FORMAL TORSORS UNDER REDUCTIVE GROUP SCHEMES

BENEDICTUS MARGAUX

Laboratoire de Recherche “Princesse Stéphanie” Monte Carlo 51840, Monaco. Email: benedictus.margaux@gmail.com

Abstract. We consider the algebraization problem for torsors over a proper formal scheme under a reductive group scheme. Our results apply to the case of semisimple group schemes (which is addressed in detail).

Keywords: Torsors, formal schemes, reductive group schemes, algebraization.

MSC: 14D15, 14L30.

1. Introduction

Throughout this paper R will be a complete noetherian local ring with maximal ideal 𝔪. We put R_{n} = R/𝔪^{n+1} for each n ≥ 0. The natural map R → \underleftarrow{lim}R_{n} is a ring isomorphism and we will henceforth identify these two rings.

For the theory of formal schemes over R, we refer the reader to [EGAI, §10], [Ha, §II.9] and [St, Tag 0AHW, §79].^{1} Let X be a proper R–scheme, and let X̂ be the associated formal scheme. Grothendieck’s existence theorem provides an equivalence of categories between the category of coherent sheaves over X and the category of coherent sheaves on the formal scheme X̂ [EGAIII, 5.1.4], [I, §8.4]. The restriction to locally trivial coherent sheaves of constant rank r yields a natural equivalence between the category of GL_{r}–torsors over X and the category of ĜL_{r}– torsors over X̂.

The purpose of the paper is to extend this statement to a larger class of affine group schemes over X which includes semisimple group schemes. This question has been also studied by Baranovsky [B, §3], but only for group schemes arising from R–group schemes by base change.

Conventions on vector groups and linear groups.

We use mainly the terminology and notation of Grothendieck-Dieudonné [EGAI, §9.4 and 9.6], which agrees with that of Demazure-Grothendieck used in [SGA3, Exp. I.4]

Let S be a scheme and let E be a quasi-coherent sheaf over S. For each morphism f : T → S, we denote by E_{(T)} = f^{∗}(E) the inverse image of E by the morphism f.

1Since the numbering of the Stacks Project [St] evolves over time, we also provide the relevant tags.

Recall that the S–scheme V(E) = Spec(Sym^{•}(E)) is affine over S and represents the S–functor T ↦ Hom_{O_{T}} (E_{(T)}, O_{T}) [EGAI, 9.4.9].

We assume now that E is locally free of finite rank and denote by E^{∨} its dual. In this case the affine S–scheme V(E) is of finite presentation (ibid, 9.4.11); also the S–functor T ↦ H^{0}(T, E_{(T)}) = Hom_{O_{T}} (O_{T} , E_{(T)}) is representable by the affine S–scheme V(E^{∨}) which is also denoted by W(E) [SGA3, I.4.6].

The above considerations apply to the locally free coherent sheaf End(E) = E^{∨} ⊗_{O_{S}} E over S so that we can consider the affine S–scheme V(End(E)) which is an S–functor in associative commutative and unital algebras [EGAI, 9.6.2]. Now we consider the S–functor T ↦ Aut_{O_{T}} (E_{(T)}). It is representable by an open S– subscheme of V(End(E)) which is denoted by GL(E) (loc. cit., 9.6.4).

We set GL_{r,S} = GL(O_{S}^{r}) for each r ≥ 1. If S = Spec(A) is affine, then E = O_{S}^{r} corresponds to the A-module E = A^{r}. In this case we will use the notation GL_{r}(E) instead of GL_{r,S}. Finally, for scheme morphisms Y → X → S, we denote by ∏_{X/S}(Y/X) the S–functor defined by

( ∏_{X/S}(Y/X))(T) = Y (X ×_{S} T)

for each S–scheme T. Recall that if ∏_{X/S}(Y/X) is representable by an S-scheme,

this scheme is called the Weil restriction of Y to S.

Acknowledgement: My sincere gratitude to the referee for his/her careful reading of the original version, and the valuable corrections and feedback.

2. Formal torsors

Let R be as above, and let X be a proper R–scheme. We start with the following key observation about limits.

