Smooth affine group schemes over the dual numbers

We provide an equivalence between the category of affine, smooth group schemes over the ring of generalized dual numbers $k[I]$, and the category of extensions of the form $1 $\rightarrow$ Lie(G, I) $\rightarrow$ E $\rightarrow$ G $\rightarrow$ 1$ where G is an affine, smooth group scheme over k. Here k is an arbitrary commutative ring and $k[I] = k $\oplus$ I$ with $I^2 = 0$. The equivalence is given by Weil restriction, and we provide a quasi-inverse which we call Weil extension. It is compatible with the exact structures and the $O\_k$-module stack structures on both categories. Our constructions rely on the use of the group algebra scheme of an affine group scheme; we introduce this object and establish its main properties. As an application, we establish a Dieudonn{\'e} classification for smooth, commutative, unipotent group schemes over $k[I]$.


Introduction
Throughout this article, we fix a commutative ring k and a free k-module I of finite rank r 1. We consider the ring of (generalized) dual numbers k[I] := k ⊕ I with I 2 = 0. We write h : Spec(k[I]) → Spec(k) the structure map and i : Spec(k) → Spec(k[I]) the closed immersion. Also we denote by O k the Spec(k)-ring scheme such that if R is a k-algebra then O k (R) = R with its ring structure. To explain the idea, let G be an affine, flat, finitely presented k-group scheme. It is shown in Illusie's book [Il72] that the set of isomorphism classes of deformations of G over k [I] is in bijection with the cohomology group H 2 (BG, ∨ G ⊗ I), see chap. VII, thm 3.2.1 in loc. cit. Here, the coefficients of cohomology are the derived dual of the equivariant cotangent complex G ∈ D(BG), tensored (in the derived sense) by I viewed as the coherent sheaf it defines on the fpqc site of BG, also equal to the vector bundle V (I ∨ ). If we assume moreover that G is smooth, then the augmentation G → ω 1 G to the sheaf of invariant differential 1-forms is a quasi-isomorphism and it follows that ∨ (2) If 1 → G → G → G → 1 is an exact sequence in Gr/k[I], then 1 → h * G → h * G → h * G is exact in Ext(I)/k. If moreover G is smooth then 1 → h * G → h * G → h * G → 1 is exact. In particular, h * is an exact equivalence between the subcategories of smooth objects endowed with their natural exact structure.
(3) The equivalence h * is a morphism of O k -module stacks fibred over Gr/k, i.e. it transforms the addition and scalar multiplication of deformations of a fixed G ∈ Gr/k into the Baer sum and scalar multiplication of extensions.
(4) Let P be one of the properties of group schemes over a field: of finite type, smooth, connected, unipotent, split unipotent, solvable, commutative. Say that a group scheme over an arbitrary ring has property P if it is flat and its fibres have P . Then G ∈ Gr/k[I] has property P if and only if the k-group scheme E = h * G has P .
In order to show that h * is an equivalence, we build a quasi-inverse h + which we call Weil extension. The construction and study of this functor is the hardest part of the proof.
As an application of the Theorem, we prove a Dieudonné classification for smooth, commutative, unipotent group schemes over the generalized dual numbers of a perfect field k. This takes the form of an exact equivalence of categories with a category of extensions of smooth, erasable Dieudonné modules.
Here is the precise statement (we refer to Section 6 for the definition of all undefined terms). Comments. An important tool in many of our arguments is the group algebra scheme. It provides a common framework to conduct computations in the groups and their tangent bundles simultaneously. It allows us to describe conveniently the Weil restriction of a group scheme, and is essential in the proof of Theorem A. Since we are not aware of any treatment of the group algebra scheme in the literature, we give a detailed treatment in Section 2. We point out that this concept is useful in other situations; in particular it allows to work out easily the deformation theory of smooth affine group schemes, as we show in Subsection A.3.
Let us say a word on the assumptions. The choice to work with differentially flat group schemes instead of simply smooth ones is not just motivated by the search for maximal generality or aesthetic reasons. It is also extremely useful because when working with an affine, smooth group scheme G, we use our results also for the group algebra O k [G] in the course of proving the main theorem; and the group algebra O k [G] is differentially flat and rigid, but usually infinite-dimensional and hence not smooth.
There are at least two advantages to work over generalized dual numbers k[I] rather than simply the usual ring k[ε] with ε 2 = 0. The first is that in order to prove that our equivalence of categories respects the O k -module stack structure, we have to introduce the ring k[I] with the two-dimensional k-module I = kε + kε . Indeed, this is needed to describe the sum of deformations and the Baer sum of extensions. The other advantage is that since arbitrary local Artin k-algebras are filtered by Artin k-algebras whose maximal ideal has square zero, our results may be useful in handling deformations along more general thickenings.

