Exposé XV. Complements on sub-tori of a group scheme. Application to smooth groups

by M. Raynaud ¹

0. Introduction

This Exposé complements and partially recasts Exposés XI and XII; the contents of Exposés XIII and XIV are not indispensable. Continuing the effort undertaken in XII, we shall work with $S$-preschemes in groups that are not necessarily affine and not necessarily separated over $S$.

Sections 1, 2, 3, 4 are devoted to the study of sub-tori of a group prescheme. We obtain theorems of infinitesimal lifting (§ 2) and global lifting (§ 4), in which an essential role is played by points of finite order (§ 1).

Sections 5, 6 and 7 are independent of the preceding ones. The consideration of infinitesimal neighborhoods leads to the representability of the functor of smooth subgroups equal to their connected normalizer (§ 5). In §§ 6 and 7, we turn more specifically to Cartan subgroups.

Finally, in § 8 we give a necessary and sufficient condition for the functor of sub-tori of a smooth group, or that of maximal tori, to be representable.

1. Lifting of finite subgroups

1.1. Finite, smooth and central subgroups of multiplicative type

Proposition 1.1. Let $S$ be an affine scheme, $S_0$ a closed subscheme of $S$ defined by an ideal of square zero, $G$ an $S$-prescheme in groups, $H_0$ a subgroup scheme of $G_0=G{\times }_S{S}_0$ which is smooth over $S_0$, of finite type, and of multiplicative type. Then, in order that there exist a subgroup scheme of $G$, of multiplicative type, which lifts $H_0$, it is necessary and sufficient that there exist a subscheme $H^{\prime }$ of $G$, flat over $S$, which lifts $H_0$.

The necessity of the condition is clear; let us prove sufficiency. The group $H_0$ of multiplicative type is quasi-isotrivial (X 4.5); by Exp. X 2.1, there exist an $S$-group $H$ of multiplicative type and an $S_0$-isomorphism of groups:

u₀ : H ×_S S₀ ⥲ H₀.

Since $H^{\prime }$ is flat over $S$ and $H^{\prime }{\times }_S{S}_0$ is of finite presentation over $S_0$ (Exp. IX 2.1 b)), $H^{\prime }$ is of finite presentation over $S$; moreover, its fibers are smooth, so $H^{\prime }$ is smooth over $S$ (EGA IV 17.5.1). Since $S$ is affine, $u_0$ therefore lifts to an $S$-morphism of preschemes:

$$u:H⟶H^{\prime }.$$

It then follows from Exp. III 2.1 and Exp. IX 3.1 that the composed morphism $v_0:H{\times }_{S_0}S\stackrel{\sim }{\to }{H}_0\to G_0$

also lifts to an $S$-morphism of groups:

$$v:H⟶G.$$

Since $v_0$ is an immersion, so is $v$. The image of $H$ by $v$ is therefore a subgroup scheme of $G$, of multiplicative type, which lifts $H_0$.

Proposition 1.2. Let $S$ be a prescheme, $S_0$ a subprescheme of $S$ defined by a locally nilpotent sheaf of ideals, $G$ an $S$-prescheme in groups, flat and of finite presentation over $S$, and $H_0$ a subgroup scheme of $G_0=G{\times }_S{S}_0$ which is smooth, finite over $S_0$, of multiplicative type and central. Then there exists a unique subgroup scheme $H$ of $G$, of multiplicative type, which lifts $H_0$. Moreover $H$ is central. (See XVII App. III, 1).

Proposition 1.2 bis. Let $S$, $G$, $S_0$, $G_0$ be as above, $H$ an $S$-group scheme of multiplicative type, smooth and finite over $S$, and $u_0:H_0=H{\times }_S{S}_0\to G_0$ a central homomorphism. Then $u_0$ lifts uniquely to a homomorphism $u:H\to G$. Moreover $u$ is central.

The existence of the lifting $u$ in 1.2 bis is easily deduced from 1.2 by considering the graph of $u_0$. The lifting $u$ is unique and central by Exp. IX 3.4 and Exp. IX 5.1.

Proof of 1.2. The uniqueness of $H$ and the fact that $H$ is central follow from Exp. IX 5.6 b) and Exp. IX 3.4 bis. Given uniqueness, in order to prove the existence

of $H$ we may assume $S$ affine and $S_0$ defined by an ideal of square zero, and it suffices (1.1) to find a subscheme of $G$, flat over $S$, which lifts $H_0$.

Since $H_0$ is smooth and finite over $S_0$, we may assume — possibly after restricting $S$ — that there exists an integer $n>0$, invertible on $S$, such that $H_0={}_n{H}_0$. Consider the $n$-th power morphism in $G$:

u : G ⟶ G,    x ↦ xⁿ.

We still denote by ${}_{n}G$ the "kernel of $u$", that is, the inverse image under $u$ of the unit section of $G$ (N.B. $u$ is not in general a group morphism). Granting for a moment the following lemma:

Lemma 1.3. Let $k$ be a field, $G$ a group scheme locally of finite type over $k$, $n$ an integer prime to the characteristic of $k$, $u$ the $n$-th power morphism in $G$. Then $u$ is étale at every point $x$ of $G$ belonging to the center of $G$.

Since $G$ is flat and of finite presentation over $S$, it follows from the preceding lemma and from EGA IV 17.8.2 that if $x$ is a point of $G$ projecting to $s$ in $S$ and belonging to the center of $G_s$, the morphism $u$ is étale at $x$. If moreover $x$ is a point of ${}_{n}G$, then ${}_{n}G$ is étale over $S$ at $x$. By hypothesis, the group $H_0$ is central and contained in ${}_n{G}_0$, so it is in fact contained in the largest open subset $V$ of ${}_{n}G$ which is étale over $S$. Since $H_0$ and $V{\times }_S{S}_0=V_0$ are étale over $S_0$, $H_0$ is an open subset of $V_0$ (EGA IV 17.8.7 and 17.9.1).

But then the open subprescheme of $V$ having the same underlying space as $H_0$ is a subscheme of $G$, flat over $S$, which lifts $H_0$.

It remains to prove Lemma 1.3. For this, note that the largest open subset of $G$ on which $u$ is étale is invariant under base field extension (EGA IV 17.8.2); this allows us to reduce to the case where $x$ is rational over $k$. Let $t_x$ denote translation by $x$, which is a $k$-automorphism of the scheme $G$. Since $x$ is in the center of $G$, we have the commutative diagram:

G ──u──→ G
│        │
tₓ       tₓⁿ
│        │
▼        ▼
G ──u──→ G.

It therefore suffices to show that $u$ is étale at the origin, but this was seen in VII_A 8.4.

1.2. Global lifting of finite groups

Lemma 1.4. Let $A$ be a local ring, separated and complete for the topology defined by its maximal ideal $\mathfrak{m}$, let $S=\mathrm{Spec}(A)$, $S_n=\mathrm{Spec}(A/{\mathfrak{m}}^n)$. Then for every prescheme $X$ (resp. every $S$-prescheme $X$), the canonical map

(*)    Hom(S, X) ⟶ lim_{←n} Hom(Sₙ, X)

(resp.

(**)   Γ(X/S) ⟶ lim_{←n} Γ(Xₙ/Sₙ),    where Xₙ = X ×_S Sₙ)

is bijective.

Statement (**) is an easy consequence of (). Let us prove ().

Let $u_n:S_n\to X$ ($n\in \mathbb{N}$) be a coherent system of morphisms. The image $y$ of the closed point of $S_n$ is therefore independent of $n$, and $u_n$ factors through $\mathrm{Spec}({\mathcal{O}}_y)$. The morphisms $u_n$ define, by passage to the projective limit, a ring morphism

ũ : 𝒪_y ⟶ lim_{←n} (A/𝔪ⁿ) = A.

This shows that (*) is surjective; it is injective as soon as $A$ is separated for the $\mathfrak{m}$-adic topology.

Corollary 1.5. Let $A$ be a complete noetherian local ring, $\mathfrak{m}$ its maximal ideal, $S=\mathrm{Spec}(A)$, $S_n=\mathrm{Spec}(A/{\mathfrak{m}}^n)$, $X$ a finite scheme over $S$ and $Y$ an $S$-prescheme. Then the canonical map

Hom_S(X, Y) ⟶ lim_{←n} Hom_{Sₙ}(Xₙ, Yₙ)

(where $X_n=X{\times }_S{S}_n$ and similarly $Y_n=\cdots$) is bijective.