Lemma 2.1. Let f : Y → X be a separated morphism of finite type. Then the natural map

( ∏_{X/R}(Y/X))(R) → \underleftarrow{lim}_{n} ( ∏_{X/R}(Y/X))(R_{n}) = \underleftarrow{lim}_{n} ( ∏_{X_{n}/R_{n}}(Y_{n}/X_{n}))(R_{n})

is bijective.

Proof. The last equality follows from the fact that ∏_{X/S}(Y/X) commutes with base

change. Consider the commutative diagram

[figure: Hom_{R}(X, Y); \underleftarrow{lim}_{n} Hom_{R_{n}}(X_{n}, Y_{n}); ( ∏_{X/R}(Y/X))(R); \underleftarrow{lim}_{n}( ∏_{X_{n}/R_{n}}(Y_{n}/X_{n}))(R_{n})]

According to [St, Tag 0898, 29.28.3], the top horizontal map is bijective so that the bottom horizontal map is injective. Let (s_{n} : X_{n} → Y_{n})_{n≥0} be a coherent family of sections. It lifts to a (unique) morphism s : X → Y. Then the morphism g = f ∘ s : X → X is such that g_{n} = id_{X_{n}} for all n ≥ 0. Since the map Hom_{R}(X, X) → \underleftarrow{lim}_{n} Hom_{R_{n}}(X_{n}, X_{n}) is bijective, we conclude that g = id_{X} whence s is a section of Y → X. We have shown the surjectivity of the bottom map. □

Let 𝔊 be an affine X–group scheme of finite presentation. We set X_{n} = X ×_{R}R_{n} and 𝔊_{n} = 𝔊 ×_{X} X_{n} for each n ≥ 0. We denote by 𝔊̂ = (𝔊_{n})_{n≥0} the formal group scheme over X̂ attached to 𝔊.

A formal 𝔊̂-torsor 𝔓̂ is the data of a 𝔊_{n}–torsor 𝔓_{n} over X_{n} for each n ≥ 0 together with compatible 𝔊_{n+1}–isomorphisms θ_{n} : 𝔓_{n+1} ×_{R_{n+1}} R_{n} ⟶^{∼} 𝔓_{n}. If 𝔓 is 𝔊–torsor, 𝔓̂ is formal 𝔊̂-torsor and this assignment is faithful in the following sense.

Lemma 2.2. Let 𝔓, 𝔔 be two 𝔊–torsors. The natural map Isom_{𝔊}(𝔓, 𝔔) → Isom𝔊̂(𝔓̂, 𝔔̂) is bijective.

Proof. Up to replacing 𝔊 (resp. 𝔔) by the twisted R–group scheme ^{𝔓}𝔊 (resp. 𝔓^{op}∧^{𝔊}𝔔), we may assume that 𝔓 = 𝔊. In this case, we have Isom_{𝔊}(𝔓, 𝔔) = 𝔔(X) so that our original question is reduced to showing that the natural map

𝔔(X) → \underleftarrow{lim}_{n}𝔔_{n}(X_{n})

is bijective. Locally for the fppf topology, 𝔔 is isomorphic to 𝔊. According to the permanence properties of faithfully flat descent 𝔔 is affine of finite presentation over X [EGAIV, 2.7.1.(vi) and (xiii)]. So Lemma 2.1 applies and shows that the above map is bijective. □

2.1. Algebraizable torsors. We say that a formal 𝔊̂–torsor 𝔓̂ is algebraizable if it arises from a 𝔊–torsor 𝔓. Lemma 2.2 shows that if such a 𝔓 exists, it is unique up to isomorphism.

Lemma 2.3. Let 𝔊 and 𝔊′ be two X-group schemes which are affine and of finite presentation. Assume that 𝔊 is flat and that i : 𝔊 → 𝔊′ is a monomorphism of X-group schemes with the property that the fppf quotient 𝔊′/𝔊 is representable by an affine X–scheme 𝔔.

Let 𝔉̂ be a 𝔊̂–torsor and denote by 𝔉̂′ = i_{∗}(𝔉̂) the corresponding 𝔊̂′–torsor. Then 𝔉̂ is algebraizable if and only if 𝔉̂′ is algebraizable.

