Z_i$ by the base change $S_0\to S$. Then, if the family $(Z_i^{\prime }{)}_{i\in I}$ is schematically dense in X_0, the family $(Z_i{)}_{i\in I}$ is so in $X$.

Suppose $J^{n+1}=0$ (where $n⩾0$); let us argue by induction on $n$, the assertion being trivial for $n=0$. Defining $S_m,X_m,Z_m^i$ by reduction modulo $J^{m+1}$ as is customary, the induction hypothesis already implies that $(Z_{n-1}^i{)}_{i\in I}$ is schematically dense in $X_{n-1}$. Replacing $X$ by an induced open if necessary, we are reduced to proving that every section $f$ of O_X which vanishes on the $Z_i$ is zero.

Now the section $f_{n-1}$ of $O_{X_{n-1}}=O_X/J^n{O}_X$ defined by $f$ vanishes on the $Z_{n-1}^i$, hence is zero, hence $f$ is a section of $J^n{O}_X$. Since $X$ is flat over $S$, one has

J^n O_X ⥲ E ⊗_{O_{S_0}} O_{X_0},          where E = J^n = J^n/J^{n+1}.

Likewise, since each $Z_i$ is flat over $S$, the restriction $f_i$ of $f$ to $Z_i$ may be regarded as a section of:

J^n O_{Z_i} ⥲ E ⊗_{O_{S_0}} O_{Z'_i}.

By hypothesis, the $f_i$ are zero. Now $E$ is locally free by hypothesis, hence so is $F=E{\otimes }_{O_{S_0}}{O}_{X_0}$.

restriction to $Z_i^{\prime }$ is zero. Since $(Z_i^{\prime }{)}_{i\in I}$ is schematically dense in X_0, it follows at once that $f$ is zero. QED.

Corollary 4.3. Let $X$ be a locally noetherian $S$-prescheme flat over $S$, $(Z_i{)}_{i\in I}$ a family of subpreschemes of $X$ flat over $S$. Suppose that for every $s\in S$, the family $(Z_{i,s}{)}_{i\in I}$ of fibers at $s$ is schematically dense in $X_s$. Then the family $(Z_i{)}_{i\in I}$ is schematically dense in $X$.

Replacing $X$ by an induced open if necessary, one is reduced to proving that every section $f$ of O_X whose restrictions to the $Z_i$ are zero is itself zero.¹⁵ For this, it suffices to prove that for every $x\in X$, the image of $f$ in $O_{X,x}$ is zero. Denote by $m_x$ the maximal ideal of $O_{X,x}$, and by $m_s$ that of $O_{S,s}$, where $s$ is the image of $x$ in $S$. Since $O_{X,x}$ is noetherian, one has ${\bigcap }_{n\in \mathbb{N}}{m}_x^n=0$, hence a fortiori ${\bigcap }_{n\in \mathbb{N}}{O}_{X,x}{m}_s^n=0$.

So it suffices to show that, for every integer $n$, the section induced by $f$ on $X{\times }_S\mathrm{Spec}(O_{S,s}/m_s^{n+1})$ is zero. By hypothesis, the family of fibers $Z_{i,s}=Z_i{\otimes }_{O_{S,s}}\kappa (s)$ is schematically dense in the fiber $X_s=X{\otimes }_{O_{S,s}}\kappa (s)$. This reduces us to the case where $S$ is the spectrum of a local ring whose maximal ideal $J$ is nilpotent, $S_0=\mathrm{Spec}\kappa (s)$, and the hypotheses of 4.2 are satisfied; hence the conclusion.

Lemma 4.4. Let $S$ be a locally noetherian prescheme, $X$ a locally noetherian $S$-prescheme flat over $S$, $(Z_i{)}_{i\in I}$ a family of subpreschemes of $X$ flat over $S$. Suppose that for every $s\in S$, the family $(Z_{i,s}{)}_{i\in I}$ of fibers at $s$ is schematically dense in $X_s$.