Indeed, it follows from EGA II 6.2.5 that $X$ is a finite sum of local $S$-schemes finite over $S$. This reduces us to the case where $X$ itself is the spectrum of a complete noetherian local ring. But ${\mathrm{Hom}}_S(X,Y)=\Gamma (Z/X)$ where $Z$ is the $X$-prescheme $Y{\times }_{S}X$, and we apply the preceding proposition.

Proposition 1.6. Let $A$, $S$, $S_n$ be as above, and let $G$ and $M$ be two $S$-preschemes in groups, with $M$ finite over $S$. Then:

a) The canonical map

Hom_{S-gr}(M, G) ⟶ lim_{←n} Hom_{Sₙ-gr}(Mₙ, Gₙ)

is bijective.

b) If $M$ is of multiplicative type and $G$ is smooth over $S$, the canonical map

φ : Hom_{S-gr}(M, G) ⟶ Hom_{S₀-gr}(M₀, G₀)

is surjective. Moreover, if $\varphi (u)=\varphi (u^{\prime })=u_0$, then $u$ and $u^{\prime }$ are conjugate by an element of $G(S)$ reducing to the unit element of $G(S_0)$.

c) If $M$ is of multiplicative type and smooth over $S$, if $G$ is flat of finite type over $S$, and if $u_0:M_0\to G_0$ is a central homomorphism, then $u_0$ lifts uniquely to a homomorphism

$$u:M⟶G.$$

Moreover $u$ is central if $G$ has connected fibers.

Proof. a) Follows from 1.5, from the fact that $M{\times }_{S}M$ is finite over $S$, and from the characterization of group morphisms as those rendering commutative the well-known diagram:

M ×_S M ──u×u──→ G ×_S G
   │                │
   │                │
   ▼                ▼
   M  ────u───────→ G.

b) By Exp. IX 3.6, one can construct a coherent system of homomorphisms $u_n:M_n\to G_n$ lifting a homomorphism $u_0:M_0\to G_0$. Hence the first assertion of b), in view of a).

If now $u$ and $u^{\prime }$ are two liftings of $u_0$, then $u_n$ and $u_n^{\prime }$ are conjugate

by an element $g_n$ of $G(S_n)$ lifting the unit element of $G(S_0)$ (Exp. IX 3.6); loc. cit. also implies that one may choose the $g_n$ in a coherent way, and hence (1.4) coming from a section $g$ of $G(S)$. The morphisms $u$ and $int(g)\cdot u^{\prime }$ then coincide modulo ${\mathfrak{m}}^n$ for every $n$, so they coincide (1.5).

c) The existence and uniqueness of $u$ follow from a) and 1.2 bis. If $G$ has connected fibers, $u$ is central by Exp. IX 5.6 a).

2. Infinitesimal lifting of sub-tori

2.1. Statement of the theorem

We shall give a theorem of infinitesimal lifting of sub-tori of a group prescheme which does not appeal to smoothness hypotheses (in contrast to Exp. IX 3.6 bis) and which moreover answers a very natural question²: does it suffice to be able to lift "enough" points of finite order of a sub-torus $T_0$ in order to be assured of being able to lift $T_0$ (infinitesimally, of course)?

Theorem 2.1. Let $S$ be a noetherian affine scheme, $S_0$ a closed subscheme of $S$ defined by an ideal $J$ of square zero, $G$ an $S$-prescheme in groups of finite type, $G_0=G{\times }_S{S}_0$, $T_0$ a sub-torus of $G_0$, $q$ an integer > 0 invertible on $S$. Suppose that for every integer $n$ equal to a power of $q$, there exists a subscheme $M_n$ of $G$, flat over $S$, such that $M_n{\times }_S{S}_0={}_n{T}_0$. Then there exists a sub-torus $T$ of $G$ such that $T{\times }_S{S}_0=T_0$.

Theorem 2.1 will be useful to us through the following two corollaries:

Corollary 2.2. Let $S$ be a locally noetherian prescheme, $S_0$ a closed subprescheme of $S$ defined by a locally nilpotent sheaf of ideals, $G$ an $S$-prescheme in groups of finite type, $T_0$ a sub-torus of $G_0=G{\times }_S{S}_0$, $q$ an integer > 0 invertible on $S$; finally, with the integer $n$ ranging over powers of $q$, let $(M_n)$ be a coherent system

of $S$-subgroup schemes of $G$, of multiplicative type, which lifts the ${}_n{T}_0$ (N.B. The system of subgroups of multiplicative type $(M_n)$ is said to be coherent if $M_m={}_m(M_n)$ whenever the integer $m$ divides $n$.) Then there exists one and only one sub-torus $T$ of $G$ such that $T{\times }_S{S}_0=T_0$ and ${}_{n}T=M_n$ for every $n$.

Corollary 2.3. Let $G$ be an $S$-prescheme in groups, flat and of finite presentation over $S$, $S_0$ a closed subprescheme of $S$ defined by a sheaf of ideals of finite type and locally nilpotent, $T_0$ a central torus of $G_0=G{\times }_S{S}_0$. Then there exists one and only one sub-torus $T$ of $G$ lifting $T_0$. Moreover $T$ is central.

Remark 2.4. We leave to the reader the task of formulating the analogue of statements 2.1, 2.2, 2.3 in which, instead of lifting a sub-torus of $G_0$, one is given a torus $T$ over $S$ and one proposes to lift a morphism

$$u_0:T_0⟶G_0$$

(one reduces to the preceding cases by considering the graph of $u_0$).

Let us show how Corollaries 2.2 and 2.3 are deduced from 2.1.

Proof of Corollary 2.2. The uniqueness of $T$ follows from Exp. IX 4.8 b) and Exp. IX 4.10. To prove the existence of $T$, we may therefore reduce to the case where $S$ is affine, hence noetherian, and where $S_0$ is defined by an ideal of square zero.

Lemma 2.5. Let $G$ be an $S$-prescheme in groups, of finite presentation, and $H$ a subgroup scheme of $G$, of multiplicative type, smooth over $S$. Then $Cent{r}_G(H)$ is representable by a subgroup prescheme of $G$, of finite presentation.

The lemma follows from Exp. VIII 6.5 e), without smoothness hypothesis on $H$, when $G$ is separated over $S$. In the present case, one notes that the assertion to be proved is local for the fpqc topology, which allows us to assume $H$ diagonalizable, hence of the form $D_S(M)$. We may also assume $S$ affine, then $S$ noetherian thanks to EGA IV 8. Since $H$ is smooth over $S$ and of finite type, the order $q$ of the torsion subgroup of $M$ is invertible on $S$ (Exp. VIII 2.1 e)). It is then immediate (cf. Exp. IX 4.10) that the subgroups ${}_{n}H$, where $n$ ranges over powers of $q$, are schematically dense in $H$ (Exp. IX 4.1). But ${}_{n}H$ is a completely decomposed covering of $S$ (i.e. is isomorphic to a finite direct sum of copies of $S$), so $Cent{r}_G({}_{n}H)=Z_n$ is representable as the intersection of the centralizers in $G$ of the sections of ${}_{n}H$ over $S$. It then suffices to apply the lemma:

Lemma 2.5 bis. Let $S$ be a noetherian prescheme, $G$ an $S$-prescheme in groups of finite type, $H$ a subgroup of $G$ of multiplicative type, $(H_i)$ an increasing filtered family of subgroups of $G$ of multiplicative type, and suppose that $Z_i=Cent{r}_G(H_i)$ is representable by a subgroup prescheme of $G$. Then the family of Zᵢ is stationary.

If moreover the Hᵢ are schematically dense in $H$, one has $Z_i=Cent{r}_G(H)$ for $i$ large enough.

To see that the family of Zᵢ is stationary, it suffices to show that the family of underlying sets $ens(Z_i)$ is stationary. Indeed, the stationary value will then be a closed subset of an open subset $U$ of $G$; and, possibly replacing $G$ by $U$, we are reduced to studying a decreasing filtered family of closed subpreschemes of a noetherian prescheme. An easy constructibility argument reduces us to the case where $S$ is integral. We must then show that the family of $ens(Z_i)$ is stationary above some non-empty open subset of $S$. Now the generic fiber of $G$ is separated (Exp. VI_A 0.3), so, possibly restricting $S$, we may assume $G$ separated over $S$ (EGA IV 8). But then Zᵢ is closed in $G$ (Exp. VIII 6.5 e)).

To establish the last assertion of the lemma, denote by $Z$ the stationary value of the family Zᵢ. It is clear that $Cent{r}_G(H)$ is a subfunctor of $Z$; let us show that $Z$ centralizes $H$. Let $E$ be the subprescheme of $H{\times }_{S}Z$ which is the kernel of the pair of morphisms:

H ×_S Z ⇒ G
(h, c) ↦ c
(h, c) ↦ hch⁻¹.