Proof. It is clear that if 𝔉̂ is algebraizable then so is 𝔉̂′. Conversely, assume that the 𝔊̂′–torsor 𝔉̂′ is algebraizable i.e. it arises from a 𝔊′–torsor 𝔉′. We consider the affine X-scheme ℨ = 𝔉′/𝔊 := 𝔉′ ∧^{𝔊′} (𝔊′/𝔊); the reduction of 𝔉′ to 𝔊 defined by faithfully flat descent. According to [SGA3, VI_{B}.9.2.(xiii).b], the X-scheme 𝔊′/𝔊 is of finite presentation. Since ℨ is locally isomorphic to 𝔊′/𝔊 with respect to the fppf topology, the permanence properties of faithfully flat descent show that ℨ is affine of finite presentation over X [EGAIV, 2.7.1.(vi) and (xiii)]. According to Lemma 2.1, the map ℨ(X) → \underleftarrow{lim}_{n} ℨ_{n}(X_{n}) is bijective.

Each 𝔉_{n} defines a point z_{n} ∈ ℨ(R_{n}) in a coherent way so that we get a point z ∈ ℨ(R). That point defines a reduction of the 𝔊′–torsor 𝔉′ to a 𝔊–torsor 𝔉 [Gd, III.3.2.1]. Since z maps to z_{n}, we have 𝔉_{R_{n}} = 𝔉_{n} for each n ≥ 0. Thus 𝔉̂ is algebraizable. □

3. Representations of group schemes

3.1. The Chevalley case. Let G be a reductive split ℤ–group scheme and we denote by G_{ad} its adjoint quotient. We remind the reader that the functor of automorphisms of G is representable by a smooth ℤ–group scheme Aut(G) [SGA3, XXIV.1]. Furthermore there is an exact sequence of ℤ–group schemes

1 → G_{ad} ⟶^{int} Aut(G) ⟶^{π} Out(G) → 1

where Out(G) is a constant group scheme. In other words, Out(G) is the ℤ–group scheme attached to the abstract group Out(G)(ℤ). In the semisimple case Out(G) is finite (and in particular Aut(G) is affine). This is not the case in general. For example Aut(𝔾_{m}^{2}) is the constant ℤ–group scheme attached to the abstract group GL_{2}(ℤ).

Let Γ be a finite subgroup of Out(G)(ℤ). We get a monomorphism of ℤ–group schemes Γ_{ℤ} → Out(G) and consider the ℤ–group scheme

Aut_{Γ}(G) = Aut(G) ×_{Out(G)} Γ_{ℤ}.

obtained by pullback. The above yields the exact sequence

1 → G_{ad} → Aut_{Γ}(G) ⟶^{π} Γ_{ℤ} → 1.

Since Γ_{ℤ} and G_{ad} are smooth affine over ℤ, so is Aut_{Γ}(G) [SGA3, VI_{B}9.2.(viii)].

Lemma 3.1. There exists a free ℤ–module of finite type E, and a closed immersion ℤ–group scheme homomophism i : G ⋊ Aut_{Γ}(G) → GL(E) such that the fppf quotient sheaf GL(E)/G (resp. GL(E)/(G ⋊ Aut_{Γ}(G)), GL(E)/G_{ad}) is representable by a smooth affine ℤ–scheme.

Proof. Since G ⋊ Aut_{Γ}(G) is an affine smooth ℤ–group scheme, there exists a free ℤ-module of finite rank E and a faithful linear representation ρ : G ⋊ Aut_{Γ}(G) → GL(E) which is a closed immersion [BT, 1.4.5].

The fppf sheaf GL(E)/(G ⋊ Aut_{Γ}(G)) is representable by a ℤ-scheme [A, Th. IV.4.B] which is smooth and separated [SGA3, VI_{B}.9.2.(x) and (xii)]. The ℤ– group scheme G ⋊ G_{ad} is reductive. According to [CTS, 6.12.ii], the fppf sheaf GL(E)/(G ⋊ G_{ad}) is representable by an affine smooth ℤ–scheme and so are GL(E)/G and GL(E)/G_{ad}. Since the map GL(E)/(G ⋊ G_{ad}) → GL(E)/(G ⋊ Aut_{Γ}(G)) is a Γ_{ℤ}-torsor, it is a finite étale cover. It follows that GL(E)/(G ⋊ Aut_{Γ}(G)) is affine [St, Tag 01YN, lemma 29.13.3]. Similarly the ℤ-scheme GL(E)/Aut_{Γ}(G) is affine. □

