Exposé XXIV. Automorphisms of reductive groups

by M. Demazure

¹

The first part of this Exposé (nos 1 to 5) is a direct consequence of the existence, for a reductive group, of "sufficiently many outer automorphisms", a result which is itself a consequence of the weakest form of the isomorphism theorem for pinned groups. The second part (nos 6 and 7) presents two applications of the more precise results of the preceding Exposé; in particular, no 7 uses the theorem of generators and relations in its explicit form. Finally, we have given in an appendix (no 8) some results on "Galois" cohomology used in the text.

Let us specify our cohomological notation: if $S$ is a scheme and $G$ an $S$-group scheme, we write $H^1(S,G)$ for the first cohomology set of $S$ with coefficients in $G$, computed for the (fpqc) topology; this is also the set of isomorphism classes of (fpqc) principal homogeneous sheaves under $G$. We write $H_{\acute{e}}^{1}t(S,G)$ for the corresponding set for the étale topology; this is therefore the part of $H^1(S,G)$ formed by classes of homogeneous sheaves under $G$ which are quasi-isotrivial (= locally trivial for the étale topology). We write $Fib(S,G)$ for the part of $H^1(S,G)$ formed by classes of representable sheaves (principal homogeneous bundles). We thus have the inclusions

H¹_ét(S, G) ⊂ H¹(S, G),
Fib(S, G) ⊂ H¹(S, G).

If every principal homogeneous sheaf under $G$ is representable (for example if $G$ is quasi-affine over $S$, cf. SGA 1, VIII 7.9), then $Fib(S,G)=H^1(S,G)$.

If $S^{\prime }\to S$ is a covering morphism for the (fpqc) topology, we write $H^1(S^{\prime }/S,G)$ for the kernel of the canonical map $H^1(S,G)\to H^1(S^{\prime },G_{S^{\prime }})$. It is known that $H^1(S^{\prime }/S,G)$ can be computed simplicially (TDTE I, § A.4), which implies that when $S^{\prime }\to S$ is covering for the étale topology, $H^1(S^{\prime }/S,G)$ is also the kernel of $H_{\acute{e}}^{1}t(S,G)\to H_{\acute{e}}^{1}t(S^{\prime },G_{S^{\prime }})$.

Finally, following Exp. VIII, 4.5, we call "theorem 90" the following assertion: "every principal homogeneous sheaf under $G_{m,S}$ is representable and locally trivial", an assertion equivalent to "$H^1(S,G_{m,S})=\mathrm{Pic}(S)$", or again to "$H^1(S,G_{m,S})=0$ for $S$ local (or more generally semi-local)".

1. The automorphism scheme of a reductive group

1.0.

It is convenient first to make certain definitions of the preceding Exposé more precise. Let $S$ be a nonempty scheme, $G$ an $S$-reductive group, $R=(M,M\ast ,R,R\ast ,\Delta )$ a pinned reduced root datum (Exp. XXIII 1.5). We call pinning of $G$ of type $R$, or $R$-pinning of $G$, the datum:

(i) of an isomorphism of $D_S(M)$ onto a maximal torus $T$ of $G$ (or, what amounts to the same, of a monomorphism $D_S(M)\to G$ whose image is a maximal torus $T$ of $G$), identifying $R$ with a root system of $G$ relative to $T$ (Exp. XIX, 3.6) and $R\ast$ with the corresponding set of coroots,

(ii) for each $\alpha \in \Delta$, of an $X_{\alpha }\in \Gamma (S,g_{\alpha }{)}^{\times }$.

For $G$ to possess an $R$-pinning, it is necessary and sufficient that it be splittable and of type $R$ (Exp. XXII, 2.7).

If $u:G\to G^{\prime }$ is an isomorphism of $S$-reductive groups, to every $R$-pinning $E$ of $G$ there corresponds by "transport of structure" an $R$-pinning $u(E)$ of $G^{\prime }$. If $v:R^{\prime }\to R$ is an isomorphism of pinned root data, to every $R^{\prime }$-pinning $E$ of $G^{\prime }$ there corresponds by transport of structure an $R$-pinning $v(E)$ of $G$.

Let us call a pinned group a triple $(G,R,E)$ where $G$ is an $S$-reductive group, $R$ is a pinned reduced root datum, and $E$ is a pinning of $G$ of type $R$. We call an isomorphism of the pinned group $(G,R,E)$ onto the pinned group (G', R', E') a pair $(u,v)$ where $u$ is an isomorphism $u:G\to G^{\prime }$ and $v$ is an isomorphism of pinned root data $v:R^{\prime }\to R$, such that $u(E)=v(E^{\prime })$.²

N. B. If $S$ is nonempty, $v$ is uniquely determined by $u$, and we shall also say by abuse of language that $u$ is an isomorphism of pinned groups. In particular, if $(G,R,E)$ is a pinned group, an automorphism of $(G,R,E)$ is therefore an automorphism $u$ of $G$ such that there exists an automorphism $v$ of $R$ with $u(E)=v(E)$; it is therefore an automorphism of $G$ normalizing $T$, inducing on $T$ an automorphism of the form $D_S(h)$ where $h$ is an automorphism of $M$, and permuting among themselves the elements $X_{\alpha }$, $\alpha \in \Delta$. (As is easily seen, the preceding conditions moreover characterize the automorphisms of $(G,R,E)$.)

One has an obvious contravariant functor

Φ :  (G, R, E) ↦ R,   (u, v) ↦ v

and the principal result of the preceding Exposé (Exp. XXIII, 4.1) shows us that this is a fully faithful functor (we shall moreover see in the next Exposé that it is an equivalence of categories). It follows in particular that the group of automorphisms of $(G,R,E)$ is canonically isomorphic to the group of automorphisms of the pinned root datum $R$ (cf. Exp. XXIII, 5.5).

1.1.

Let $S$ be a scheme; endow $(Sch)/S$ with the (fpqc) topology and consider the $S$-group sheaf ${\mathrm{Aut}}_{S-gr.}(G)$, where $G$ is an $S$-group scheme. One has an exact sequence of $S$-group sheaves

1 → Centr(G) → G ─int→ Aut_{S-gr.}(G)

which defines a monomorphism

j : G / Centr(G) → Aut_{S-gr.}(G).

The image sheaf of $j$ is the sheaf of inner automorphisms of $G$; for an automorphism $u$ of $G$ to be inner, it is necessary and sufficient that there exist a covering family $S_i\to S$ and for each $i$ a $g_i\in G(S_i)$ such that $int(g_i)=u_{S_i}$. In this case, if $v$ is another automorphism of $G$, one sees at once that $int(v)u=vu{v}^{-1}$ is the inner automorphism defined by the family $g_i^{\prime }=v(g_i)$. It follows that the image of $j$ is normal in ${\mathrm{Aut}}_{S-gr.}(G)$. The quotient group sheaf, denoted $Autext(G)$, is the sheaf of outer automorphisms of $G$. One thus has an exact sequence

1 → G / Centr(G) → Aut_{S-gr.}(G) → Autext(G) → 1.

The preceding definitions are all compatible with base change. They are naturally valid in any site.

1.2.

Let $S$ be a scheme and $(G,R,E)$ a pinned $S$-reductive group. Let $E$ be the (abstract) group of automorphisms of the pinned root datum $R$, i.e. the group of automorphisms of $R$ normalizing $\Delta$. By Exp. XXIII, 5.5, one has a canonical monomorphism

$$E\to {\mathrm{Aut}}_{S-gr.}(G)$$

which associates to $h\in E$ the unique automorphism $u$ of the pinned group $G$ such that $\Phi (u)=h$. This monomorphism canonically defines a monomorphism of sheaves

$$(\ast )a:E_S\to {\mathrm{Aut}}_{S-gr.}(G).$$

For an automorphism $u$ of $G$ to be a section of the image sheaf of $a$, it is necessary and sufficient that the following conditions be satisfied:

(i) $u$ normalizes $T$. One then knows that $u$ permutes the roots of $G$ relative to $T$. If $\alpha \in R$, then $u(\alpha ):t\mapsto \alpha (u^{-1}(t))$ is locally on $S$ of the form $t\mapsto \beta (t)$, with $\beta \in R$. The second condition is then written as follows:

(ii) If $\alpha \in \Delta$ and if $U$ is an open set of $S$ such that $u(\alpha {)}_U\in R$, then $u(\alpha {)}_U\in \Delta$ and

$$Lie(u_U)(X_{\alpha }{)}_U=(X_{u(\alpha )}{)}_U.$$

It follows at once from the definitions that the sections of $a(E_S)$ normalize the subgroups of $G$ defined by the pinning: $T$, $B$, $B^{-}$, $U$, $U^{-}$.

