Exposé XXV. The existence theorem

by M. Demazure

¹ For the sake of completeness, we give in this Exposé a proof of the existence theorem for split groups. Like Chevalley's original proof ("Sur certains schémas de groupes semi-simples", Séminaire Bourbaki, May 1961, no. 219), it rests on the existence of complex semisimple algebraic groups of all possible types. The principle of a more satisfactory proof, establishing directly the existence of a simply connected split semisimple $\mathbb{Z}$-group corresponding to a given Cartan matrix, was given by Cartier (unpublished).² Let us point out, however, that the difficulty is not to give an explicit construction of a group scheme, but to verify that the group thus constructed really meets the required conditions, that is, essentially, that its fibers are indeed smooth and reductive.

1. Statement of the theorem

Theorem 1.1. Let $S$ be a non-empty scheme. The functor

$$G\mapsto R(G)$$

is an equivalence of the category of pinned $S$-reductive groups with the category of pinned reduced root data.

By virtue of the uniqueness theorem (Exp. XXIII, 4.1), the preceding statement is equivalent to:

Corollary 1.2 (Existence of split groups). For every reduced root datum $R$, there exists a split reductive $\mathbb{Z}$-group $G$ such that $R(G)\simeq R$.³

In particular:

Corollary 1.3. Let $k$ be a field. For every splittable reductive $k$-group $G_k$, there exists a reductive $\mathbb{Z}$-group $G$ such that $G{\otimes }_{\mathbb{Z}}k\simeq G_k$.

Conversely, let us remark first that to prove 1.2 it suffices, by Exp. XXII 4.3.1 and Exp. XXI 6.5.10, to consider the case where the root datum $R$ is simply connected (and even irreducible if one insists, by Exp. XXI, 7.1.6).

On the other hand, under the conditions of 1.3, the group scheme $G$ is of constant type (since $\mathrm{Spec}(\mathbb{Z})$ is connected), hence of type $R(G_k)$; by Exp. XXIII, 5.9, it follows that the validity of 1.3 for a given group $G_k$ entails the existence of a split $\mathbb{Z}$-group of type $R(G_k)$.⁴

To prove 1.2, and hence 1.1, it therefore suffices to prove 1.3 when $k$ is of characteristic zero (for example $k=\mathbb{C}$) and $G_k$ is simply connected (and in particular semisimple), as well as:

Proposition 1.4. For every simply connected reduced root datum $R$, there exists a semisimple algebraic $\mathbb{C}$-group of type $R$.

One may prove 1.4 in the following way. One knows first that there exists a complex semisimple Lie algebra $g$ of type $R$, cf. for example N. Jacobson, Lie Algebras, ch. VII, Th. 5.⁵ Then $G=\mathrm{Aut}(g{)}^0$ is a semisimple algebraic $\mathbb{C}$-group of type $ad(R)$.⁶ By Bible, § 23.1, Prop. 1, one deduces from this the existence of a semisimple $\mathbb{C}$-group of type $R$.

The rest of this Exposé is devoted to the proof of 1.3 for $k$ of characteristic zero and $G_k$ semisimple. It will be carried out in two stages: construction of a "piece of $\mathbb{Z}$-group scheme" (n° 2), and study of the group obtained by application of the "theorem of Weil" (n° 3).

To avoid confusion, we shall not use in n° 2 the abbreviated notation $X_k$ to denote the $k$-scheme $X{\otimes }_{\mathbb{Z}}k$, where $X$ is a $\mathbb{Z}$-scheme.

2. Existence theorem: construction of a piece of group

2.1.

Let us choose, once and for all, a splitting of $G_k$, denoted

(G_k, T_k, M, R)

(cf. Exp. XXII, 1.13), a system of simple roots $\Delta$ of $R$ (defining the system of positive roots $R^{+}$), a Chevalley system $(X_{\alpha ,k}{)}_{\alpha \in R}$ of $G_k$ (Exp. XXIII, 6.1 and 6.2) satisfying the following supplementary condition (cf. XX 2.6): for every $\alpha \in R$, one has

$$X_{\alpha ,k}{X}_{-\alpha ,k}=1.$$

Finally, let us choose on the subgroup of $M$ generated by $R$ a total order relation compatible with the group structure, such that the roots > 0 are the elements of $R^{+}$. We then denote the roots

−α_n < −α_{n−1} < ⋯ < −α_1 < α_1 < α_2 < ⋯ < α_n.

