Exposé XI. Representability criteria. Applications to multiplicative-type subgroups of affine group schemes

by A. Grothendieck

0. Introduction

As we have already seen examples in Exp. X, N°s 4, 5, the representability of certain functors, such as certain functors of type ${\mathrm{Hom}}_S(X,Y)$ and various variants, plays an important role in many questions concerning preschemes in groups.

Among the results particularly useful in this theory, let us mention (in addition to the questions of representability of quotients, studied in Exposés V and VI_B and in Exp. VIII 5) the question of the representability of functors of the form ${\prod }_{X/S}Y/X$ (where $Y$ is a subobject of $X$), studied in Exp. VIII 6 in a very elementary case, of which we shall give variants in N° 6 of the present Exposé; these results give us the representability of various centralizers, normalizers, transporters.

Less elementary representability criteria, using results that will appear in EGA VI, are indicated in 6.12 and in Exp. XV, XVI, where we shall give a criterion of representability of quotients $G/H$ in cases not covered by the preceding Exposés (a criterion that was not developed in the oral Exposés).

Our principal object in the present Exposé is the proof of theorems 4.1 and 4.2, which furnish a typical example of a non-projective construction technique (close to the one that will be developed in EGA VI). It has indeed appeared, since the oral Exposé and the writing of the present text, that the affine hypotheses made in 4.1 and 4.2 can to a large extent be eliminated (cf. XV), and that, on the other hand, one can, for the essential part of the theory developed in the following Exposé, do without 4.1 and 4.2. It might finally be interesting to prove the analogue of these results for a general reductive (for instance semisimple) group prescheme instead of a group of multiplicative type, in which case 4.1 and 4.2 will doubtless be the key result for the proof.

1. Reminders on smooth, étale, and unramified morphisms

The reader is referred to EGA IV §§ 17 & 18, and pending its publication, to SGA 1, I, II, III (where it is however appropriate to replace certain noetherian hypotheses, troublesome in applications, by hypotheses of finite presentation).

Definition 1.1. Let $S$ be a prescheme, $F$ a functor $(Sch{)}^{\circ }/S\to (Ens)$. One says that $F$ is formally smooth (resp. formally unramified ¹, resp. formally étale*) if for every $S$-prescheme $S^{\prime }$, affine (in the absolute sense), and every subscheme $S^{\prime \prime }$ of $S^{\prime }$ defined by a nilpotent ideal $J$, the map*

$$F(S^{\prime })⟶F(S^{\prime \prime })$$

is surjective (resp. injective, resp. bijective). A prescheme $X$ over $S$ is said to be formally smooth over $S$ (resp. formally unramified over $S$, resp. formally étale over $S$) if the corresponding functor is formally smooth (resp. formally unramified, resp. formally étale); one says that $X$ is smooth over $S$ (resp. unramified over $S$, resp. étale over $S$) if it satisfies the preceding condition and if moreover $X$ is locally of finite presentation over $S$.

The interest of these definitions for $X$ resides in the fact that, on the one hand, they are expressed in a remarkably simple way in terms of the functor represented by $X$ (and in practice, $X$ is often given as the object over $S$ representing some explicit functor), and that, on the other hand, they are also expressed by remarkable properties concerning the local structure of $X$, which we shall recall in the following statements (for the proof, we refer to loc. cit.).

Proposition 1.2. Let $X$ be a prescheme locally of finite presentation over $S$. Then:

(i) For $X$ to be smooth over $S$, it is necessary and sufficient that $X$ be flat over $S$, and that its geometric fibers $X{\otimes }_S\mathrm{Spec}(\kappa (s))$ be regular schemes. More generally, for $X$ to be smooth over $S$ in a neighborhood of the point $x\in X$ (one then says that $X$ is smooth over $S$ at $x$), it is necessary and sufficient that $X$ be flat over $S$ at $x$ and $X_s$ be smooth over $\kappa (s)$ at $x$, i.e. $X_s{\otimes }_{\kappa (s)}\bar{\kappa }(s)$ be regular at the points (or simply, a point) above $x$.

(ii) Suppose $S$, and hence $X$, locally noetherian; let $x\in X$ and let $s\in S$ be its image in $S$. Then the smooth nature of $X$ over $S$ at $x$ is detected on the local homomorphism $A=O_{S,s}\to B=O_{X,x}$ of noetherian local rings (or even on the local homomorphism $\hat{A}\to \hat{B}$ of their completions), by the following characteristic property: $B$ is flat over $A$, and $B{\otimes }_{A}k$ is geometrically regular over $k$ (the residue field of $A$), i.e. for every finite extension $k^{\prime }$ of $k$, $(B{\otimes }_{A}k){\otimes }_k{k}^{\prime }=B{\otimes }_A{k}^{\prime }$ is a regular semi-local ring. When the residual extension $k(B)/k(A)$ is trivial, these conditions are equivalent also to the following: $\hat{B}$ is isomorphic as Â-algebra to an algebra of formal power series $\hat{A}[[t_1,\cdots ,t_n]]$.

Thus, from the "formal" point of view, the structure of $X$ over $S$ is that of the typical affine space $S[t_1,\cdots ,t_n]$ over $S$.

Proposition 1.3. Let $X$ be a prescheme locally of finite presentation over $S$. Then:

(i) The following conditions are equivalent:

a) $X$ is unramified over $S$.

b) The diagonal morphism $X\to X{\times }_{S}X$ is an open immersion.

c) ${\Omega }_{X/S}^1=0$.

d) The geometric fibers of $X/S$ are discrete and reduced, i.e. isomorphic to sums of copies of the base field.

e) For every $s\in S$, the fiber $X{\otimes }_S\mathrm{Spec}\kappa (s)=X_s$ is unramified over $\kappa (s)$, or equivalently is isomorphic to a sum of spectra of finite separable extensions of $\kappa (s)$.

(ii) One has analogous conditions of pointwise nature for $X$ to be unramified over $S$ at a given point $x$ (i.e. in a neighborhood of the said point); for example, it is necessary and sufficient that $O_{X_s,x}$ be a finite separable extension of $\kappa (s)$, which is expressed also in terms of the local homomorphism $A=O_{S,s}\to B=O_{X,x}$ by the condition that $B{\otimes }_{A}k$ be a finite separable extension of $k$ (residue field of $A$), i.e. $r(A)B=r(B)$ and $k(B)$ is a finite separable extension of $k(A)$. When $A$, and hence also $B$, is noetherian, and $k(A)\stackrel{\sim }{\to }k(B)$, this means also that $\hat{A}\to \hat{B}$ is surjective.

Thus, from the "formal" point of view, to say that $X$ is unramified over $S$ means that $X$ is essentially a subprescheme of $S$. Observe also that conditions d) and e) are expressed solely in terms of the fibers of $X/S$.

Proposition 1.4. Let $X$ be a prescheme locally of finite presentation over $S$. For $X$ to be étale over $S$, it is necessary and sufficient that it be smooth over $S$ and unramified over $S$ (trivial by definition), which permits applying criteria 1.2 and 1.3. One finds in particular:

*(i) For $X$ to be étale over $S$, it is necessary and sufficient that it be flat over $S$ and unramified over $S$.

(ii) Suppose moreover $S$, and hence $X$, locally noetherian. Then the fact that $X$ is étale over $S$ at a point $x$ is detected on the local homomorphism $A=O_{S,s}\to B=O_{X,x}$ (and even on the local homomorphism $\hat{A}\to \hat{B}$) by the following characteristic property: $B$ is flat over $A$, and $B{\otimes }_{A}k$ is a finite separable extension of $k$ (residue field ² of $A$). When $k(A)\stackrel{\sim }{\to }k(B)$, this condition means simply that $\hat{A}\to \hat{B}$ is an isomorphism.

Thus, from the "formal" point of view, to say that $X$ is étale over $S$ means simply that $X$ is locally isomorphic to $S$.

Remark 1.5. When $X$ is locally of finite presentation over $S$ and one is given a point $x\in X$, then the fact that $X$ is smooth (resp. unramified, resp. étale) over $S$ at $x$, i.e. in a neighborhood of $x$, is detected on the functor $X:(Sch{)}^{\circ }/S\to (Ens)$ by the following property: for every $S^{\prime }$ over $S$, $S^{\prime }$ the spectrum of a local ring, every subscheme $S^{\prime \prime }$ of $S^{\prime }$ defined by a nilpotent ideal, and every $S$-morphism $u^{\prime \prime }:S^{\prime \prime }\to X$ sending the closed point $s^{\prime \prime }$ of $S^{\prime \prime }$ to $x$, there exists at least one (resp. at most one, resp. exactly one) $S$-morphism $u:S^{\prime }\to X$ which extends it. This statement shows in particular that in definition 1.1 one may restrict to $S^{\prime }$ which are local schemes (provided that the functor in question is representable by a prescheme locally of finite presentation over $S$). On the other hand, when $S$ is locally noetherian, one may even, in the preceding pointwise criterion, restrict to $S^{\prime }$ which are local artinian schemes, and one may thus impose the same restriction on $S^{\prime }$ in definition 1.1 (provided that $S$ is locally noetherian and the functor in question is represented by a prescheme locally of finite type over $S$).