These definitions being set, one has:

Theorem 1.3. Let $S$ be a scheme, $G$ an $S$-reductive group. Consider the canonical exact sequence of $S$-group sheaves³

1 → ad(G) → Aut_{S-gr.}(G) ─p→ Autext(G) → 1.

(i) ${\mathrm{Aut}}_{S-gr.}(G)$ is representable by a smooth and separated $S$-scheme.

(ii) $Autext(G)$ is representable by a finitely generated twisted constant $S$-scheme (Exp. X, 5.1).

(iii) If $G$ is splittable, the preceding exact sequence splits. More precisely, for every pinning of $G$, the morphism (cf. 1.2 (*))

p ∘ a : E_S → Autext(G)

is an isomorphism.

Let us first show how the theorem follows from the following lemma:

Lemma 1.4. Under the hypotheses of (iii), ${\mathrm{Aut}}_{S-gr.}(G)$ is the semi-direct product $a(E_S)\cdot ad(G)$.⁴

The lemma immediately implies the theorem when $G$ is splittable. Since $G$ is locally splittable for the étale topology (Exp. XXII, 2.3), hence also for the (fppf) topology, and since the latter is "of effective descent" for the fibered category of twisted constant morphisms (Exp. X, 5.5), one deduces (ii) in the general case (cf. Exp. IV, 4.6.8). To deduce (i), one notes that $ad(G)$ is affine over $S$, hence the morphism $p$ is affine when ${\mathrm{Aut}}_{S-gr.}(G)$ is representable, and one concludes by descent of affine schemes.⁵

It therefore remains only to prove 1.4. For this, it suffices to prove:

Lemma 1.5. If $(R,E)$ and (R', E') are two pinnings of the $S$-reductive group $G$, there exists a unique inner automorphism $u$ of $G$ over $S$ transforming one pinning into the other (i.e. such that there exists $v:R^{\prime }\stackrel{\sim }{\to }R$ with $u(E)=v(E^{\prime })$, cf. 1.0).

1.5.1. Uniqueness.

It suffices to prove that if $G$ is a pinned $S$-group and if $int(g)$ is an automorphism of pinned group ($g\in G(S)$), then $int(g)=id$. Now one has first $int(g)T=T$, $int(g)B=B$, so $g\in Nor{m}_G(T)(S)\cap Nor{m}_G(B)(S)=T(S)$ (cf. for example Exp. XXII, 5.6.1). It follows that $int(g)$ normalizes each $U_{\alpha }$ and that

Lie(int(g)) X_α = Ad(g) X_α = α(g) X_α

for every $\alpha \in \Delta$. One thus has $\alpha (g)=1$ for $\alpha \in \Delta$, so $g\in {\bigcap }_{\alpha \in \Delta }(Ker\alpha )(S)=Centr(G)(S)$ (Exp. XXII, 4.1.8). QED

1.5.2. Existence.

It suffices to prove this locally for the (fpqc) topology. Let

(T, M, R, Δ, (X_α)_{α ∈ Δ})    and    (T', M', R', Δ', (X'_{α'})_{α' ∈ Δ'})

be the two pinnings. By conjugacy of maximal tori, one may assume $T=T^{\prime }$. Up to restricting $S$, one may assume that the isomorphism $D_S(M)\simeq D_S(M^{\prime })$ comes from an isomorphism $M\simeq M^{\prime }$ carrying $R$ onto $R^{\prime }$, and one is reduced to the situation $T=T^{\prime }$, $M=M^{\prime }$, $R=R^{\prime }$. Since the systems of simple roots are conjugate by the Weyl group (Exp. XXI, 3.3.7), one may also assume $\Delta ={\Delta }^{\prime }$. There then exists for each $\alpha \in \Delta$ a scalar $z_{\alpha }\in G_m(S)$ such that $X_{\alpha }^{\prime }=z_{\alpha }{X}_{\alpha }$, and it suffices to construct locally for (fpqc) a section $t$ of $T$ such that $\alpha (t)=z_{\alpha }$ for each $\alpha \in \Delta$. But the morphism $T\to (G_{m,S}{)}^{\Delta }$ with components $\alpha ,\alpha \in \Delta$ is the dual of an injection ${\mathbb{Z}}^{\Delta }\to M$, hence faithfully flat, which completes the proof of 1.5.2 and so of 1.3.

Corollary 1.6. Let $S$ be a scheme, $G$ an $S$-reductive group. The following conditions are equivalent:

(i) ${\mathrm{Aut}}_{S-gr.}(G)$ is affine (resp. of finite type, resp. of finite presentation, resp. quasi-compact) over $S$.

(ii) $Autext(G)$ is finite over $S$.

(iii) For every $s\in S$, one has $r{g}_{red}(G_s)-r{g}_{ss}(G_s)⩽1$.

Indeed, since $ad(G)$ is affine, flat and of finite presentation over $S$, the morphism $p:{\mathrm{Aut}}_{S-gr.}(G)\to Autext(G)$ is affine, faithfully flat and of finite presentation.

If $Autext(G)$ is finite over $S$, it is affine over $S$, hence so is ${\mathrm{Aut}}_{S-gr.}(G)$, which proves (ii) ⇒ (i). If ${\mathrm{Aut}}_{S-gr.}(G)$ is quasi-compact over $S$, it is of finite presentation over $S$ (being in any case locally of finite presentation and separated over $S$); by Exp. V, 9.1, $Autext(G)$ is then of finite presentation over $S$, hence finite, which proves (i) ⇒ (ii). Finally, to prove the equivalence of (ii) and (iii), one may assume $G$ split, and one is reduced to Exp. XXI, 6.7.8.

Corollary 1.7. Let $S$ be a scheme and $G$ an $S$-reductive group. Then ${\mathrm{Aut}}_{S-gr.}(G{)}^0\simeq ad(G)$.

Corollary 1.8. Let $S$ be a scheme, $G$ an $S$-reductive group, $H$ an $S$-group scheme smooth, affine and with connected fibers over $S$. Then the $S$-functor

$$Iso{m}_{S-gr.}(G,H)$$

is representable by a smooth and separated $S$-scheme (which is affine over $S$ if $G$ is semisimple).

Indeed, let $U$ be the set of points $s$ of $S$ such that $H_s$ is reductive; this is open (Exp. XIX, 2.6); if $S^{\prime }$ is an $S$-scheme, $H_{S^{\prime }}$ is reductive if and only if $S^{\prime }\to S$ factors through $U$. It follows that the canonical morphism $Iso{m}_{S-gr.}(G,H)\to S$ factors through $U$. One may therefore assume $S=U$ and one is reduced to:

Corollary 1.9. Let $S$ be a scheme, $G$ and $G^{\prime }$ two $S$-reductive groups. Then

$$F=Iso{m}_{S-gr.}(G,G^{\prime })$$

is representable by a smooth and separated $S$-scheme (affine if $G$ or $G^{\prime }$ is semisimple). Moreover, $S$ decomposes as a sum of two open subschemes S_1 and S_2 such that $F_{S_1}=\varnothing$ and such that $F_{S_2}$ is a principal homogeneous bundle on the left (resp. right) under ${\mathrm{Aut}}_{S_2-gr.}(G_{S_2}^{\prime })$ (resp. ${\mathrm{Aut}}_{S_2-gr.}(G_{S_2})$).

Indeed, let S_2 be the set of points of $S$ where $G$ and $G^{\prime }$ are of the same type, and let S_1 be its complement.