Then, for every base change $S^{\prime }\to S$ ($S^{\prime }$ not necessarily locally noetherian), the family $(Z_i^{\prime }{)}_{i\in I}$ is schematically dense in $X^{\prime }$ (i.e. the family $(Z_i{)}_{i\in I}$ is universally schematically dense in $X$ relative to $S$, cf. EGA IV₃, Def. 11.10.8).

Proof.¹⁶ Let $f^{\prime }$ be a section of $O_{X^{\prime }}$ over an open $U^{\prime }$ of $X^{\prime }$, vanishing on the $Z_i^{\prime }$; we must show that $f^{\prime }$ is zero. Let $s^{\prime }\in S^{\prime }$; it suffices to prove that $f^{\prime }$ is zero at every point of $U^{\prime }$ above $s^{\prime }$. Let $s$ be the image of $s^{\prime }$ in $S$; replacing $S,S^{\prime }$ by the spectra of the local rings at $s,s^{\prime }$ if necessary, one may suppose $S,S^{\prime }$ local with $s,s^{\prime }$ as closed points. One may moreover suppose $X$ affine, hence

S = Spec(A),    S′ = Spec(A′),    X = Spec(B),    X′ = Spec(B′),    B′ = B ⊗_A A′

where $A,A^{\prime }$ are local, $A\to A^{\prime }$ is a local homomorphism, and A, B are noetherian. One may also suppose $U^{\prime }$ affine, of the form $X_{g^{\prime }}^{\prime }$, with $g^{\prime }\in B^{\prime }$.

For every sub-$A$-algebra $A^{\prime \prime }$ of $A^{\prime }$, consider $S^{\prime \prime }=\mathrm{Spec}(A^{\prime \prime })$ and $X^{\prime \prime },Z_{n^{\prime \prime }}^i$ deduced from $X,Z_i$ by the base change $S^{\prime \prime }\to S$, giving the diagram:

$$Z_i\leftarrow Z_{n^{\prime \prime }}^i\leftarrow Z_i^{\prime }\downarrow \downarrow \downarrow X\leftarrow X^{\prime \prime }\leftarrow X^{\prime }\downarrow \downarrow \downarrow S\leftarrow S^{\prime \prime }\leftarrow S^{\prime }.$$

We restrict in what follows to those $A^{\prime \prime }$ which are localizations of finite-type sub-$A$-algebras $A_1^{\prime \prime }$ of $A^{\prime }$ at $m^{\prime }\cap A_1^{\prime \prime }$ (where $m^{\prime }$ is the maximal ideal of $A^{\prime }$), so that the homomorphisms $A\to A^{\prime \prime }\to A^{\prime }$ are local. Note that these $A^{\prime \prime }$ form an increasing filtered family of subalgebras whose union is $A^{\prime }$, hence their direct limit is also $A^{\prime }$. One has therefore likewise $B^{\prime }={\lim }_{A^{\prime \prime }}B{\otimes }_A{A}^{\prime \prime }$. Hence there exist an $A^{\prime \prime }$ and an element $g^{\prime \prime }\in B^{\prime \prime }=B{\otimes }_A{A}^{\prime \prime }$ whose image in $B^{\prime }$ is $g^{\prime }$.

¹⁷ By Lemma 4.5 below, the property that the family of fibers $(Z_{i,s}{)}_{i\in I}$ be schematically dense in $X_s$ is preserved by every base change $S^{\prime \prime }\to S$; consequently, since each $S^{\prime \prime }$ is locally noetherian, if one replaces $S$ by an appropriate $S^{\prime \prime }$, and $X,Z_i$ by $X^{\prime \prime },Z_i^{\prime \prime }$, the hypotheses of 4.4 will be preserved.