Remark 1.6. Of course, given a morphism $f:X\to Y$ of preschemes, one will say that $f$ is smooth (resp. …) if $f$ makes $X$ into a smooth (resp. …) $Y$-prescheme. When $X$ and $Y$ are $S$-preschemes and $f$ an $S$-morphism of finite presentation, then these properties of $f$ are expressed immediately in terms of the morphism of functors $F,G:(Sch{)}^{\circ }/S\to (Ens)$ defined by $f$: $f$ is smooth (resp. unramified, resp. étale) if for every $S$-prescheme $S^{\prime }$, $S^{\prime }$ affine (in the absolute sense), and every subscheme $S^{\prime \prime }$ of $S^{\prime }$ defined by a nilpotent ideal $J$, the map

F(S′) ⟶ F(S″) ×_{G(S″)} G(S′)

deduced from the commutative square

$$F(S^{\prime })⟶G(S^{\prime })\downarrow \downarrow F(S^{\prime \prime })⟶G(S^{\prime \prime })$$

is surjective (resp. injective, resp. bijective), i.e. for every commutative square of morphisms of functors over $S$ (where $S^{\prime }$, $S^{\prime \prime }$ are as above and $i:S^{\prime \prime }\to S^{\prime }$ is the canonical immersion):

        i
   S″ ─────→ S′
   │          │
u″ │          │ v
   ↓    f     ↓
   F ──────→ G                            (Q)

there exists at least one (resp. at most one, resp. exactly one) morphism $u:S^{\prime }\to F$ making the two corresponding triangles commutative: $u^{\prime \prime }=ui$ and $v=fu$.

This property for a homomorphism of functors $f:F\to G$ over $S$ keeps its meaning even when the functors in question are not representable; one will say (if it is satisfied) that the homomorphism of functors $f:F\to G$ is formally smooth (resp. formally unramified, resp. formally étale). One will note that this depends only on the homomorphism of functors $(Sch{)}^{\circ }\to (Ens)$ defined by $F$, $G$ (cf. I 1.4.1), and not on the structural morphisms $F\to S$ and $G\to S$. An equivalent way of expressing the preceding definition, somewhat more manageable in applications, is the following: the morphism of functors $F\to G$ over $S$ is said to be formally smooth (resp. formally unramified, resp. formally étale) when, for every $S^{\prime }$ over $S$ and every morphism $S^{\prime }\to G$, the functor $F^{\prime }=F{\times }_G{S}^{\prime }$ over $S^{\prime }$ is formally smooth (resp. formally unramified, resp. formally étale).

Remark 1.7. Under the conditions of 1.6, when one is given a "point of $F$" with values in a field $k$, i.e. an element $x$ of $F(\mathrm{Spec}(k))$ or, what amounts to the same, a morphism $\mathrm{Spec}(k)\to F$, one says likewise that $f$ is formally smooth (resp. formally unramified, resp. formally étale) at $x$, if the condition of the preceding definition is satisfied whenever $S^{\prime }$ is a local scheme and the morphism $S^{\prime \prime }\to F$ in the diagram (Q) above is "compatible with the marked points" in the following sense: if $k^{\prime }$ denotes the residue field of $S^{\prime }$ and of $S^{\prime \prime }$ at their closed point, the diagram

Spec(k′)        Spec(k)
   │              │
 j ↓              ↓ x
   S″ ──────────→ F

(where $j:\mathrm{Spec}(k^{\prime })\to S^{\prime \prime }$ denotes the canonical immersion) can be completed in a commutative diagram

              Spec(k″)
             ╱        ╲
           ╱            ╲
         ↓                ↓
   Spec(k′)             Spec(k)
       │                  │
       ↓                  ↓
       S″ ──────────────→ F           ,

where $k^{\prime \prime }$ is the spectrum of a field. When $F$, $G$ are representable by $S$-preschemes $X$, $Y$ and the $S$-morphism $f:X\to Y$ is locally of finite presentation, this condition means precisely (by virtue of 1.5) that $f:X\to Y$ is smooth at the point of $X$ image of $\mathrm{Spec}(k)$ by $x:\mathrm{Spec}(k)\to X$.

Remark 1.8. When the condition of the preceding definition is satisfied while restricting to local artinian $S^{\prime }$, one will say that $F\to G$ is infinitesimally smooth (resp. infinitesimally unramified, resp. infinitesimally étale) at $x$, and one says that $F\to G$ is infinitesimally smooth (resp. …) if it is so at every point $x$, in other terms if the condition envisaged in 1.6 is satisfied whenever $S^{\prime }$ is local artinian. This variant of the preceding notions is technically useful, since it is often easier to verify, being a weaker notion, while being frequently sufficient (for example if $F\to G$ is a morphism locally of finite presentation, with $G$ representable by a prescheme locally noetherian…) to entail the strong condition.

Smooth morphisms of preschemes behave in a remarkably simple way with respect to differential calculus. We confine ourselves here to recalling the following property:

Proposition 1.9. Let $f:X\to S$ be a smooth morphism of preschemes. Then ${\Omega }_{X/S}^1$ is a locally free module of finite type over $X$, and its rank at a point $x\in X$ equals the dimension of the fiber $X_s$ (where $s=f(x)$) in a neighborhood of the point $x$.

This dimension is called the relative dimension of $X$ over $S$ at $x$. One will note that it is zero (when $f$ is smooth at $x$) if and only if $f$ is étale at $x$. This dimension is computed in practice still in terms of the functor $F$ represented by $X$, in the following way. Let $\xi$ be a point of $F$ with values in a field $k$, "localized at $x$", i.e. a morphism $\mathrm{Spec}(k)\to F=X$ whose image is $x$. Consider the algebra $D(k)=k[t]/(t^2)$ of dual numbers over $k$, considered as an $S$-prescheme, and consider the map

$$F(D(k))⟶F(k)$$

deduced from the augmentation $D(k)\to k$, and let finally $F(D(k),\xi )$ be the inverse image of $\xi \in F(k)$ under this map. Then this set is naturally endowed with a structure of vector space over $k$ (in fact, it is the vector space dual of ${\Omega }_{X/S}^1{\otimes }_{O_{X,x}}k$), whose dimension is the relative dimension of $X$ over $S$ at $x$.

To make explicit the vector law on $F(D(k),\xi )$, it is convenient to introduce more generally, as in Exp. II, for every vector space $V$ over $k$, the algebra $D_k(V)=k+V$ (with $V$ an ideal of square zero), and to consider $F(D_k(V),\xi )$, the inverse image of $\xi$ by $F(D_k(V))\to F(k)$, as a covariant functor in $V$, with values in (Ens). It then suffices that this functor commute with products of two factors (which means that $F$ transforms certain amalgamated sums of a very special type into fibered products, compare Exp. II — a condition always satisfied when $F$ is representable), to conclude that the $F(D_k(V),\xi )$ and in particular $F(D(k),\xi )$ are endowed with vector structures over $k$. One can thus define the relative dimension of $F$ over $X$ at the "point" $\xi$, under conditions appreciably broader than the representability of $F$.

In the present Exposé, the fact that certain functors which we shall make explicit are representable by preschemes smooth over $S$ will serve us mainly through the intermediary of the following result, which will be for us the technical intermediary to pass from constructions on the completion of the local ring $A$ of a point $s$ of a noetherian scheme $S$ to a local ring $A^{\prime }$ over $A$ étale over $A$, which will yield in particular a means of passing from $s$ to neighboring points:

Proposition 1.10. Let $f:X\to S$ be a smooth morphism of preschemes, $s$ a point of $S$, and $x$ a point of $X$ above $s$ such that $\kappa (x)$ be a finite separable extension of $\kappa (s)$. Then there exists a subscheme $S^{\prime }$ of $S$, étale over $S$, passing through $x$. Hence one can find an étale morphism $S^{\prime }\to S$, a point $s^{\prime }$ of $S^{\prime }$ above $s$ of residue extension equal to $\kappa (x)/\kappa (s)$, and an $S$-morphism $S^{\prime }\to X$ sending $s^{\prime }$ to $x$.

To construct $S^{\prime }$, one simply takes a system $g_1,\cdots ,g_n$ of sections of O_X on a neighborhood of $x$, which induce at $x$ a regular system of parameters of the local ring of the fiber $X_s$ at $x$; the subscheme $S^{\prime }$ defined by the $g_i$ is then étale over $S$ at $x$, and provided one shrinks $S^{\prime }$, it will then be étale over $S$.