Since the type of a reductive group is a locally constant function, S_1 and S_2 are open. It is clear that $F_{S_1}=\varnothing$, and one may assume $S=S_2$. By Exp. XXIII, 5.6, $F$ is a principal homogeneous sheaf under ${\mathrm{Aut}}_{S-gr.}(G)$, locally trivial for the étale topology. It follows that $F_0=F/ad(G)$ is a principal homogeneous sheaf under $Autext(G)$, locally trivial for the étale topology, hence representable (Exp. X, 5.5). Since $ad(G)$ is affine, $F$ is therefore also representable.

Let us remark that in the course of the proof, we have obtained:

Corollary 1.10. Let $S$ be a scheme, $G$ and $G^{\prime }$ two $S$-reductive groups of the same type at each $s\in S$. Then $ad(G)$ operates freely (on the right) in $Iso{m}_{S-gr.}(G,G^{\prime })$, the quotient sheaf

Isomext(G, G') = Isom_{S-gr.}(G, G') / ad(G)

is representable by a twisted constant $S$-scheme, which is a principal homogeneous bundle under $Autext(G)$ (and which is therefore finite over $S$ if $G$ is semisimple). Moreover, the isomorphism

Isom_{S-gr.}(G, G') ≃ Isom_{S-gr.}(G', G)

defined by $u\mapsto u^{-1}$ induces an isomorphism

Isomext(G, G') ≃ Isomext(G', G).

Remark 1.11. If $Isomext(G,G^{\prime })(S)\ne \varnothing$, one says that $G$ is an inner twisted form of $G^{\prime }$; then $G^{\prime }$ is an inner twisted form of $G$; one can then reduce the structure group of $Iso{m}_{S-gr.}(G,G^{\prime })$ to $ad(G)$. More precisely, let $u\in Isomext(G,G^{\prime })(S)$, considered as a section $u:S\to Isomext(G,G^{\prime })$. Write

$$Isomin{t}_u(G,G^{\prime })$$

for the inverse image under the canonical morphism $Iso{m}_{S-gr.}(G,G^{\prime })\to Isomext(G,G^{\prime })$ of the closed subscheme of $Isomext(G,G^{\prime })$ image of $u$. The natural operation of $ad(G)$ on $Isomin{t}_u(G,G^{\prime })$ endows this scheme with a structure of principal homogeneous bundle; by extension of the structure group⁶ $ad(G)\to \mathrm{Aut}(G)$, $Isomin{t}_u(G,G^{\prime })$ recovers $Iso{m}_{S-gr.}(G,G^{\prime })$.

By Hensel's lemma (Exp. XI, 1.11), 1.8 gives at once:

Corollary 1.12. Let $S$ be a henselian local scheme, $G$ an $S$-reductive group, $G^{\prime }$ a smooth affine $S$-group with connected fibers, $s$ the closed point of $S$. If $G_s$ and $G_s^{\prime }$ are isomorphic $\kappa (s)$-algebraic groups, then $G$ and $G^{\prime }$ are isomorphic. More precisely, every $\kappa (s)$-isomorphism $G_s\simeq G_s^{\prime }$ comes from an $S$-isomorphism $G\simeq G^{\prime }$.

Applying now 1.7 (resp. 1.12) to the scheme of dual numbers over a field, one deduces from Exp. III, 2.10 (resp. 3.10) point (i) (resp. (ii)) of the following corollary.

Corollary 1.13. Let $k$ be a field and $G$ a $k$-reductive group.

(i) If $G$ is adjoint, one has $H^1(G,Lie(G/k))=0$.⁷

(ii) One has $H^2(G,Lie(G/k))=0$.

Remark 1.14. (i) The assertion concerning $H^1$ was known (Chevalley); the one concerning $H^2$ had been proved in most cases of the classification by Chevalley.

(ii) In fact, the conjunction of 1.13 and the uniqueness theorem over an algebraically closed field is essentially equivalent to the uniqueness theorem. A direct proof of 1.13 would therefore give a way to deduce the general uniqueness theorem from Chevalley's uniqueness theorem over a field.

The existence of reductive groups of all types over all schemes (Exp. XXV) shows that the obstructions to lifting a $k$-reductive group $G$ over Artinian rings with residue field $k$ (which by Exp. III, 3.8 are elements of

H³(G, Lie(G/k) ⊗ V) ≃ H³(G, Lie(G/k)) ⊗ V,

where $V$ is a certain $k$-vector space (equipped with the trivial action of $G$)) are zero. This seems to suggest that $H^3(G,Lie(G/k))=0$. Here too, a direct proof of this fact (if it is true) would doubtless give a way to deduce the general existence theorem from the existence theorem over a field (Chevalley's Tôhoku).

Corollary 1.15. Let $k$ be a field,⁸ $G$ a $k$-reductive group. Consider $k$ as a trivial $G$-module. Then

H¹(G, k) = H²(G, k) = 0.

⁹ Since $H^i(G_{\bar{k}},\bar{k})=H^i(G,k)\otimes \bar{k}$, one may assume $k$ algebraically closed. An element of $H^1(G,k)$ is nothing but a morphism of $k$-groups $\varphi :G\to G_{a,k}$. Then $\varphi (G)=G/Ker(\varphi )$ is a smooth, connected and reductive subgroup (cf. XIX 1.7) of $G_{a,k}$, hence trivial. So $H^1(G,k)=0$.

Consider now the $k$-reductive group $H=G{\times }_k{G}_{m,k}$. One has $Lie(H/k)=Lie(G/k)\oplus k$, a decomposition stable under $H$. For any $H$-module $V$, one has

Hⁱ(H, V) = Hⁱ(G, H⁰(G_{m, k}, V))

(this follows from the characterization of $H^i(H,-)$ as derived functors of $H^0(H,-)=H^0(G,H^0(G_{m,k},-))$, and from the fact that the functor $H^0(G_{m,k},-)$ is exact, cf. Exp. I, 5.3.1 and 5.3.3). In particular, one has for every $i$

Hⁱ(H, Lie(H/k)) = Hⁱ(G, Lie(G/k)) ⊕ Hⁱ(G, k)

whence $H^2(G,k)=0$ by applying 1.13 (ii) to the reductive group $H$.

Corollary 1.15.1. ¹⁰ Let $k$ be a field, $G$ a $k$-reductive group, $Z$ its center and $R=(M,M\ast ,R,R\ast )$ the type of $G$. Then one has

H¹(G, Lie(G/k)) ≃ Ext¹_ℤ(M / Γ₀(R), k).

In particular, $H^1(G,Lie(G/k))=0$ if and only if $Z$ is smooth over $k$.

Indeed, let $g$ (resp. $z$, resp. $g_{ad}$) be the Lie algebra of $G$ (resp. $Z$, resp. $G_{ad}=G/Z$). It follows from 1.15 (and from its proof) that

H¹(G, g) = H¹(G_ad, g) ≃ H¹(G_ad, g/z).

Set $C=Coker(g\to g_{ad})$; one has an exact sequence

$$0\to g/z\to g_{ad}\to C\to 0.$$

Since $H^1(G_{ad},g_{ad})=0$ (1.13) and $H^0(G_{ad},g_{ad})=0$ (cf. Exp. II, 5.2.3), one obtains

H¹(G_ad, g/z) ≃ H⁰(G_ad, C).

To compute the right-hand term, one may assume $k$ algebraically closed. Let $(G,T,M,R)$ be a splitting of $G$; then $Z=D_k(M/{\Gamma }_0(R))$ and $C$ is identified with

Coker( Hom_ℤ(M, k) → Hom_ℤ(Γ₀(R), k) ) ≃ Ext¹_ℤ(M / Γ₀(R), k),

equipped with the trivial action of $G_{ad}$. The corollary follows, since $Z$ is not smooth over $k$ if and only if $char(k)=p>0$ and $M/{\Gamma }_0(R)$ has $p$-torsion (Exp. IX, 2.1).

Definition 1.16. Let $S$ be a scheme, $G$ an $S$-reductive group. We call a form of $G$ over $S$ an $S$-group scheme $G^{\prime }$ locally isomorphic to $G$ for the (fpqc) topology (it amounts to the same to say (cf. Exp. XXIII, 5.6) that $G^{\prime }$ is locally isomorphic to $G$ for the étale topology, or again that $G^{\prime }$ is an $S$-reductive group of the same type as $G$ at every point of $S$).

