Vanishing of Hochschild Cohomology

for Affine Group Schemes

and Rigidity of Homomorphisms

between Algebraic Groups

Benedictus Margaux

Received: April 2, 2009 Revised: November 17, 2009

Communicated by Wolfgang Soergel

Abstract. Let k be an algebraically closed field. If G is a linearly reductive k–group and H is a smooth algebraic k–group, we establish a rigidity property for the set of group homomorphisms G → H up to the natural action of H(k) by conjugation. Our main result states that this set remains constant under any base change K/k with K algebraically closed. This is proven as consequence of a vanishing result for Hochschild cohomology of affine group schemes.

2000 Mathematics Subject Classification: 20G05

Keywords and Phrases: Group schemes, representations, linearly reductive group, deformation theory.

Contents

1. Introduction. 653 2. Generalities on Hochschild cohomology 654 2.1. Definition and basic properties 655 3. Vanishing of Hochschild cohomology 657 3.1. Linearly reductive groups 657 3.2. Finiteness considerations. 659 3.3. Proof of Theorem ?? 660 4. Rigidity and deformation theory 663 4.1. Locally finitely presented S–functors 663 4.2. Formal étalness 664 5. Applications 668 5.1. Rigidity 668 5.2. Lie algebras 669 6. Appendix: Affine group schemes which are essentially free 670 References 671 1. Introduction.

Our original goal was to prove a strong version of a rigidity principle for homomorphisms between algebraic groups which is part of the area’s folklore. The general philosophy is that if G and H are algebraic groups over an algebraically closed field k, then the set Hom_{k−gr}(G, H) modulo the adjoint action of H should remain constant under any base change K/k with K algebraically closed. Our result is as follows.

Theorem 1.1. Let k be an algebraic closed field. Let G be a linearly reductive (affine) algebraic k–group, and H a smooth algebraic k–group scheme. Then for every algebraically closed field extension K/k, the natural map

Hom_{k−gr}(G, H)/H(k) → Hom_{K−gr}(G_{K}, H_{K})/H(K)

is bijective.

When k is of characteristic 0 and G and H are both reductive this result has been established by Vinberg [19, prop. 10] by reducing to the case where G = GL_{N} and H is connected. Our proof is very different in spirit than Vinberg’s, and the main result more general. The proof we give is based on the deformation theory à la Demazure-Grothendieck described in [17], which is itself linked to the analytic viewpoint later taken by Richardson on similar problems [12] [13] [16]. The main auxiliary statement we use is case (i) of the following Theorem, a vanishing result for Hochschild cohomology of affine group schemes which is of its own interest.

Theorem 1.2. Let R be a commutative ring. Let G be a flat affine group scheme over Spec(R). Assume that the fibers of G over all closed points of Spec(R) are linearly reductive groups (as affine groups over the corresponding residue fields. See §3.1 below for the relevant definitions and references). Let L be a G-R–module (see §2.1 below). Assume that one of the following two conditions holds:

(i) R is noetherian, (ii) the group G is of finite presentation as an R-scheme, and L is a direct limit G-R–modules which are finitely presented as R-modules.

Then

H^{i}(G, L) = 0 for all i > 0.

This result extends a theorem of Grothendieck for R–groups of multiplicative type [17, IX.3.1].

At this point we recall some standard notation that will be used throughout the paper. Let S be scheme, and G a group scheme over S. For all scheme morphism S → T we will denote as it is customary the T -group G ×_{S} T by G_{T} . If T = Spec(R) we write G_{R} instead of G_{T} , and G(R) instead of G(T). Group schemes over a given scheme S will for brevity and convenience sometimes be refereed to simply as S–groups, or R–groups in the case when S = Spec(R).

2. Generalities on Hochschild cohomology

In this paper, we deal with Hochschild cohomology of a flat affine group scheme G over an affine base 𝔛 = Spec(R), and their corresponding G-O_{𝔛}–modules [17, I 4.7]. This set up is equivalent to that of G-R-modules as we now explain. Let G = Spec(R[G]). The group structure of G gives the R-algebra R[G] a coassociative and counital Hopf algebra structure. We have thus a comultiplication Δ_{G} : R[G] → R[G] ⊗_{R} R[G], a counit ϵ : R[G] → R and an antipode map ι : R[G] → R[G]. For any ring homomorphism R → S recall that the S-group G×_{R}S obtained by base change is denoted by G_{S}. This is an affine S–group with S[G] = S⊗_{R}R[G] as its Hopf algebra. Similarly, for any R–module L we denote the S–module L ⊗_{R} S by L_{S}.

2.1. Definition and basic properties. Let L be an R-module, and ρ : G → GL(L) a linear representation of G. This amounts to give for each R–algebra S an S–linear representation ρ_{S} of the abstract group G(S) on the S–module L_{S} in such a way that the family (ρ_{S}) is “functorial on S.” We also then say that L is a (left) G-R–module. Because G is affine, to give L a G-R–module structure is equivalent to give L a (right) R[G]-comodule structure, that is an R–linear map

Δ_{L} : L → L ⊗_{R} R[G]

satisfying the two following natural axioms:

(CM1) The following diagram is commutative

L  ⟶^{Δ_{L}}  L ⊗_{R} R[G]
Δ_{L}↓  ^{id_{L}⊗Δ_{G}}↓
L ⊗_{R} R[G] ⟶^{Δ_{L}⊗id_{R[G]}} L ⊗_{R} R[G] ⊗_{R} R[G]

(CM2) The composite map

L ⟶^{Δ_{L}} L ⊗_{R} R[G] ⟶^{id_{L}⊗ϵ} L

is the identity map id_{L}.

The flatness condition on G/R is natural within the present context since the category of G-R–modules is then abelian. See [15, prop. 2].^{1} Recall that the fixed points of L under G are defined by

L^{G} := {f ∈ L | Δ_{L}(f) = f ⊗ 1}.

This is an R–submodule of L. Because of the assumption on flatness, the Hochschild cohomology groups H^{n}(G, L) are the derived functors of the “fixed point” functor G-R − mod → R − mod given by L → L^{G} [17, I 5.3.1]. The H^{n}(G, L) can thus be computed as the cohomology groups of the complex [4, II §3.3.1]

1The existence of the unit section of G, more precisely of the counit ϵ, shows that R[G] is in fact a faithfully flat R-algebra.

(2.1)  L →^{∂_{0}} L ⊗_{R} R[G] →^{∂_{1}} L ⊗_{R} R[G^{2}] →^{∂_{2}} L ⊗_{R} R[G^{3}] → ⋯

where as usual R[G^{n}] = R[G × ⋯ × G] ≃ R[G] ⊗_{R} ⋯ ⊗_{R} R[G], and both products and tensor products are taken n-times. We denote as it is customary ker(∂_{i}) by Z^{i}(G, L) (the cocycles), and Im(∂_{i−1}) by B^{i}(G, L) (the coboundaries). In particular we have the exact sequence

(2.2) 0 → L^{G} = H^{0}(G, L) → L →^{∂_{0}} Z^{1}(G, L) → H^{1}(G, L) → 0.