We shall use 1.10 when $\kappa (x)=\kappa (s)$, i.e. $x$ is rational over $\kappa (s)$, i.e. $x$ can be considered as a section of $X_s$ over $\mathrm{Spec}(\kappa (s))$. Then 1.10 is in the nature of a theorem of extension of sections (after étale extension of the base). It takes a particularly simple form in the following special case:

Corollary 1.11. Under the conditions of 1.10, suppose that $S$ is the spectrum of a henselian local ring, and that $\kappa (x)=\kappa (s)$. Then there exists a section of $X$ over $S$ passing through $x$ (uniquely determined if $X$ is in fact étale over $S$ at $x$).

Indeed, $S$ being henselian, it follows under the conditions of 1.10 that $S^{\prime }$ contains an open subscheme which is finite over $S$ and whose fiber at $s$ is reduced to $x$. As it is étale over $S$, it follows that it is isomorphic to $S$, whence the conclusion. — One will note that when $S$ is the spectrum of a complete local ring, 1.10 or 1.11 is more or less the equivalent of the classical "Hensel's lemma", and one sometimes refers to it by this name.

2. Examples of formally smooth functors drawn from the theory of multiplicative-type groups

We are going to interpret, in the language introduced in the preceding N°, the results stated in IX 3 concerning infinitesimal extensions of a homomorphism of a group of multiplicative type (consequences of the vanishing of the Hochschild cohomology of such a group, established in Exposé I).

Proposition 2.1. Let $S$ be a prescheme, $H$ a group of multiplicative type over $S$, $G$ an $S$-prescheme in groups smooth over $S$; consider the functor over $S$

$$M_H={\mathrm{Hom}}_{S-gr.}(H,G)$$

(whose value at $S^{\prime }$ over $S$ is the set of homomorphisms of $S^{\prime }$-groups from $H_{S^{\prime }}$ to $G_{S^{\prime }}$). This functor is formally smooth over $S$.

Cf. IX 3.6. More generally:

Corollary 2.2. Let $S$, $G$ be as above, and consider a homomorphism $u:H_1\to H_2$ of group schemes of multiplicative type over $S$, whence with the preceding notation a morphism of functors over $S$:

M_{H_2} ⟶ M_{H_1}        (M_{H_i} = Hom_{S-gr.}(H_i, G)),

given by $w\mapsto w\circ u$. This homomorphism is formally smooth.

Indeed, by virtue of definitions 1.6, this is equivalent to the following statement: when $S$ is affine, $S^{\prime }$ a subscheme defined by a nilpotent ideal, let

$$v:H_1⟶G$$

be a homomorphism of $S$-groups, and

$$w^{\prime }:(H_2{)}_{S^{\prime }}⟶G_{S^{\prime }}$$

homomorphism of $S$-groups

$$w:H_2⟶G$$

extending $w^{\prime }$, and such that $wu=v$. To see this, one begins by extending $w^{\prime }$ to a homomorphism of $S$-groups $w_0:H_2\to G$, which is possible by 2.1; consider then $v^{\prime }=w_{0}u:H_1\to G$. It is such that $v_{S^{\prime }}^{\prime }=v_{S^{\prime }}$ by hypothesis on $w^{\prime }$, hence by virtue of VIII 3.6 there exists an element $g$ of $G(S)$, whose image in $G(S^{\prime })$ is the unit element, and such that $v=int(g)v^{\prime }$, whence $v=(int(g)w_0)u$, and it will therefore suffice to take $w=int(g)w_0$.

Corollary 2.3. With the notation of 2.1, consider $M_H=M$ as a functor with group of operators $G$ ($G$ operating by $(v,g)\mapsto int(g)\circ v$). Then the corresponding morphism

R : G ×_S M ⟶ M ×_S M

defined by $(g,v)\mapsto (int(g)\circ v,v)$ is a formally smooth morphism.

By means of a base change $S^{\prime }\to S$, this is equivalent to the following statement:

Corollary 2.4. With the notation of 2.1, let $v_1,v_2:H\to G$ be two morphisms of $S$-groups, and let $Transp(v_1,v_2)$ be the subfunctor of $G$ formed of the $g$ such that $int(g)\circ v_1=v_2$. Then this functor is formally smooth over $S$. In particular (if $v_1=v_2=v$) the functor $Centr(v)$, subgroup of $G$ formed of the $h$ such that $int(h)v=v$, is formally smooth over $S$.

(N.B. The pair $(v_1,v_2)$ can be considered as a section over $S$ of the second member in the morphism $R$ of 2.3, and $Transp(v_1,v_2)$ as the inverse image functor of the said section under $R$.) Statement 2.4 itself is equivalent to the following: when $S$ is affine and $S^{\prime }$ is a subscheme of $S$ defined by a nilpotent ideal, for every $g_0\in G(S^{\prime })$ such that $int(g_0)(v_1{)}_{S^{\prime }}=(v_2{)}_{S^{\prime }}$, $g_0$ extends to a $g\in G(S)$ such that $int(g)v_1=v_2$. To prove this, one begins by extending $g_0$ to a section $g^{\prime }$ of $G$ over $S$, which is possible since $G$ is smooth over $S$; one sets $v_2^{\prime }=int(g^{\prime })v_1$, one notes that $v_2$ and $v_2^{\prime }$ have the same restriction over $S^{\prime }$, hence by the result already invoked IX 3.6 there exists a $g^{\prime \prime }\in G(S)$, inducing the unit section over $S^{\prime }$, and such that $v_2=int(g^{\prime \prime })v_2^{\prime }$, whence v_2 = int(g″) int(g′) v_1 = int(g″ g′) v_1, so that it suffices to take $g=g^{\prime \prime }{g}^{\prime }$.

Proposition 2.1 bis. Let $S$ be a prescheme, $G$ an $S$-prescheme in groups smooth over $S$, and consider the functor $M:(Sch{)}^{\circ }/S\to (Ens)$:

M(S′) = set of multiplicative-type subgroups of G_{S′}.

Then $M$ is formally smooth over $S$.

Cf. IX 3.6 bis.

Corollary 2.2 bis. Let $n$ be an integer, and consider the morphism of functors

$${\varphi }_n:M⟶M$$

defined by

φ_n(H) = ₙH = Ker(n · id_H).

Then ${\varphi }_n$ is a formally smooth morphism. If for every integer $p>0$, $M_p$ denotes the subfunctor of $M$ such that $M_p(S^{\prime })$ is the set of multiplicative-type subgroups $H$ of $G_{S^{\prime }}$ such that ${}_{p}H=H$, then the morphism induced by ${\varphi }_n$:

$$M_{np}⟶M_n$$

is formally smooth.

The second assertion is trivially contained in the first and is included only for the convenience of a later reference. The proof of the first is analogous to that of 2.2, invoking this time IX 3.6 bis.

Corollary 2.3 bis. With the notation of 2.1 bis, consider $M$ as a functor with group of operators $G$ ($G$ operating by $(g,H)\mapsto int(g)(H)$). Then the corresponding morphism

R : G ×_S M ⟶ M ×_S M

defined by $(g,v)\mapsto (int(g)v,v)$ is formally smooth.

This is equivalent to the following statement:

Corollary 2.4 bis. With the notation of 2.1 bis, let $H_1,H_2$ be two multiplicative-type subgroups of $G$, and let $Trans{p}_G(H_1,H_2)$ be the subfunctor of $G$ formed of the $g$ such that $int(g)H_1=H_2$. Then this functor is formally smooth over $S$. In particular, if $H_1=H_2=H$, the subfunctor $Nor{m}_G(H)$ of $G$, normalizer of $H$, is formally smooth over $S$.

The proof is analogous to that of 2.4, invoking again IX 3.6 bis.

Proposition 2.5. Let $S$ be a prescheme, $G$ an $S$-prescheme in groups smooth over $S$, $K$ a sub-$S$-prescheme in groups, smooth over $S$ or of multiplicative type, $H$ an $S$-prescheme in groups of multiplicative type, $u:H\to G$ a homomorphism of $S$-groups, and denote by $Trans{p}_G(u,K)$ the subfunctor of $G$ whose value, at an $S^{\prime }$ over $S$, is formed of the $g\in G(S^{\prime })$ such that $int(g)u_{S^{\prime }}:H_{S^{\prime }}\to G_{S^{\prime }}$ factors through $K_{S^{\prime }}$. Then this functor is formally smooth over $S$.

The proof is analogous to that of 2.2, invoking IX 3.6 and X 2.1 (the latter in the case $K$ of multiplicative type).