For $\alpha \in R$, we denote by $U_{\alpha ,k}$ the vector group corresponding to the root $\alpha$, and by

$$p_{\alpha ,k}:{\mathbb{G}}_{a,k}\stackrel{\sim }{\to }{U}_{\alpha ,k}$$

the isomorphism of vector groups defined by $X_{\alpha ,k}$.

2.2.

The splitting of $G_k$ includes in particular an isomorphism of $k$-groups

$$T_k\simeq D_k(M).$$

Put $T=D(M)$; this is a $\mathbb{Z}$-torus, and the previous isomorphism may be regarded as an isomorphism

T_k ≃ T ⊗_ℤ k.

One has

$${\mathrm{Hom}}_{\mathbb{Z}-gr.}(T,{\mathbb{G}}_m)=M,$$

and one will regard the elements of $R\subset M$ as characters of $T$. Likewise, one will regard the elements of $R\ast$ as morphisms of $\mathbb{Z}$-groups ${\mathbb{G}}_m\to T$.

2.3.

For each $\alpha \in R^{+}$, let ${\mathbb{G}}_a(\alpha )$ be a copy of the group ${\mathbb{G}}_a$; consider the $\mathbb{Z}$-scheme

$$U={\mathbb{G}}_a({\alpha }_1)\times \cdots \times {\mathbb{G}}_a({\alpha }_n).$$

If $U_k$ denotes the unipotent part of the Borel group $B_k$ of $G_k$ defined by $R^{+}$, denote by

a : U ⊗_ℤ k ⥲ U_k

the isomorphism of $k$-schemes defined by

a(x_1, …, x_n) = p_{α_1,k}(x_1) ⋯ p_{α_n,k}(x_n).

2.4.

The group law of $U_k$ translates into relations of the form

a(x_1, …, x_n) · a(y_1, …, y_n) = a(z_1, …, z_n),

where each $z_h$ ($h=1,\cdots ,n$) is expressed as a polynomial

z_h = x_h + y_h + Q_h(x_1, …, x_{h−1}, y_1, …, y_{h−1}),

the coefficients of $Q_h$ being integers (Exp. XXII, 5.5.8 and Exp. XXIII, 6.4). Moreover $Q_h(x_1,\cdots ,x_{h-1},0,\cdots ,0)=0$.

Let us equip $U$ with the composition law defined by the preceding formulas (which are indeed "defined over $\mathbb{Z}$"). Since this law induces on $U_k$ its group law, it is associative, and (0) is a unit element (indeed, the two preceding assertions are expressed by relations between the polynomials $Q_h$, and $\mathbb{Z}\to k$ is injective). Let us show that this is a group law: if $(x_i)$ is a section of $U$ (over some $S$), one computes the inverse $(y_i)$ of $(x_i)$ by the recurrence formulas

y_i = −x_i − Q_i(x_1, …, x_{i−1}, y_1, …, y_{i−1})

which are still "defined over $\mathbb{Z}$".

In summary, we have constructed on $U$ a group law such that the previous isomorphism $a$ is an isomorphism of groups.

For each $\alpha \in R^{+}$, consider the morphism

$$p_{\alpha }:{\mathbb{G}}_a\to U$$

defined by $p_{\alpha }(x)=(x_i)$ where

x_i = x  if α_i = α,
x_i = 0  if α_i ≠ α.

This is a closed immersion and a group homomorphism; we denote its image by $U_{\alpha }$. One has $(x_i)=p_{{\alpha }_1}(x_1)\cdots p_{{\alpha }_n}(x_n)$, which proves that $U$ is identified with the product

U = U_{α_1} · U_{α_2} ⋯ U_{α_n}.

2.5.

Let $T=D(M)$ act on each $U_{\alpha }$ through the character $\alpha$; one verifies at once that this defines an action of $T$ on the group $U$, and one can construct the semi-direct product $B=T\cdot U$. One has a canonical isomorphism of $k$-groups

B ⊗_ℤ k ⥲ B_k.