Hence, replacing $S$ by $S^{\prime \prime }$ (and $X,Z_i$ by $X^{\prime \prime },Z_i^{\prime \prime }$) if necessary, one may suppose that $g^{\prime }$ comes from $g\in B$. Replacing $X$ by the open $X_g=U$ if necessary, one may therefore suppose $U^{\prime }=X^{\prime }$. One sees likewise that one may suppose that $f^{\prime }$ comes from a section $f$ of O_X over $X$.

Let $Y=V(f)$ be the subscheme of $X$ defined by $f$, and $Y_i=Z_i{\times }_{X}Y$ its intersection with $Z_i$, which is a closed subscheme of $Z_i$, equal to $V(f_i)$, where $f_i$ is the section of $O_{Z_i}$ induced by $f$. Denoting by $Y^{\prime },Y_i^{\prime }$ the $S^{\prime }$-schemes deduced from the preceding by base change, one will again have

Y′ = V(f′),    Y'_i = Z'_i ×_{X′} Y′ = V(f'_i),

and one has the analogous relations for $Y^{\prime \prime },Y_i^{\prime \prime }$. The hypothesis that $f^{\prime }$ vanishes on the $Z_i^{\prime }$ is expressed by the relations $Y_i^{\prime }=Z_i^{\prime }$ for every $i$.¹⁸

Let $m$ be the maximal ideal of $A$. For every integer $n⩾0$, introduce the subscheme $S_n=\mathrm{Spec}(A/m^{n+1})$ of $S$, and the schemes $X_n,Y_n,Z_n^i,Y_n^i$ deduced from $X,Y,Z_i,Y_i$ by the base change $S_n\to S$. In general, for every $S$-prescheme $P$, we shall set $P_n=P{\times }_S{S}_n$.

For every $n$, and $i\in I$, consider the functor

F^i_n = ∏_{Z^i_n/S_n} Y^i_n/Z^i_n : (Sch)°/S_n ⟶ (Ens)

defined by (cf. VIII, 6.4): for every $P$ over $S_n$,

F^i_n(P) = Γ((Y^i_n)_P / (Z^i_n)_P) =  ∅       if (Y^i_n)_P ≠ (Z^i_n)_P;
                                       {id}    if (Y^i_n)_P = (Z^i_n)_P.

Since $S_n$ is local artinian, and $Z_n^i$ flat hence essentially free over $S_n$, then, by VIII, 6.4, each $F_n^i$ is representable by a closed subprescheme $T_n^i$ of $S_n$.

¹⁹ The completion Â of the local ring $A$ is noetherian since $A$ is, and it is the projective limit of the $A_n$. Denote by $K_n^i$ the ideal of $A_n=A/m^{n+1}$ defining $T_n^i$ and $K^i$ the projective limit of the $K_n^i$; it is an ideal of Â.

For $i$ fixed, $m⩾n$ and every $S_n$-prescheme $P$, one has

(Y^i_n)_P = Y_i ×_S S_n ×_{S_n} P = Y_i ×_S P = (Y^i_m)_P,

and likewise $(Z_n^i{)}_P=(Z_m^i{)}_P$; it follows that $F_n^i$ is the restriction $F_m^i{\times }_{S_m}{S}_n$ of $F_m^i$ to $(Sch)/S_n$, whence $T_n^i=T_m^i{\times }_{S_m}{S}_n$. One therefore has a commutative diagram with exact rows:

$$0\to K_m^i\to A_m\to O(T_m^i)\to 0\downarrow \downarrow \downarrow 0\to K_n^i\to A_n\to O(T_n^i)\to 0$$

where all the vertical arrows are surjective. This has the following consequences: on the one hand, the projective system $(K_n^i{)}_{n\in \mathbb{N}}$ satisfies the Mittag-Leffler condition, hence the projective limit of the $O(T_n^i)$ identifies with the topological ring quotient $\hat{A}/K^i$ (cf. EGA 0_III, § 13.2). On the other hand, the map $K^i\to K_n^i$ is surjective (cf. [BEns], III, § 7.4, Prop. 5), whence it follows that $K_n^i\simeq (K^i+m^{n+1}\hat{A})/m^{n+1}\hat{A}$ and $O(T_n^i)\simeq (\hat{A}/K^i){\otimes }_A(A/m^{n+1})$.