3. Auxiliary representability results

Proposition 3.1. Let $F\to S$ be a functor over the prescheme $S$. The following conditions are equivalent:

(i) $F$ is representable, and $F\to S$ is an open immersion (one also says simply that $F\to S$ is an open immersion).

(ii) $F$ is a sheaf for the faithfully flat quasi-compact topology, "commutes with inductive limits of rings", $F\to S$ is a monomorphism, and finally the following condition is satisfied: for every local prescheme $S^{\prime }$ over $S$, of residue field $k$, and every $S$-morphism $\mathrm{Spec}(k)=S^{\prime \prime }\to F$, there exists an $S$-morphism $S^{\prime }\to S$ which extends it.

(iii) (When $S$ is locally noetherian.) $F$ is a sheaf for the faithfully flat quasi-compact topology, "commutes with inductive limits of rings", "commutes with adic projective limits of local artinian rings", $F\to S$ is a monomorphism, and is infinitesimally étale (cf. 1.8).

Let us first make precise two points of terminology.

Remark 3.2. One says that a functor ³ $F$ over $S$ "commutes with inductive (understood: filtered) limits of rings" if for every filtered projective system $(S_i^{\prime }{)}_{i\in I}$ of $S$-affine schemes above an affine open of $S$, with rings $A_i^{\prime }$, the natural homomorphism

(*)    lim_→ F(S′_i) → F(S′)   (where S′ = Spec A′, A′ = lim_→ A′_i)

is bijective. One will note that $S^{\prime }$ is none other than the projective limit of $(S_i^{\prime })$ in the category of preschemes (and even of all ringed spaces), so that the condition envisaged is in the nature of a right-exactness condition (commutation with certain inductive limits in $(Sch{)}^{\circ }/S$), just as the condition of being a sheaf for some topology. One will pay attention to the fact that the condition envisaged is essentially relative, i.e. involves the morphism $F\to S$ and not only the functor $F:(Sch{)}^{\circ }\to (Ens)$; more precisely, in (*), $F(S_i^{\prime })$ and $F(S^{\prime })$ denote ${\mathrm{Hom}}_S(S_i^{\prime },F)$, ${\mathrm{Hom}}_S(S^{\prime },F)$. Thus, when $F$ is representable, the condition envisaged means that $F$ is locally of finite presentation over $S$. (And we have used several times, in the last two Exposés, the fact that a functor represented by an $S$-prescheme locally of finite presentation commutes with inductive limits of rings.)

Remark 3.3. One says that a functor ³ $F$ over $S$ commutes with adic projective limits of local artinian rings if for every $S^{\prime }$ over $S$ which is the spectrum of a complete noetherian local ring $A^{\prime }$, setting $S_n^{\prime }=\mathrm{Spec}(A^{\prime }/rad(A^{\prime }{)}^{n+1})$, the natural map

$$(\ast \ast )F(S^{\prime })⟶\underleftarrow{\lim }F(S_n^{\prime })$$

is bijective. One will note that this condition, which is in the nature of a left-exactness condition, is satisfied whenever $F$ is representable. One sees easily that, contrary to the condition of commutation with inductive limits of rings, it is intrinsic to $F$ as an element of $Ob(\hat{S}ch)$, i.e. it does not involve the morphism $F\to S$.

Remark 3.4. Let $F$ be a functor over $S$ which is a sheaf for the Zariski topology, or as one also says, which is "of local nature". (It suffices for this that $F$ be a sheaf for a finer topology, such as the faithfully flat quasi-compact topology.) Let $(S_i)$ be a covering of $S$ by opens; then one easily verifies (by a method of gluing pieces) that $F$ is representable if and only if the $F_i=F{\times }_S{S}_i$ are, which allows for example reduction to the case where $S$ is affine. Suppose that the functor $F$ of local nature commutes with inductive limits of rings. Then, for $F$ to be representable, it is necessary and sufficient that its restriction to the category of preschemes locally of finite presentation over $S$ be representable. The "necessary" was pointed out in 3.2; the "sufficient" amounts to this: if $X$ is a prescheme locally of finite presentation over $S$ and $X\to F$ a morphism such that, for every $S^{\prime }$ locally of finite presentation over $S$, the induced morphism

Hom_S(S′, X) ⟶ Hom_S(S′, F)

is bijective, then $X\to F$ is an isomorphism. Now this results easily from the fact that $X$ and $F$ are two functors of local nature which commute with inductive limits of rings.

Let us now prove 3.1. The implications (i) ⇒ (ii) and (i) ⇒ (iii) are evident; let us prove the inverse implications.

One has (ii) ⇒ (i). Let $U$ indeed be the set of $s\in S$ such that the canonical monomorphism $\mathrm{Spec}(\kappa (s))\to S$ factors through $F$. By virtue of the last condition (ii), for every $s\in U$, the canonical monomorphism $\mathrm{Spec}(O_{S,s})\to U$ factors through $F$. Noting that $O_{S,s}$ is the inductive limit of the rings of affine neighborhoods of $s$, it follows from the fact that $F$ commutes with inductive limits of rings that for every $s\in S$, there exists an open neighborhood $U_s$ such that the canonical immersion $U_s\to S$ factors through $F$. This implies $U_s\subset U$, so $U$ is open. As $F\to S$ is a monomorphism, and $F$ is of local nature, the $S$-morphisms $U_s\to F$ glue on the intersections $U_s\cap U_{s^{\prime }}$ ($s,s^{\prime }\in U$), hence come from an $S$-morphism $U\to F$. It remains to prove that this is an isomorphism, hence that every $S$-morphism $S^{\prime }\to F$ factors uniquely through $U$ (where $S^{\prime }$ is an $S$-prescheme). As $F\to S$ and $U\to S$ are monomorphisms, it amounts to the same to say that the structural morphism $S^{\prime }\to S$ factors through $U$, which reduces us to the case where $S^{\prime }$ is the spectrum of a field, hence reduced to a single point $s^{\prime }$. Let $s$ be the point of $S$ below $s^{\prime }$; I say that the $S$-morphism $S^{\prime }\to F$ factors through $\mathrm{Spec}(\kappa (s))=S_0\to F$ (which implies $s\in U$ and will prove what we want).

It amounts to the same, since $S^{\prime }\to S_0$ is covering for fpqc and $F$ is a sheaf for this topology, that the two composites

S″ = S′ ×_{S_0} S′ ⇉ S′ → F

are the same, which follows from the fact that $F\to S$ is a monomorphism.

One has (iii) ⇒ (ii) (when $S$ is locally noetherian). It suffices to prove the last condition of (ii), and moreover (by virtue of the preceding proof) it suffices to do so when $S^{\prime }$ is of the form $\mathrm{Spec}(O_{S,s})$, with $s\in S$. Let $A=O_{S,s}$, $A_n=A/m^{n+1}$, $S_n=\mathrm{Spec}(A_n)$. Then it follows from the hypothesis that $F\to S$ is infinitesimally smooth, that the given morphism $S_0\to F$ extends to morphisms $S_n\to F$. As $F\to S$ is a monomorphism, one thus obtains an element of $\underleftarrow{\lim }F(S_n)$, and since $F$ commutes with adic projective limits of local artinian rings, the $S_n\to F$ come from a morphism $\mathrm{Spec}(\hat{A})={\hat{S}}^{\prime }\to F$. As $F\to S$ is a monomorphism, $F$ a sheaf for fpqc, and ${\hat{S}}^{\prime }\to S^{\prime }$ covering for the said topology, this morphism ${\hat{S}}^{\prime }\to F$ factors through $S^{\prime }\to F$, which completes the proof.

Proposition 3.5. Let $S$ be a locally noetherian prescheme, $F\to S$ a functor over $S$, $(X_i,u_i{)}_{i\in I}$ a family of $S$-morphisms $u_i:X_i\to F$, where the $X_i$ are preschemes locally of finite type over $S$. Suppose the following conditions are satisfied:

a) $F$ is a sheaf for the faithfully flat quasi-compact topology, commutes with inductive limits of rings, commutes with adic projective limits of local artinian rings.

b) The $u_i:X_i\to F$ are monomorphisms, and are infinitesimally étale (cf. 1.8).

c) The family of the $u_i$ is "set-theoretically surjective".

Under these conditions, $F$ is representable by a prescheme locally of finite type over $S$ (and the $u_i$ are open immersions, which make the family of the $X_i$ into an open covering of $F$).