If we now take any order on $R^{+}$, the morphism

$$\prod _{\alpha \in R^{+}}{U}_{\alpha }\to U$$

defined by the product in $U$ is still an isomorphism. Indeed, since both sides are flat $\mathbb{Z}$-schemes of finite presentation, it suffices to verify the assertion on geometric fibers; one is then reduced to Lazard's theory (Bible, § 13.1, Prop. 1): one considers $U$ as a group with operators $T$, and one uses the fact that the $U_{\alpha }$ are pairwise non-isomorphic as groups with operators (since the characters $\alpha \in R^{+}$ of $T$ are pairwise distinct on each fiber).

2.6.

Replacing $R^{+}$ by $R^{-}=-R^{+}$, one constructs similarly the groups $U^{-}$, $B^{-}$, $U_{\alpha }$ ($\alpha \in R^{-}$) and the isomorphisms

p_α : 𝔾_a ⥲ U_α,  α ∈ R⁻.

Let us finally introduce the product scheme

Ω = U⁻ × T × U;

one has a canonical isomorphism of $k$-schemes

Ω ⊗_ℤ k ⥲ U⁻_k ×_k T_k ×_k U_k ≃ Ω_k,

where ${\Omega }_k$ is the "big cell" of $G_k$ (Exp. XXII, 4.1.2).

From now on, we identify $\Omega {\otimes }_{\mathbb{Z}}k$ with ${\Omega }_k$ by the previous isomorphism; we regard $U^{-}$, $T$, $U$ as subschemes of $\Omega$, through their unit sections. We denote by $e=((0),e,(0))$ the "unit section" of $\Omega$.

Our goal now is to put a piece-of-group law on $\Omega$.

Lemma 2.7. Let $\alpha \in \Delta$, and let $w_{\alpha ,k}$ be the element of $Nor{m}_{G_k}(T_k)(k)$ defined by $X_{\alpha ,k}$ (recall that by definition

w_{α,k} = p_{−α,k}(−1) p_{α,k}(1) p_{−α,k}(−1)).

There exist an open subset $V_{\alpha }$ of $\Omega$ containing the section $e$, and a morphism

$$h_{\alpha }:V_{\alpha }\to \Omega ,$$

satisfying the following conditions:

(i) $h_{\alpha }(e)=e$,

(ii) $(h_{\alpha }){\otimes }_{\mathbb{Z}}k$ coincides with the restriction of $int(w_{\alpha ,k})$ to $V_{\alpha }{\otimes }_{\mathbb{Z}}k\subset G_k$.

(iii) One has $T\subset V_{\alpha }$, and $h_{\alpha }$ sends $T$ into $T$. For every $\beta \in R$, one has $U_{\beta }\subset V_{\alpha }$ and $h_{\alpha }$ sends $U_{\beta }$ into $U_{s_{\alpha }(\beta )}$.

By virtue of the definition of a Chevalley system (Exp. XXIII, 6.1), there exists for each $\beta \in R$ an integer $e_{\beta }=\pm 1$ such that

int(w_{α,k}) p_{β,k}(x) = p_{s_α(β),k}(e_β x)

for every $x\in {\mathbb{G}}_a(S)$, $S\to \mathrm{Spec}(k)$.

Let $S$ be any scheme, and write an arbitrary element of $\Omega (S)$ in the form

u = ( ∏_{β∈R⁻, β≠−α} p_β(x_β) · p_{−α}(x_{−α}),  t,  p_α(x_α) · ∏_{β∈R⁺, β≠α} p_β(x_β) ),

where one has chosen some (arbitrary) order on $R^{-}--\alpha$ and $R^{+}-\alpha$ (cf. 2.5).

One defines a morphism $d:\Omega \to \mathrm{Spec}(\mathbb{Z})$ by

d(u) = α(t) + x_α x_{−α}.

Let $V_{\alpha }$ be the open subset ${\Omega }_d$ (that is, the open subset of $\Omega$ defined by "$d(u)$ invertible"); it contains $e$, $T$, and each $U_{\beta }$, $\beta \in R$. Let

$$h_{\alpha }:V_{\alpha }\to \Omega$$

be the morphism defined by $h_{\alpha }(u)=(a(u),b(u),c(u))$ where

a(u) = ( ∏_{β∈R⁻, β≠−α} p_{s_α(β)}(e_β x_β) ) · p_{−α}(−x_α d(u)⁻¹),

b(u) = t · α*(d(u)),

c(u) = p_α(−x_{−α} d(u)⁻¹) · ( ∏_{β∈R⁺, β≠α} p_{s_α(β)}(e_β x_β) ).