The following four properties easily follow from the resolution (2.1).

Lemma 2.1. Let L be a G-R–module.

(1) Let I be an ideal of R which annihilates L. Then L_{R/I} = L ⊗_{R} R/I is naturally a G_{R/I}-R/I–module, and H^{n}(G, L) ⟶^{∼} H^{n}(G_{R/I}, L_{R/I}) for all n ≥ 0.

(2) If S/R is a flat extension of rings, then

H^{n}(G, L) ⊗_{R} S ⟶^{∼} H^{n}(G_{S}, L_{S}) for all n ≥ 0.

(3) Let L = \underrightarrow{lim}_{i} L_{i} be the inductive limit of G-R–modules. Then

\underrightarrow{lim}_{i}H^{n}(G, L_{i}) ⟶^{∼} H^{n}(G, L) for all n ≥ 0.

(4) Let S = \underrightarrow{lim}_{α} S_{α} be an inductive limit of R–rings. Then

\underrightarrow{lim}_{α}H^{n}(G_{S_{α}}, L_{S_{α}}) ⟶^{∼} H^{n}(G_{S}, L_{S}) for all n ≥ 0.

Proof. (1) The natural map L → L ⊗_{R} R/I is an isomorphism of both R and R/I–modules. We have R and R/I–module isomorphisms

L ⊗_{R} R[G^{n}] ≃ L ⊗_{R} R/I ⊗_{R} R[G^{n}] ≃ L ⊗_{R} R/I[G^{n}] ≃ ≃ L ⊗_{R} R/I ⊗_{R/I} R/I[G^{n}] ≃ L_{R/I} ⊗_{R/I} R/I[G^{n}].

Now (1) follows from the fact that H^{n}(G, L) and H^{n}(G_{R/I}, L_{R/I}) are computed by the cohomology of the same complex. This is also a special case of [17, III 1.1.2].

(2) See [10, I.4.13].

(3) See [10, I.4.17].

(4) The terms of the complex (2.1) for the G_{S}-S–module L_{S} are

L_{S} ⊗_{S} S[G^{n}] = (L ⊗_{R} S) ⊗_{S} (S ⊗_{R} R[G^{n}]) ≃(L ⊗_{R} R[G^{n}]) ⊗_{R} S

So this complex reads

L ⊗_{R} S → (L ⊗_{R} R[G]) ⊗_{R} S → (L ⊗_{R} R[G^{2}]) ⊗_{R} S →(L ⊗_{R} R[G^{3}]) ⊗_{R} S ⋯

which is the inductive limit over the S_{α} of the complexes

L⊗_{R}S_{α} → (L⊗_{R}R[G])⊗_{R}S_{α} → (L⊗_{R}R[G^{2}])⊗_{R}S_{α} →(L⊗_{R}R[G^{3}])⊗_{R}S_{α} ⋯ whence the statement. □

The third property in the last Proposition is useful in view of the following fact.

Proposition 2.2. (Serre) Assume that one of the following hypothesis holds.

(i) R is noetherian, (ii) G is essentially free over R (see §6).

Let L be a G-R–module. Then L is the inductive limit of its G-R submodules which are of finite type as R–modules.

Proof. (i) See [14, prop. 2]. (ii) See [17, VI_{B} 11.10].

□

We also recall the following application of erasing functors.

Lemma 2.3. Let d > 0 be a positive integer such that H^{d}(G, L) = 0 for all G-R–modules L. Then H^{d+i}(G, L) = 0 for all G-R–modules L and for all i ≥ 0.

Proof. It is enough to prove the vanishing for d + 1. Let e_{R} be the trivial R– group, and view L as a (necessarily trivial) e_{R}-R–module. We also view L as a trivial G-R–module which we denote by L^{0} to avoid any possible confusion. Now we embed L into the induced G-R–module ind_{e_{R}}^{G}(L) = L^{0} ⊗_{R} R[G] via the comodule map Δ_{L}, and denote by Q the resulting quotient. We know that the Shapiro lemma holds [10, I.4.6], namely that

H^{i}(G, ind_{e_{R}}^{G}(L)) ⟶^{∼} H^{i}(e_{R}, L) = 0 ∀ i > 0.

The long exact sequence for cohomology for 0 → L → ind_{1}^{G}(L) → Q → 0 yields an isomorphism H^{d}(G, Q) ⟶^{∼} H^{d+1}(G, L), whence the result. □

3. Vanishing of Hochschild cohomology

The proof of Theorem 1.2 proceeds by considering successively the cases of fields, artinian rings, complete noetherian rings and local rings. We begin by recalling and collecting a few facts about linearly reductive groups.

3.1. Linearly reductive groups. Let k be a field. A k–group G is linearly reductive if it is affine and its corresponding category Rep_{k}(G) of finite dimensional linear representations is semisimple. We recall the following criterion.

Proposition 3.1. Let G be an affine k-group. Then the following are equivalent:

(1) G is linearly reductive.

(2) Every linear representation of G is semisimple.

(3) H^{1}(G, V) = 0 for any finitely dimensional G-k–module V.

(4) H^{1}(G, V) = 0 for any G-k–module V.

(5) H^{i}(G, V) = 0 for any G-k–module V and all i > 0.

(6) k is a direct summand of the G-k–module structure on k[G] corresponding to the right regular representation.

(7) k is an injective G-k–module.

Proof. For the equivalence of the first five assertions, see [4, II prop. 3.3.7].

(2) ⟹ (6): This follows from the fact that k is a submodule of k[G].

(6) ⟹ (7): See [10, I.3.10].

(7) ⟹ (3): If k is an injective G-k–module, the group H^{1}(G, V) = Ext_{G}^{1}(k, V) vanishes for each finite dimensional G-k–module V (by duality). □

The property of being linearly reductive behaves well with respect to base change.

Proposition 3.2. Let G be an affine algebraic k–group. Let K/k be a field extension. For the K-group G_{K} to be linearly reductive it is necessary and sufficient that G be linearly reductive. In particular, if \overline{k} is an algebraic closure of k and G\overline{k} is a linearly reductive \overline{k}-group, then G is linearly reductive.

This result is certainly known. We give three different proofs for the sake of completeness.

Proof. (1) As observed by S. Donkin in §2 of [5], G is linearly reductive if and only if the injective envelope E_{G}(k) of the trivial G-k–module k coincides with k. One also knows [ibid. eq. (1)] that E_{G_{K}} (K) = E_{G}(k)⊗_{k} K. The proposition follows.

(2) Assume that the k–group G is linearly reductive. By the criterion (6) of Proposition 3.1, k is a direct summand of k[G]. Hence K is a direct summand of K[G] and therefore G_{K} is linearly reductive. Conversely if G_{K} is linearly reductive and V is a G-k–module, then by Lemma 2.1.2 we have H^{1}(G, V) ⊗_{k} K ≃ H^{1}(G_{K}, V_{K}) = 0.