Remark 3.6. Proceeding as at the end of Remark 1.7, one defines, for every functor $F:(Sch{)}^{\circ }\to (Ens)$, an "underlying set" $set(F)$ as a quotient set of the set of points of $F$ with values in fields (for the equivalence relation made precise in 1.7). When $F$ is representable by $X$, one recovers the underlying set of $X$. Evidently $set(F)$ depends functorially on $F$, so that if $G\to F$ is a morphism of functors, one will say that this morphism is set-theoretically surjective if the induced map $set(G)\to set(F)$ is surjective. This means therefore also that every point of $F$ with values in a field $k$ "comes from" a point of $G$ with values in a suitable extension of $k$. This definition extends immediately to the case of a family of morphisms $G_i\to F$, which makes precise the meaning of c).

Let us prove 3.5. For this, let us introduce, for $(i,j)\in I\times I$,

X_{i,j} = X_i ×_F X_j,

and consider the projections

v_{i,j} : X_{i,j} ⟶ X_i    and   w_{i,j} : X_{i,j} ⟶ X_j.

I say that these last are representable by open immersions. To see it, one applies criterion 3.1 (iii): $X_{i,j}$ satisfies the three exactness conditions (being a sheaf, commuting with $\underrightarrow{\lim }$ of rings and with adic $\underleftarrow{\lim }$ of local artinian rings), since $F$, $X_i$, $X_j$ satisfy them, and these conditions are stable under finite projective limits, in particular under fibered products; since $X_i\to F$ is a monomorphism, so is $v_{i,j}:X_{i,j}\to X_i$ deduced from it by base change $X_j\to F$, and symmetrically $v_{j,i}$ is a monomorphism; finally the "infinitesimally étale" condition is also preserved by base change. This proves that one is under the conditions of 3.1 (iii).

We may now use the $X_i$, $X_{i,j}$, $v_{i,j}$ and $w_{i,j}$ to construct in the usual way an $S$-prescheme $X$ such that the $X_i$ are identified with opens of $X$, the $X_{i,j}$ with the intersections $X_i\cap X_j$, and the $v_{i,j}$, $w_{i,j}$ with the canonical immersions. Note that $X$ is also the quotient of

$$Y={⨿}_i{X}_i$$

by the equivalence relation $R={⨿}_{i,j}{X}_{i,j}$ (the two projections v, w : R ⇉ Y being defined by the $v_{i,j}$ resp. the $w_{i,j}$). More precisely, $F$ being a sheaf for fpqc, the $u_i:X_i\to F$ come from a $u:Y\to F$, and $R$ is none other than the equivalence relation defined by $u$, $R=Y{\times }_{F}Y$; finally the quotient $X=Y/R$ is also a quotient in the category of fpqc-sheaves (and even, in the category of sheaves for the Zariski topology): it suffices to use the definitions of "quotient" and "sheaf" to convince oneself. Consequently, $u$ factors uniquely through a morphism

$$\bar{u}:X⟶F,$$

and this morphism is a monomorphism. It remains to show that it is an isomorphism. As $F$ is of local nature, one may suppose $S$ affine, and as moreover $F$ commutes with inductive limits of rings, it suffices to verify that for every $T$ affine of finite type over $S$, every morphism $T\to F$ factors through $X$ (cf. 3.4). For this, consider $G=X{\times }_{F}T\to T$. It suffices to prove that this is an isomorphism. Now, $T$ being noetherian, one sees as above that this is an open immersion (N.B. $X\to F$ is infinitesimally étale, as follows at once from the fact that the induced morphisms $u_i:X_i\to F$ are). Now by hypothesis $X\to F$ is set-theoretically surjective, and one sees at once that this is a condition stable under base change, hence $G\to T$ is set-theoretically surjective, hence an isomorphism since it is an open immersion. QED.

Proposition 3.7. Let $S$ be a locally noetherian prescheme, $I$ a filtered increasing index set, $(T_i{)}_{i\in I}$ a projective system of $S$-preschemes locally of finite type, $T=\underleftarrow{\lim }{T}_i$ the projective limit functor, $F$ a functor over $S$, $u:F\to T$ an $S$-morphism. Suppose the following conditions are satisfied:

a) $F$ is a sheaf for the faithfully flat quasi-compact topology, commutes with inductive limits of rings, and with adic projective limits of local artinian rings.

b) The morphism $u:F\to T$ is a monomorphism.

b′) The morphism $u:F\to T$ is infinitesimally étale.

c) For every point $\xi$ of $F$ with values in the spectrum of a field $k$, denoting by ${\xi }_i\in T_i(\mathrm{Spec}(k))$ its image and by $t_i$ the corresponding element of $T_i$, there exists an $i\in I$ such that for $j⩾i$ the transition morphism $p_{i,j}:T_j\to T_i$ is étale at $t_j$.

d) For every prescheme $X$ locally of finite type over $S$, and every $S$-morphism $X\to F$, the set of $x\in X$ at which this morphism is infinitesimally étale is open.

Under these conditions, $F$ is representable by a prescheme locally of finite type over $S$.

Let us note immediately that in the case which will occupy us in the following N°, one will verify conditions c) and d) through the following corollary:

Corollary 3.8. Granting a), b), b′), conditions c) and d) are implied by the following:

c′) The $T_i$ are smooth over $S$, and the transition morphisms $p_{i,j}:T_j\to T_i$ are smooth.

d′) For every point $\xi$ of $F$ with values in the spectrum of a field $k$, let $t_i(\xi )$ be the element of $T_i$ defined by $\xi$, $d_i(\xi )$ the relative dimension of $T_i$ over $S$ at $t_i(\xi )$, and $d(\xi )=\sup d_i(\xi )$. Then:

1°) for every $\xi$ as above, one has $d(\xi )<+\infty$, and

2°) for every prescheme $X$ locally of finite type over $S$ and every $S$-monomorphism $v:X\to F$, the function $x\mapsto d({\xi }_x)$ on $X$ is locally constant (where for $x\in X$ one denotes by ${\xi }_x$ the point of $F$ with values in $\kappa (x)$ induced by $v$).

Let us prove 3.7. Let us place ourselves under the conditions of c); let $t$ be the element of $set(F)$ defined by $\xi$ (cf. 3.6) and set

O_t = lim_→ O_{T_i, t_i}.

Then using the condition stated in c), one sees easily that $O_t$ is a noetherian local ring (EGA 0_IV 10.3.1.3). Its residue field $\kappa (t)$ is the inductive limit of the residue fields $\kappa (t_i)$, and $k$ is an extension of it. If $\eta$ denotes the spectrum of $\kappa (t)$, $\xi$ that of $k$, one has a commutative diagram

ξ ─→ F
↓     ↓
η ─→ T          ,

whose definition is evident and left to the reader. As $\xi \to \eta$ is covering for the fpqc topology, $F$ is a sheaf for it by virtue of a), and $F\to T$ is a monomorphism by virtue of b), one sees at once that the morphism $\eta \to T$ factors (uniquely) through a morphism $\eta \to F$. Let us note now the

Lemma 3.9. Under the conditions of 3.7, let $T_0=\mathrm{Spec}(A)$ be a noetherian local scheme, $T_0^{\prime \prime }=\mathrm{Spec}(k(A))$, $i:T_0^{\prime \prime }\to T_0$ the canonical immersion, and suppose given a commutative square of morphisms

T_0″ ─→ F
  │      │
i ↓      ↓
T_0  ─→ T       .

Then there exists a unique $T$-morphism $T_0\to F$.

The proof is that of 3.1 (iii) ⇒ (ii) (where the fact that the $S$ of the cited statement be representable was not used), using that $F$ is a sheaf for the fpqc topology, commutes with adic projective limits of local artinian rings (in this case the $A/rad(A{)}^{n+1}$), and that $F\to T$ is a monomorphism and is infinitesimally étale.

We shall apply lemma 3.9 to the case where $T_0=\mathrm{Spec}(O_t)$, hence $T_0^{\prime \prime }=\mathrm{Spec}(\kappa (t))$, noting that we have just constructed $T_0^{\prime \prime }\to F$, and that the canonical homomorphisms $O_{T_i,t_i}\to O_t$ define a projective system of morphisms $\mathrm{Spec}(O_t)\to T_i$, whence a canonical morphism $\mathrm{Spec}(O_t)\to T$. The commutativity of the corresponding square (×) is trivial (since by definition of $T$, it suffices to verify it with $T$ replaced by $T_i$), whence a unique $T$-morphism

v′ : T_0 = Spec(O_t) ⟶ F.

As $F$ commutes with inductive limits of rings, this morphism factors through

v′_i : T′_i = Spec(O_{T_i, t_i}) ⟶ F

for $i$ large. For such an $i$, one has

T_0 ⥲ T′_i      i.e.        O_{T_i, t_i} ⥲ O_t.