In other words, $(T_n^i{)}_n$ is an inductive system of affine artinian schemes, and the inductive limit $T^i={\lim }_n{T}_n^i$ is a closed formal subscheme of the formal scheme $\hat{S}=Spf(\hat{A})$ (cf. EGA I, § 10), whose reduction modulo $m^{n+1}$ is $T_n^i$.

Let $T$ be the closed formal subscheme of Ŝ intersection of the $T^i$, that is, defined by $K={\sum }_i{K}^i$. Since Â is noetherian, there exists a finite part $J$ of $I$ such that one has $K={\sum }_{i\in J}{K}^i$ (i.e. $T={\bigcap }_{i\in J}{T}^i$). Note then that for every $n$, one has $K_n={\sum }_{i\in J}{K}_n^i$ where $K_n$ denotes the image of $K$ in $A_n$.

Recall that $f_i$ denotes the image of $f\in B=O(X)$ in $O(Z_i)$, and $f_i^{\prime }$ its image in $O(Z_i^{\prime })=O(Z_i){\otimes }_A{A}^{\prime }$; the hypothesis $Y_i^{\prime }=Z_i^{\prime }$ is equivalent to the vanishing of $f_i^{\prime }$. Since $O(Z_i){\otimes }_A{A}^{\prime }$ is the inductive limit of the subalgebras $O(Z_i){\otimes }_A{A}^{\prime \prime }$ (where $A^{\prime \prime }$ satisfies the conditions made explicit above), there exists therefore an $A^{\prime \prime }$ such that $Y_i^{\prime \prime }=Z_i^{\prime \prime }$. A priori, $A^{\prime \prime }$ depends on $i$, but one can find an $A^{\prime \prime }$ that works for all $i\in J$, since $J$ is finite. Set $S^{\prime \prime }=\mathrm{Spec}(A^{\prime \prime })$ and $S_{(n)}^{\prime \prime }=S^{\prime \prime }{\times }_S{S}_n$.²⁰

Since $Y_i^{\prime \prime }{\times }_{S^{\prime \prime }}{S}_{(n)}^{\prime \prime }=(Y_n^i{)}_{S_{(n)}^{\prime \prime }}$ equals $Z_i^{\prime \prime }{\times }_{S^{\prime \prime }}{S}_{(n)}^{\prime \prime }=(Z_n^i{)}_{S_{(n)}^{\prime \prime }}$ for every $i\in J$, it follows from the definition of the $T_n^i$ that $S_{(n)}^{\prime \prime }\to S_n$ factors through $T_n^i$ for every $i\in J$, hence also through $T_n={\bigcap }_{i\in J}{T}_n^i$, hence also through $T_n^i$ for every $i\in I$. Denoting by $f^{\prime \prime }$ the image of $f$ in $B^{\prime \prime }=B{\otimes }_A{A}^{\prime \prime }$, and $f_{(n)}^{\prime \prime }$ its image in $B_{(n)}^{\prime \prime }=B^{\prime \prime }/m^{n+1}{B}^{\prime \prime }$, this means that for every $n$, $f_{(n)}^{\prime \prime }$ vanishes on the $Z_i{\times }_S{S}_{(n)}^{\prime \prime }$.