The prescheme $E$ majorizes $H_i{\times }_{S}Z$ for every $i$. On the other hand, the Hᵢ are flat over $S$, so (EGA IV 11.10.9) for every point $s$ of $S$, the $(H_i{)}_s$ are schematically dense in $H_s$ and the $H_i{\times }_{S}Z$ are schematically dense in $H{\times }_{S}Z$. Since $G_s$ is separated, $E_s$ is closed in $(H{\times }_{s}Z{)}_s$ and therefore equal to it. But then $E$ is closed in

$H{\times }_{S}Z$, so equals $H{\times }_{S}Z$. This says that $Z$ centralizes $H$.

Let us return to the proof of 2.2. By 2.5, $Z_n=Cent{r}_G(M_n)$ is representable, and by 2.5 bis the decreasing family of subpreschemes $Z_n$ is stationary; let $Z$ be its stationary value. The group $Z$ majorizes $T_0$ and the $M_n$. Possibly replacing $G$ by $Z$, we may therefore assume the $M_n$ central.

We are then in the conditions of application of Theorem 2.1, and there exists a sub-torus $T$ of $G$ lifting $T_0$. The groups ${}_{n}T$ and $M_n$ are then two liftings of ${}_n{T}_0$, hence are conjugate (Exp. IX 3.2 bis) and consequently coincide, $M_n$ being central. The torus $T$ answers the question.

Proof of Corollary 2.3. The uniqueness of $T$ follows from Exp. IX 5.1 bis, and the fact that $T$ is central follows from IX 5.6 b). This remark allows us, by the usual procedure, to reduce to the case where $S$ is affine (so $S_0$ is defined by a nilpotent ideal of finite type), then to the case where $S$ is noetherian. Possibly restricting $S$, we may assume that there exists an integer $q$ invertible on $S$. Corollary 2.3 is then a consequence of 2.2 and of 1.2.

Remark 2.6. One easily shows that Corollary 2.3 remains true if one replaces the torus $T_0$ by a smooth central subgroup of multiplicative type of $G_0$.

2.2. Proof of 2.1

a) Reduction to the case $T_0=G_{m,S_0}$.

Thanks to 1.1, we may assume that $M_n$ is a subgroup of multiplicative type. Using Exp. IX 3.2 bis, we may assume that the family of the $M_n$ is coherent (2.2). The centralizer $Z_n$ of $M_n$ in $G$ is representable (2.5), and the filtered family of $Z_n$ is stationary (2.5 bis). Possibly replacing $G$ by $Z_n$ for $n$ large enough, we may therefore assume $T_0$ and the $M_n$ central. The uniqueness of the lifting of $T_0$ is then assured (Exp. IX 5.1 bis).

Proceeding as in the proof of 1.1, we may assume that there exist an $S$-torus $T$ and an $S_0$-isomorphism

u₀ : T ×_S S₀ ⥲ T₀,

and it is equivalent to lift $u_0$ or to lift $T_0$. In view of uniqueness, it suffices to prove the existence of a lifting of $u_0$ after performing a faithfully flat affine extension $S^{\prime }\to S$ of finite type (fpqc descent), which allows us to assume $T=G_{m,S}^r$ (Exp. X 4.5). If the restriction of $u_0$ to each factor $G_m$ lifts to an $S$-morphism — necessarily central — one immediately deduces a lifting of $u_0$. In short, we may assume $T_0=G_{m,S_0}$.

b) Definition of the obstruction to the existence of a lifting of $T_0$.

To prove 2.1, it suffices by 1.1 to find a subscheme of $G$, flat over $S$, which lifts $T_0$. We shall see that one can define the obstruction to the existence of such a lifting as an element of a certain $Ex{t}^1(\cdot ,\cdot )$ of sheaves of modules.

Let $U$ be an open subset of $G$ such that $T_0$ is closed in $U$, and let us still denote by $U$ (resp. $U_0$) the open subscheme of $G$ (resp. $G_0$) having $U$ as underlying space. The sheaf ${\mathcal{O}}_{T_0}$, viewed as a sheaf on $U$, is therefore a quotient of ${\mathcal{O}}_{U_0}$. Let $h$ be the canonical epimorphism:

$$h:{\mathcal{O}}_{U_0}⟶{\mathcal{O}}_{T_0}.$$

Lemma 2.7. The canonical map

h̃ = id_J ⊗ h : J ⊗_{S₀} 𝒪_{U₀} ⟶ J ⊗_{S₀} 𝒪_{T₀}

factors (evidently uniquely) as $i\circ j_U$, where $j_U$ is the canonical epimorphism

J ⊗_{S₀} 𝒪_{U₀} ⟶ J𝒪_U ≃ J𝒪_{U₀}.

We must show that $\tilde{h}(K)=0$, where $K$ is the kernel of $j_U$. Now for every integer $n$ equal to a power of $q$, we have an epimorphism

$$h_n:{\mathcal{O}}_{T_0}⟶{\mathcal{O}}_{{}_n{T}_0}$$

and since ${}_n{T}_0$ lifts to a scheme $M_n$ flat over $S$, the canonical morphism

jₙ : J ⊗_{S₀} 𝒪_{ₙT₀} ⟶ J𝒪_{Mₙ}

is an isomorphism.

The commutative diagram below:

K ──→ J ⊗_{S₀} 𝒪_{U₀} ──j_U──→ J𝒪_U ⊂ 𝒪_U
            │             ⤡
            │ h̃           i
            ▼            ↗
       J ⊗_{S₀} 𝒪_{T₀}
            │
            │ h̃ₙ
            ▼            ≅
      J ⊗_{S₀} 𝒪_{ₙT₀} ──jₙ──→ J𝒪_{Mₙ} ⊂ 𝒪_{Mₙ}

shows that $h(K)$ is contained in $Ker{h}_n$ for every $n$, hence is contained in ${\bigcap }_{n}Ker{\tilde{h}}_n$, and it suffices to show that this last intersection is zero. Now the sheaf ${\mathcal{O}}_{T_0}$ is equal to the sheaf ${\mathcal{O}}_{S_0}[\mathbb{Z}]$, the algebra of the group $\mathbb{Z}$ with coefficients in ${\mathcal{O}}_{S_0}$, while ${\mathcal{O}}_{{}_n{T}_0}$ is the quotient algebra ${\mathcal{O}}_{S_0}[\mathbb{Z}/n\mathbb{Z}]$.

Let $a={\Sigma }_{m\in \mathbb{Z}}{a}_m\otimes m$ be an element of $J{\otimes }_{S_0}{\mathcal{O}}_{T_0}$. The $a_m$ are then sections of $J$, almost all zero. Take $n$ large enough that the indices $m$ for which $a_m$ is non-zero have distinct images in $\mathbb{Z}/n\mathbb{Z}$. Then if $a\in Ker{h}_n$, one necessarily has $a=0$. This proves that ${\bigcap }_{n}Ker{h}_n=0$, and proves 2.7.

Let then $K_0$ be the kernel of $h$; consider the following diagram:

                          0
                          │
                          ▼
       J𝒪_U              K₀
        ≅│                │
         ▼                ▼
   0 → J𝒪_{U₀} ────→ 𝒪_U ────→ 𝒪_{U₀} → 0
         │ i               h    │
         ▼                      ▼
  J ⊗_{S₀} 𝒪_{T₀}            𝒪_{T₀}
                                │
                                ▼
                                0.

The sheaf ${\mathcal{O}}_U$ defines an element $\Phi$ of the group $Ex{t}_{{\mathcal{O}}_U}^1({\mathcal{O}}_{U_0},J{\mathcal{O}}_{U_0})$. Let $\Psi$ be the element of

$Ex{t}_{{\mathcal{O}}_U}^1(K_0,J{\otimes }_{S_0}{\mathcal{O}}_{T_0})$ deduced from $\Phi$ by bifunctoriality of $Ex{t}^1(\cdot ,\cdot )$ through the morphisms $K_0\to {\mathcal{O}}_{U_0}$ and $i:J{\mathcal{O}}_{U_0}\to J{\otimes }_{S_0}{\mathcal{O}}_{T_0}$. It follows from Exp. III 4.1 and from the infinitesimal flatness criterion (cf. Exp. III 4.3) that there exists a subscheme of $U$, flat over $S$, which lifts $T_0$, if and only if $\Psi$ is zero.