Indeed, as $T_0\to T_i^{\prime }$ is faithfully flat quasi-compact, hence an effective epimorphism, it suffices to see that it is a monomorphism. Now if one has two morphisms R ⇉ T_0 (with $R$ representable) having the same composite with $T_0\to T_i^{\prime }$, they have the same composite with $T_0\to T$ (which factors through $T_0\to T_i^{\prime }$), hence the same composite with $T_0\to T_j$ for every $j$, hence with $T_0\to T_j^{\prime }$ for every $j$, hence are equal because T_0 is the projective limit of the $T_j^{\prime }$ in the category (Sch).

Consider the diagram

              v′
        T_0 ───────→ F
         │          ↗
         │       ↗ v′_i
         ↓     ↗
       T′_i ──────→ T_i           ,

where the unspecified arrows are the obvious ones. The square is commutative and so is the upper triangle, hence as $T_0\to T_i^{\prime }$ is an epimorphism, it follows that the lower triangle is commutative. Now $O_{T_i,t_i}$ is the inductive limit of the affine rings of affine open neighborhoods of $t_i$ in $T_i$, hence as $F$ commutes with inductive limits of rings, $v_i^{\prime }$ comes from a morphism

$$v_t:U_t⟶F$$

from an open neighborhood $U_t$ of $t_i$ in $T_i$. One sees at once that, restricting this neighborhood if necessary, $v_t$ is necessarily a $T_i$-morphism ($T_i$ being locally of finite type over $S$, hence also commuting with inductive limits of rings). Then the composite of $v_t$ with $F\to T_i$ is a monomorphism, hence $v_t$ is a monomorphism. I say that it is infinitesimally étale at $t_i$. Since $F\to T$ is infinitesimally étale, it amounts to the same to say that the composite $T_i\to F\to T$ is infinitesimally étale at $t_i$, or again that the induced morphism $T_i^{\prime }\to T$ is infinitesimally étale, or finally (since $T_0\stackrel{\sim }{\to }{T}_i^{\prime }$) that $T_0\to T$ is infinitesimally étale, which is immediate, this morphism being the projective limit of the infinitesimally étale morphisms $T_i^{\prime }\to T_i$.

Let us now apply condition d), which had not yet been used; it follows that $v_t$ is infinitesimally étale in a neighborhood of $t$, hence, restricting $U_t$ to a smaller open if necessary, one may suppose that $v_t$ is a monomorphism infinitesimally étale.

For $t\in set(F)$ variable, the family of morphisms $v_t$ is amenable to 3.5, which implies the conclusion of 3.7.

Let us prove now 3.8, supposing satisfied conditions c′) and d′) of 3.8. Then with the notation of d′), one will have $d_i(\xi )=$ constant for $i$ large, hence the relative dimension of $T_j$ over $T_i$ at $t_j$, equal to $d_j(\xi )-d_i(\xi )$, is zero, hence $T_j\to T_i$ is étale at $t_j$, which proves condition c). Consequently the preceding proof applies to give, for each $t\in set(F)$, an index $i$, an open neighborhood $U_t$ of $t_i$, and a morphism $U_t\to F$ which is a monomorphism, infinitesimally étale at $t_i$, and all reduces to proving that this morphism is infinitesimally étale in a neighborhood of $t_i$. Now with the notation of d′) 2°), where one takes $X=U_t$, one may suppose, replacing $U_t$ by the connected component of $t_i$ in $U_t$ if necessary, that for every $x\in U_t$, one has

d(ξ_x) = d(ξ_{t_i}) for every x ∈ U_t.

Now $d({\xi }_{t_i})$ = relative dimension of $T_i$ over $S$ at $t_i$, and as the relative dimension of $T_i$ over $S$ remains constant on $U_t$, one will have also, for every $x\in U_t$:

(*)    d(ξ_x) = relative dimension of U_t over S at x.

On the other hand, with the notation of the proof of 3.7, one sees at once that for every local noetherian prescheme $R_0=\mathrm{Spec}(A)$, setting $R_0^{\prime \prime }=\mathrm{Spec}k(A)$, every morphism $R_0\to F$ such that the induced morphism $R_0^{\prime \prime }\to F$ factors through $F$ (i.e. sends the closed point of R_0 to $t\in set(F)$) factors (uniquely in an obvious way) through T_0. (The proof is that of 3.1 or 3.9: one uses that $T_0\to F$ is a monomorphism infinitesimally étale at $t$, and that T_0 is a sheaf for fpqc commuting with adic $\underleftarrow{\lim }$ …). Applying this result to the morphism $\mathrm{Spec}(O_{U_t,x})=R_0\to F$ induced by $v_t$ and to the point $t_x\in set(F)$ image of the closed point of R_0, i.e. image of $x$ by $v_t$, one finds a factorization

Spec(O_{U_t, x}) ⟶ Spec(O_{t_x}) ⟶ F

and as the second arrow is infinitesimally étale at $t_x$, to prove that the composite is so at $x$, it suffices to prove that the first is so at $t_x$. Now thanks to formula (×) above, it is an $S$-homomorphism of localizations, at two points $x$, $t_x$, of smooth $S$-preschemes $U_t$, $U_{t_x}$ of the same relative dimension $d$ over $S$ at $t$, $t_x$. This morphism is induced by an $S$-morphism

$$w:U⟶V$$

where $U$ is an open neighborhood of $x$ in $U_t$, and $V=U_{t_x}$ (EGA I 6.5.1). It all reduces to proving that this morphism is étale at $x$. Moreover, $U$ and $V$ are equipped with monomorphisms $U\to F$, $V\to F$, and one sees at once that, restricting $U$ further if necessary, $w$ is an $F$-morphism, hence $w$ is a monomorphism. It now suffices to prove the

Lemma 3.10. Let $U$, $V$ be two smooth $S$-preschemes, of the same relative dimension $d$ over $S$, and $w:U\to V$ an $S$-morphism which is a monomorphism; then $w$ is an open immersion (and a fortiori is étale).

By virtue of SGA 1, I 5.7, one is reduced to the case where $S$ is the spectrum of a field, which one may suppose algebraically closed. By virtue of SGA 1, I 5.1, it suffices to prove that $w$ is étale, and it suffices to prove this at the closed points of $U$. Let $x$ be such a point, $y=w(x)$; then, taking a regular system of parameters $f_1,\cdots ,f_d$ of $O_{V,y}$, one sees that $A=O_{U,x}/\sum f_i{O}_{U,x}$ is the trivial extension of $k=O_{V,y}/\sum f_i{O}_{V,y}$ (for $w$ being a monomorphism, so is the structural morphism $\mathrm{Spec}(A)\to \mathrm{Spec}(k)$ deduced from it by base change). As $O_{U,x}$ is a regular local ring of dimension $d$, it follows that the $f_i$ form a regular system of parameters of this ring, which immediately implies that $w$ is étale at $x$ and completes the proof of 3.10, hence of 3.8.

Corollary 3.11. Under the conditions of 3.8, for every quasi-compact open $U$ of $F$ separated over $S$, there exists an $i\in I$ such that for every $j⩾i$ the morphism $u_j|U:U\to T_j$ is an open immersion. In particular, if the $T_i$ are quasi-affine over $S$, then every open $U$ of $F$ quasi-compact over $S$, i.e. of finite type over $S$, is quasi-affine over $S$.

The proof of 3.7 shows that for every $t\in F$, there exists an $i\in I$ such that $u_i:F\to T_i$ is a local isomorphism at $t$, and then $u_j$ is a local isomorphism at $x$ for every $j⩾i$. By reason of quasi-compactness, one may choose $i$ independent of $x\in U$. It remains to prove that for $i$ large, $u_i|U:U\to T_i$ is a monomorphism. Now as $U\to T$ is a monomorphism, one sees that the intersection of the equivalence relations $U{\times }_{T_i}U\subset U{\times }_{S}U$ is reduced to the diagonal, and as $U{\times }_{S}U$ is a noetherian prescheme and the $U{\times }_{T_i}U$ closed subpreschemes, it follows that one of these $U{\times }_{T_i}U$ is already reduced to the diagonal, i.e. $u_i|U$ is a monomorphism. This proves the first assertion in 3.11, and the second is an immediate consequence of it.

Proposition 3.12. Let $S$ be a prescheme, $G$ an affine $S$-prescheme in groups.

a) Let $F$ be the functor $(Sch{)}^{\circ }/S\to (Ens)$ such that, for every $T$ over $S$,

F(T) = set of multiplicative-type subgroups of G_T which are finite over T.

Suppose $S$ locally noetherian or $G$ of finite presentation over $S$. Then the functor $F$ is representable and is affine over $S$. If $G$ is of finite presentation over $S$, then $F$ is locally of finite presentation over $S$.

b) Let $H$ be an $S$-prescheme in groups of multiplicative type, and finite over $S$. Then ${\mathrm{Hom}}_{S-gr.}(H,G)$ is representable. It is affine over $S$, and if $G$ is of finite type (resp. of finite presentation) over $S$, the same holds for ${\mathrm{Hom}}_{S-gr.}(H,G)$.