Since the morphisms $A\to A^{\prime \prime }\to A^{\prime }$ are local, the image $s^{\prime \prime }$ of $s^{\prime }$ in $S^{\prime \prime }$ is the closed point of $S^{\prime \prime }$, corresponding to the maximal ideal $m^{\prime \prime }$ of $B^{\prime \prime }$, and the image of $s^{\prime \prime }$ in $S$ is $s$. Fix $n$ and denote now by $S_n^{\prime \prime }$ the $n$-th infinitesimal neighborhood of $s^{\prime \prime }$ in $S^{\prime \prime }$, that is, $\mathrm{Spec}(A^{\prime \prime }/m^{\prime \prime n+1})$. Set $B_n^{\prime \prime }=B^{\prime \prime }/m^{\prime \prime n+1}{B}^{\prime \prime }$ and $Z_{n,i}^{\prime \prime }=Z_i{\times }_S{S}_n^{\prime \prime }=\mathrm{Spec}(B_n^{\prime \prime })$. Then the image $f_n^{\prime \prime }$ of $f_{(n)}^{\prime \prime }$ in $B_n^{\prime \prime }$ vanishes on $Z_{n,i}^{\prime \prime }$, for every $i$. Now, by Lemma 4.5 below, the family of fibers

Z_i''' = Z″_{n,i} ×_{S″_n} κ(s″) = Z_{i,s} ×_{κ(s)} κ(s″)

is schematically dense in the fiber $X_0^{\prime \prime \prime }=X_n^{\prime \prime }{\times }_{S_n^{\prime \prime }}\kappa (s^{\prime \prime })=X_s{\times }_{\kappa (s)}\kappa (s^{\prime \prime })$. Hence $S_n^{\prime \prime },X_n^{\prime \prime },(Z_{n,i}^{\prime \prime }{)}_{i\in I}$, and $S_0^{\prime \prime \prime }=\mathrm{Spec}\kappa (s^{\prime \prime })$ satisfy the hypotheses of Lemma 4.2; it follows therefore that $f_n^{\prime \prime }=0$, i.e. $f^{\prime \prime }\in m^{\prime \prime n+1}{B}^{\prime \prime }$, for every $n$.

Since $B$ is noetherian and $B^{\prime \prime }=B{\otimes }_A{A}^{\prime \prime }$ is a localization of a finite-type $B$-algebra, $B^{\prime \prime }$ is noetherian, hence its local rings are separated for their usual topology. It follows that $f^{\prime \prime }$ is zero at the points $x^{\prime \prime }$ of $X^{\prime \prime }$ such that $m^{\prime \prime }{B}^{\prime \prime }$ is contained in the maximal ideal of $O_{X^{\prime \prime },x^{\prime \prime }}$, i.e. at the points of $X^{\prime \prime }$ above $s^{\prime \prime }$. A fortiori, $f^{\prime }$ is zero at the points $x^{\prime }$ of $X^{\prime }$ above $s^{\prime }$. QED.

Lemma 4.5.0.²¹ Let $X$ be an $S$-prescheme, $(Z_i{)}_{i\in I}$ a family of subpreschemes of $X$, and $S^{\prime }\to S$ a faithfully flat morphism. If the family $(Z_i^{\prime }{)}_{i\in I}$ is schematically dense in $X^{\prime }$, then $(Z_i{)}_{i\in I}$ is schematically dense in $X$.

This follows from EGA IV₃, 11.10.5 (i) and 11.9.10 (i). It remains to give the proof of:

Lemma 4.5. Let $X$ be a locally noetherian prescheme²² over a field $k$, $(Z_i{)}_{i\in I}$ a family of subpreschemes of $X$, $k^{\prime }$ an extension of $k$, $X^{\prime }$ and $Z_i^{\prime }$ the preschemes deduced from $X$ and $Z_i$ by the base change $k^{\prime }/k$. For $(Z_i{)}_{i\in I}$ to be schematically dense in $X$, it is necessary and sufficient that $(Z_i^{\prime }{)}_{i\in I}$ be so in $X^{\prime }$.

The "if" follows from 4.5.0; let us prove the "only if".²³ First, one may suppose $X=\mathrm{Spec}(B)$ affine. For every $i$, let $(Z_{ij}{)}_{j\in J_i}$ be a covering of $Z_i$ by affine opens; replacing $(Z_i{)}_{i\in I}$ by the family $(Z_{ij}{)}_{(i,j)\in J}$, where $J={\coprod }_{i\in I}{J}_i$, one may also suppose the $Z_i$ affine.