(3) The argument depends on the characteristic of k. One uses [4] IV prop. 3.3 in characteristic 0, and Nagata’s theorem (ibid. théorème 3.6) if the characteristic is positive.

□

Remark 3.3. Let G be an affine algebraic group over a field k. Let S be a scheme over k, and consider the S-group scheme G_{S} = G ×_{k} S. The fibers of G_{S} are then affine algebraic groups over the corresponding residue fields. It follows from the previous proposition that if any of the fibers is linearly reductive, then all fibers are linearly reductive.

The following useful statement seems to have gone unnoticed in the literature.

Proposition 3.4. Let 1 → G_{1} → G_{2} → G_{3} → 1 be an exact sequence of affine algebraic k–groups. Then the following are equivalent:

(1) G_{2} is linearly reductive ,

(2) G_{1} and G_{3} are linearly reductive.

Proof. (1) ⟹ (2): Since G_{2}/G_{1} is affine, we know that the induction functor indG_{2} G_{1} is exact [10, I.5.13], and therefore Shapiro’s lemma hence holds (ibid. I.4.6). Thus

H^{∗}(G_{2}, ind\begin{matrix}G & 2 \\ G & 1 \end{matrix}(V_{1})) ⟶^{∼} H^{∗}(G_{1}, V_{1})

for any G_{1}-k–module V_{1}. Thus H^{i}(G_{1}, V_{1}) = 0 for i > 0 and Proposition 3.1 shows that G_{1} is linearly reductive. Since the functor indG_{2} G_{1} is exact we can use the Hochschild-Serre spectral sequence in this framework (ibid. I.6.6.) Given a finite dimensional representation V_{3} of G_{3}, this spectral sequence reads as follows

E_{p,q}^{2} = H^{p}(G_{3}, H^{q}(G_{1}, V_{3})) ⟹ H^{p+q}(G_{2}, V_{3}).

Since G_{1} is linearly reductive H^{q}(G_{1}, V_{3}) vanishes for all q ≥ 1, hence H^{n}(G_{3}, V_{3}) ⟶^{∼} H^{n}(G_{2}, V_{3}) for all n ≥ 0. Since H^{1}(G_{2}, V_{3}) = 0, H^{1}(G_{3}, V_{3}) = 0 and we conclude that G_{3} is linearly reductive by Proposition 3.1.

(2) ⟹ (1): Assume that G_{1} and G_{3} are linearly reductive. Let us check that G_{2} is linearly reductive by again appealing to Proposition 3.1. Let V_{2} be a finitely dimensional representation of G_{2}. Again we can use the Hochschild-Serre spectral sequence which now reads as follows

E_{p,q}^{2} = H^{p}(G_{3}, H^{q}(G_{1}, V_{2})) ⟹ H^{p+q}(G_{2}, V_{2}).

The only non zero E_{2}-term is H^{0}(G_{3}, H^{0}(G_{1}, V_{2})) = H^{0}(G_{2}, V_{2}). Hence H^{i}(G_{2}, V_{2}) = 0 for i > 0. Thus G_{2} is linearly reductive. □

Note that Proposition 3.4 agrees with Nagata’s theorem characterizing linearly reductive groups over an algebraically closed field [11].

Proposition 3.5. Let G be an affine algebraic k–group which admits a composition series where each of the factors is of one of the following types:

(i) algebraic k-groups of multiplicative type,

(ii) finite étale k-group whose order is invertible in k,

(iii) reductive k-group if k is of characteristic zero.

Then G is linearly reductive.

Proof. By Proposition 3.4 we are reduced to verifying the result for each of the given types. Proposition 3.2 permits us to assume that the base field k is algebraically closed. Case (i) is then that of a diagonalizable k–group [17, th. I.5.3.3]. Case (ii) is the case of a finite constant group of invertible order (Maschke’s theorem, see [11]). Case (iii) is a classical result due to H. Weyl (see [18, th. 27.3.3]). □

3.2. Finiteness considerations. Recall that for arbitrary groups schemes G and H over a scheme S the functor Hom_{S−gp}(G, H) : Sch/S → Sets is defined by

T ↦ Hom_{S−gr}(G, H)(T) = Hom_{T −gr}(G_{T} , H_{T})

for all schemes T/S. The following observations will be repeatedly used in the proofs of our main results. For the remainder of this section we assume that G and H are finitely presented group schemes over S. Assume that T = Spec(B) is an affine scheme (in the absolute sense) over S. In what follows we will encounter ourselves several times in the situation where B is given to us as an inductive limit

(3.1)  B = \underrightarrow{lim}_{λ∈Λ} B_{λ}

over some directed set Λ. Note that the Spec(B_{λ}) do not in general have any natural structure of schemes over S. Under these assumptions the group schemes G_{B} and H_{B} are defined over some B_{μ} by [17, V I_{B} 10.10.3], i.e. there exists μ ∈ Λ and finitely presented B_{μ}–group schemes G_{μ} and H_{μ} such that

(3.2)  G_{B} = G_{μ} ×_{B_{μ}} B and H_{B} = H_{μ} ×_{B_{μ}} B.

Furthermore if either G is affine (resp. flat, smooth), so is G_{μ} by [9] 8.10.5 (resp. 11.2.6, 17.7.8). Similarly for H. It follows from the very definition that (3.3)

Hom_{S−gp}(G, H)(B) = Hom_{B−gp}(G_{B}, H_{B}) = Hom_{B_{μ}−gp}(G_{μ}, H_{μ})(B)

For all λ ≥ μ define G_{λ} = G_{μ} ×_{B_{μ}} B_{λ} and H_{λ} = H_{μ} ×_{B_{μ}} B_{λ}. Then the canonical map

(3.4)  \underrightarrow{lim}_{λ≥μ} Hom_{B_{λ}−gr}(G_{λ}, H_{λ}) → Hom_{B−gr}(G_{B}, H_{B}).

is bijective by [17, V I_{B} 10.10.2] (see also [9, théorème 8.8.2]).

Remark 3.6. From the foregoing it follows that if u, v : G → H are two homomorphisms of S-group schemes, then there exist μ ∈ Λ such that u_{B} and v_{B} are obtained by the base change B_{μ} → B from group homomorphisms u_{μ}, v_{μ} ∈ Hom_{B_{μ}−gp}(G_{μ}, H_{μ}).

Lemma 3.7. Let L be a G_{B}-B-module which is of finite presentation as a Bmodule. Then there exists an index μ and a G_{μ}-B_{μ}-module L_{μ} which is finitely presented as a B_{μ}–module such that L = L_{μ} ⊗_{B_{μ}} B.

Proof. According to (3.2) and proposition 8.9.1 (ii) of [9] we can find an index α, a B_{α}–group G_{α} and a finitely presented B_{α}-module L_{α} such that G_{α} ×_{B_{α}} B = G_{B} and L_{α}⊗_{B_{α}}B = L. For λ ≥ α, we set G_{λ} = G_{α}×_{B_{α}}B_{λ} and L_{λ} = L⊗_{B_{α}} B_{λ}.