Further developments.
Our results have several desirable generalizations. Here are the two most natural directions. First, one may wish to relax the assumptions on the group schemes and consider nonaffine or non-smooth group schemes; second one may wish to consider more general thickenings than that given by the dual numbers. Let us explain how our personal work indicates a specific axis for research. In previous work of the authors with Ariane Mézard [MRT13], we studied models of the group schemes of roots of unity µ p n over p-adic rings. As a result, we raised a conjecture which says in essence that every such model can be equipped with a cohomological theory that generalizes the Kummer theory available on the generic fibre. In the process of trying to prove the conjecture, we encountered various character groups of smooth and finite unipotent group schemes over truncated discrete valuation rings. In order to compute these, it is therefore desirable to obtain a statement similar to Theorem A in this context. The present paper can be seen as the first part of this programme, carried out in the simplest case; we plan to realize the second part of the programme by using the derived Weil restriction or derived Greenberg functor in place of the usual Weil restriction.
1.1.5. Plan of the article. The present Section 1 ends with material of preliminary nature on the description of the O k -module stack structure of the categories Gr/k[I] and Ext(I)/k and on Weil restriction. In Section 2 we introduce group algebra schemes. In Section 3 we describe the functor h * : Gr/k[I] → Ext(I)/k, in Section 4 we construct a functor h + : Ext(I)/k → Gr/k[I], while in Section 5 we prove that these functors are quasi-inverse and we complete the proof of Theorem A. Finally, in Section 6 we derive the Dieudonné classification for smooth commutative unipotent group schemes over the dual numbers. In the Appendices, we review notions from differential calculus (tangent bundle, Lie algebra and exponentials) and module categories (Picard categories with scalar multiplication) in the level of generality needed in the paper.
Again, the verification of the axioms of an O k -module stack is tedious but not difficult.

Weil restriction generalities
We briefly give the main definitions and notations related to Weil restriction; we refer to [BLR90,§ 7.6] for more details. Let h : Spec(k ) → Spec(k) be a finite, locally free morphism of affine schemes. Let (Sch /k) be the category of k-schemes and (Fun /k) the category of functors (Sch /k) • → (Sets). The Yoneda functor embeds the former category into the latter. By sending a morphism of functors f : X → Spec(k ) to the morphism h • f : X → Spec(k) we obtain a functor h ! : (Fun /k ) → (Fun /k). Sometimes we will refer to h ! X as the k -functor X viewed as a k-functor and the notation h ! will be omitted. The pullback functor h * : The functors h ! and h * preserve the subcategories of schemes. The same is true for h * if h is radicial, a case which covers our needs (see [BLR90,§ 7.6] for refined representability results). We write for the unit and counit of the (h * , h * )-adjunction. If X is a separated k-scheme then α X : X → h * h * X is a closed immersion. If X is a k -group (resp. algebra) functor (resp. scheme), then also h * X is a k-group (resp. algebra) functor (resp. scheme). If moreover X → Spec(k ) is smooth of relative dimension n, then h * X → Spec(k) is smooth of relative dimension n[k : k] where [k : k] is the locally constant rank of h. Quite often, it is simpler to consider functors defined on the subcategory of affine schemes; the functors h ! , h * , h * are defined similarly in this context.

Group algebras of group schemes
Let G be an affine k-group scheme. In this subsection, we explain the construction of the group algebra , which is the analogue in the setting of algebraic geometry of the usual group algebra of abstract discrete groups. Note that for a finite constant group scheme, the set O k [G](k) of k-rational points of the group algebra and the usual group algebra k[G] are isomorphic, but for other groups they do not have much in common in general; this will be emphasized below.
There is a canonical S-morphism ν X : X −→ V (X/S) which is initial among all S-morphisms from X to a vector bundle. We call it the vector bundle envelope of X/S. If X is affine over S, the map ν X is a closed immersion because it is induced by the surjective morphism of algebras For later use, we give some complements on the case where k = k[I] is the ring of generalized dual numbers, for some finite free k-module I. We will identify I and its dual I ∨ = Hom k (I, k) with the coherent O Spec(k) -modules they define, thus we have the vector bundle V (I ∨ ). For each k-algebra R, we have R[I] = R ⊕ I ⊗ k R where IR I ⊗ k R has square 0. This decomposition functorial in R gives rise to a direct sum decomposition of O k -module schemes: It is natural to use the notation O k [V (I ∨ )] for the O k -algebra scheme on the right-hand side, however we will write more simply