But note that $T_0$ is affine, so it suffices (Exp. III 4.5 and 4.6) to show that there locally on $U$ exists a subscheme flat over $S$ lifting $T_0$. In short, it suffices to show that the image of $\Psi$ in the sheaf

ℰ = Ext¹_{𝒪_U}(K₀, J ⊗_{S₀} 𝒪_{T₀})

is zero. We shall still denote by $\Psi$ this element of $\Gamma (U,\mathcal{E})$.

c) Reduction to the case $S$ local artinian with algebraically closed residue field.

Since $K_0$ and $J{\otimes }_{S_0}{\mathcal{O}}_{T_0}$ are coherent sheaves, so is the sheaf $\mathcal{E}$; moreover $\mathcal{E}$ has its support in $T_0$, since this is the case for $J{\otimes }_{S_0}{\mathcal{O}}_{T_0}$. To show that the section $\Psi$ of $\mathcal{E}$ is zero, it suffices to see that for every point $u$ of $T_0$, the image of

$\Psi$ in the fiber of $\mathcal{E}$ at the point $u$ is zero. But the formation of the $Ex{t}^i(\cdot ,\cdot )$ of coherent sheaves commutes with flat extensions of the base³, so we are reduced to proving the existence of a lifting of $T_0\cap \mathrm{Spec}{\mathcal{O}}_u$ flat over $S$. Let $s$ be the projection of $u$ on $S$; we may then replace $S$ by $\mathrm{Spec}{\mathcal{O}}_s$ and $G$ by $G{\times }_S\mathrm{Spec}{\mathcal{O}}_s$.

Possibly again making a faithfully flat extension, we may assume that ${\mathcal{O}}_s$ has an algebraically closed residue field (EGA 0_III, 10.3.1).

Let $\mathfrak{m}$ be the maximal ideal of ${\mathcal{O}}_s$ and $S_n=\mathrm{Spec}{\mathcal{O}}_s/{\mathfrak{m}}^n$. Suppose we have shown that for every $n>0$ the obstruction to lifting $T_0{\times }_S{S}_n=(T_0{)}_n$ to a subscheme of $G_n=G{\times }_S{S}_n$, flat over $S_n$, is zero, and let $T_n$ be the unique flat lifting of $(T_0{)}_n$ over $S_n$ which is a sub-torus of $G_n$. It is clear that if $n>n^{\prime }$, one has

(Tₙ) ×_{Sₙ} Sₙ′ = Tₙ′.

If then $u$ is a point of $T_0$ projecting onto $s$, it follows from the lemma below, applied to the coherent system of liftings $(T_n\cap \mathrm{Spec}{\mathcal{O}}_u)$ of $(T_0{)}_n\cap \mathrm{Spec}{\mathcal{O}}_u$, that there indeed exists a lifting of $T_0\cap \mathrm{Spec}{\mathcal{O}}_u$ flat over ${\mathcal{O}}_s$. We are therefore reduced to proving that $\Psi$ is zero when $S=S_n$ is the spectrum of an artinian local ring with algebraically closed residue field, and to proving:

Lemma 2.8. Let $A\to B$ be a local homomorphism of noetherian local rings, $\mathfrak{m}$ the maximal ideal of $A$, $J$ an ideal of square zero of $A$, $M$ a $B$-module of finite type, $A_0=A/J$, $B_0=B/JB$, $M_0=M/JM$, $N_0$ a quotient $B_0$-module of $M_0$ which is flat over $A_0$. For every integer $n>0$, let $A_n$, $A_0{,}_n$, etc., be the objects obtained by base extension

$A\to A/{\mathfrak{m}}^n=A_n$, and let $J_n$ be the image of $J$ in $A_n$. For every integer $n>0$, let $N_n$ be a quotient $B_n$-module of $M_n$, flat over $A_n$, lifting $N_0{,}_n$, and suppose that for $n⩾n^{\prime }$, $N_n^{\prime }$ is obtained from $N_n$ by base extension $A_n\to A_n^{\prime }$. Then there exists a $B$-module $N$, quotient of $M$, flat over $A$, lifting $N_0$.

Proof of 2.8. Let $P_0$ be the kernel of the epimorphism $M_0\to N_0$. For every $n>0$, we have the following commutative diagram in which the horizontal rows are exact:

                                       0
                                       │
                                       ▼
                                     P₀,ₙ
                                       │
                                       ▼
   0 ──→ Jₙ Mₙ ──→ Mₙ ──→ M₀,ₙ ──→ 0
            ↗
        Jₙ ⊗_{A₀,ₙ} M₀,ₙ
            ↘
                         ▼      ▼       ▼
   0 ──→ Jₙ ⊗_{A₀,ₙ} N₀,ₙ ──→ Nₙ ──→ N₀,ₙ ──→ 0
                                       ▼     ▼
                                       0     0

*(ₙ)

Moreover, by hypothesis, the diagram $(\ast {)}_{n+1}$ reduces modulo ${\mathfrak{m}}^n$ to $(\ast {)}_n$.

The Artin–Rees lemma (Bourbaki, Algèbre commutative, Chap. 3 § 3 cor. 1) shows that the filtration defined on JM (resp. $J{\otimes }_{A_0}{M}_0$ and $J{\otimes }_{A_0}{N}_0$) by the kernels of the morphisms

JM ⟶ Jₙ Mₙ,   (resp.   J ⊗_{A₀} M₀ ⟶ Jₙ ⊗_{A₀,ₙ} M₀,ₙ   and   J ⊗_{A₀} N₀ ⟶ Jₙ ⊗_{A₀,ₙ} N₀,ₙ)

is $\mathfrak{m}B$-good, so that, passing to the projective limit on the diagrams $(\ast {)}_n$ and denoting by $\hat{Q}$ the separated completion of a $B$-module $Q$ for the $\mathfrak{m}B$-adic topology, one obtains the following commutative diagram, where the two horizontal rows are still exact:

                                       0
                                       │
                                       ▼
                                      P̂₀
                                       │
                                       ▼
   0 ──→ ĴM ──→ M̂ ──→ M̂₀ ──→ 0
           ↗
        Ĵ ⊗_{A₀} M₀
           ↘
   0 ──→ Ĵ ⊗_{A₀} N₀ ──→ lim_{←n} Nₙ ──→ N̂₀ ──→ 0
                                       ▼      ▼
                                       0      0

(ˆ)

Now $(B,\mathfrak{m}B)$ is a Zariski ring and $J{\otimes }_{A_0}{N}_0$ is of finite type over $B$, so it is separated for the $\mathfrak{m}B$-adic topology. The diagram $(\hat{\ast })$ then shows that the morphism

J ⊗_{A₀} M₀ ⟶ J ⊗_{A₀} N₀

deduced from the epimorphism $M_0\to N_0$ factors through JM:

J ⊗_{A₀} M₀ ──can.──→ JM
       ↘            ↙
        J ⊗_{A₀} N₀.

Under these conditions, it follows from Exp. III 4.1 and Exp. III 4.3 that there exists a $B$-module quotient $N$ of $M$, flat over $A$, lifting $N_0$, if and only if a certain element $g$ of $E=Ex{t}_B^1(P_0,J{\otimes }_{A_0}{N}_0)$ is zero. More precisely, the exact sequence

$$0⟶JM⟶M⟶M_0⟶0$$

defines an element $f$ of $F$, where $F$ is the $B$-module $Ex{t}_B^1(M_0,JM)$, and $g$ is the image of $f$

under the natural morphism $F\to E$ arising by bifunctoriality from the morphisms

P₀ ⟶ M₀    and    JM ⟶ J ⊗_{A₀} N₀.

It follows from the diagram $(\hat{\ast })$ and from Exp. III 4.1 that the image of $g$ in Ê, canonically identified with $Ex{t}_{\hat{B}}^1({\hat{P}}_0,\hat{J}{\otimes }_{A_0}{N}_0)$, is zero. But $E$ is of finite type over $B$,

so is separated for the $\mathfrak{m}B$-adic topology, and consequently $g$ is indeed equal to 0, which completes the proof of 2.8.

d) Study of $\mathcal{E}$. We therefore suppose that $S$ is the spectrum of a local artinian ring $A$ with algebraically closed residue field $k$. Let $A_0$ be the ring of $S_0$, ${\mathfrak{m}}_0$ the maximal ideal of $A_0$. Since $S_0$ is artinian, $T_0$ is closed in $G$ (Exp. VI_B 1.4.2); we may therefore take the open subset $U$ equal to $G$, so that

ℰ = Ext¹_{𝒪_G}(K₀, J ⊗_{S₀} 𝒪_{T₀}).