By [9] 8.5.2.2, we have an isomorphism

\underrightarrow{lim}_{λ≥α} Hom_{B_{λ}}(L_{λ}, L_{λ} ⊗_{B_{λ}} B_{λ}[G_{λ}]) ⟶^{∼} Hom_{B}(\underrightarrow{lim}_{λ≥α} L_{λ}, \underrightarrow{lim}_{λ≥α} L_{λ} ⊗_{B_{λ}} B_{λ}[G_{λ}]) =

= Hom_{B}(L, L ⊗_{B} B[G]).

It follows that there exists λ ≥ α such that the B–module homomorphisms Δ_{L} : L → L⊗_{B}B[G] is obtained by the base change B_{λ} → B from a B_{λ}–module homomorphism Δ_{L_{λ}} : L_{λ} → L_{λ} ⊗_{B_{λ}} B_{λ}[G_{λ}]. The same reasoning applied to Hom_{B}(L, L ⊗_{B} B[G] ⊗_{B} B[G]) and Hom_{B}(L, L) show that there exists μ ≥ λ such that Δ_{L_{λ}} satisfies conditions (CM1) and (CM2) after applying the base change B_{λ} → B_{μ}. □

3.3. Proof of Theorem 1.2. We assume throughout that i > 0. Case (i) R a noetherian ring: The proof is a classical dévissage argument [8, §7.2.7)].

Case of R a field: The result follows from Proposition 3.1.

Case of R local artinian: Let 𝔪 be the maximal ideal of R, and k the corresponding residue field. By our assumption on the closed fibers of G the k–group G_{k} = G ×_{R} k is linearly reductive. Fix an integer e ≥ 2 such that 𝔪^{e} = 0. Thus there exists a smallest integer j = j(L) such that 0 < j ≤ e and 𝔪^{j}L = 0. We reason by induction on j. If j = 1 then 𝔪L = 0. By Lemma 2.1.1, we have H^{i}(G, L) ≅ H^{i}(G_{k}, L_{k}) for all i, and H^{i}(G_{k}, L_{k}) vanishes since G_{k} is linearly reductive. Assume now that H^{i}(G, M) = 0 for all G-R–modules M satisfying 𝔪^{j}M = 0. If 𝔪^{j+1}L = 0, we consider the exact sequence

0 → 𝔪L → L → L′ → 0

of G-R–modules. Observe that 𝔪^{j}(𝔪L) = 0 and that 𝔪L′ = 0. This sequence gives rise to the long exact sequence of cohomology [10, I.4.2]

⋯ → H^{i}(G, 𝔪L) → H^{i}(G, L) → H^{i}(G, L′) → ⋯

We have H^{i}(G, L′) = 0 by the case j = 1 and H^{i}(G, 𝔪L) = 0 by the induction hypothesis. Thus H^{i}(G, L) = 0 as desired.

Case of R local and complete: We denote by 𝔪 the maximal ideal of R, and set R_{n} = R/𝔪^{n+1} for all n ≥ 0. By Lemma 2.3 it will suffice to establish the case i = 1. Furthermore, Proposition 2.2 together with Lemma 2.1.3 allows us to assume that L is finitely generated over R. By the Artin-Rees lemma [7, cor. 0.7.3.3 ] we have a natural isomorphism

L ⟶^{∼} \underleftarrow{lim}_{n}L_{n}

where L_{n} = L ⊗_{R} R_{n}. We are given a cocycle z ∈ Z^{1}(G, L) and our goal is to show by using approximation that z is a coboundary. Since R_{n} is a local artinian ring we have H^{1}(G_{n}, L_{n}) = 0 where G_{n} = G ×_{R} R_{n}.^{2} We consider the exact commutative diagram

0 ⟶ H^{0}(G, L) ⟶ L ⟶^{∂_{0}} Z^{1}(G, L) ⟶ H^{1}(G, L) → 0
↓  ↓  ↓  ↓
0 ⟶ H^{0}(G_{n}, L_{n}) ⟶ L_{n} ⟶^{∂_{0,n}} Z^{1}(G_{n}, L_{n}) ⟶ H^{1}(G_{n}, L_{n}) = 0.

Then the images z_{n} of z in the Z^{1}(G_{n}, L_{n}) define elements b_{n} ∈ B^{1}(G_{n}, L_{n}) ⊂ Z^{1}(G, L_{n}). We look now

0 ⟶ H^{0}(G_{n}, L_{n}) ⟶ L_{n} ⟶^{∂_{0,n}} B^{1}(G_{n}, L_{n}) ⟶ 0.

Since H^{0}(G_{n}, L_{n}) is a finitely generated R_{n}-module, it is artinian. Hence the system (H^{0}(G, L_{n}))_{n≥0} satisfies the Mittag-Leffler condition [7, cor. 0.13.2.2 ]. We get then an exact sequence (ibid. prop. 13.2.2)

0 ⟶ \underleftarrow{lim}_{n} H^{0}(G_{n}, L_{n}) ⟶ \underleftarrow{lim}_{n} L_{n} ⟶ \underleftarrow{lim}_{n} B^{1}(G_{n}, L_{n}) ⟶ 0.

It follows that there exists l ∈ L such that z = ∂_{0}(l) modulo 𝔪^{n+1} for all n ≥ 0. Thus z = ∂_{0}(l) and therefore the image of z in H^{1}(G, L) vanishes.

Case of R local: We know that the completion R̂ of R is local noetherian and faithfully flat over R [7, cor. 0.7.3.5]). By Lemma 2.1.2, we have

H^{i}(G, L) ⊗_{R} R̂ ⟶^{∼} H^{i}(GR̂, LR̂).

The right hand side vanishes by the local complete case, hence H^{i}(G, L) = 0 by faithfully flat descent.

Case of R arbitrary noetherian: By the same reasoning used in the previous case we have H^{i}(G, L) ⊗_{R} R_{𝔪} = 0 for any maximal ideal 𝔪 of R. Thus H^{i}(G, L) = 0.

Case (ii) The group G is finitely presented as an R-scheme and L is a direct limit of G-R-modules which are finitely presented as R–modules: By Lemma 2.1.3 we may assume that L is a finitely presented R–module. The same reasoning used in the final step of the noetherian case allows us to assume that R is a local ring. Let 𝔪 be the maximal ideal of R and k its residue field. We consider the standard filtration R = \underrightarrow{lim}_{λ} R_{λ} of R by its finitely generated (hence noetherian) ℤ–subalgebras. For each λ, we consider the prime ideal 𝔭_{λ} := 𝔭 ∩ R_{λ} of R_{λ}, and the corresponding local ring R′_{λ} := (R_{λ})_{𝔭_{λ}} whose maximal ideal 𝔭_{λ}R′_{λ} we denote by 𝔪_{λ}. Note that the residue field k_{λ} of R′_{λ} is a subfield of k. We have R = \underrightarrow{lim}_{λ} R′_{λ} and the following commutative diagram

2One of course verifies that the R_{n}-groups G_{n} satisfy the assumptions of the theorem.

