Documenta Math. 653 Vanishing of Hochschild Cohomology for Affine Group Schemes and Rigidity of Homomorphisms between Algebraic Groups

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


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 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).

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 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 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 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 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] 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 The following four properties easily follow from the resolution (2.1). 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 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].
(4) The terms of the complex (2.1) for the G S -S-module L S are which is the inductive limit over the S α of the complexes 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 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 Rgroup, 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 via the comodule map ∆ L , and denote by Q the resulting quotient.We know that the Shapiro lemma holds [10, I.4.6],namely that The long exact sequence for cohomology for 0 , whence the result.

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.
(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].
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 k is an algebraic closure of k and G k is a linearly reductive 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 (2) Assume that the k-group G is linearly reductive.By the criterion (6 (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.
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 ind G2 G1 is exact [10, I.5.13], and therefore Shapiro's lemma hence holds (ibid.I.4.6).Thus shows that G 1 is linearly reductive.Since the functor ind G2 G1 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 (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 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, Then G is linearly reductive.
Furthermore if either G is affine (resp.flat, smooth), so is G µ by [9] is bijective by [  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 By [9] 8.5.2.2, we have an isomorphism It follows that there exists λ ≥ α such that the B-module homomorphisms 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 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 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 2.1.3allows 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 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 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) 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 R of R is local noetherian and faithfully flat over R [7, cor.0.7.3.5]).By Lemma 2.1.2,we have ).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

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 2.1.3we 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 = 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 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 We also have by the transitivity of base change that 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 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.

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 -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 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.
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 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 4.2.Formal étalness.We recall the following crucial statement of deformation theory for group scheme homomorphisms due to Demazure.(2) If H 1 G 0 , Lie(H 0 ) ⊗ OS 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 Hom S−gr (G, H) : Sch/S → Sets by 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 We begin with an easy observation.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 lim (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 × Bµ B λ and H × Bµ 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 which is already known to be surjective.For the injectivity, we are given . 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 × Bα id Bγ = int(h γ )(φ ′ α × Bα id Bγ ), 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 Bγ −gr (G γ , H γ )/H γ (B γ ), hence define the same element in the inductive limit lim −→ Hom B λ −gr (G λ , H λ )/H λ (B λ ).We conclude that 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 Bα (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) Injectivity is clear since for all λ ≥ µ we have 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.
(  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, ( H i G 0 , Lie(H 0 ) ⊗ B0 I = 0 for all i > 0.
Write B = lim −→ B λ where the limit is taken over all finitely generated Zsubalgebras (hence noetherian) B λ of B. Then J λ := I ∩ B λ is an ideal of B λ such that J 2 λ = 0 and I = lim −→ J λ .Consider the trivial G 0 -B 0 module 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 Because H 0 is a smooth B 0 -group Lie(H 0 ) is a finitely presented B 0 -module (see [4] 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 B0−gr (G 0 , H 0 ) induced by the base change  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.The assumption that G be linearly reductive is not superfluous.Recall (see §3.1) that H 1 (G, V ) = Ext 1 G (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-kmodule V such that H 1 (G, V ) = 0.This implies that Theorem 5.1 fails for Hom k−gr (G, GL n ) if n = dim(V ) + 1. Indeed Hom k−gr (G, GL n )(k) measures the equivalence classes of n-dimensional linear representations of G.We know that Ext 1 G (k, V ), which by assumption is a non-trivial k-space, can be identified with the subset of 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 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 Hom k−gr (G, H)(k) → 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.This allows us to translate the relevant information within [2] to the group H(K).

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

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 ⊗ Bµ B.
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]

Theorem 4 . 3 .( 1 )
([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 If H 2 G 0 , Lie(H 0 ) ⊗ OS 0 I = 0 the homomorphism f 0 lifts to an Sgroup homomorphism f : G → H.

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), Hom S−gr (G, H) and Transp(u, v) are locally of finite presentation.

Remark 4 . 6 .
The assumption on the fibers of G is not superfluous.Let  B = C[ǫ] be the ring of dual numbers over C, and let S = Spec(B).If I = Cǫ, 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 + zǫx.Since H is abelian the functors Hom B−gr (G, H) and Hom B−gr (G, H) coincide.The above discussion shows that 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

Corollary 4 . 8 .
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.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 Remark 5.2.
Fix an n-tuple m = (m 1 , • • • , m n ) of positive integers, and let F m be the finite constant Q-group corresponding to the finite group Z/m 1Z × • • • × 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 mi i = 1 are parametrized by Hom K−gr (F m,K , H K )(K).By Theorem 1.1 we have natural bijectionsHom K−gr (F m,K , H K )(K) ≃ Hom Q−gr (F m , H Q )(Q)) ≃ ≃ Hom C−gr (F m,C , H C )(C).
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 3.2.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 byT → Hom S−gr (G, H)(T ) = Hom T −gr (G T , H T )for all schemes T /S.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) B and v B are obtained by the base change B µ → B from group homomorphisms u µ , v µ ∈ Hom Bµ−gp (G µ , H µ ).
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 The same reasoning applied toHom B (L, L ⊗ B B[G] ⊗ B B[G]) and Hom B (L, L) show that there exists µ ≥ λ such that ∆ L λ satisfies conditions (CM1) and (CM2) after applying the base change B λ → B µ .
) The functor 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 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 Hom S−gr II §4.8).Since the tensor product of finitely presented modules is finitely presented, Lie(H 0 ) ⊗ B0 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 is possible since H is smooth), and setu ′ = int(h ′ )v B .Then u ′ and u B map to the same element of Hom B0−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 .