Corollary 1.17. Let $S$ be a scheme, $G$ an $S$-reductive group.

(i) The functor

$$G^{\prime }\mapsto Iso{m}_{S-gr.}(G,G^{\prime })$$

is an equivalence between the category of forms of $G$ over $S$ and the category of principal homogeneous bundles under ${\mathrm{Aut}}_{S-gr.}(G)$.

(ii) If $S^{\prime }\to S$ is a covering morphism, forms of $G$ trivialized by $S^{\prime }$ and bundles trivialized by $S^{\prime }$ correspond.

(iii) Every principal homogeneous sheaf under ${\mathrm{Aut}}_{S-gr.}(G)$ is representable and quasi-isotrivial (i.e. locally trivial for the étale topology).

The first assertion is formal in the category of sheaves (for (fpqc) for example). On the other hand, every sheaf locally isomorphic to $G$ (for (fpqc)) is representable (since $G$ is affine over $S$) and locally isomorphic to $G$ for the étale topology. Finally, for every form $G^{\prime }$ of $G$, the $S$-sheaf $Iso{m}_{S-gr.}(G,G^{\prime })$ is representable, by 1.8. The corollary follows at once.

Corollary 1.18. The set of isomorphism classes of forms of the reductive group $G$ over $S$ is isomorphic to

H¹(S, Aut_{S-gr.}(G)) = H¹_ét(S, Aut_{S-gr.}(G)) = Fib(S, Aut_{S-gr.}(G)).

If $S^{\prime }\to S$ is a covering morphism, the subset formed by forms trivialized by $S^{\prime }$ is isomorphic to $H^1(S^{\prime }/S,{\mathrm{Aut}}_{S-gr.}(G))$.

Corollary 1.19. Let $S$ be a scheme, $R$ a pinned reduced root datum such that $\acute{E}{p}_S(R)$¹¹ exists (condition automatically satisfied, cf. Exp. XXV). Write

A_S(R) = Aut_{S-gr.}(Ép_S(R)) = ad(Ép_S(R)) · E(R)_S.

(i) The set of isomorphism classes of $S$-reductive groups of type $R$ (Exp. XXII, 2.7) is isomorphic (by Exp. XXIII, 5.12) to

H¹(S, A_S(R)) = H¹_ét(S, A_S(R)) = Fib(S, A_S(R)).

(ii) If $S^{\prime }\to S$ is a covering morphism, the subset formed by the classes of groups splittable over $S^{\prime }$ is isomorphic to $H^1(S^{\prime }/S,A_S(R))$.

Remark 1.20. With the preceding notation, to every $S$-reductive group of type $R$ is canonically associated a right principal homogeneous bundle under $A_S(R)$:

$$Iso{m}_{S-gr.}(\acute{E}{p}_S(R),G)=P.$$

Note that $P$ is to be interpreted as the "scheme of pinnings of $G$ of type $R$" (cf. 1.0). Moreover $P$ is also a principal homogeneous bundle (on the left) under ${\mathrm{Aut}}_{S-gr.}(G)$, a structure that appears at once in the description above.

Proposition 1.21. Let $S$ be a henselian local scheme, $s$ its closed point. The functor

$$G\mapsto G_s$$

induces a bijection of the set of isomorphism classes of $S$-reductive groups onto the set of isomorphism classes of $\kappa (s)$-reductive groups.

In particular, for every $S$-reductive group $G$, there exists a finite surjective étale morphism $S^{\prime }\to S$ such that $G_{S^{\prime }}$ is splittable.

Using the existence of the $\acute{E}{p}_S(R)$ (Exp. XXV), one is reduced to proving that if one writes $H=A_S(R)$, the canonical map

Fib(S, H) → Fib(κ(s), H_s)

is bijective (and that every element of $Fib(S,H)$ has the property indicated above). Now, every finite part of $H$ is contained in an affine open set (it is indeed trivial for a constant group, and $H$ is affine above a constant group); one may therefore use the result proved in the appendix (8.1).

2. Automorphisms and subgroups

Let us introduce a notation: if $H={\mathrm{Aut}}_{S-gr.}(G)$, and if $X$ is a subfunctor of $G$, we write

Aut_{S-gr.}(G, X) = Norm_H(X),    Aut_{S-gr.}(G, id_X) = Centr_H(X).

If $Y$ is a second subfunctor of $G$, one defines similarly Aut_{S-gr.}(G, X, Y) = Aut_{S-gr.}(G, X) ∩ Aut_{S-gr.}(G, Y), and if $G^{\prime }$ is a second $S$-group and $X^{\prime }$ a subfunctor of $G^{\prime }$, one writes $Iso{m}_{S-gr.}(G,X;G^{\prime },X^{\prime })$ for the subfunctor of $Iso{m}_{S-gr.}(G,G^{\prime })$ defined by: for every $S^{\prime }\to S$,

Isom_{S-gr.}(G, X; G', X')(S') = { u ∈ Isom_{S-gr.}(G, G')(S') | u(X_{S'}) = X'_{S'} }

and one defines similarly $Iso{m}_{S-gr.}(G,X,Y;G^{\prime },X^{\prime },Y^{\prime })$, etc.¹²

Proposition 2.1. Let $S$ be a scheme, $G$ an $S$-reductive group, $T$ a maximal torus of $G$ (resp. $B$ a Borel subgroup of $G$, resp. $B\supset T$ a Killing couple of $G$). Write $T_{ad}$ (resp. $B_{ad}$) for the maximal torus (resp. Borel subgroup) of $ad(G)$ corresponding to $T$ (resp. $B$):

B_ad ≃ B / Centr(G) = B / Centr(B),
T_ad ≃ T / Centr(G).

Then ${\mathrm{Aut}}_{S-gr.}(G,T)$ (resp. ${\mathrm{Aut}}_{S-gr.}(G,B)$, resp. ${\mathrm{Aut}}_{S-gr.}(G,B,T)$) is representable by a closed subscheme of ${\mathrm{Aut}}_{S-gr.}(G)$, smooth over $S$, and the exact sequence of Theorem 1.3 induces exact sequences:

1 → Norm_{ad(G)}(T_ad) → Aut_{S-gr.}(G, T) → Autext(G) → 1;
1 → B_ad → Aut_{S-gr.}(G, B) → Autext(G) → 1;
1 → T_ad → Aut_{S-gr.}(G, B, T) → Autext(G) → 1.

By descent of closed subschemes,¹³ one is at once reduced to the case where $G$ is pinned and where $B\supset T$ is its canonical Killing couple (cf. Exp. XXII, 5.5.5 (iv)). Since the group $E$ of 1.2 normalizes $B$ and $T$, the result follows at once from the normalization theorems in $ad(G)$ (Exp. XXII, 5.3.12 and 5.6.1).

Using now the conjugation theorems (cf. Exp. XXIII, 5.12), and arguing as in no 1, one deduces:

Corollary 2.2. Let $S$ be a scheme, $G$ and $G^{\prime }$ two $S$-reductive groups of the same type at each point. Let $B\supset T$ (resp. $B^{\prime }\supset T^{\prime }$) be a Killing couple of $G$ (resp. $G^{\prime }$).

(i) The $S$-functor $Iso{m}_{S-gr.}(G,T;G^{\prime },T^{\prime })$ is representable by a closed smooth subscheme of $Iso{m}_{S-gr.}(G,G^{\prime })$ which is principal homogeneous under ${\mathrm{Aut}}_{S-gr.}(G,T)$.

Moreover, $Nor{m}_{ad(G)}(T_{ad})$ operates freely on this scheme, and one has a canonical isomorphism

Isom_{S-gr.}(G, T; G', T') / Norm_{ad(G)}(T_ad) ≃ Isomext(G, G').

(ii) The $S$-functor $Iso{m}_{S-gr.}(G,B;G^{\prime },B^{\prime })$ is representable by a closed smooth subscheme of $Iso{m}_{S-gr.}(G,G^{\prime })$ which is principal homogeneous under ${\mathrm{Aut}}_{S-gr.}(G,B)$.

Moreover, $B_{ad}$ operates freely on this scheme and one has a canonical isomorphism

Isom_{S-gr.}(G, B; G', B') / B_ad ≃ Isomext(G, G').