Similar considerations apply to the reductions that follow.

R′_{λ}  ⟶  R
↓  ↓
k_{λ} = R′_{λ}/𝔪_{λ} ⟶ R/𝔪 = k.

We now apply the considerations of §3.2 to the case when S = Spec(R), B = R and B_{λ} = R′_{λ}. This yield the existence of an R′_{μ}, an affine, flat and finitely presented R′_{μ}– group scheme G_{μ} and a G_{μ}-R′_{μ}-module L_{μ} such that G = G_{μ} ×_{R′_{μ}} R and L = L_{μ} ⊗_{R′_{μ}} R. By Lemma 2.1.4, we have

(3.5)  H^{i}(G, L) = \underrightarrow{lim}_{λ≥μ} H^{i}(G_{μ} ×_{R′_{μ}} R′_{λ}, L_{μ} ⊗_{R′_{μ}} R′_{λ}).

We also have by the transitivity of base change that

(3.6) (G_{μ} ×_{R′_{μ}} k_{μ}) ×_{k_{μ}} k ≃ G_{μ} ×_{R′_{μ}} k ≃ (G_{μ} ×_{R′_{μ}} R) ×_{R} k = G ×_{R} k.

From our assumptions on the R–group G it follows that the k–group G ×_{R} k is affine algebraic and linearly reductive. It then follows from (3.6) and Proposition 3.2 that the k_{μ}–algebraic group G_{μ} ×_{R′_{μ}} k_{μ} is linearly reductive as well. This shows that the R′_{μ}–group G_{μ} satisfies the assumption of the first part of the theorem. Similar considerations apply to the R′_{λ}–group G_{μ} ×_{R′_{μ}} R′_{λ} for all λ ≥ μ. Thus the noetherian case that we have already established shows, with the aid of (3.5), that H^{i}(G, L) = 0. □

4. Rigidity and deformation theory

4.1. Locally finitely presented S–functors. Let S be a scheme and F : Sch/S → Sets a contravariant functor. We recall the following definitions:

- F is locally of finite presentation over S if for every filtered inverse system of affine S-schemes Spec(B_{i}), the canonical morphism

\underrightarrow{lim}F (B_{i}) → F (\underrightarrow{lim}B_{i})

is an isomorphism [3, §8.3].^{3}

- F is formally smooth (resp. formally unramified, formally étale) if for any affine scheme Spec(B) over S and any subscheme Spec(B_{0}) of Spec(B) defined by a nilpotent ideal I of B, the map

F (B) → F (B_{0})

is surjective (resp. injective, bijective) [17, XI.1.1]. Note that all these definitions are stable by an arbitrary base change T → S. In the second definition, we can require furthermore that I^{2} = 0. The following lemma is elementary.

3This reference has assumptions on the nature of S related to Artin’s approximation theorem which are relevant to their work, but not to ours. As it is customary, given an affine scheme Spec(B) over S, we write F (B) instead of F ( Spec(B)).

Lemma 4.1. Assume that F is locally of finite presentation over S. Consider a field extension K/k over S, that is morphisms Spec(K) → Spec(k) → S. Assume that k is separably closed and K is a separable field extension of k. Then the map F (k) → F (K) is injective.

Remark 4.2. If k is algebraically closed, any field extension K/k is separable, hence the Lemma applies.

Proof. We may assume without loss of generality that S = Spec(k). We are given two elements α, β ∈ F (k) with same image in F (K). Since K is the inductive limit of its finitely generated subalgebras, there exists a finitely generated k–algebra A such that α and β have same image in F (A). Since K/k is separable, the finitely generated k–subalgebras of K are separable over k. Hence A is integral and absolutely reduced (i.e. A ⊗_{k} \overline{k} is reduced), and therefore the affine variety Spec(A) admits a k–point [1, AG.13.3]. In other words, the ring homomorphim k → A admits a section. This, in turn, induces a section of the group homomorphism F (k) → F (A), hence α = β in F (k). □

4.2. Formal étalness. We recall the following crucial statement of deformation theory for group scheme homomorphisms due to Demazure.

Theorem 4.3. ([17, cor. III.2.6]) Let G and H be group schemes over a scheme S. Assume that G is affine (in the absolute sense) and flat, and that H is smooth. Let S_{0} be a closed subscheme of S defined by an ideal I of O_{S} such that I^{2} = 0. We set G_{0} = G ×_{S} S_{0} and H_{0} = H ×_{S} S_{0}. Let f_{0} : G_{0} → H_{0} be a homomorphism of S_{0}-groups, and let G_{0} act on Lie(H_{0}) via f_{0} and the adjoint representation of H_{0}. Then

(1) If H^{2}(G_{0}, Lie(H_{0}) ⊗_{O_{S_{0}}} I) = 0 the homomorphism f_{0} lifts to an S– group homomorphism f : G → H.

(2) If H^{1}(G_{0}, Lie(H_{0}) ⊗_{O_{S_{0}}} I) = 0, then any two liftings f and f′ of f_{0} as in (1) are conjugate under an element of ker (H(S) → H(S_{0})). More precisely f′ = int(h)f for some h ∈ ker (H(S) → H(S_{0})). □

Combined with the vanishing result given by Theorem 1.2 we are very close to the completion of the proof of our main result. The missing ingredient is some detailed information pertaining to the nature of certain functors related to homomorphisms between group schemes. Let G and H be group schemes over a scheme S. The functor Hom_{S−gp}(G, H) was already defined in §3.2. Any element h ∈ H(T) defines an inner automorphism int(t) ∈ Aut_{T −gr}(H_{T}), and this last group acts naturally on the set Hom_{S−gr}(G, H)(T). This allows us to define a new functor \overline{Hom}_{S−gr}(G, H) : Sch/S → Sets by

T ↦ \overline{Hom}_{S−gr}(G, H)(T) = Hom_{T −gr}(G_{T} , H_{T})/H(T).

The final functor which is relevant to us is the transporter of two elements of Hom_{S−gr}(G, H). Let u, v : G → H two homomorphisms of S-group schemes.

Recall the subfunctor Transp(u, v) of H defined by

T → Transp(u, v)(T) = {h ∈ H(T) | u_{T} = int(h) v_{T} }.

We begin with an easy observation.

Lemma 4.4. Let G and H be finitely presented group schemes over S, and let u, v ∈ Hom_{S−gr}(G, H). The S-functors Hom_{S−gr}(G, H), \overline{Hom}_{S−gr}(G, H) and Transp(u, v) are locally of finite presentation.