Let $B_0$ be the ring of the affine $S_0$-scheme $T_0=G_{m,S_0}$, and $B_0^{\prime }$ the ring of the special fiber $T_0{\times }_{S_0}\mathrm{Spec}(k)=G_{m,k}$ of $T_0$. The sheaf $\mathcal{E}$ is a coherent ${\mathcal{O}}_{T_0}$-module, so is defined by a $B_0$-module of finite type which we shall denote $E$.

Consider the graded module associated to $E$ and to the ${\mathfrak{m}}_0{B}_0$-adic filtration:

Eᵣ = 𝔪₀ʳ E / 𝔪₀^{r+1} E.

Each Eᵣ is therefore a $B_0^{\prime }$-module of finite type, and $E_r=0$ for $r$ large enough, since $S_0$ is artinian.

Let then $g$ be a section of $G$ above $S$ which on $S_0$ induces a section $g_0$ of $T_0$. Translation (on the left, to fix ideas) by the element $g$ defines an "automorphism of the situation" from the viewpoint of the obstruction problem under consideration. In particular, to $g$ corresponds an $S_0$-automorphism of the sheaf $\mathcal{E}$ leaving the obstruction $\Psi$ fixed. More precisely, $g$ defines a semi-linear automorphism of the $B_0$-module $E$ (relative to the $A_0$-automorphism of $B_0$ defined by translation by $g_0$ in the group $T_0$). By reduction modulo ${\mathfrak{m}}_0^{r+1}$, $g$ then defines a semi-linear automorphism of Eᵣ (relative to the $k$-automorphism of $B_0^{\prime }$ defined by translation by $g_0{\times }_{S_0}\mathrm{Spec}(k)$ in $T_0{\times }_{S_0}\mathrm{Spec}(k)$).

Lemma 2.9. For every integer $r⩾0$, Eᵣ is a locally free $B_0^{\prime }$-module.

Let $x$ be a point of $T_0$, $\kappa (x)$ its residue field, $(E_r{)}_x$ "the fiber" of Eᵣ at $x$, equal to $E_r{\otimes }_{B_0^{\prime }}\kappa (x)$, $\ell (x)$ the rank of $(E_r{)}_x$ over $\kappa (x)$, $\ell$ the maximum value of $\ell (x)$ as $x$ ranges over the points of $T_0$. Let $L$ be the largest closed subscheme of $\mathrm{Spec}{B}_0^{\prime }=G_{m,k}$ above which Eᵣ is locally free of rank $\ell$ (TDTE IV Lemma 3.6). Let $\beta$ be a point of $L(k)$ (there is one, $L$ being of finite type over $k$ algebraically closed) and let $\alpha$ be a point of $G_{m,k}(k)$ of order equal to a power $n$ of $q$. The point $\alpha$ is therefore rational

over $k$, and since by hypothesis ${}_n{T}_0$ lifts to a subscheme $M_n$ étale over $S$, there exists a section $a$ of $G$ above $S$ which lifts $\alpha$ and which, above $S_0$, is a section of $T_0$. The remarks made above then show that the fibers of Eᵣ are $k$-isomorphic at the points $\beta$ and $\alpha \beta$ of $G_{m,k}(k)$. But the points of order a power of $q$ are dense in $G_{m,k}$, and similarly their translates by $\beta$. Since $L$ is closed in $G_{m,k}$ and $G_{m,k}$ is reduced, $L$ equals $G_{m,k}$ and Eᵣ is locally free over $B_0^{\prime }$ of rank $\ell$.

e) End of the proof of Theorem 2.1.

Let $K_0(n)$ be the sheaf of ideals of ${\mathcal{O}}_{G_0}$ defining the closed subscheme ${}_n{T}_0$. The sheaf $K_0$⁴ is therefore a subsheaf of $K_0(n)$. Set, for simplicity,

R = J ⊗_{S₀} 𝒪_{T₀}    and    R(n) = J ⊗_{S₀} 𝒪_{ₙT₀}.

The sheaf $R(n)$ is therefore a quotient of $R$, and one has the diagram of morphisms:

                K₀
                │
                ▼
              K₀(n)
                │
                ▼
0 ──→ J𝒪_G ──→ 𝒪_G ──→ 𝒪_{G₀} ──→ 0
        │
        ▼
        R
        │
        ▼
      R(n)

Using then the bifunctoriality of $Ex{t}_{{\mathcal{O}}_G}^1(\cdot ,\cdot )$, one obtains the following commutative diagram:

Ext¹_{𝒪_G}(𝒪_{G₀}, J𝒪_G)
        ↘
         Ext¹_{𝒪_G}(K₀(n), R) ──→ Ext¹_{𝒪_G}(K₀, R) = ℰ
               │                          │
               ▼                          ▼
         Ext¹_{𝒪_G}(K₀(n), R(n)) ──→ Ext¹_{𝒪_G}(K₀, R(n)).

Let us again denote by $\Phi$ the element of $Ex{t}_{{\mathcal{O}}_G}^1({\mathcal{O}}_{G_0},J{\mathcal{O}}_G)$ defined by the exact sequence

$$0⟶J{\mathcal{O}}_G⟶{\mathcal{O}}_G⟶{\mathcal{O}}_{G_0}⟶0,$$

so that $\Psi$ is the image of $\Phi$ in $\mathcal{E}$. Since ${}_n{T}_0$ lifts to a subscheme $M_n$ of $G$, flat over $S$, the image of $\Phi$ in the sheaf $Ex{t}_{{\mathcal{O}}_G}^1(K_0(n),R(n))$ is zero (Exp. III 4.1); a fortiori, the image of $\Phi$ in $Ex{t}_{{\mathcal{O}}_G}^1(K_0,R(n))$, which is also the image of $\Psi$, is zero.

Lemma 2.10. The canonical morphism

Ext¹_{𝒪_G}(K₀, R) ⊗_{B₀} 𝒪_{ₙT₀} ⟶ Ext¹_{𝒪_G}(K₀, R(n))

is injective for every integer $n$.

Indeed, the affine scheme $T_0=G_{m,S_0}$ has ring

$$B_0=A_0[T,T^{-1}].$$

The subscheme ${}_n{T}_0$ is defined by the vanishing of the following section of $B_0$:

$$h(n)=T^n-1,$$

which is regular (EGA 0_IV 15.2.2) and remains regular after any base change $S_0^{\prime }\to S_0$. We therefore have an exact sequence of sheaves:

0 ⟶ 𝒪_{T₀} ──h(n)──→ 𝒪_{T₀} ⟶ 𝒪_{ₙT₀} ⟶ 0.

Since ${}_n{T}_0$ is flat over $S$, one obtains, by tensoring with $J$ over ${\mathcal{O}}_{S_0}$, the exact sequence

0 ⟶ R ──h(n)──→ R ⟶ R(n) ⟶ 0,

then the exact sequence of Ext:

⋯ ⟶ Ext¹_{𝒪_G}(K₀, R) ──h(n)──→ Ext¹_{𝒪_G}(K₀, R) ⟶ Ext¹_{𝒪_G}(K₀, R(n)),

which completes the proof of the lemma.

The foregoing shows that for every integer $n$ equal to a power of $q$, the image of $\Psi$ in $\mathcal{E}/h(n)\mathcal{E}$ is zero. To show that $\Psi$ is zero, it suffices to see that if $\Psi \in {\mathfrak{m}}_0^r\mathcal{E}$, then $\Psi \in {\mathfrak{m}}_0^{r+1}\mathcal{E}$. Let $\bar{\Psi }$ be the image of $\Psi$ in Eᵣ. There exists an element $\Psi (n)$ of $\mathcal{E}$ such that one has

Ψ = Ψ(n) · h(n)    (n equal to a power of q).

We noted that the image $\bar{h}(n)$ of $h(n)$ in $B_0^{\prime }$ is again $B_0^{\prime }$-regular. Since $E_s$ is locally free over $B_0^{\prime }$ for every $s$ (2.9), multiplication by $\bar{h}(n)$ in Eᵣ is injective. One deduces immediately that $\Psi (n)$ is in ${\mathfrak{m}}_0^r\mathcal{E}$. Let $\bar{\Psi }(n)$ be its image in Eᵣ, so that one has the relation

Ψ̄ = h̄(n) Ψ̄(n)    (n equal to a power of q).

This shows that the set $F$ of points of $G_{m,k}$ at which $\bar{\Psi }$ takes the value 0 contains the dense set of points of order a power of $q$. Moreover $F$ is a closed set (since Eᵣ is locally free over $B_0^{\prime }$) and $G_{m,k}$ is reduced, so $\bar{\Psi }$ is zero.