Let $g\in B^{\prime }=B{\otimes }_k{k}^{\prime }$ and $t^{\prime }\in B_g^{\prime }$ be a section of $O_{X^{\prime }}$ on the affine open $U^{\prime }=X_g^{\prime }$, vanishing on the $Z_i^{\prime }\cap U^{\prime }$, i.e. whose image in each $(O(Z_i){\otimes }_k{k}^{\prime }{)}_g$ is zero. There exists a finite-type sub-$k$-algebra $A$ of $k^{\prime }$ such that $g\in B_A=B{\otimes }_{k}A$ and $t^{\prime }\in (B_A{)}_g$. The map $O(Z_i){\otimes }_{k}A\to O(Z_i){\otimes }_k{k}^{\prime }$ being injective, so is the map

(O(Z_i) ⊗_k A)_g ⟶ (O(Z_i) ⊗_k k′)_g;

given the hypothesis, this implies that the image of $t^{\prime }$ in each $(O(Z_i){\otimes }_{k}A{)}_g$ is zero. This reduces us to showing that the family $(Z_{iA}{)}_{i\in I}$ is schematically dense in X_A,²⁴ which follows from EGA IV₃, 11.9.10 b).

Let us note in passing the following result, which will serve in a later Exposé:²⁵

Corollary 4.6. Let $X$ be a flat $S$-prescheme, $U$ an open subset of $X$. Suppose that for every $s\in S$, $U_s$ is schematically dense in $X_s$, and that $X$ is locally noetherian, or locally of finite presentation over $S$. Then $U$ is schematically dense in $X$.

Suppose for simplicity $U$ retrocompact in $X$ (cf. EGA IV₃, 11.10.10 and 11.9.17 for the general case). The case $X$ locally noetherian is included only for memory; it is contained in 4.3.

In the second case envisaged, one may evidently suppose $S$ and $X$ affine; then $X$ and $U$ are of finite presentation over $S=\mathrm{Spec}(A)$. The ring $A$ is the inductive limit of its finite-type sub-$\mathbb{Z}$-algebras $A_i$. The "patented procedure" already used (cf. EGA IV₃, 8.8.2 and 8.10.5 (iii)) shows that there exist an $i$, an affine scheme $X_i$ over $S_i=\mathrm{Spec}(A_i)$ and an open $U_i$ in $X_i$, from which X, U are deduced by base change $S\to S_i$. Let $E_i$ be the part of $S_i$ consisting of $s\in S_i$ such that $(U_i{)}_s$ is schematically dense in $(X_i{)}_s$.

²⁶ By EGA IV₂, 5.9.9 and 5.10.2, $E_i$ is the set of $s\in S_i$ such that $(U_i{)}_s$ contains the set $AssO(X_i{)}_s$ of points "associated" with the structure sheaf of $(X_i{)}_s$, and by EGA IV₃, 11.9.17.1 this condition is of constructible nature, i.e. $E_i$ is a constructible part of $S_i$.

²⁶ By 4.5, the inverse image of $E_i$ by $S_j\to S_i$ (resp. by $S\to S_i$) is $E_j$ (resp. the set $E$ of $s\in S$ such that $U_s$ is schematically dense in $X_s$). Moreover, by hypothesis, $E=S$, which is also the inverse image by each $S\to S_i$ of $S_i$. By EGA IV₃, 8.3.11, this implies that there exists $j⩾i$ such that $E_j=S_j$, i.e. such that for every $s\in S_j$, $(U_j{)}_s$ is schematically dense in $(X_j{)}_s$.

Then, by 4.4 applied to $(S_j,X_j,U_j)$ and to the base change $S\to S_j$, it follows that $U$ is schematically dense in $X$. One will note moreover that we use 4.4 here only in the case of a family with finite index set (in fact, reduced to one element), in which case the proof of 4.4 simplifies considerably, as the reader will check.