Remark 3.13. Except for the precision that ${\mathrm{Hom}}_{S-gr.}$ is affine, and in the case where $G$ is of finite presentation over $S$ (which will suffice for us), 3.12 is an immediate consequence of the theory of Hilbert schemes (A. Grothendieck, Techniques de construction et théorèmes d'existence en Géométrie Algébrique: IV Les Schémas de Hilbert, Séminaire Bourbaki, May 1961, N° 221). It even suffices that $G$ be quasi-projective over $S$; in case a), one may also represent the larger functor

F′(T) = { set of subpreschemes in groups of G_T,
          flat, proper and of finite presentation over T }

(the canonical monomorphism $F\to F^{\prime }$ is an "open immersion", as follows from criterion 3.1 and from X 4.7 b), so that the representability of $F^{\prime }$ entails that $F$ is representable by an open of $F^{\prime }$); in case b), one may restrict to supposing that $H$ is projective and of finite presentation over $S$. In both cases, one obtains a functor locally of finite presentation over $S$. In the present Exposé, 3.12 is only a technical lemma for proving a key result in the following N°, so we shall sketch an easy direct proof of 3.12, not using Hilbert schemes.

Let us first prove b). It will suffice to prove that ${\mathrm{Hom}}_S(H,G)$ is representable (independently of any group structure on $H$, $G$), and has the supplementary properties stated for ${\mathrm{Hom}}_{S-gr.}(H,G)$, since using also the same result for ${\mathrm{Hom}}_S(H{\times }_{S}H,G)$, one makes explicit the subfunctor ${\mathrm{Hom}}_{S-gr.}(H,G)$ of ${\mathrm{Hom}}_S(H,G)$ by a finite projective limit (in fact, with the help of fibered products) involving $G$, ${\mathrm{Hom}}_S(H,G)$ and ${\mathrm{Hom}}_S(H{\times }_{S}H,G)$, which we leave to the reader to make explicit. On the other hand, one will have

$$H=\mathrm{Spec}(B),$$

where $B$ is a sheaf of algebras over $S$ which is locally free as a sheaf of modules (this is the only hypothesis on $H$ that we shall need to retain). As the representability question envisaged is local on $S$, we may suppose $S$ affine with ring $A$. On the other hand, one will have $G=\mathrm{Spec}(C)$, where $C$ is an $A$-algebra. When $G=S[t]=G_0$, $t$ an indeterminate, the functor Hom is none other than

$$T\mapsto \Gamma (T,B_T),$$

which is representable ($B$ being locally free) by the vector bundle $V(B^{\vee })$, where $B^{\vee }$ is the sheaf of modules dual to $B$. When $G=S[(t_i)]$, with $(t_i)$ a (not necessarily finite) family of indeterminates, one will have therefore $G=G_0^I$ (product over $S$ of a family of copies of G_0), which is representable by the affine scheme

Hom_S(H, G)^I = V(B^∨)^I = V(B^{∨(I)}).

In the general case, $G$ will be isomorphic to a closed subscheme of a scheme of the form $S[(t_i)]$, i.e. $C$ will be a quotient of an $A$-algebra of the form $A[(t_i)]$. Let $(F_j)$ be a system of generators of the ideal by which one divides. Suppose $B$ free of rank $n$, which is permissible by covering $S$ by smaller affine opens. Choose a basis $(e_k{)}_{1⩽k⩽n}$ of $B$; then writing the $n$ components in this basis of $F_j((x_i))$, for $x_i=\sum x_{ik}{e}_k$ ($x_{ik}$ indeterminate coefficients, taken in an unspecified algebra $A^{\prime }$ over $A$), one finds, for each $F_j$, $n$ polynomials $F_{j,k}$ in the $(x_{i,k}{)}_{i\in I,1⩽k⩽n}$, with coefficients in $A$. One verifies at once that ${\mathrm{Hom}}_S(H,G)$ is represented by the spectrum of the quotient of the polynomial ring $A[(x_{i,k})]$ by the ideal generated by the $F_{j,k}$. This proves at once b).

Let us prove a). For every ordinary finite commutative group $M$, let F_M be the subfunctor of $F$ obtained by restricting to subgroups of G_T which are of multiplicative type and of type $M$. One sees easily, by a gluing argument like the one which served in 3.5 (which we should have stated by unscrewing a bit more!), that it suffices to verify that the F_M are representable; then $F$ will be representable by the prescheme sum of the F_M, where $M$ runs over the set of classes of finite commutative groups up to isomorphism. ($F$ is in fact the sum of the F_M in the category of sheaves…).

From now on we suppose $M$ fixed, and we shall write $F$ instead of F_M. Let $H=D_S(M)$; consider the subfunctor $F^{\prime }=Im{m}_{S-gr.}(H,G)$ of ${\mathrm{Hom}}_{S-gr.}(H,G)$ whose value at $T$ is the set of homomorphisms of $T$-groups $H_T\to G_T$ which are closed immersions. We know already that ${\mathrm{Hom}}_{S-gr.}(H,G)$ is representable by an affine $S$-prescheme by virtue of b), and using IX 6.8, one sees at once that $F^{\prime }$ is representable by an open and closed subprescheme of the latter, hence it is also affine over $S$. Consider finally the canonical morphism $F^{\prime }\to F$, which associates to each monomorphism $H_T\to G_T$ the image group. One verifies at once, by virtue of the definitions, that in this way $F^{\prime }$ becomes a principal homogeneous bundle (in the category of fpqc sheaves) over $F$, with group ${\Gamma }_F={\Gamma }_S{\times }_{S}F$, where $\Gamma ={\mathrm{Aut}}_{gr.}(M)$, whence it follows "by descent" that this morphism is representable (i.e. for every morphism $T\to F$, with $T$ representable, $T{\times }_F{F}^{\prime }$ over $T$ is representable — in fact, representable by a principal galois bundle under $\Gamma$). Hence by virtue of IV 4.6.6, $F$ is representable if and only if the quotient $F^{\prime }/F^{\prime \prime }$, where $F^{\prime \prime }=F^{\prime }{\times }_F{F}^{\prime }$, exists in (Sch) and is universally effective for faithfully flat quasi-compact morphisms, or what amounts to the same, if and only if the quotient $F^{\prime }/\Gamma$ exists and is universally effective for the said morphisms. Now as $F^{\prime }$ is affine over $S$, one has seen in V 4.1 that the condition in question is indeed satisfied. This proves the representability of $F$ in a).

As for the complement, relative to the case where one supposes $G$ of finite presentation over $S$, it follows at once from the preceding proof, taking into account that by virtue of b), $F^{\prime }$ is then locally of finite presentation over $S$, so that one may apply Exposé V⁴.

4. The scheme of multiplicative-type subgroups of a smooth affine group

The principal result of the present Exposé is constituted by

Theorem 4.1. Let $S$ be a prescheme, $G$ an $S$-prescheme in groups smooth and affine over $S$, $F$ the functor $(Sch{)}^{\circ }/S\to (Ens)$ defined by

F(T) = set of multiplicative-type subgroups of G_T.

Then the functor $F$ is representable, and is smooth and separated over $S$.

Let us signal at once the following variant:

Corollary 4.2. Let $G$, $H$ be two $S$-preschemes in groups, with $G$ smooth and affine over $S$, $H$ of multiplicative type and of finite type. Then ${\mathrm{Hom}}_{S-gr.}(H,G)$ is representable, and is smooth and separated over $S$.

Indeed, when $H$ is smooth over $S$, so is $H{\times }_{S}G$, and one may apply 4.1 to this latter. By the consideration of graph subgroups, ${\mathrm{Hom}}_{S-gr.}(H,G)$ becomes a subfunctor $F^{\prime }$ of the functor $F$ of "multiplicative-type subgroups of $H{\times }_{S}G$", namely $F^{\prime }(T)$ = set of multiplicative-type subgroups $K$ of $(H{\times }_{S}G{)}_T=H_T{\times }_T{G}_T$ such that the homomorphism induced by the first projection

$$K⟶H_T$$

is an isomorphism. By virtue of IX 2.9, the canonical morphism $F^{\prime }\to F$ is an open immersion, hence $F$ being representable, so is $F^{\prime }$, which will be representable by an open of $F$; and $F$ being separated and smooth over $S$, so will $F^{\prime }$ be. In the case where $H$ is finite over $S$, it suffices to apply 3.12 b) for the representability of ${\mathrm{Hom}}_{S-gr.}(H,G)$. When $H$ is a product $H_1{\times }_S{H}_2$, with H_1 smooth over $S$ and H_2 finite over $S$, then ${\mathrm{Hom}}_{T-gr.}(H_T,G_T)$ is identified with the subset of ${\mathrm{Hom}}_{T-gr.}((H_1{)}_T,G_T)\times {\mathrm{Hom}}_{T-gr.}((H_2{)}_T,G_T)$ formed of pairs $(u_1,u_2)$ such that $u_1\cdot u_2=u_2\cdot u_1$, whence it follows that ${\mathrm{Hom}}_{S-gr.}(H,G)$ is representable by a closed subprescheme of ${\mathrm{Hom}}_{S-gr.}(H_1,G){\times }_S{\mathrm{Hom}}_{S-gr.}(H_2,G)=X$, as one sees by applying VIII 6.5 b), where one takes $Y=H$, $Z=G$, $q_1$ and $q_2$ being defined respectively by $(u_1,u_2)\mapsto u_1\cdot u_2$ and $(u_1,u_2)\mapsto u_2\cdot u_1$.