This completes the proof of Theorem 2.1.

3. Characterization of a sub-torus by its underlying set

3.1. Statement of the theorem

Notation. If $X$ is a prescheme, $ens(X)$ denotes the underlying set of $X$. If $X$ and $S^{\prime }$ are two $S$-preschemes, $X_{S^{\prime }}=X{\times }_S{S}^{\prime }$, $u:X_{S^{\prime }}\to X$ the canonical morphism, $E$ a subset of $ens(X)$, one denotes by $E_{S^{\prime }}$ (or $E{\times }_S{S}^{\prime }$) the subset of $ens(X_{S^{\prime }})$ equal to $u^{-1}(E)$.

Theorem 3.1. Let $S$ be a locally noetherian prescheme, $G$ an $S$-prescheme in groups of finite type, $q$ an integer > 0 invertible on $S$, $E$ a subset of $ens(G)$. Consider the following assertions concerning $E$:

• (i) The set $E$ is the underlying set of a sub-torus $T$ of $G$.
• (ii) a) For every point $s$ of $S$, there exists a sub-torus $T_s$ of $G_s$ such that $E\cap ens(G_s)=E_s$ is the underlying set of $T_s$.
• • b) As the integer $n$ ranges over powers of $q$, there exists a coherent family (cf. 2.2) $M_n$ of subgroup schemes of $G$, of multiplicative type, such that for every point $s$ of $S$ one has*

$$(M_n{)}_s={}_n{T}_s.$$

• (iii) a) As in (ii) a).
• • b) The set $E$ is locally closed in $ens(G)$, and the dimension of the fibers of $E$ over $S$ is locally constant.*
• (iv) a) As in (ii) a).
• • b) For every $S$-scheme $S^{\prime }$ which is the spectrum of a complete discrete valuation ring with algebraically closed residue field, $E_{S^{\prime }}$ is the underlying set of a sub-torus of $G_{S^{\prime }}$.*

Then one has the following implications:

• A) (i) ⇔ (ii) ⇒ (iii) ⇒ (iv).
• B) If $G$ is separated over $S$, one has (iii) ⇔ (iv), and moreover $E$ is closed.
• C) Conditions (i), (ii), (iii) (and also (iv) if $G$ is separated over $S$) are equivalent in the following two cases:
• • 1st case: a) The prescheme $S$ is reduced, or $G$ is flat over $S$, and*

• • b) For every point `s` of `S`, `Tₛ` is a central torus of `Gₛ`.*
    

    - - 2nd case: `S` is normal.\*
    

Moreover, in the two cases above, the torus $T$ with underlying set $E$ is unique.

Remarks 3.2. a) When $S$ is reduced, it is unnecessary in (ii) to assume that the family $M_n$ is coherent, this condition being entailed by the other hypotheses. Indeed, if the integer $m$ divides $n$, the subgroups ${}_m{M}_n$ and $M_m$ are étale over $S$, hence reduced, and have the same underlying space, so they coincide.

b) If $S$ is not assumed normal, it is no longer true in general that (iii) ⇒ (i), even when $S$ is reduced, geometrically unibranched and $G$ is a smooth group scheme over $S$. Indeed, consider the Borel subgroup of $S{L}_{2,S}$ formed by matrices of the form

$$(tu)(0t^{-1}),$$

where $S$ is the affine curve over a field $k$ with ring

$$k[x,y]/(y^2-x^3).$$

Consider then the set $E$ obtained as follows: above the "cusp of $S$" ($x=y=0$) we take the diagonal torus ($u=0$). Above the complementary open subset ($x\ne 0$) we take the torus deduced from the diagonal torus by conjugation by the element

$$(1y/x)(01).$$

The set $E$ so obtained satisfies (iii) a); on the other hand it is closed in $G$, and the reduced subscheme having $E$ as underlying set has equations

$$xu+y(t-t^{-1})=0u^2-x(t-t^{-1}{)}^2=0.$$

This is therefore not a sub-torus of $G$, since the fiber above the cusp is not reduced.

Plan of the proof of 3.1. In a first part we shall establish the following "easy" implications:

                  (ii)
              ↗        ↘
         (i)              (iv)
              ↘        ↗
                  (iii)

and [(iv) ⇒ (iii) and E closed if G is separated over S].

The proof of the more delicate implications will be carried out in three stages:

• I) Reduction of the implication (iii) ⇒ (i) (under the hypotheses of C), 1st case) to the case where $S$ is normal.
• II) (iii) ⇒ (ii) when $S$ is normal.
• III) (ii) ⇒ (i).

3.2. Proof of the "easy" assertions contained in 3.1

(i) ⇒ (ii) and (iii) is trivial.

(iii) ⇒ (iv). Possibly replacing $S$ by $S^{\prime }$, we may assume that $S$ is the spectrum of a discrete valuation ring. Let $t$ be the generic point of $S$ and $s$ the closed point.

Since $E$ is locally closed, there exists a subprescheme of $G$ which is reduced and whose underlying space is $E$; we shall denote it $E$. The generic fiber $E_t$ of $E$ is therefore equal to the sub-torus $T_t$ of rank $r$ of $G_t$ (by (iii) a)). Let $E^{\prime }$ be the schematic closure of $E_t$ in $E$. The prescheme

$E^{\prime }$ is irreducible, and its closed fiber $E_s^{\prime }$ is non-empty (it contains the unit section of $G_s$), so $E_s^{\prime }$ is of dimension $r$ (EGA IV 14.3.10). But then $E_s^{\prime }$ is a closed subscheme of $E_s$, and the latter is of dimension $r$ (by (iii) b)) and irreducible (since it has the same underlying space as $T_s$), so $E_s^{\prime }$ has the same underlying space as $E_s$, and consequently $E^{\prime }=E$, which proves that $E$ is flat over $S$.

Let now $E^{\prime \prime }$ be the schematic closure of $E_t$ in $G$. Then $E^{\prime \prime }$ is a subprescheme in groups of $G$, flat over $S$ (Exp. VIII 7.1), which majorizes $E^{\prime }$, hence $E$. The closed

fiber $E_s^{\prime \prime }$ is an algebraic subgroup of $G_s$ of dimension $r$ (loc. cit.). Since $T_s$ is a closed irreducible subset of $G_s$ of dimension $r$, $E_s$ has the same underlying set as the connected component $(E_s^{\prime \prime }{)}^0$ of $E_s^{\prime \prime }$. Let $(E^{\prime \prime }{)}^0$ be the "connected component" of $E^{\prime \prime }$, i.e. the open subgroup of $E^{\prime \prime }$ complementary to the union of the irreducible components of $E_s^{\prime \prime }$ not containing the origin. One then has

$$E=ens[(E^{\prime \prime }{)}^0].$$

Since $E$ and $E^{\prime \prime }$ are reduced, one has even $E=(E^{\prime \prime }{)}^0$. Finally $E$ is a subgroup prescheme of $G$, flat and of finite type over $S$, with connected fibers, hence separated (Exp. VI_B 5.2), whose generic fiber is a torus $T_t$, and whose reduced closed fiber is a torus $T_s$; but then $E$ is a torus (Exp. X 8.8).

(ii) ⇒ (iv). One may again assume that $S$ is the spectrum of a discrete valuation ring, and we keep the notation introduced above. The schematic closure in $G$ of $T_t$ is a subgroup prescheme $T$ of $G$, flat over $S$, which majorizes $M_n$ for every integer $n$ equal to a power of $q$. Consequently the closed fiber of $T$ is a closed subscheme of $G_s$ majorizing $(M_n{)}_s$ for every $n$, hence majorizing $T_s$. For dimension reasons, the "connected component" $T^0$ of $T$ has $E$ as underlying set, and one concludes as above that $T^0$ is a sub-torus of $G$.

(iv) ⇒ [(iii) and $E$ closed], if $G$ is separated over $S$. Let us show that $E$ is closed.⁵ First let us prove the lemma:

Lemma 3.3. If the conditions stated in 3.1 (iv) are satisfied, $E$ is a locally constructible part of $ens(G)$.