Theorem 4.7. Let $S$ be a prescheme, $G$ a group of multiplicative type and of finite type over $S$. For every integer $n>0$, let ${}_{nG}$ be the kernel of $n\cdot i{d}_G$. Then the family of subpreschemes $({}_{nG}{)}_{n>0}$ of $G$ is schematically dense (Def. 4.1).

We distinguish two cases.

a) Case $S$ locally noetherian. Then by 4.3 one is reduced to the case where $S$ is the spectrum of a field $k$. By 4.5.0, one may suppose $k$ algebraically closed and $G$ diagonalizable, i.e. of the form $D_k(M)$, where $M$ is a commutative group of finite type. Then $M$ is of the form $\Gamma \times {\mathbb{Z}}^r$, with $\Gamma$ finite, hence $G$ is of the form $G_1\times G_2$, with $G_1=D(\Gamma )$ and $G_2=G_m^r$; and for $n$ large multiplicatively (namely $n$ a multiple of the order of $\Gamma$) one will have _nG = G_1 × _nG_2 since one will have _nG_1 = G_1.

Applying again 4.3 to the projection $G\to G_1$, one is reduced to the case of G_2, i.e. to the case where $G=G_m^r$. Since $G$ is then reduced, it amounts to the same to say that $({}_{nG}{)}_{n>0}$ is schematically dense in $G$, or that the union of the ${}_{nG}$ is dense in $G$ for the ordinary topology. Since _nG = (_nG_m)^r, one is reduced to the case of $G=G_m$, hence $G$ irreducible of dimension 1. Then this follows from the fact that the union of the ${}_{nG}$ (equal to the set of roots of unity in $k$) is infinite.

b) General case. For every point $s$ of $S$, there exist an open neighborhood $U$ of $s$ and a faithfully flat quasi-compact morphism $S^{\prime }\to U$ such that $G^{\prime }=G_{S^{\prime }}$ is diagonalizable, i.e. of the form $D_{S^{\prime }}(M)$. By 4.5.0 again, one may reduce to the case of $G^{\prime }$, so one may suppose $G$ diagonalizable; hence it comes from the absolute group $H=D_{\mathbb{Z}}(M)$ over $\mathrm{Spec}(\mathbb{Z})$ by base change $S\to \mathrm{Spec}(\mathbb{Z})$. By the proof of a), for every $s\in \mathrm{Spec}(\mathbb{Z})$, the family (_nH_s)_{n > 0} is schematically dense in $H$; it now suffices to apply 4.4.²⁷

Corollary 4.8. a) Let $u,v:H\to G$ be two homomorphisms of $S$-preschemes in groups, with $H$ of multiplicative type and²⁸ of finite type, and suppose that for every integer $n>0$, the restrictions of $u$ and $v$ to ${}_{nH}$ are identical. Then $u=v$.

b) Let $H_1,H_2$ be two subgroups of multiplicative type and of finite type of a prescheme in groups $G$, and suppose that for every integer $n>0$, one has _nH_1 = _nH_2; then $H_1=H_2$.

The first assertion follows from the second, by consideration of the subgroups H_1 and H_2 of $H\times G$, graphs of $u$ and $v$. To prove b), let $H=H_1\cap H_2$; this is a subprescheme in groups of $H_i$ ($i=1,2$); one must show that it is identical to $H_i$. Now the hypothesis means that it majorizes the _nH_i. One is therefore reduced to proving the:

Corollary 4.9. Let $G$ be an $S$-group of multiplicative type and of finite type, and $H$ a subprescheme in groups which majorizes the ${}_{nG}$, $n>0$. Then $H=G$.

By 4.7, one is reduced to proving that the subprescheme $H$ is closed, or again that it equals $G$ set-theoretically. This reduces us to the case where $S$ is the spectrum of a field; but then every subprescheme in groups of $G$ is closed (VI_B 1.4.2), whence the conclusion.