Since $s_{\alpha }$ permutes the positive roots (resp. negative roots) distinct from $\alpha$ (resp. $-\alpha$), $c(u)$ (resp. $a(u)$) is a section of $U$ (resp. $U^{-}$) and the previous morphism is well-defined. It satisfies (i) and (iii) trivially. As for (ii), this follows at once from the definition of the $e_{\beta }$, $\beta \in R$, and from Exp. XX, 3.12.

Lemma 2.8. There exist open subsets $V$ and $V^{\prime }$ of $\Omega$ and morphisms

h : V → Ω,  h′ : V′ → Ω,

satisfying the following conditions:

(i) $V$ and $V^{\prime }$ contain $e$, and $h(e)=h^{\prime }(e)=e$.

(ii) The morphism induced by $h^{\prime }\circ h:h^{-1}(V^{\prime })\to \Omega$ is the restriction of the identity morphism.

(iii) The $k$-morphisms $h{\otimes }_{\mathbb{Z}}k$ and $h^{\prime }{\otimes }_{\mathbb{Z}}k$ are the restrictions to $V{\otimes }_{\mathbb{Z}}k$ and $V^{\prime }{\otimes }_{\mathbb{Z}}k$ of automorphisms of the group $G_k$.

(iv) $V$ and $V^{\prime }$ contain $U$, $T$ and $U^{-}$; $h$ and $h^{\prime }$ send $U$ into $U^{-}$, $U^{-}$ into $U$, and $T$ into $T$.

Let $w^0$ be the element of the Weyl group of $G_k$ that transforms $R^{+}$ into $R^{-}$. Write

w⁰ = s_{α_n} ⋯ s_{α_1},  α_i ∈ Δ

(no relation to the numbering of 2.1). Put

w_0 = w_{α_n,k} ⋯ w_{α_1,k} ∈ Norm_{G_k}(T_k)(k).

Define by recursion on $i\le n$ an open subset $V_i$ of $\Omega$ and a morphism $h_i:V_i\to \Omega$ by $V_0=\Omega$, $h_0=id$, and, for $i=0,\cdots ,n-1$,

V_{i+1} = h_i⁻¹(V_{α_{i+1}}),  h_{i+1} = h_{α_{i+1}} ∘ h_i,

where the notations $V_{{\alpha }_j}$ and $h_{{\alpha }_j}$ are those of 2.7.

Take $V=V_n$ and $h=h_n$. The conditions in (i), (iii) and (iv) bearing on $V$ and $h$ are indeed verified; for (i) and (iii) this follows at once from 2.8, for (iv) from the fact that $h{\otimes }_{\mathbb{Z}}k$ is the restriction of $int(w_0)$ to $V{\otimes }_{\mathbb{Z}}k$.

Since $(w^0{)}^2=1$, one also has

w⁰ = s_{α_1} ⋯ s_{α_n} = (s³_{α_1}) ⋯ (s³_{α_n}).

Setting

$$w_0^{\prime }=(w_{{\alpha }_1,k}{)}^3\cdots (w_{{\alpha }_n,k}{)}^3,$$

and carrying out the same construction as above, one deduces from it an open subset $V^{\prime }$ and a morphism $h^{\prime }$ also satisfying (i), (iii), (iv). Moreover, $h^{\prime }{\otimes }_{\mathbb{Z}}k$ is the restriction of $int(w_0^{\prime })$ to $V^{\prime }{\otimes }_{\mathbb{Z}}k$. But for each simple root $\alpha \in \Delta$, one has $(w_{\alpha ,k}{)}^4=e$ (cf. Exp. XX, 3.1), hence $w_0^{\prime }\cdot w_0=e$, which shows that $h^{\prime }\circ h$ induces the identity morphism on a non-empty open subset of $\Omega {\otimes }_{\mathbb{Z}}k$. But since $\Omega$ is smooth and of finite presentation over $\mathbb{Z}$, $\Omega {\otimes }_{\mathbb{Z}}k$ is schematically dense in $\Omega$, which proves (ii).