(iii) The $S$-functor $Iso{m}_{S-gr.}(G,B,T;G^{\prime },B^{\prime },T^{\prime })$ is representable by a closed smooth subscheme of $Iso{m}_{S-gr.}(G,G^{\prime })$, principal homogeneous under ${\mathrm{Aut}}_{S-gr.}(G,B,T)$.

Moreover, $T_{ad}$ operates freely on this scheme and one has a canonical isomorphism

Isom_{S-gr.}(G, B, T; G', B', T') / T_ad ≃ Isomext(G, G').

Arguing again as in no 1, one deduces:

Corollary 2.3. Let $S$ be a scheme, $G$ an $S$-reductive group, $B\supset T$ a Killing couple of $G$. The functor

(G', T') ↦ Isom_{S-gr.}(G, T; G', T'),

resp.

(G', B') ↦ Isom_{S-gr.}(G, B; G', B'),

resp.

(G', B', T') ↦ Isom_{S-gr.}(G, B, T; G', B', T'),

is an equivalence between the category of pairs (G', T') (resp. of pairs (G', B'), resp. of triples (G', B', T')), where $G^{\prime }$ is a form of $G$ and $T^{\prime }$ a maximal torus of $G^{\prime }$ (resp. $B^{\prime }$ a Borel subgroup of $G^{\prime }$, resp. $B^{\prime }\supset T^{\prime }$ a Killing couple of $G^{\prime }$), and the category of principal homogeneous bundles under the $S$-group $H$, where $H={\mathrm{Aut}}_{S-gr.}(G,T)$ (resp. $H={\mathrm{Aut}}_{S-gr.}(G,B)$, resp. $H={\mathrm{Aut}}_{S-gr.}(G,B,T)$).

Moreover, every principal homogeneous sheaf under $H$ is representable and quasi-isotrivial, so that one has

H¹(S, H) = H¹_ét(S, H) = Fib(S, H).

Remark 2.4. Under the conditions of 2.2, the morphism noted set-theoretically $u\mapsto u(T)$ (resp. $u\mapsto u(B)$, resp. $u\mapsto (u(B),u(T))$) induces an isomorphism

Isom_{S-gr.}(G, G') / Aut_{S-gr.}(G, T) ≃ Tor(G')

resp.    Isom_{S-gr.}(G, G') / Aut_{S-gr.}(G, B) ≃ Bor(G'),

resp.    Isom_{S-gr.}(G, G') / Aut_{S-gr.}(G, B, T) ≃ Kil(G').

The proof is immediate: it suffices to do it locally for (fpqc), so one may assume $G\simeq G^{\prime }$, and one is reduced to Exp. XXII, 5.8.3 (iii).

Remark 2.5. The preceding results are at once interpreted in terms of restriction of the structure group: if $G^{\prime }$ is a form of $G$, corresponding to the principal bundle $Iso{m}_{S-gr.}(G,G^{\prime })$, to give a restriction of the structure group of this bundle to ${\mathrm{Aut}}_{S-gr.}(G,T)$ amounts to giving a maximal torus $T^{\prime }$ of $G^{\prime }$, the bijections suggested above being that of 2.4 on the one hand, the map $T^{\prime }\mapsto Iso{m}_{S-gr.}(G,T;G^{\prime },T^{\prime })$ on the other. Similarly for Borel subgroups and Killing couples.

Proposition 2.6. Let $S$ be a scheme, $G$ and $G^{\prime }$ two $S$-reductive groups of the same type at each point, $T$ (resp. $T^{\prime }$) a maximal torus of $G$ (resp. $G^{\prime }$). Then $T_{ad}$ operates freely on $Iso{m}_{S-gr.}(G,T;G^{\prime },T^{\prime })$, the quotient

P = Isom_{S-gr.}(G, T; G', T') / T_ad

is representable; it is a principal homogeneous bundle under

A = Aut_{S-gr.}(G, T) / T_ad,

where $A$ is representable by a twisted constant $S$-scheme, extension of $Autext(G)$ by $W_{ad(G)}(T_{ad})=Nor{m}_{ad(G)}(T_{ad})/T_{ad}$. Moreover, if one makes $A$ operate on $T$ in the obvious way, the bundle associated to $P$ is none other than $T^{\prime }$.¹⁴

The first part of the proposition follows at once from the preceding results. To prove the second, one notes that there is an obvious morphism $P{\times }_{S}T\to T^{\prime }$ (defined by $(u,t)\mapsto u(t)$); to show that after passage to the quotient by $A$ it induces an isomorphism, one may once again assume $(G,T)\simeq (G^{\prime },T^{\prime })$, in which case it is immediate.

In an entirely analogous way, one has:

Proposition 2.7. Let $S$ be a scheme, $G$ and $G^{\prime }$ two $S$-reductive groups of the same type at each point, $B\supset T$ (resp. $B^{\prime }\supset T^{\prime }$) a Killing couple of $G$ (resp. $G^{\prime }$). If one makes ${\mathrm{Aut}}_{S-gr.}(G,B,T)/T_{ad}\simeq Autext(G)$ operate in the obvious way on $T$, the bundle associated to $Isomext(G,G^{\prime })$ is none other than $T^{\prime }$.

Corollary 2.8. Let $G$ and $G^{\prime }$ be two $S$-reductive groups that are inner twisted forms of each other; let $B\supset T$ (resp. $B^{\prime }\supset T^{\prime }$) be a Killing couple of $G$ (resp. $G^{\prime }$). Then $T$ and $T^{\prime }$ are isomorphic.¹⁵

Remark 2.9. It is not true in general that $B$ and $B^{\prime }$ are isomorphic; they are however inner twisted forms of each other (cf. no 5).

One can develop "Isomint" variants¹⁶ of the preceding results. Let us point out one:

Proposition 2.10. Let $S$ be a scheme, $G$ and $G^{\prime }$ two $S$-reductive groups of the same type at each point, $B\supset T$ (resp. $B^{\prime }\supset T^{\prime }$) a Killing couple of $G$ (resp. $G^{\prime }$). Let $u\in Isomext(G,G^{\prime })(S)$; consider

Isomint_u(G, B, T; G', B', T') = Isom_{S-gr.}(G, B, T; G', B', T') ∩ Isomint_u(G, G').

It is a smooth and affine $S$-scheme which is a principal homogeneous bundle under $T_{ad}$. In particular, $Isomin{t}_u(G,B,T;G^{\prime },B^{\prime },T^{\prime })(S)\ne \varnothing$ if and only if the corresponding element of $H^1(S,T_{ad})$ is zero.

To conclude this section, let us prove two results which will be useful later:

Proposition 2.11. Let $S$ be a scheme, $G$ an $S$-reductive group, $T$ a maximal torus of $G$. The obvious morphism

$$T_{ad}\stackrel{\sim }{\to }{\mathrm{Aut}}_{S-gr.}(G,i{d}_T)$$

is an isomorphism.

This follows from the preceding statements; moreover, one has given a direct proof in the course of the proof of 1.5.2.

Corollary 2.12. Under the preceding conditions, there exists an equivalence between the category of pairs (G', f), where $G^{\prime }$ is a form of $G$ and $f$ is an isomorphism of $T$ onto a maximal torus of $G^{\prime }$, and the category of principal homogeneous bundles under $T_{ad}$.

Corollary 2.13. If $H^1(S,T_{ad})=0$, and if $G^{\prime }$ is a form of $G$ possessing a maximal torus isomorphic to $T$, then $G^{\prime }$ is isomorphic to $G$.

Corollary 2.14. Let $S$ be a scheme such that $\mathrm{Pic}(S)=0$, and $G$ an $S$-reductive group of constant type. For $G$ to be splittable, it is necessary and sufficient that $G$ possess a split maximal torus.

Let $G$ be a reductive group, $rad(G)$ its radical (Exp. XXII, 4.3.9); since $rad(G)$ is central and characteristic in $G$, one has a canonical morphism

q : Autext(G) → Aut_{S-gr.}(rad(G)).