Proof. For every filtered inverse system of affine schemes Spec(B_{λ}) over S based on some directed set Λ, and we have to show that the canonical morphisms

\underrightarrow{lim} Hom_{S−gr}(G,H)(B_{λ}) → Hom_{S−gr}(G,H)(\underrightarrow{lim}B_{λ}) = Hom_{S−gr}(G,H)(B),

\underrightarrow{lim} \overline{Hom}_{S−gr}(G,H)(B_{λ}) → \overline{Hom}_{S−gr}(G,H)(\underrightarrow{lim}B_{λ}) = \overline{Hom}_{S−gr}(G,H)(B),

and

\underrightarrow{lim} Transp(u, v)(B_{λ}) → Transp(u, v)(\underrightarrow{lim}B_{λ}) = Transp(u, v)(B)

are bijective. Taking into account (3.2), (3.3) and (4.1) we may replace S by Spec(B_{μ}) for some suitable index μ ∈ Λ, and replace Λ by the subset of Λ consisting of all indices λ ≥ μ. Denote G ×_{B_{μ}} B_{λ} and H ×_{B_{μ}} B_{λ} by G_{λ} and H_{λ} respectively, just as we did in §3.2. Then (3.4) shows that Hom_{S−gr}(G, H) is locally of finite presentation. As for the second assertion, we look in view of (3.3) at the map

\underrightarrow{lim} Hom_{B_{λ}−gr}(G_{λ}, H_{λ})/H_{λ}(B_{λ}) → Hom_{B−gr}(G_{B}, H_{B})/H_{B}(B)

which is already known to be surjective. For the injectivity, we are given ϕ_{α}, ϕ′_{α} ∈ Hom_{B_{α}−gr}(G_{α}, H_{α}) for some α ≥ μ whose images ϕ, ϕ′ in Hom_{B−gr}(G_{B}, H_{B}) are conjugated under H_{B}(B) = H(B). Since H is of finite presentation \underrightarrow{lim}H(B_{λ}) ⟶^{∼} H(B). So there exists β ≥ α and h_{β} ∈ H(B_{β}) such that ϕ = int(h)ϕ′ where h stands for the image of h_{β} in H(B). By (3.4) there exists γ ≥ β such that ϕ_{α} ×_{B_{α}} id_{B_{γ}} = int(h_{γ})(ϕ′_{α} ×_{B_{α}} id_{B_{γ}}), where h_{γ} is the image of h_{β} in H(B_{γ}). In other words, ϕ_{α}, ϕ′_{α} map to the same element of Hom_{B_{γ}−gr}(G_{γ}, H_{γ})/H_{γ}(B_{γ}), hence define the same element in the inductive limit \underrightarrow{lim} Hom_{B_{λ}−gr}(G_{λ}, H_{λ})/H_{λ}(B_{λ}). We conclude that \overline{Hom}_{S−gr}(G, H) is locally of finite presentation. Finally we look at the case of the transporter. Assume that h ∈ H(B) is such that u_{B} = int(h)v_{B}. Since H is finitely presented there exists α ≥ μ and an element h_{α} ∈ H(B_{α}) whose image in H(B) is h. Then the two elements u_{α} and int(h_{α})v_{α} of Hom_{B_{α}}(G_{α}, H_{α}) map to the same element of Hom_{B}(G_{B}, H_{B}). By (3.4) there exists β ≥ α such that u_{β} = int(h_{β})v_{β} (where the subindex β denotes the image of the element in question under the map B_{α} → B_{β}). This shows that our map is surjective. Note that from the definition of the transporter it follows that

(4.1)  Transp(u, v)(B) = Transp(u_{B}, v_{B})(B) = Transp(u_{μ}, v_{μ})(B)

Injectivity is clear since for all λ ≥ μ we have Transp(u_{μ}, v_{μ})(B_{λ}) ⊂ H_{μ}(B_{λ}) and H_{μ} is of finite presentation. □

Theorem 4.5. Let S be a scheme and let G and H be finitely presented group schemes over S. Assume that G is affine (in the absolute sense) and flat, and that H is smooth. Assume that for all s ∈ S the fiber G_{s} is linearly reductive (as an affine algebraic group over the residue field κ(s) of s). Then

(1) The functor Hom(G, H) is formally smooth.

(2) The functor \overline{Hom}(G, H) is formally étale.

(3) If u, v : G → H are two homomorphisms of S-group schemes, the subfunctor Transp(u, v) of H is formally smooth.

The case when G is of multiplicative type is an important result of Grothendieck [17, XI prop. 2.1]. If S is of characteristic zero and G is reductive, the first statement is due to Demazure [17, XXIV prop. 7.3.1.a].

Proof. We note that if Hom_{S−gr}(G, H) is formally smooth, then \overline{Hom}_{S−gr}(G, H) is formally smooth as well. As a consequence, to establish (1) and (2) it will suffice to prove that Hom_{S−gr}(G, H) is formally smooth and that \overline{Hom}_{S−gr}(G, H) is formally unramified. We are given an affine scheme Spec(B) over S, and a closed subscheme Spec(B_{0}) defined by an ideal I of B satisfying I^{2} = 0, and we need to show that

(I) Hom_{S−gr}(G, H)(B) → Hom_{S−gr}(G, H)(B_{0}) is surjective, (II) \overline{Hom}_{S−gr}(G, H)(B) → \overline{Hom}_{S−gr}(G, H)(B_{0}) is injective, and (III) Transp(u, v)(B) → Transp(u, v)(B_{0}) is surjective.

Proof of (I) and (II): By the first equality of (3.3) we may assume with no loss of generality that S = Spec(B). We claim that, with the obvious adaptations to the notation of Theorem 4.3,

(4.2)  H^{i}(G_{0}, Lie(H_{0}) ⊗_{B_{0}} I) = 0 for all i > 0.

Write B = \underrightarrow{lim}B_{λ} where the limit is taken over all finitely generated ℤ– subalgebras (hence noetherian) B_{λ} of B. Then J_{λ} := I ∩ B_{λ} is an ideal of B_{λ} such that J_{λ}^{2} = 0 and I = \underrightarrow{lim}J_{λ}. Consider the trivial G_{0}-B_{0} module

I_{λ} := J_{λ} ⊗_{B_{λ}} B_{0}.

Since J_{λ} is a B_{λ}-module of finite presentation, I_{λ} is a B_{0}-module of finite presentation. We have an isomorphism of B_{0}-modules

\underrightarrow{lim}I_{λ} ⟶^{∼} I

hence an isomorphism of G_{0}-B_{0}–modules

\underrightarrow{lim}(Lie(H_{0}) ⊗_{B_{0}} I_{λ}) ⟶^{∼} Lie(H_{0}) ⊗_{B_{0}} I.