Proposition 2.9. There exist an open subset V_1 of $\Omega \times \Omega$, an open subset V_2 of $\Omega$, and morphisms

π : V_1 → Ω,  σ : V_2 → Ω,

having the following properties:

(i) If $x\in \Omega (S)$, then $(e,x)$ and $(x,e)$ are sections of V_1, and

π(e, x) = π(x, e) = x.

(ii) V_2 contains $e$ and $\sigma (e)=e$.

(iii) ${\pi }_k$ and ${\sigma }_k$ are the restrictions of the morphisms $G_k{\times }_k{G}_k\to G_k$ and $G_k\to G_k$ defined by the product (resp. the inverse).

Proof. Let $(v,t,u)$ and $(v^{\prime },t^{\prime },u^{\prime })$ be two sections of $\Omega$. Then $h(u)$ is a section of $U^{-}$, $h(v^{\prime })$ is a section of $U$ by 2.8 (iv), and one can therefore regard $(h(u),e,h(v^{\prime }))$ as a section of $\Omega$. Let V_1 be the open subset of $\Omega \times \Omega$ defined by the condition:

((v, t, u), (v′, t′, u′)) ∈ V_1(S) ⟺ (h(u), e, h(v′)) ∈ V′(S)

(notation of 2.8). If $((v,t,u),(v^{\prime },t^{\prime },u^{\prime }))$ is a section of V_1, then $h^{\prime }(h(u),e,h(v^{\prime }))$ is defined; it is a section of $\Omega$ which one can decompose:

h′(h(u), e, h(v′)) = (v′′, t′′, u′′).

One then sets

π((v, t, u), (v′, t′, u′)) = (v · ᵗv′′ t⁻¹, t t′′ t′, t′⁻¹ u′′ t′ · u′).

The verification of (i) is immediate (by 2.8 (ii)). To verify the condition in (iii) bearing on $\pi$, one sees that

h′(h(u), e, h(v′)) = u v′ = v′′ t′′ u′′

when $u\in U(S)$, $v\in U^{-}(S)$, $S\to k$, by virtue of 2.8 (iii) and (ii). One constructs $\sigma$ similarly: if $(v,t,u)$ is a section of $\Omega$, $h(u^{-1})$ is a section of $U^{-}$,

$h(v^{-1})$ is a section of $U$, $h(t^{-1})$ is a section of $T$, so $(h(u^{-1}),h(t^{-1}),h(v^{-1}))$ is a section of $\Omega$ and one can define an open subset V_2 of $\Omega$ by

(v, t, u) ∈ V_2(S) ⟺ (h(u⁻¹), h(t⁻¹), h(v⁻¹)) ∈ V′(S)

and a morphism $\sigma :V_2\to \Omega$ by

σ(v, t, u) = h′(h(u⁻¹), h(t⁻¹), h(v⁻¹)).

One verifies the conditions on $\sigma$ as above.

Corollary 2.10. $\pi$ is "generically associative" and $\sigma$ is a "generic inverse": if $x,y,z\in \Omega (S)$ and if the expressions below are defined (which always happens over an open subset of $\Omega$ containing the unit section), one has:

π(x, π(y, z)) = π(π(x, y), z),  π(x, σ(x)) = e = π(σ(x), x).

Indeed, the two sides of each of these formulas define morphisms between smooth $\mathbb{Z}$-schemes of finite presentation, which coincide on the generic fibers, by 2.10 (iii).

Corollary 2.11. Let $\alpha \in R$. For all $S$ and all $x,y\in {\mathbb{G}}_a(S)$ such that $(p_{\alpha }(x),p_{-\alpha }(y))\in V_1(S)$ and $1+xy\in {\mathbb{G}}_m(S)$ (which defines an open subset of ${\mathbb{G}}_{a,S}^2$ containing the section (0, 0)), one has:

π(p_α(x), p_{−α}(y)) = ( p_{−α}( y / (1 + xy) ), α*(1 + xy), p_α( x / (1 + xy) ) ).

The proof is the same as before, by Exp. XX, 2.1.