Proposition 2.15. Let $G$ be an $S$-reductive group. The following sequence is exact:

1 → ad(G) → Aut_{S-gr.}(G, id_{rad(G)}) ─p→ Autext(G) ─q→ Aut_{S-gr.}(rad(G))

and $Ker(q)=Im(p)$ is an open and closed subscheme of $Autext(G)$, finite over $S$.

One may assume $G$ split. The first assertion is immediate; the second follows from Exp. XXI, 6.7.5 and 6.7.7.

Writing $H={\mathrm{Aut}}_{S-gr.}(G,i{d}_{rad(G)})$, one deduces:

Corollary 2.16. There exists an equivalence between the category of pairs (G', f), where $G^{\prime }$ is a form of $G$ and $f$ is an isomorphism of $rad(G)$ onto the radical of $G^{\prime }$, and the category of principal bundles under a certain $S$-group scheme $H$, where $H$ is such that there exists an exact sequence

$$1\to ad(G)\to H\to F\to 1,$$

where the $S$-group $F$ is étale and finite over $S$.

3. Dynkin scheme of a reductive group. Quasi-split groups

3.1.

Recall (Exp. XXI, 7.4.1) that a Dynkin diagram is a finite set endowed with the structure defined by a set of pairs of distinct elements (bonds) and a map to {1, 2, 3} (lengths). To each pinned reduced root datum $R$ is associated a Dynkin diagram $\Delta (R)$, whose underlying set is the set of simple roots.

3.2.

Let $S$ be a scheme. An $S$-Dynkin scheme is a finite twisted constant $S$-scheme $\Delta$, equipped with the structure defined by a subscheme $L$ of $\Delta {\times }_S\Delta$ having empty intersection with the diagonal, and by a morphism $\Delta \to {1,2,3}_S$. If $S^{\prime }$ is a connected¹⁷ $S$-scheme, $\Delta (S^{\prime })$ is naturally endowed with a structure of Dynkin diagram.

One defines at once the following notions: isomorphism of two Dynkin schemes, base change of a Dynkin scheme, constant Dynkin scheme associated to a Dynkin diagram.

Every descent datum on a Dynkin scheme for the étale topology is effective.

3.3.

We propose to associate to each $S$-reductive group $G$ an $S$-Dynkin scheme. Suppose first that $G$ is splittable over $S$; for every pinning $E$ of $G$, write $\Delta (E)$ for the constant Dynkin scheme associated to the pinned root datum defined by $E$; if $E$ and $E^{\prime }$ are two pinnings of $G$, there exists by 1.5 a unique inner automorphism of $G$ over $S$ transforming $E$ into $E^{\prime }$; this automorphism of $G$ defines an isomorphism $a_{E{E}^{\prime }}:\Delta (E)\stackrel{\sim }{\to }\Delta (E^{\prime })$; the $a_{E{E}^{\prime }}$ evidently form a transitive system, so that one may identify the $\Delta (E)$ (i.e. take the inductive limit); the result is a constant Dynkin scheme denoted $Dyn(G)$. If now $G$ is an arbitrary $S$-reductive group, there exists a covering family for the étale topology $S_i\to S$ such that $G_{S_i}$ is splittable. Arguing as previously, one therefore has a canonical descent datum on the $Dyn(G_{S_i})$, allowing one to construct by descent an $S$-Dynkin scheme $Dyn(G)$.

3.4.

This construction satisfies the following properties (which moreover essentially characterize it):

(i) To each $S$-reductive group is associated a Dynkin scheme $Dyn(G)$; to every isomorphism $u:G\stackrel{\sim }{\to }{G}^{\prime }$ is functorially associated an isomorphism Dyn(u) : Dyn(G) ⥲ Dyn(G').

(ii) If $S^{\prime }$ is an $S$-scheme and $G$ an $S$-reductive group, one has

Dyn(G ×_S S') ≃ Dyn(G) ×_S S'.

(iii) If $E$ is a pinning of $G$, defining the pinned root datum with Dynkin diagram $\Delta$, one has

$$Dyn(G)\simeq {\Delta }_S.$$

(iv) If $u$ is an inner automorphism of $G$, $Dyn(u)$ is the identity automorphism of $Dyn(G)$.

3.5.

Let $S$ be a scheme, $G$ an $S$-reductive group. It is clear that the functor ${\mathrm{Aut}}_{Dyn}(Dyn(G))$ of automorphisms of $Dyn(G)$ for the structure of Dynkin scheme is representable by a finite twisted constant $S$-scheme. By 3.4 (i) and (ii), one has a canonical morphism

Aut_{S-gr.}(G) → Aut_{Dyn}(Dyn(G)),

which, by (iv), factors through a morphism

Autext(G) → Aut_{Dyn}(Dyn(G)).

More generally, if $G$ and $G^{\prime }$ are two $S$-reductive groups, one has a canonical morphism

Isomext(G, G') → Isom_{Dyn}(Dyn(G), Dyn(G'));

in particular, if $G^{\prime }$ is an inner twisted form of $G$ (1.11), the Dynkin schemes $Dyn(G)$ and $Dyn(G^{\prime })$ are isomorphic.

3.6.

If $G$ is semisimple (resp. adjoint or simply connected), the morphism

Autext(G) → Aut_{Dyn}(Dyn(G))

is a monomorphism (resp. an isomorphism). Indeed, one may assume $G$ pinned and one is reduced to the corresponding result for pinned reduced root data (cf. Exp. XXI, 7.4.5).

One has an analogous result for Isom's; whence it follows in particular that two semisimple adjoint (resp. simply connected) $S$-groups are inner twisted forms of each other if and only if their Dynkin schemes are isomorphic.

3.7.

One can give a different construction of the Dynkin scheme associated to a reductive group. Let $R$ be a pinned reduced root datum, $G$ an $S$-reductive group of type $R$; write $\Delta (R)$ for the Dynkin diagram defined by the root datum $R$. One has (3.5) a canonical morphism

A_S(R) = Aut_{S-gr.}(Ép_S(R)) → Aut_{Dyn}(Δ(R)_S).

The $S$-reductive group $G$ corresponds (1.17) to a bundle $Iso{m}_{S-gr.}(\acute{E}{p}_S(R),G)$, principal homogeneous under $A_S(R)$. The bundle under ${\mathrm{Aut}}_{Dyn}(\Delta (R{)}_S)$ associated corresponds to a form on $S$ of $\Delta (R{)}_S$: this is $Dyn(G)$; in other words, this associated bundle is none other than Isom_{Dyn}(Δ(R)_S, Dyn(G)). In this last form, the proof is immediate.

3.8. Dynkin scheme and Killing couples.

Let $S$ be a scheme, $G$ an $S$-reductive group, $B\supset T$ a Killing couple of $G$. There exists a canonical morphism

i : Dyn(G) → Hom_{S-gr.}(T, G_{m, S})

which identifies $Dyn(G)$ with the "scheme of simple roots of $B$ relative to $T$"; this morphism is defined at once by descent from the pinned case. Note moreover that the datum of $T$ and of $i$ allows one to reconstruct $B$ ("biunivocal correspondence between systems of simple roots and systems of positive roots").

It follows from the preceding description of $D=Dyn(G)$ that there exists a canonical root of B_D with respect to T_D: this root ${\alpha }_D$ is the image under $i(D)$ of the identity morphism of $D$. One thereby deduces a canonical invertible O_D-module $g_D$:

g_D = (g ⊗_{O_S} O_D)^{α_D}.

In the pinned case, one has

D = ⨿_{α ∈ Δ} S_α,

where each $S_{\alpha }$ is a copy of $S$, and $g_D$ is then the O_D-module which induces $g_{\alpha }$ on $S_{\alpha }$, for every $\alpha \in \Delta$.

3.9. Quasi-pinnings. Quasi-pinned groups.

If $G$ is an $S$-reductive group, we call quasi-pinning of $G$ the datum:

(i) of a Killing couple $B\supset T$ of $G$,

(ii) of a section $X\in \Gamma (Dyn(G),g_D{)}^{\times }$.

We say that an $S$-reductive group is quasi-splittable if it possesses a quasi-pinning. We call quasi-pinned group a reductive group equipped with a quasi-pinning.