3.2. An isotriviality condition. In this section, we assume that the base scheme S is noetherian and we are given a reductive S–group scheme 𝔊 of constant type. Thus, there exists a Chevalley ℤ–group scheme G such that 𝔊 is locally isomorphic to G_{S} for the étale topology [SGA3, XXII.2.3, 2.5]. Also the fppf sheaf \underline{Isom}(G_{S}, 𝔊) is representable by a Aut(G)_{S}–torsor Isom(G_{S}, 𝔊) defined in [SGA3, XXIV.1.8]. The contracted product Isomext(G_{S}, 𝔊) := Isom(G_{S}, 𝔊) ∧^{Aut(G)_{S}} Out(G)_{S} is a Out(G)_{S}–torsor (ibid, 1.10) which encodes the isomorphism class of the quasi-split form of 𝔊.

Proposition 3.2. We assume that the Out(G)_{S}–torsor Isomext(G_{S}, 𝔊) is isotrivial, i.e. there exists a finite étale cover S′/S such that Isomext(G_{S}, 𝔊)(S′) ≠ ∅. Then there exists a locally free coherent O_{S}–module E, and a closed immersion S–group scheme homomorphism i : 𝔊 → GL(E) such that the fppf quotient sheaf GL(E)/𝔊 is representable by a smooth affine S–scheme.

Remark 3.3. (a) If G is semisimple, Out(G) is a finite constant group so that the isotriviality condition is obviously satisfied.

(b) If S is a normal connected scheme, the isotriviality condition is satisfied since Isomext(G_{S}, 𝔊) → S is a Out(G)_{S}–cover [SGA3, X.6.2 and 5.14].

Proof. The noetherian assumption reduces the problem to the connected case (in particular S is non-empty by convention [St, Tag 004R, 5.7.1]). We consider the Aut(G)_{S}-torsor 𝔈 = Isom(G_{S}, 𝔊) defined above; we have 𝔊 = ^{𝔈}(G_{S}), i.e. 𝔊 is the twist of G_{S} by the Aut(G)_{S}–torsor 𝔈.

The isotriviality assumption for the Out(G)_{S}–torsor 𝔉 = 𝔈 ∧^{Aut(G)_{S}} Out(G)_{S} means that there exists a finite étale cover S′/S such that 𝔉(S′) ≠ ∅. Grothendieck’s theory of the algebraic fundamental group [SGA1] permits to assume that S′ is connected and that S′ → S is a Θ_{S}–torsor where Θ is a finite abstract group.

We have a bijection H^{1}(Θ, Out(G)(S′)) ⟶^{∼} H^{1}(S′/S, Out(G)) [Gi, end of §2.2]. Since S′ is connected, we have Out(G)(ℤ) = Out(G)(S′) so that the action of Θ on Out(G)(S′) is trivial. We have then a bijection

Hom_{gr}(Θ, Out(G)(ℤ))/Out(G)(ℤ) ⟶^{∼} H^{1}(Θ, Out(G)(S′)).

It follows that the class of the Out(G)_{S}–torsor 𝔉 is given by the conjugacy class of a homomorphism ρ : Θ → Out(G)(ℤ).

Let Γ = Im(ρ), it is a finite subgroup of Out(G)(ℤ). We consider the ℤ– group scheme Aut_{Γ}(G) = π^{−1}(Γ) as in the previous section. The isomorphism Aut(G)_{S}/Aut_{Γ}(G)_{S} ⟶^{∼} Out(G)_{S}/Γ_{S} induces an isomorphism 𝔈/Aut_{Γ}(G)_{S} ⟶^{∼} 𝔉/Γ_{S}. The reduction of the Out(G)_{S}–torsor 𝔉 to Γ_{S} defines then a reduction of the Aut(G)_{S}–torsor 𝔈 to a Aut_{Γ}(G)_{S}–torsor 𝔈_{♯} [Gd, III.3.2.1]. □

Remark 3.4. (a) If G is semisimple, we can take in the proof Γ = Out(G)(ℤ). We thus find a O_{S}–coherent sheaf E as desired which is 𝔊 ⋊ Aut(𝔊)-equivariant.