3. Existence theorem: end of the proof

To simplify language, let us pose the following definition.

Definition 3.1. Let $S$ be a scheme and $G$ an $S$-group scheme. One says that $G$ is admissible if there exists an open immersion of $S$-schemes $i:{\Omega }_S=\Omega \times S\to G$ satisfying the following conditions:

(i) The diagram

(V_1)_S ──i ×_S i──▶ Ω_S ×_S Ω_S ──i ×_S i──▶ G ×_S G
   │                                              │
   π                                              π_G
   │                                              │
   ▼          ─────────── i_S ───────────▶        ▼
  Ω_S                                             G,

where ${\pi }_G$ denotes the multiplication morphism in $G$, is commutative.

(ii) There exists a finite set of sections $a_j\in \Omega (S)$ such that the $i(a_j)\cdot i({\Omega }_S)$ cover $G$.

By Weil's "theorem" (Exp. XVIII, 3.13 (iii) and (iv)), one has:

Lemma 3.2. If for every scheme $S$ étale and of finite type over $\mathbb{Z}$, every admissible $S$-group is affine, then there exists an admissible affine $\mathbb{Z}$-group.

Now one has:

Lemma 3.3. Let $S$ be a scheme and $G$ an admissible $S$-group. Then $G$ is smooth and of finite presentation over $S$, with affine connected semisimple fibers.

Since ${\Omega }_S$ is smooth and of finite presentation over $S$, with connected fibers, the same holds for $G$, by condition (ii). To verify 3.3, one may therefore suppose that $S$ is the spectrum of a field $K$. Let us identify ${\Omega }_K$ with its image in $G$. It is clear that ${\Omega }_K$ is the product

∏_{α∈R⁻} U_{α,K} · T_K · ∏_{α∈R⁺} U_{α,K}

of the subgroups T_K and $U_{\alpha ,K}$ ($\alpha \in R$) of $G$. The Lie algebra of $G$ is therefore identified with the direct sum

$$Lie(T_K)\oplus \underset{\alpha \in R}{⨁}Lie(U_{\alpha ,K}).$$

Since the inner automorphism defined by a section of T_K acts on $U_{\alpha ,K}$, and therefore on $Lie(U_{\alpha ,K})$, through the character

α ∈ R ⊂ M ≃ Hom_{K-gr.}(T_K, 𝔾_{m,K}),

the previous decomposition of $Lie(G)$ is exactly the decomposition under the adjoint action of $T$. The roots of G_K with respect to T_K are therefore the $\alpha \in R$. Apply Exp. XIX, 1.13. Let $T_{\alpha }$ be the maximal torus of $Ker(\alpha )\subset T_K$, and let $Z_{\alpha }=Cent{r}_G(T_{\alpha })$; it suffices for us to prove that each $Z_{\alpha }$ is reductive. Now $Z_{\alpha }\cap {\Omega }_K$ is none other than

∏_{β∈R⁻, β|_{T_α} = e} U_{β,K} · T_K · ∏_{β∈R⁺, β|_{T_α} = e} U_{β,K};

but the roots vanishing on $T_{\alpha }$ are the rational multiples of $\alpha$, hence $\alpha$ and $-\alpha$; this proves

Z_α ∩ Ω_K = U_{−α,K} · T_K · U_{α,K}.

To prove that $Z_{\alpha }$ is reductive, it suffices, by Exp. XX 3.4, to prove that $U_{\alpha ,K}$ and $U_{-\alpha ,K}$ do not commute, which follows at once from 2.11.

It follows from 3.2 and 3.3 that the proof will be complete if we prove:

Lemma 3.4. If $S$ is a locally noetherian scheme of dimension $\le 1$, and if $G$ is a smooth $S$-group of finite type with affine connected semisimple fibers, then $G$ is affine (and therefore semisimple).

Nota. In Exp. XVI, it was seen that 3.4 is true without hypothesis on $S$, but the proof is relatively delicate; since here we need only the particular case 3.4, we give a direct proof of it.