Let $B\supset T$ be a Killing couple of the $S$-reductive group $G$; then $G$ is quasi-pinnable relative to this Killing couple if and only if $g_D$ possesses a non-zero section at every point, i.e. if the element of $\mathrm{Pic}(Dyn(G))$ defined by $g_D$ is zero. Suppose in particular that $S$ is semi-local; then $Dyn(G)$ is also semi-local, so $\mathrm{Pic}(Dyn(G))=0$. One deduces:

Proposition 3.9.1. Let $S$ be a semi-local scheme, $G$ an $S$-reductive group. For $G$ to be quasi-splittable, it is necessary and sufficient that it possess a Borel subgroup.

¹⁸ Indeed, $S$ is affine so, by the first assertion of Exp. XXII, 5.9.7, if $G$ possesses a Borel subgroup $B$, it also possesses a Killing couple $B\supset T$. Then, since $\mathrm{Pic}(D)=0$, $g_D$ possesses a section $X$ nowhere zero.

Let still $A$ be a semi-local ring and $S=\mathrm{Spec}(A)$; remark now that for every $S$-reductive group $G$ the morphism $Bor(G)\to S$ is surjective (since $G_s$ possesses Borel subgroups, for every $s\in S$) and smooth and projective (Exp. XXII, 5.8.3), so it possesses sections after a finite surjective étale extension of the base.¹⁹ One deduces:

Corollary 3.9.2. Let $S$ be a semi-local scheme, $G$ an $S$-reductive group. There exists an étale, finite and surjective morphism $S^{\prime }\to S$ such that $G_{S^{\prime }}$ is quasi-splittable.

Remark 3.9.3. Under the preceding conditions, let $T$ be a maximal torus of $G$ (cf. Exp. XIV, 3.20); then one may moreover assume that $G_{S^{\prime }}$ is quasi-splittable relative to $T_{S^{\prime }}$: it suffices to apply the preceding argument to the "scheme of Borel subgroups containing $T$", which is finite and étale over $S$ (Exp. XXII, 5.5.5 (ii)).

3.10.

Let $E$ and $E^{\prime }$ be two quasi-pinnings of the $S$-reductive group $G$. There exists a unique inner automorphism of $G$ transforming $E$ into $E^{\prime }$. Indeed, one is at once reduced to the split case, where the assertion has already been proved (1.5; it suffices indeed to note that for an inner automorphism of $G$ it amounts to the same to respect a pinning or the underlying quasi-pinning). One concludes as in no 1 that a

quasi-pinning of the $S$-reductive group $G$ defines a splitting $h$ of the exact sequence:

1 → ad(G) → Aut_{S-gr.}(G) ⇄^{h}_{p} Autext(G) → 1,

the image of $h$ being the subgroup of ${\mathrm{Aut}}_{S-gr.}(G)$ which leaves invariant the quasi-pinning.

Similarly if $G$ and $G^{\prime }$ are two quasi-pinned $S$-groups, one defines in a natural way the subfunctor

$$Iso{m}_{S-gr.q-\acute{e}p.}(G,G^{\prime })$$

of $Iso{m}_{S-gr.}(G,G^{\prime })$; the projection of $Iso{m}_{S-gr.}(G,G^{\prime })$ onto $Isomext(G,G^{\prime })$ induces an isomorphism

(*)    Isom_{S-gr. q-ép.}(G, G') ⥲ Isomext(G, G').

Theorem 3.11. Let $S$ be a scheme, $R$ a pinned reduced root datum such that $\acute{E}{p}_S(R)$ exists (cf. Exp. XXV), $E$ the group of its automorphisms. Consider the three following categories:

(i) The category Rev of principal Galois coverings of $S$ with group $E$ (the morphisms are isomorphisms of $E$-bundles).

(ii) The category Redext whose objects are the $S$-reductive groups of type $R$ (Exp. XXII, 2.7), the morphisms from $G$ to $G^{\prime }$ being the elements of $Isomext(G,G^{\prime })(S)$.

(iii) The category Qép of quasi-pinned $S$-reductive groups of type $R$ (the morphisms are the isomorphisms respecting the quasi-pinnings).

These three categories are equivalent: more precisely, one has a diagram of functors, commutative up to functorial isomorphisms

       qép
Rev ───────→ Qép
   ╲          ╱
rev ╲        ╱ i
     ╲      ╱
       Redext

We shall describe below these three functors, leaving to the reader the task of verifying the commutativity of the diagram.²⁰

3.11.1. The functor $i$.

It is the obvious functor: $i(G)=G$, $i(f)=$ image of $f$ under the isomorphism of 3.10 (*).

3.11.2. The functor qép.

Let $S^{\prime }$ be a principal Galois covering of $S$ with group $E$. Let $(G,T,M,R,\Delta ,(X_{\alpha }{)}_{\alpha \in \Delta })$ be a pinning of $G=\acute{E}{p}_S(R)$ and $(X_{\alpha }^{\prime }{)}_{\alpha \in \Delta }$ the corresponding pinning of $G^{\prime }=G_{S^{\prime }}=\acute{E}{p}_{S^{\prime }}(R)$. By Exp. XXIII 5.5 bis, there exists a unique morphism of groups

θ : E → Aut_{S-gr.}(Ép_S(R)) = A(R)(S)

such that, for every $h\in E$, $\theta (h)$ induces $D_S(h)$ on $T$ and $Lie(\theta (h))(X_{\alpha })=X_{h(\alpha )}$ for every $\alpha \in \Delta$. Taking into account the action of $E$ on $S^{\prime }$, one thereby obtains an action of $E$ on $G^{\prime }$, compatible with the action of $E$ on $S^{\prime }$ and respecting the canonical quasi-pinning of $G^{\prime }$.²¹ Since $S^{\prime }\to S$ is covering for the (fpqc) topology, it is an effective descent morphism for the fibered category of affine morphisms, and one writes

$$q\acute{e}p(S^{\prime })=Q-\acute{E}{p}_{S^{\prime }/S}(R)$$

for the quasi-pinned $S$-group obtained by Galois descent.²²

3.11.3. The functor rev.

Let $G$ be an $S$-reductive group of type $R$. One writes

$$rev(G)=Isomext(\acute{E}{p}_S(R),G);$$

this is a principal homogeneous bundle under E_S (cf. 1.10 and 1.19), that is, an object of Rev.

3.11.4. 23

It is clear, by the isomorphism 3.10 (*), that the functor $i$ is fully faithful. On the other hand, if G_0, $G$, $G^{\prime }$ are $S$-reductive groups, the natural morphism

Isom(G_0, G) ×_S Isom(G, G') → Isom(G_0, G')

induces a morphism

(⋆)    Isomext(G_0, G) ×_S Isomext(G, G') → Isomext(G_0, G')

since, for every $S^{\prime }\to S$, if $g_0\in G_0(S^{\prime })$, $g\in G(S^{\prime })$, $u\in Isom(G_0,G)(S^{\prime })$, and $v\in Isom(G,G^{\prime })(S^{\prime })$, one has the equality v ∘ int(g) ∘ u ∘ int(g_0) = v ∘ u ∘ int(u^{-1}(g) g_0). Taking $G_0=\acute{E}{p}_S(R)$, one obtains a map

Isomext(G, G')(S) → Hom_{Rev}( Isomext(Ép_S(R), G), Isomext(Ép_S(R), G') )

and to show that this is a bijection, one may assume by (fpqc) descent that $G$ and $G^{\prime }$ are pinned, in which case it is evident.

Similarly, if S_1, S_2 are two objects of Rev and if one writes $S_i{\times }^E\acute{E}{p}_S(R)$ for the $S$-group $q\acute{e}p(S_i)$ obtained by twisting $\acute{E}{p}_S(R)$ by the $E$-torsor $S_i$ (cf. N.D.E. (21)), one has a natural map

Hom_{Rev}(S_1, S_2) → Hom_{Qép}(S_1 ×^E Ép_S(R), S_2 ×^E Ép_S(R)),

and to show that this is a bijection, one may assume by (fpqc) descent that S_1 and S_2 are trivial $E$-bundles, in which case it is evident.