(b) Thomason has proven stronger statements than (1) for embedding group schemes in linear group schemes [T, §3].

4. Main statement

4.1. The following generalization of Grothendieck existence theorem strengthens Baranovsky’s result [B, Th. 3.1].

Theorem 4.1. Let R be a complete noetherian local ring. Let X be a proper R–scheme and let X̂ be the associated formal scheme. Let G be a Chevalley ℤ– group scheme and let 𝔊 be an X-form of G_{X}. Assume that the Out(G)_{X}–torsor Isomext(G_{X}, 𝔊) is isotrivial. Then.

(1) The assignment 𝔓 ↦ 𝔓̂ induces an equivalence of categories between the category of 𝔊–torsors of X and that of 𝔊̂–torsors over X̂.

(2) Assume that 𝔊 is semisimple. For ℌ = 𝔊, Aut(𝔊), 𝔊 ⋊ Aut(𝔊) the assignment 𝔓 ↦ 𝔓̂ induces an equivalence of categories between the category of ℌ–torsors of X and that of ℌ̂–torsors over X̂.

Proof. (1) By Lemma 2.2, we have only to show algebraization. The R–scheme X is proper, namely separated, of finite type, and universally closed. Since R is noetherian, X is locally noetherian. Also the morphism X → Spec(R) is quasicompact [St, 28.39.10] so that X is quasi-compact. The scheme X is quasi-compact and locally noetherian, hence is noetherian by definition [St, Tag 01OU, 27.5.1]. Without lost of generality we may assume that X is connected.

Proposition 3.2 provides a closed immersion i : 𝔊 → GL(E) where E is a locally free coherent O_{X}–module and such that the fppf quotient sheaf GL(E)/𝔊 is representable by a smooth affine X–scheme. Lemma 2.3 reduces the algebraization problem to the case of GL(E). Since X is connected, E is locally free of rank r. We consider the GL_{r}–torsor 𝔔 = Isom(O_{X}^{r}, E) over X. Torsion by 𝔔 (resp. 𝔔̂) induces an equivalence of categories between the category of GL_{r}–torsors (resp. ĜL_{r}–torsors) and that of GL(E)–torsors (resp. { \atop GL(E)}–torsors). It follows that the algebraization question is equivalent for GL_{r}–torsors and for GL(E)–torsors. Grothendieck’s existence theorem states that GL_{r}–torsors over X̂ are algebraizable. Thus algebraization holds for GL(E) and for 𝔊.

(2) Remark 3.4.(a) shows that the representation 𝔊 → GL(E) arises from a representation 𝔊 ⋊ Aut(𝔊) → GL(E). The same argument applies then to 𝔊 ⋊ Aut(𝔊) and Aut(𝔊). □

4.2. Examples and applications. Let d ≥ 1 be a positive integer. If we consider the case of 𝔊 = PGL_{n} and use the dictionary given in [G, §7] between PGL_{d}-torsors and Azumaya algebras of degree d, we get an algebraization statement for Azumaya algebras of degree d we obtain the following.

Corollary 4.2. There is an equivalence of categories between Azumaya algebras over X (of degree d) and formal degree d Azumaya algebras over X̂ (of degree d). □

Similarly, by considering the case of the Chevalley ℤ–group scheme of type G_{2}, we obtain an equivalence of categories octonion algebras over X and formal octonion algebras over X̂ [Co, App. B].

More generally for the group scheme Aut(G) of a semisimple Chevalley ℤ–group G we have the following fact as special case of Theorem 4.1.(2).

Corollary 4.3. There is an equivalence of categories between the groupoid of X– forms of G_{X} and that of formal X̂-forms of Ĝ_{X}. □

In particular, we obtain the following fact.

Corollary 4.4. Assume that we are given a formal X̂-group scheme 𝔊̂ such that each 𝔊_{n} is an X_{n}–form of G_{X_{n}}. Then 𝔊̂ is algebraizable in a semisimple X–group scheme 𝔊 which is a X-form of G_{X}. □

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