Consider the Lie algebra $g$ of $G$, which is a locally free ${\mathcal{O}}_S$-module, and the adjoint representation of $G$

$$Ad:G\to G{L}_{{\mathcal{O}}_S}(g).$$

To prove that $G$ is affine over $S$, it suffices to prove that the morphism Ad is affine. Since $G$ is smooth with connected fibers, it is separated over $S$ (VI_B 5.5), so the morphism Ad is separated. Using a result proved in the appendix (see 4.1), it suffices to prove that the morphism Ad is quasi-finite. One is thus reduced to the case where $S$ is the spectrum of a field; in this case $G$ is affine, hence semisimple, and one is reduced to Exp. XXII 5.7.14.

4. Appendix

We used in the course of the proof the following proposition:

Proposition 4.1. Let $S$ be a locally noetherian scheme of dimension $\le 1$, let $G$ and $H$ be two $S$-group schemes of finite type, and $f:G\to H$ a quasi-finite separated group morphism. If $G$ is flat over $S$,⁷ then $f$ is affine.

We shall give the proof only in the case where $G$ is smooth over $S$, a hypothesis which is indeed verified in the application we have made of the proposition.

4.2.

By EGA II, 1.6.4, one may suppose $S$ reduced. By the usual techniques of passage to the limit,⁸ one may suppose $S$ local. If $\dim (S)=0$, the assertion is trivial,⁹ suppose $\dim (S)=1$. By faithfully flat descent, one may suppose that $S$ is complete with algebraically closed residue field. Replacing $S$ by its normalization $\tilde{S}$ if necessary, one may (EGA II, 6.7.1 and EGA 0_IV 23.1.5) suppose $S$ normal.¹⁰ One is therefore reduced to the case where $S$ is the spectrum of a complete discrete valuation ring $A$ with algebraically closed residue field.

4.3.

Let $\eta$ (resp. $s$) be the generic (resp. closed) point of $S$. Consider the image $f_{\eta }(G_{\eta })$ of $G_{\eta }$ in $H_{\eta }$. It is a closed subgroup scheme of $H_{\eta }$. Let $H^{\prime }$ be the schematic closure in $H$ of $f_{\eta }(G_{\eta })$. Since $H^{\prime }\to H$ is affine (it is a closed immersion), one may replace $H$ by $H^{\prime }$ and so suppose $H$ flat over $S$ and $f_{\eta }$ surjective. Since $f_{\eta }$ is finite, and $G$ and $H$ are flat over $S$, one has

dim(G_s) = dim(G_η) = dim(H_η) = dim(H_s).

4.4.

Let $H_s^0,\cdots ,H_s^n$ be the irreducible components of $H_s$, where $H_s^0$ denotes the neutral component, and let $z_0,\cdots ,z_n$ be their generic points. Since each local ring ${\mathcal{O}}_{H,z_i}$ is of dimension $\le 1$, the morphism $G{\times }_H{\mathcal{O}}_{H,z_i}\to {\mathcal{O}}_{H,z_i}$ is affine because quasi-finite and separated (cf. Exp. XVI, Lemma 4.2), so $G$ is affine over $H$ in a neighborhood of $z_i$. Denoting by $V$ the largest open subset of $H$ such that $G{|}_V$ is affine over $V$, it follows

that $V$ contains all the $z_i$,¹¹ so contains at least one closed point $y_i\in H_s^i$ (and one has $\kappa (y_i)=\kappa (s)$ since $\kappa (s)$ is algebraically closed).

On the other hand, $V$ is obviously stable under the translation defined by any element $g\in G(S)$. But one has $\dim (G_s)=\dim (H_s)$ and $f_s$ is finite, so

$$f(s):G_s^0(s)\to H_s^0(s)$$

is surjective. Since $A$ is complete and $G$ smooth over $S$, the canonical map $G^0(S)\to G_s^0(s)$ is surjective; since $H_s^0(s)$ acts transitively on each $H_s^i(s)$, it follows that $V\supset H_s(s)$, hence (since $\kappa (s)$ is algebraically closed) $V\supset H_s$.¹²

Since one obviously has $V\supset H_{\eta }$, since $f_{\eta }$ is finite, one therefore has $V=H$. QED.

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Footnotes

                   g
          S′′ ────────▶ G
          │ ╲           │
          π   ╲ φ       │
          │     ╲       │
          ▼       ▼     ▼
          S′ ─── w ───▶ S