Remark 4.10. Under the conditions of 4.7, let $m$ be an integer > 0 having the following properties: for every $s\in S$, $m$ is not a power of the characteristic of $\kappa (s)$, and if $G_s$ is of type $M$, the prime divisors of the torsion of $M$ divide $m$. (N.B. This second condition is always satisfied if $G$ is a torus.) Then the proof given shows that in the statement of 4.7 and the Corollaries 4.8 and 4.9, one may restrict attention to subgroups of the form ${}_{(m^r)}G$, with $r>0$.

Moreover, one sees at once that when $G$ is smooth over $S$, then for every point $s\in S$, there exist an open neighborhood $U$ of $s$ and an integer $m>0$, prime to all the residual characteristics of $U$, satisfying the preceding conditions for G_U. (Take for example for $m$ the order of the torsion of the type of $G$ at $s$, cf. 1.4.1 a) and 2.1 e).) Under these conditions, one finds ${}_{(m^r)}G$ that are finite and étale over $S$, and whose family is schematically dense, provided one restricts to over $U$. This remark allows, in certain cases (notably those involving the theorems 3.2 and 3.2 bis, cf. XI 6, but not those involving the theorems 3.6 and 3.6 bis), to dispense with the theorem 3.1, which involves Hochschild cohomology.

5. Central homomorphisms of groups of multiplicative type

Lemma 5.0.²⁹ Let $(A,m)$ be a noetherian local ring, $S=\mathrm{Spec}(A)$, and $H$ a finite scheme over $S$, so $H=\mathrm{Spec}(B)$, where $B$ is finite over $A$. Let $K$ be a subscheme of $H$, such that by reduction modulo $m^{n+1}$ one has $K_n=H_n$ for every $n$. Then $K=H$.

Let $s$ be the closed point of $S$. Note first that $K$ is a closed subscheme of $H$. Indeed, it is a priori a closed subscheme of an open $U$ of $H$. But $K$, hence $U$, contains the fiber $K_s=H_s$; since the morphism $H\to S$ is finite, hence closed, it follows that the complement of $U$ is empty, i.e. $U=H$. Hence $K$ is defined by an ideal $I$ of $B$. The hypothesis entails that $I$ is contained in $m^{n}B$ for every $n$; since $B$ is a finite $A$-module, it is separated for the $m$-adic topology, whence $I=0$.

Theorem 5.1. Let $u,v:H\to G$ be two homomorphisms of $S$-preschemes in groups, with $H$ of multiplicative type and of finite type, and $S$ locally noetherian or $G$ of finite presentation over $S$. Let $s\in S$ be such that $u_s=v_s$, and suppose $u_s$ central. Then there exists an open neighborhood $U$ of $s$ such that $u_U=v_U$.

We distinguish the two cases:

a) $S$ locally noetherian.³⁰ Let $K=Ker(u,v)$ be the inverse image of the diagonal of $G{\times }_{S}G$ by the morphism $(u,v)$; this is a subprescheme in groups of $H$. We wish to find $U$ such that $K_U=H_U$. Note that, since $S$ is locally noetherian and $H$ of finite type over $S$, $H$ is locally noetherian, hence the immersion $K\hookrightarrow H$ is of finite type (cf. EGA I, 6.3.5). Hence $K$ is of finite type over $S$, hence of finite presentation over $S$, since $S$ is locally noetherian. Consequently, by EGA IV₃, 8.8.2.4, to show that there exists an open neighborhood $U$ of $s$ such that $K_U=H_U$, it suffices to show that $K_{S_0}=H_{S_0}$, where S_0 is the spectrum of $A=O_{S,s}$. One may therefore suppose $S$ local with closed point $s$. Replacing the noetherian local ring $A$ by its completion Â if necessary, which introduces a base change $\hat{S}\to S$ faithfully flat and quasi-compact, one may even, if one wishes, suppose³¹ $A$ complete.