Because H_{0} is a smooth B_{0}–group Lie(H_{0}) is a finitely presented B_{0}–module (see [4] II §4.8). Since the tensor product of finitely presented modules is finitely presented, Lie(H_{0}) ⊗_{B_{0}} I is a direct limit of G_{0}-B_{0}–modules which are finitely presented as B_{0}-modules. It is clear that the B_{0}-groups G_{0} and H_{0} satisfy the assumptions of Theorem 1.2.2. This shows that (4.2) holds, and we can now apply Theorem 4.3 to obtain (I) and (II) Proof of (III): For convenience we denote Transp(u, v) by T(u, v). Consider the B-group homomorphisms u_{B}, v_{B} ∈ Hom_{B−gr}(G_{B}, H_{B}) induced by the base change Spec(B) → S. By the definition of the transporter we see that T(u, v)(B) = T(u_{B}, v_{B})(B) and T(u, v)(B_{0}) = T(u_{B}, v_{B})(B_{0}) where Spec(B_{0}) → Spec(B) is the natural map. From this it follows that to establish (III) we may assume without loss of generality that S = Spec(B). Let u_{0} and v_{0} be the elements of Hom_{B_{0}−gr}(G_{0}, H_{0}) induced by the base change B → B_{0}. Let h_{0} ∈ Transp(u_{0}, v_{0})(B_{0}), so that u_{0} = int(h_{0})v_{0}. Lift h_{0} to an element h′ ∈ H(B) (which is possible since H is smooth), and set u′ = int(h′)v_{B}. Then u′ and u_{B} map to the same element of Hom_{B_{0}−gr}(G_{0}, H_{0}), namely u_{0}. By II there exists h′′ ∈ H(B) such that u_{B} = int(h′′)u′. Furthermore, because of (4.2) we may assume that h′′ ∈ ker (H(B) → H(B_{0})). Then h = h′′h′ ∈ H(B) maps to h_{0} and satisfies u_{B} = int(h)v_{B}. □

Remark 4.6. The assumption on the fibers of G is not superfluous. Let B = ℂ[ϵ] be the ring of dual numbers over ℂ, and let S = Spec(B). If I = ℂϵ, then B_{0} = ℂ. Consider now the case when G = G_{a} and H = G_{m} (the additive and multiplicative groups over B.) It is well-known that Hom_{B−gr}(G, H)(B_{0}) = Hom_{ℂ−gr}(G_{a,ℂ}, G_{m,ℂ}) is trivial. On the other hand Hom_{B−gr}(G, H)(B) is infinite; it consists of the homomorphisms {ϕ_{z} : z ∈ ℂ} which under Yoneda correspond to the B–Hopf algebra homomorphisms ϕ_{z}^{∗} : B[t^{±1}] → B[x] given by ϕ_{z}^{∗} : t ↦ 1 + zϵx. Since H is abelian the functors \overline{Hom}_{B−gr}(G, H) and Hom_{B−gr}(G, H) coincide. The above discussion shows that \overline{Hom}_{B−gr}(G, H) is not formally étale.

Lemma 4.7. If G is essentially free over S (see §6), the functor Transp(u, v) is representable by a closed S–subscheme of H

Proof. Consider the two morphisms q_{1}, q_{2} : H → Hom(G, H) which for all schemes T/S and h ∈ H(T) are given by and q_{1}(h) = u_{T} and q_{2}(h) = int(h)v_{T} . Since G is assumed essentially free over S and H is separated over S, Grothendieck’s criterion [17, VIII.6.5.b] applied to X = H, Y = G, Z = H shows the representability of Transp(u_{1}, u_{2}) by a closed S–subscheme of H. □

Corollary 4.8. Under the assumptions of Theorem 4.5, assume furthermore that G is essentially free over S. Let u, v : G → H be two homomorphisms of S-group schemes. Then the S-functor Transp(u, v) is representable by a smooth closed S-subscheme of H. In particular, if u = v, then the centralizer subfunctor Centr(u) of H is representable by a smooth closed subscheme of H.

Proof. By the last Lemma the S-functor Transp(u, v) is representable by a closed subscheme of H, which is locally of finite presentation by Lemma 4.4 and [9] 8.14.2, and formally smooth by Theorem 4.5. Thus Transp(u, v) is a smooth scheme over S. □

Corollary 4.9. Under the assumptions of Theorem 4.5, assume furthermore that G is essentially free over S and that S = Spec(B) where B is a henselian local ring of residue field k. Then the map \overline{Hom}(G, H)(B) → \overline{Hom}(G, H)(k) is injective. Thus two homomorphisms u, v : G → H of S-group schemes are conjugate under H(B) if and only if u_{k}, v_{k} : G ×_{B} k → H ×_{B} k are conjugate under H(k).

Proof. By Corollary 4.8, the B–functor Transp(u, v) is representable by a smooth B–scheme. By Hensel’s lemma [3, §2.3] the map

Transp(u, v)(B) → Transp(u, v)(k)

is surjective. Thus if u_{k}, v_{k} : G ×_{B} k → H ×_{B} k are such that u_{k} = int(h_{0})v_{k} for some h_{0} ∈ H(k), then there exists h ∈ H(B) such that u = int(h)v. □

5. Applications

5.1. Rigidity. Our first result establishes Theorem 1.1

Theorem 5.1. Let k be a field. Let G be a linearly reductive algebraic k–group and let H be a smooth algebraic k–group. Let K/k be a field extension such that k and K are both separably closed and K is separable over k (for example if both k and K are algebraically closed). Then the map

\overline{Hom}_{k−gr}(G, H)(k) → \overline{Hom}_{k−gr}(G, H)(K) = \overline{Hom}_{K−gr}(G_{K}, H_{K})(K)

is bijective.

Proof. By Lemma 4.1.1 and Lemma 4.4 the map \overline{Hom}_{k−gr}(G, H)(k) → \overline{Hom}_{k−gr}(G, H)(K) is injective. Conversely we are given an element u ∈ Hom_{K−gr}(G_{K}, H_{K}) and we want to show that there exists v_{0} ∈ Hom_{k−gr}(G, H) and h ∈ H(K) such that v_{0} ×_{k} id_{K} = int(h)u. The homomorphism u : G_{K} → H_{K} arises by base change from some A-group scheme homomorphism v ∈ Hom_{A−gr}(G_{A}, H_{A}), i.e. u = v_{K}, where A ⊂ K is a finitely generated k–algebra. Under our assumption on k we may assume, by considering a basic open affine subsheme of Spec(A) if needed, that A is smooth over k. In particular, A is normal. Since A is separable over k and k is separably closed, there exists a maximal ideal 𝔪 of A such that A/𝔪 = k. Then v gives rise to a k–homomorphism v_{0} : G → H. Denote by B the (strict) henselization of the local ring A_{𝔪}. Then B is noetherian and may be identified with a subring of a separable closure of the fraction field of A [M, I.4.10, 11]. In particular B can be assumed to embed into K. By Proposition 3.2 and Remark 3.3 the group G_{B} (which is clearly affine and free of finite rank over B) satisfies the assumption on the fibers of Theorem 4.5. By Corollary 4.9, v_{0} ×_{k} id_{B} = int(h)(v ×_{A} id_{B}) for some h ∈ H(B). Thus v_{0} ×_{k} id_{K} = int(h_{K})u as desired. □