Thus the three functors of the diagram are fully faithful. Moreover, for every object $S^{\prime }$ of Rev, one has a natural morphism

S' → Isomext(Ép_S(R), S' ×^E Ép_S(R))

and to see that this is an isomorphism, one may assume by (fpqc) descent that $S^{\prime }$ is a trivial $E$-bundle, in which case it is evident.

One thus has an isomorphism of functors $I{d}_{Rev}\to rev\circ i\circ q\acute{e}p$. Since the functor rev is fully faithful one obtains therefore, for every $S$-reductive group $H$ (not necessarily quasi-split), a bijection

Isomext(Q-Ép_{rev(H)/S}(R), H)(S) ≃ Aut_{Rev}(rev(H))

functorial in $H$. In particular, the identity morphism of $rev(H)$ corresponds to a "canonical" element $u_0$ of $Isomext(Q-\acute{E}{p}_{rev(H)/S}(R),H)(S)$; moreover, every $u\in Isomext(Q-\acute{E}{p}_{rev(H)/S}(R),H)(S)$ corresponds to an automorphism ${\varphi }_u$ of $rev(H)$, and $q\acute{e}p({\varphi }_u)$ is the unique automorphism of pinned $S$-group of $Q-\acute{E}{p}_{rev(H)/S}(R)$ such that $u_0\circ q\acute{e}p({\varphi }_u)=u$. We have thus proved theorem 3.11.

Let us develop one of the corollaries of 3.11:

Corollary 3.12. For every $S$-reductive group $G$, there exists a quasi-pinned $S$-group $G_{q-\acute{e}p.}$ and an "outer isomorphism" $u\in Isomext(G_{q-\acute{e}p.},G)(S)$. The pair $(G_{q-\acute{e}p.},u)$ is unique up to a unique isomorphism.

Indeed, one may assume $G$ of constant type $R$, and one takes $G_{q-\acute{e}p.}=Q-\acute{E}{p}_{rev(G)/S}(R)$.

3.12.1.

Note that the datum of the pair $(G_{q-\acute{e}p.},u)$ allows one to define canonically the $S$-scheme (cf. 1.11)

$$Q=Isomint(G_{q-\acute{e}p.},G),[{}^N.D.E-XXIV-23]$$

which is a principal homogeneous bundle under $ad(G_{q-\acute{e}p.})$, and whose datum "is equivalent" to that of the inner twisted form $G$ of $G_{q-\acute{e}p.}$. Moreover, $Q$ is none other than the "scheme of quasi-pinnings of $G$", a definition which accounts for its structure of principal homogeneous bundle (on the left) under $ad(G)$ (3.10) — compare with 1.20.

Proposition 3.13. Let $S$ be a scheme, $G$ a semisimple adjoint (resp. simply connected) $S$-group, $B\supset T$ a Killing couple of $G$, $Dyn(G)$ the Dynkin $S$-scheme

of $G$. There exists a canonical isomorphism of $S$-group schemes

T ⥲ ∏_{Dyn(G)/S} G_{m, Dyn(G)}.

(Recall (cf. Exp. II, § 1) that the right-hand side is by definition the $S$-functor which to $S^{\prime }\to S$ associates $G_m(Dyn(G){\times }_S{S}^{\prime })$, or, what amounts to the same, $G_m(Dyn(G_{S^{\prime }}))$.)

First proof. Let us do it for simplicity in the adjoint case. Consider the composite morphism

T → ∏_{D/S} T_D → ∏_{D/S} G_{m, D},

where the first morphism is the canonical morphism, the second is ${\prod }_{D/S}{\alpha }_D$ (we have written $D=Dyn(G)$). To verify that this morphism is an isomorphism, one may assume $G$ split; now, in this case, this is none other than the morphism $T\to (G_{m,S}{)}^{\Delta }$ with components $\alpha$, for $\alpha \in \Delta$, and the latter is an isomorphism (Exp. XXII, 4.3.8).

Second proof. By 2.8, 3.5 and 3.11, one may assume that $G=Q-\acute{E}{p}_{S^{\prime }/S}(R)$, $T$ being the canonical maximal torus. On $S^{\prime }$, one has by Exp. XXII 4.3.8, an isomorphism $T_{S^{\prime }}\stackrel{\sim }{\to }(G_{m,S^{\prime }}{)}^{\Delta }$, defined by the simple roots (resp. the simple coroots). The group $E$ operates on the right-hand side by permutation of $\Delta$. Now the bundle associated to $S^{\prime }/S$ by $E\to \mathrm{Aut}(\Delta )$ is $Dyn(G)$ (3.7), and one concludes at once.

Using Prop. 8.4 of the appendix, one deduces:

Corollary 3.14. Under the preceding conditions, one has

H¹(S, T) ⥲ H¹(Dyn(G), G_m) ⥲ Pic(Dyn(G)).

In particular, $H^1(S,T)=0$ when $S$ is semi-local.

Remark 3.15. Let $S$ be a scheme, $G$ (resp. $G^{\prime }$) an $S$-reductive group, $B\supset T$ (resp. $B^{\prime }\supset T^{\prime }$) a Killing couple of $G$ (resp. $G^{\prime }$), $u\in Isomext(G,G^{\prime })(S)$. Set (cf. 2.10)

P = Isomint_u(G, B, T; G', B', T');

this is a principal homogeneous bundle under $T_{ad}$ (by $(f,t)\mapsto f\circ int(t)$).

Let on the other hand $D=Dyn(G)=Dyn(G^{\prime })$ (identified via $u$ (3.5)), and let $L=Iso{m}_D(g_D,g_D^{\prime })$ be the principal homogeneous bundle under $G_{m,D}$ defined by the invertible O_D-module

Hom_{O_D}(g_D, g'_D) = g'_D ⊗ (g_D)^∨.

Each $f\in P(S^{\prime })$ defines an isomorphism of $g_D{\otimes }_{O_S}{O}_{S^{\prime }}$ onto $g_D^{\prime }{\otimes }_{O_S}{O}_{S^{\prime }}$, whence a canonical morphism

$$P\to \prod _{D/S}L.$$

This morphism is an isomorphism, compatible with the isomorphism of operator groups

T_ad ⥲ ∏_{D/S} G_{m, D}

defined above. Indeed, it suffices to verify it in the case where the two groups are pinned, where it is easy.

It follows in particular that in the isomorphism

$$H^1(S,T_{ad})\stackrel{\sim }{\to }\mathrm{Pic}(Dyn(G))$$

of 3.14, the class of the bundle $P$ is transformed into $c\ell (g^{\prime D})-c\ell (g_D)$. The bundle $P$ is therefore trivial if and only if $g^{\prime D}$ and $g_D$ are isomorphic.

If $(G,B,T)$ is quasi-splittable, for example if one takes for $G$ the group $G_{q-\acute{e}p.}^{\prime }$, with its canonical Killing couple, it follows that the image of the class of $P$ is none other than the obstruction to the quasi-splitting of $G^{\prime }$ defined in 3.9.

3.16. Symmetries

3.16.1.

Let $S$ be a scheme, $G$ an $S$-reductive group, $B\supset T$ a Killing couple of $G$. Write $D=Dyn(G)$. Recall (Exp. XXII, 5.9.1) that there exists a unique Borel subgroup $B^{-}$ of $G$ such that $B\cap B^{-}=T$. If $X\in \Gamma (D,g_D{)}^{\times }$ defines a quasi-pinning of $G$ relative to $(B,T)$ (3.9), then $Y=-X^{-1}\in \Gamma (D,(g_D{)}^{\vee }{)}^{\times }$ defines a quasi-pinning of $G$ relative to $(B^{-},T)$; one says that this is the opposite quasi-pinning.

If $R$ is a pinned reduced root datum and if $E$ is an $R$-pinning of the $S$-reductive group $G$, one defines an $R$-pinning $E^{-}$ said to be opposite to $E$ in the following way: one keeps the same maximal torus $T$, one takes the opposite of the isomorphism $D_S(M)\stackrel{\sim }{\to }T$, and one "pins" by $Y_{\alpha }=-X_{\alpha }^{-1}\in \Gamma (S,g^{-\alpha }{)}^{\times }$, for $\alpha \in \Delta$. The quasi-pinning underlying $E^{-}$ is the quasi-pinning opposite to the quasi-pinning underlying $E$.