In the general case, the question being local on $S$ for the Zariski topology, one may suppose that $S$ is affine, and $H$ of constant type over $S$. Then $H$ being quasi-isotrivial (X 4.5), one can find an étale surjective morphism $S^{\prime }\to S$, $S^{\prime }$ affine, that splits $H$, i.e. such that $H^{\prime }=H_{S^{\prime }}$ is diagonalizable. Then the preceding result applies, since a diagonalizable group is the product of a diagonalizable torus by a diagonalizable finite group. It remains to see that the descent datum obtained on the $S^{\prime }$-prescheme ${\mathrm{Hom}}_{S^{\prime }-gr.}(H^{\prime },G^{\prime })=X^{\prime }$ is effective. This is seen by the reasoning of X 5.4, noting that in loc. cit., the hypothesis that $X^{\prime }\to S^{\prime }$ was separated and locally quasi-finite served only to ensure that every open part of $X^{\prime }$ quasi-compact over $S^{\prime }$ was quasi-affine over $S^{\prime }$. Now this property is still verified in the present case, as follows easily from the fact that this is so for the functor $F$ of 4.1 (which will be seen in the course of the proof of 4.1). This (or any other variant of this little unscrewing) establishes the representability of ${\mathrm{Hom}}_{S-gr.}(H,G)$, and at the same time the fact that it is separated over $S$. It is locally of finite presentation over $S$, as one sees for example (as was pointed out in 3.2) thanks to the fact that this functor "commutes with inductive limits of rings". Finally, this functor being formally smooth over $S$ (by virtue of 2.1), it is smooth over $S$.

Remark 4.3. We have here deduced 4.2 from 4.1, which is really immediate only when $H$ is also smooth over $S$. For the deduction to be made without contortions for the general case, the representability result 4.1 would have to be established without supposing $G$ smooth over $S$, but only affine of finite presentation over $S$. (Of course, then $F$ will no longer be smooth in general over $S$!) There is little doubt that 4.1 remains true under these more general hypotheses, but the proof appears to have to be more delicate (failing the possibility of invoking 3.8)[^XI-4-1]⁵. Let us point out however that when $G$ is a closed subgroup of an affine and smooth group G_0 over $S$, then the functor $F$ representing the multiplicative-type subgroups of $G$ is representable by a closed subprescheme of the prescheme representing the analogous functor F_0 for G_0 (amenable to 4.1), as one sees easily by applying VIII 6.4. This raises also the question: is an affine $S$-group scheme $G$, which is affine and of finite presentation over $S$, isomorphic to a subgroup scheme of some $GL(n{)}_S$, $n$ suitable? This is true when $S$ is the spectrum of a field, cf. VI_B 11.11, but unfortunately false in general, even for tori, cf. 4.6. Finally, note that one could also prove 4.2 directly by exactly the same method as 4.1.

Let us now prove 4.1. Since the functor $F$ is evidently of local nature, one may suppose $S$ affine, so that $S=\mathrm{Spec}(A)$, $A$ a ring. Considering $A$ as the inductive limit of its subrings of finite type over $\mathbb{Z}$, and noting that $G$ comes from a smooth and affine group over such a subring (EGA IV 8), one is reduced to the case where $S$ is noetherian. For every integer $n>0$, let $T_n$ be the functor defined as $F$, but restricting to subgroups $H$ of multiplicative type of G_T such that $n\cdot i{d}_H=0$, i.e. such that ${}_{n}H=H$. Order the set $I$ of integers > 0 by the divisibility relation. When $m$ is a multiple of $n$, define

$$p_{n,m}:T_m⟶T_n$$

functors over $S$. By virtue of 3.12 a), the functors $T_n$ are representable, affine and of finite type over $S$. Define likewise morphisms

$$u_n:F⟶T_n,$$

by the relation $u_n(H)={}_{n}H$. In this way, one obtains a morphism

u : F ⟶ T = lim_← T_n,

where the $\underleftarrow{\lim }$ is taken in the category of functors over $S$. But let us point out at once that, the $T_n$ being representable and affine over $S$, so is $T$ (it will be the spectrum of the inductive limit of the quasi-coherent algebras on $S$ which define the $T_n$). Of course, $T$ is not in general of finite type over $S$.

We are going to apply 3.7 and are reduced to verifying conditions a) to d) of 3.7, which will imply that $F$ is representable by a prescheme locally of finite type over $S$. It then follows from 2.1 bis that $F$ is even smooth over $S$, and as $F$ is a subfunctor of $T$ which is affine over $S$, it follows that $F$ is separated over $S$ (being separated over $T$, which is separated over $S$). Let us prove at once the complement invoked above, namely that every open $U$ of $F$ quasi-compact over $S$ is quasi-affine over $S$: this follows from 3.11 and from the fact that the $T_n$ are affine over $S$.

Let us verify therefore the conditions of 3.7.

a) $F$ is a sheaf for the faithfully flat quasi-compact topology, by descent theory SGA 1, VIII, which applies here since multiplicative-type groups over $T$ are affine over $S$. It commutes with inductive limits of rings by the general carpet ⁶ EGA IV 8. Let us show that it commutes with adic projective limits of local artinian rings. When dealing with, instead of the functor $F$, the analogous functor envisaged in 4.2, this property is none other than that of IX 7.1 in the special case where $A$ is a complete noetherian local ring, equipped with an ideal of definition for its usual topology (N.B. This is exactly where the hypothesis $G$ affine intervenes in an essential way). In the present case, we are reduced to proving the

Lemma 4.4. Let $A$ be a complete noetherian local ring, with maximal ideal $m$, $G$ a group scheme affine over $S=\mathrm{Spec}(A)$. For every integer $n⩾0$, let $S_n=\mathrm{Spec}(A/m^{n+1})$, $G_n=G{\times }_S{S}_n$. Let for every $n$, $H_n$ be a subgroup of multiplicative type and of finite type of $G_n$, such that for $m⩾n$, $H_n$ is deduced from $H_m$ by reduction. Under these conditions, there exists a unique multiplicative-type subgroup $H$ in $G$ which reduces along the $H_n$.

By virtue of X 3.2 there exists a group of multiplicative type $H$ over $S$, necessarily of finite type and isotrivial, determined up to unique isomorphism, equipped with an isomorphism $H{\times }_S{S}_0\simeq H_0$ (using the fact that H_0 is isotrivial, being of finite type over a field, cf. X 1.4). By virtue of X 2.1, for every $n$, the isomorphism $H{\times }_S{S}_0\simeq H_0$ lifts to a unique isomorphism $H{\times }_S{S}_n\simeq H_n$. Having said this, by virtue of IX 7.1 already cited, the homomorphisms $H_n\to G_n$ come from a unique homomorphism of $S$-groups $u:H\to G$. By virtue of IX 6.6, this last is a monomorphism since $u_0:H_0\to G_0$ is one. This completes the proof of 4.4.

b) The morphism $u:F\to T$ is a monomorphism. This follows from the density theorem IX 4.7, in the form of corollary 4.8 b). One will pay attention to the fact that it is essential, for the application we make of it here, to dispose of this result over a not necessarily noetherian base. (N.B. As the functor $T$ does not commute with inductive limits of rings, it is not possible to reduce to that a priori.)

b′) The morphism $u:F\to T$ is infinitesimally étale; in other terms one has the

Lemma 4.5. Let $A$ be an artinian local ring of residue field $k$, $S=\mathrm{Spec}(A)$, $I$ an ideal $\subset rad(A)$, $S_0=\mathrm{Spec}(A/I)$, $G$ an $S$-prescheme in groups smooth over $S$, $G_0=G{\times }_S{S}_0$; let us give ourselves a multiplicative-type subgroup H_0 of G_0 and for every integer $n>0$, a multiplicative-type subgroup $H(n)$ of $G$ such that

1°) for every multiple $m$ of $n$, $H(n)={}_{n}H(m)$, and