By the usual method, we are reduced to studying the case where $S$ is noetherian, integral, with generic point $\eta$. Possibly restricting $S$, we may assume (Exp. VI_B 10.10) that there exists a subgroup scheme $H$ of $G$, flat over $S$, with connected fibers, whose generic fiber $H_{\eta }$ is equal to $T_{\eta }$. To prove 3.3 it then suffices to show that $ens(H)=E$. Now if $s$ is a point of $S$, there exists, by EGA II 7.1.9, an $S$-scheme $S^{\prime }$, the spectrum of a discrete valuation ring, that one may assume complete with algebraically closed residue field, whose generic point $t^{\prime }$ projects onto $\eta$ and whose closed point $s^{\prime }$ projects onto $s$. By (iv) b), there exists a sub-torus $T_{S^{\prime }}$ of $G_{S^{\prime }}$ having $E_{S^{\prime }}$ as underlying space. The two subprescheme in groups $T_{S^{\prime }}$ and $H_{S^{\prime }}=H{\times }_S{S}^{\prime }$ of $G_{S^{\prime }}$ are flat over $S^{\prime }$, have connected fibers, and the same generic fiber $T_{t^{\prime }}$, so they coincide with the connected component of the schematic closure

of $T_{t^{\prime }}$ in $G_{S^{\prime }}$. Consequently $ens(H_s)=E_s$, so $ens(H)=E$, which proves the lemma.

This being so, knowing that $E$ is locally constructible, in order to see that $E$ is closed it suffices to show that every specialization in $G$ of a point of $E$ is a point of $E$. By the usual technique we are reduced to the case where $S$ is the spectrum of a complete discrete valuation ring with algebraically closed residue field. But then the sub-torus of $G$ of underlying space $E$ ((iv) b)) is closed in $G$ since $G$ is separated (Exp. VIII 7.12).

3.3. Continuation of the proof of 3.1

I) Reduction of (iii) ⇒ (i) (C, 1st case) to the case where $S$ is normal.

a) Reduction to the case $S$ affine reduced. We therefore assume that for every point $s$ of $S$, $E_s$ is the underlying space of a central sub-torus of $G_s$. The uniqueness of $T$ then follows from Exp. IX 5.1 bis, and moreover $T$ will necessarily be central (loc. cit.). In view of uniqueness, to prove the existence of $T$ we may assume $S$ noetherian, affine of ring $A$. If $S$ is not reduced, by hypothesis $G$ is flat over $S$. By 2.3 it then suffices to solve the problem for $S_{red}$ and $G{\times }_S{S}_{red}$. We may therefore assume in addition that $S$ is reduced.

b) Reduction to the case where the ring $A$ is of finite type over $\mathbb{Z}$. Let us first prove two lemmas:

Lemma 3.4. Let $k$ be a field, $G$ a $k$-algebraic group, $E$ a subset of $G$, $k^{\prime }$ an extension of $k$, $T_{k^{\prime }}$ a sub-torus of $G_{k^{\prime }}$ having $E_{k^{\prime }}$ as underlying space. Then, if $T_{k^{\prime }}$ is central or if $k$ is perfect, $E$ is the underlying set of a sub-torus of $G_k$.

Indeed, by fpqc descent it suffices to show that the two inverse images of $T_{k^{\prime }}$ in $G_{k^{\prime \prime }}$, where $k^{\prime \prime }=k^{\prime }{\otimes }_k{k}^{\prime }$, coincide. Now they have the same underlying space, namely the inverse image of $E$. If $T_{k^{\prime }}$ is central, the lemma is a consequence of Exp. IX 5.1 bis. If $k$ is perfect, $k^{\prime \prime }$ is reduced and the two inverse images of $T_{k^{\prime }}$, being smooth over $k^{\prime \prime }$, are reduced, hence coincide.

Remark 3.5. It follows from the preceding lemma that in the statement of 3.1 (iv), property (iv) a) is a consequence of (iv) b) in the two following cases:

1. One assumes that the residue fields of the points of $S$ are perfect.
2. For every $S^{\prime }$ as in (iv) b), one assumes that the torus with underlying space $E_{S^{\prime }}$ is central in $G_{S^{\prime }}$.

Lemma 3.6. Let $S$ be a prescheme projective limit of affine schemes Sᵢ (cf. EGA IV 8), $H$ an $S$-group scheme of multiplicative type and of finite type. Then there exist an index $i$, an Sᵢ-group scheme Hᵢ of multiplicative type and of finite type, and an $S$-isomorphism

Hᵢ ×_{Sᵢ} S ⥲ H.

If moreover $H$ is isotrivial, one may assume Hᵢ isotrivial.

Since $H$ is of finite type over $S$, $H$ is in fact of finite presentation over $S$ (Exp. IX 2.1 b)); there therefore exist an index $\ell$ and an $S_{\ell }$-group scheme $H_{\ell }$ such that $H_{\ell }{\times }_{S_{\ell }}S$ is isomorphic to $H$ (Exp. VI_B 10). Setting $H_i=H_{\ell }{\times }_{S_{\ell }}{S}_i$, one therefore has $H\cong H_i{\times }_{S_i}S$ for every $i⩾\ell$.

Since $H$ is of finite type over $S$, $H$ is quasi-isotrivial (Exp. X 4.5), hence trivialized by an étale surjective morphism $S^{\prime }\to S$. Using the quasi-compactness of $S$, one easily sees that there exist a covering of $S$ by a finite number of affine open subsets $S_{\alpha }$, and for every $\alpha$ an étale, surjective and finitely presented morphism $S_{\alpha }^{\prime }\to S_{\alpha }$ trivializing $H|S_{\alpha }$. This covering $(S_{\alpha })$ of $S$ then comes from a covering $(S_{i,\alpha })$ of Sᵢ for $i$ large enough (EGA IV 8). Possibly replacing Sᵢ by $S_{i,\alpha }$ and $S$ by $S_{\alpha }$, we may therefore assume that $H$ is trivialized by an étale surjective morphism $S^{\prime }\to S$ of finite presentation. For $i$ large enough, there then exist a prescheme $S_i^{\prime }$, an étale surjective morphism $S_i^{\prime }\to S_i$ of finite presentation, and an $S$-isomorphism $S_i^{\prime }{\times }_{S_i}S\to S^{\prime }$ (EGA IV 17.16). Set then for $j$ large enough:

S′_j = S′ᵢ ×_{Sᵢ} S_j,    H′_j = H_j ×_{S_j} S′_j,    H′ = H ×_S S′.

Given the choice of $S^{\prime }$, there exist a finitely generated abelian group $M$ and an $S^{\prime }$-isomorphism of group schemes $D_{S^{\prime }}(M)\stackrel{\sim }{\to }{H}^{\prime }$. Since the $S_i^{\prime }$ are quasi-compact and $S^{\prime }=\lim S_i^{\prime }$, it follows from Exp. VI_B 10 that there exist an index $j$ and an $S_j^{\prime }$-isomorphism of group schemes

$$D_{S_j^{\prime }}(M)\stackrel{\sim }{\to }{H}_j^{\prime }.$$

But this says that $H_j$ is a quasi-isotrivial group of multiplicative type.

When $H$ is isotrivial, one proceeds analogously, using a trivialization of $H$ by a finite étale morphism $S^{\prime }\to S$.

This being so, we can carry out the announced reduction b). The ring $A$ of $S$ is a filtered inductive limit of its subrings Aᵢ of finite type over $\mathbb{Z}$. Let $S_i=\mathrm{Spec}{A}_i$, $u_j:S\to S_j$ and $u_{jk}:S_k\to S_j$ ($k⩾j$) the transition morphisms. Since $S$ is noetherian and $G$ is of finite presentation over $S$, there exist an index $i$ and an Sᵢ-prescheme in groups Gᵢ, of finite type over Sᵢ, such that $G$ is $S$-isomorphic to $G_i{\times }_{S_i}S$. Similarly, since $E$ is locally closed in $G$, we may assume that there exists a locally closed subset Eᵢ of Gᵢ such that $E=E_i{\times }_{S_i}S$ (EGA IV 8.3.11). For every $j>i$, let $G_j=G_i{\times }_{S_i}{S}_j$ and $E_j=E_i{\times }_{S_i}{S}_j$, and let $Q_j$ be the set of points $s$ of $S_j$ such that $(E_j{)}_s$ is the underlying set of a central sub-torus of $(G_j{)}_s$.

It follows from 3.4 that $Q_k=u_{jk}^{-1}(Q_j)$ for $k⩾j$, and by hypothesis $ens(S)=u_j^{-1}(Q_j)$ for $j⩾i$. Moreover, I claim that $Q_j$ is ind-constructible (EGA IV 1.9.4). Indeed, since $S_j$ is noetherian, it suffices (EGA IV 1.9.10) to see that if $S$ is a noetherian integral scheme with generic point $\eta$, and if $E_{\eta }$ is the underlying set of a central sub-torus $T_{\eta }$ of $G_{\eta }$, there exists a neighborhood $U$ of $\eta$ such that for every point $s$ of $U$, $E_s$ has the same property. Now, possibly restricting $S$, we may assume that the

center $Z$ of $G$ is representable and that there exists a subgroup scheme $T$ of $G$ whose generic fiber is $T_{\eta }$ (VI_B § 10). But then $ens(T)$ and $E$ are two locally closed subsets of $ens(G)$ which coincide on the generic fiber; one may therefore find a neighborhood $U$ of $\eta$ such that, above $U$, $ens(T)=E$ (EGA IV 8.3.11). For the same reasons, one may assume that above $U$, $T$ is central, since $T$ is a torus (3.6).