Remark 5.2. The assumption that G be linearly reductive is not superfluous. Recall (see §3.1) that H^{1}(G, V) = Ext_{G}^{1}(k, V). Assume that k is algebraically closed of positive characteristic, and let K/k be an arbitrary field extension. One knows from Nagata’s work that for each non-trivial semisimple k-group G there exists a non-trivial finite dimensional irreducible G-k– module V such that H^{1}(G, V) ≠ 0. This implies that Theorem 5.1 fails for \overline{Hom}_{k−gr}(G, GL_{n}) if n = dim(V) + 1. Indeed \overline{Hom}_{k−gr}(G, GL_{n})(k) measures the equivalence classes of n–dimensional linear representations of G. We know that Ext_{G}^{1}(k, V), which by assumption is a non-trivial k-space, can be identified with the subset of \overline{Hom}_{k−gr}(G, GL_{n})(k) that corresponds to those representations of G that are extensions of k by V. Similar considerations apply to \overline{Hom}_{K−gr}(G_{K}, GL_{n,K})(K). Since H^{1}(G, V) ⊗_{k} K = H^{1}(G_{K}, V_{K}) the foregoing discussion shows that the natural map \overline{Hom}_{k−gr}(G, H)(k) → \overline{Hom}_{k−gr}(G, H)(K) is not surjective whenever k ≠ K.

Remark 5.3. Let H be a simple Chevalley ℤ–group of adjoint type. In [2] Borel, Friedman and Morgan provide a considerable amount of information about the set of conjugacy classes of n-tuples x = (x_{1}, ⋯, x_{n}) of commuting elements of finite order of H(ℂ).^{4} The methods used in [2] are topological and analytic in nature, and do not immediately translate to other algebraically closed fields of characteristic 0. One of the reasons why this problem is relevant is because of its applications to infinite dimensional Lie theory. The interested reader can consult [6] for details and further references about this topic. Fix an n-tuple m = (m_{1}, ⋯, m_{n}) of positive integers, and let F_{m} be the finite constant \overline{ℚ}–group corresponding to the finite group ℤ/m_{1}ℤ × ⋯ × ℤ/m_{n}ℤ. Because of the nature of our base field the group F_{m} is diagonalizable, hence linearly reductive. Let K be an algebraically closed field of characteristic 0. The conjugacy classes of n-tuples x = (x_{1}, ⋯, x_{n}) of commuting elements of H(K) where the x_{i} satisfy x_{i}^{m_{i}} = 1 are parametrized by \overline{Hom}_{K−gr}(F_{m,K}, H_{K})(K). By Theorem 1.1 we have natural bijections

\overline{Hom}_{K−gr}(F_{m,K}, H_{K})(K) ≃ \overline{Hom}_{\overline{ℚ}−gr}(F_{m}, H\overline{ℚ})(\overline{ℚ})) ≃
≃ \overline{Hom}_{ℂ−gr}(F_{m,ℂ}, H_{ℂ})(ℂ).

This allows us to translate the relevant information within [2] to the group H(K).

5.2. Lie algebras. Assume henceforth that the base scheme S is of “characteristic zero”, i.e. that S is a scheme over ℚ. Let G/S be a semisimple group scheme and let H/S be an affine smooth group scheme. In this case, we already know that the functor Hom_{S−gp}(G, H) is representable by a smooth affine Sscheme of finite presentation [17, XXIV.7.3.1]. Furthermore, if G/S is simply connected, we have an S-scheme isomorphism

Hom_{S−gr}(G, H) ⟶^{∼} Hom_{S−Lie}(Lie(G), Lie(H)).

4When lifted to the simply connected cover of H(ℂ) the n-tuples will be comprised of “almost commuting” elements.

From this and Theorem 4.5 it follows that the functor

T ↦ Hom_{T −Lie}(Lie(G_{T}), Lie(H_{T}))/H(T)

is formally étale.

Corollary 5.4. Let k be an algebraically closed field of characteristic zero. Let 𝔤 be a finite dimensional semisimple Lie algebra over k. Let H be a smooth algebraic k–group. If K is an algebraically closed field extension of k, then the map

Hom_{k−Lie}(𝔤, Lie(H))/H(k) → Hom_{K−Lie}(𝔤 ⊗_{k} K, Lie(H) ⊗_{k} K)/H(K)

is bijective. □

6. Appendix: Affine group schemes which are essentially free

Definition 6.1. [17, VIII 6] A morphism of schemes X/S is essentially free if there exists an open covering of S by affine schemes S_{i} = Spec(A_{i}), and for all i an faithfully flat extension S′_{i} = Spec(A′_{i}) → S_{i} such that each X ×_{S} S′_{i} admits an open covering by affine schemes (Spec(B′_{i,j})) such that B′_{i,j} is a free A′_{i}–module for all j.

Note that an essentially free morphism is flat. Furthermore this property is stable by arbitrary base change and is local with respect to the Zariski and the fpqc topology. Recall that that a sequence

1 → G_{1} → G_{2} → G_{3} → 1

of S–group schemes is said to be exact if it is exact as a sequence of fpqc-sheaves over S [17, VI_{B} 9].

Lemma 6.2. (1) Let G/S be a flat group scheme which is essentially free over S. Let X → S be a G–torsor which is locally trivial for the fpqc– topology. Then X is essentially free over S.

(2) Let 1 → G_{1} → G_{2} → G_{3} → 1 be an exact sequence of S–group schemes. If G_{1} and G_{3} are essentially free over S, then G_{2} is essentialy free over S.

Proof. (1) Since we can reason locally for the Zariski and for the fpqc topology, we can assume that X is the trivial torsor, namely X = G.

(2) Similarly, we can assume that S = Spec(A) and that G_{1} (resp. G_{3}) is covered by open affines subschemes Spec(B_{j}) (resp. Spec(C_{l})) for the fpqc topology such that the B_{j} and the C_{l} are free A–modules. Up to refining the second fpqc covering, we can furthermore assume that

G_{2} ×_{G_{3}} Spec(C_{l}) ⟶^{∼} G_{1} ×_{S} Spec(C_{l}).

It follows that the Spec(B_{l}⊗_{A}C_{l})’s form a fpqc–cover of G_{2} where the B_{l}⊗_{A}C_{l} are free A–modules. Thus G_{2} is essentially free over S as desired. □

Several well-known affine group schemes are essentially free.

Proposition 6.3. Let G/S be an affine S–group scheme which admits locally for the fpqc topology a composition series with factors of the following kind:

(i) S-group schemes of multiplicative type,

(ii) twisted finite constant S–group schemes,

(iii) smooth S-group schemes with connected geometric fibers.

Then G is essentially free over S.

Note that the last case includes reductive group schemes over S and their parabolic subgroups.

Proof. By Lemma 6.2 it suffices to verify each of the three cases locally for the fpqc topology. Case (i) is then the case of diagonalizable groups which are essentially free over S by definition. Case (ii) is that of finite constant S–group schemes which are also essentially free over S by definition. Case (iii) has been noticed by Seshadri [15, Lemma 1 p. 230] using a result of Raynaud. □

Acknowledgements: The author would like to thank the referee for his/her thorough reading of the original submission. He would also like to express his gratitude to BIRS for their hospitality.

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