Knowing now that $Q_j$ is ind-constructible, it follows from EGA IV 8.3.4 that $Q_j=ens(S_j)$ for $j$ large enough. We may then replace $S$, $G$, $E$ by $S_j$, $G_j$, $E_j$, which reduces us to the case where $S$ is an affine reduced scheme of finite type over $\mathbb{Z}$.

c) Reduction to the case where $S$ is the spectrum of a complete reduced noetherian local ring. Owing to the uniqueness of the torus of underlying set $E$, the usual technique (EGA IV 8) and Lemma 3.6 allow us to replace $S$ by the spectrum of the local ring $A$ of a point of $S$. Let Ŝ be the spectrum of the completion Â of $A$ for the topology defined by the maximal ideal. Since $A$ is the localization of a finitely generated algebra over $\mathbb{Z}$, Ŝ is still

reduced (EGA IV 7.6.5). I claim that it suffices to solve the problem after the base change $\hat{S}\to S$. Indeed if $\hat{T}$ is the sub-torus of $G_{\hat{S}}$ with underlying space $E_{\hat{S}}$, its two inverse images in $G_{\hat{S}}{\times }_S\hat{S}$ are two central sub-tori with the same underlying space, so they coincide (Exp. IX 5.1 bis), and by fpqc descent, $\hat{T}$ comes from a torus $T$ of $G$ which answers the question (cf. 3.4).

d) A descent lemma. Let us recall the following properties of finite morphisms which were noted in TDTE I: Let $S$ and $S^{\prime }$ be two preschemes and $u:S^{\prime }\to S$ a finite morphism. Then:

1. The morphism $u$ is an epimorphism if and only if the canonical morphism of sheaves ${\mathcal{O}}_S⟶u_{\ast }({\mathcal{O}}_{S^{\prime }})$ is injective.
2. The morphism $u$ is an effective epimorphism (Exp. IV 1.3) if and only if the canonical diagram $0⟶{\mathcal{O}}_S⟶u_{\ast }({\mathcal{O}}_{S^{\prime }})\Rightarrow u_{\ast }({\mathcal{O}}_{S^{\prime }}){\otimes }_{{\mathcal{O}}_S}{u}_{\ast }({\mathcal{O}}_{S^{\prime }})$ is exact.
3. If moreover $S$ is noetherian and if $u$ is an epimorphism, $u$ is the composite of a finite sequence of effective finite epimorphisms.

We are then in a position to prove the following lemma, whose proof uses a technique of non-flat descent:

Lemma 3.7. Let $S$ be a locally noetherian prescheme, $G$ an $S$-prescheme in groups of finite type, $S^{\prime }$ a prescheme and $u:S^{\prime }\to S$ a finite morphism. For every $S$-prescheme $T$, let $M(T)$ denote the set of subgroups of multiplicative type of G_T

(resp. the set of central subgroups of multiplicative type of G_T), so that $M$ is in a natural way a contravariant functor defined on $Sch/S$. Then, if $u$ is an effective epimorphism (resp. an epimorphism), the diagram of sets

(*)    M(S) ⟶ M(S′) ⇒ M(S′ ×_S S′)

is exact.

Proof. i) Reduction to the case where $u$ is an effective epimorphism. We are then interested in the functor of central subgroups of multiplicative type of $G$. The injectivity of $M(S)\to M(S^{\prime })$ is a local question on $S$, and, this injectivity being granted, the exactness of (*) becomes a local problem on $S$. We may therefore assume $S$ affine noetherian.

Let us study the case where $u:S^{\prime }\to S$ is the composite of two finite epimorphisms:

S′ ──v──→ S″ ──w──→ S.

I claim that if the two diagrams

(*)′    M(S″) ⟶ M(S′) ⇒ M(S′ ×_{S″} S′)
(*)″    M(S) ⟶ M(S″) ⇒ M(S″ ×_S S″)

are exact, then so is (*).

Indeed the injectivity of $M(S)\to M(S^{\prime })$ is clear. If now $T^{\prime }$ is an element of $KerM(S^{\prime })\Rightarrow M(S^{\prime }{\times }_S{S}^{\prime })$, a fortiori $T^{\prime }$ belongs to $KerM(S^{\prime })\Rightarrow M(S^{\prime }{\times }_{S^{\prime \prime }}{S}^{\prime })$, so by exactness of (*)′ comes from a unique element $T^{\prime \prime }$ of $M(S^{\prime \prime })$. It suffices

to show that $T^{\prime \prime }$ belongs to $KerM(S^{\prime \prime })\Rightarrow M(S^{\prime \prime }{\times }_S{S}^{\prime \prime })$, since, by exactness of (*)″, $T^{\prime \prime }$ will descend to $S$. Let $T_1^{\prime \prime }$ and $T_2^{\prime \prime }$ be the two inverse images of $T^{\prime \prime }$ in $M(S^{\prime \prime }{\times }_S{S}^{\prime \prime })$. Since these are two central subgroups of multiplicative type of $G_{S^{\prime \prime }{\times }_S{S}^{\prime \prime }}$, to show that $T_1^{\prime \prime }=T_2^{\prime \prime }$ it suffices to see that they have the same fibers (Exp. IX 5.1 bis). Consider the commutative diagram

S′ ←──── S′ ×_S S′
│          │
v          v × v
▼          ▼
S″ ←──── S″ ×_S S″.

The morphism $v$ is a finite epimorphism, hence is dominant (property (a) above) and closed, hence is surjective, and the same is true of $v\times v$. Let $x^{\prime \prime }$ be a point of $S^{\prime \prime }{\times }_S{S}^{\prime \prime }$ and $x^{\prime }$ a point of $S^{\prime }{\times }_S{S}^{\prime }$ above $x^{\prime \prime }$. It follows from commutativity of the diagram above that the two inverse images of $(T_1^{\prime \prime }{)}_{x^{\prime \prime }}$ and $(T_2^{\prime \prime }{)}_{x^{\prime \prime }}$ in $G_{x^{\prime }}$ coincide, so $(T_1^{\prime \prime }{)}_{x^{\prime \prime }}=(T_2^{\prime \prime }{)}_{x^{\prime \prime }}$ (EGA IV 2.2.15).

What precedes, and an immediate induction on the number of factors of a factorization of $u:S^{\prime }\to S$ into effective epimorphisms (property (c) recalled above), show that to prove the exactness of (*) in the case of central subgroups of multiplicative type, we may restrict to the case where $u$ is an effective epimorphism. Finally, using once again Exp. IX 5.1 bis, one sees that it suffices to prove 3.7 when $M$ is the functor of subgroups of multiplicative type of $G$ and $u$ an effective epimorphism.

ii) Injectivity of $M(S)\to M(S^{\prime })$. Since $u:S^{\prime }\to S$ is an epimorphism, the canonical

morphism ${\mathcal{O}}_S⟶u_{\ast }({\mathcal{O}}_{S^{\prime }})$ is injective; moreover an $S$-group of multiplicative type is flat over $S$; the injectivity of $M(S)\to M(S^{\prime })$ is therefore a consequence of the more general following lemma:

Lemma 3.8. Let $f:X\to S$ and $g:S^{\prime }\to S$ be two morphisms of preschemes, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-module, $X^{\prime }=X{\times }_S{S}^{\prime }$, ${\mathcal{F}}^{\prime }=\mathcal{F}{\otimes }_{{\mathcal{O}}_S}{\mathcal{O}}_{S^{\prime }}$, $Q(\mathcal{F})$ the set of quotient ${\mathcal{O}}_X$-modules $\mathcal{G}$ of $\mathcal{F}$ which are quasi-coherent and flat over $S$, $Q({\mathcal{F}}^{\prime })$ the analogue relative to ${\mathcal{F}}^{\prime }$, $X^{\prime }$ and $S^{\prime }$. Suppose $g$ is quasi-compact and ${\mathcal{O}}_S\to g_{\ast }({\mathcal{O}}_{S^{\prime }})$ injective; then the canonical map