O k -Algebra schemes.
Here we define a category (O k -Alg) of linear O k -algebra schemes and give a summary of elementary properties. For us an O k -algebra scheme is a k-scheme D endowed with two internal composition laws +, × : D × D → D called addition and multiplication possessing two neutral sections 0, 1 : Spec(k) → D, and an external law · : O k × D → D, such that for each k-algebra R the tuple (D(R), +, ×, 0, 1, ·) is an associative unitary R-algebra. In particular (D, +, 0) is a commutative group scheme, and (D(R), ×, 1) is a (possibly noncommutative) monoid. We say that D is a linear O k -algebra scheme if its underlying O k -module scheme is a vector bundle. In this case F can be recovered as the "dual bundle" sheaf F = Hom O k -Mod (D, O k ), the Zariski sheaf over S whose sections over an open U are the morphisms 9.4.9] between the lines). For an affine O k -algebra scheme in our sense, the comultiplication is a map ∆ : S(F ) → S(F )⊗S(F ) and the bilinearity of m implies that this map is induced from a map ∆ 0 : F → F ⊗F . Finally we point out two constructions on linear O k -algebra schemes. The first is the tensor product D ⊗ O k D , which as a functor is defined as The second construction is the group of units. We observe that for any linear O k -algebra scheme D, the subfunctor D × ⊂ D of (multiplicative) units is the preimage under the multiplication D × D → D of the unit section 1 : Spec(k) → D and is therefore representable by an affine scheme. This gives rise to the group of is the category of affine k-group schemes.

Remark.
We do not know if an O k -algebra scheme whose underlying scheme is affine over S is always of the form V (F ).

Group algebra: construction and examples
Let G = Spec(A) be an affine k-group scheme. We write (u, v) → u v or sometimes simply (u, v) → uv the multiplication of G. This operation extends to the vector bundle envelope V (G/k), as follows. Let ∆ : A → A ⊗ k A be the comultiplication. For each k-algebra R, we have V (G/k)(R) = Hom k-Mod (A, R). If u, v : A → R are morphisms of k-modules, we consider the composition u v := (u ⊗ v) • ∆: Here the map u ⊗ v :

Definition.
The group algebra of the k-group scheme G: is the vector bundle V (G/k) endowed with the product just defined. We write ν G : G → O k [G] for the closed immersion as in paragraph 2.1.1.
We check below that O k [G] is a linear O k -algebra scheme. Apart from G(R), there is another noteworthy subset inside O k [G](R), namely the set Der G (R) of k-module maps d : A → R which are u-derivations for some k-algebra map u : A → R (which need not be determined by d); a more accurate notation would be Der(O G , O k )(R) but we favour lightness. Here are the first basic properties coming out of the construction.

Proposition.
Let G be an affine k-group scheme. Let O k [G] and Der G be as described above.
(2) As a k-scheme, O k [G] is k-flat (resp. has k-projective function ring) iff G has the same property.
If moreover G is the finite constant k-group scheme defined by a finite abstract group Γ , then O k [G] is isomorphic to the algebra scheme defined by the abstract group algebra k [Γ ], that is to say functorially in R.

Example 2: the additive group. Let
be the R-algebra of divided power formal power series in one variable t, with t If k is a ring of characteristic p > 0, let H = α p be the kernel of Frobenius. The algebra O k [H](R) is identified with the R-subalgebra of R t generated by t, which is isomorphic to R[t]/(t p ) because t p = pt [p] = 0.
2.2.6. Example 3: the multiplicative group. Let G = G m = Spec(k[x, 1/x]). Let R Z be the product algebra, whose elements are sequences with componentwise addition and multiplication. We have an More generally, for any torus T with character group X(T ),

Properties: functoriality and adjointness
Here are some functoriality properties of the group algebra. (1) is functorial in G and faithful; (2) commutes with base change k /k; (4) is compatible with Weil restriction: if h : Spec(k ) → Spec(k) is a finite locally free morphism of schemes, (2) The isomorphism (3) Write G = Spec(A) and H = Spec(B). To a pair of k-module maps u : . This defines an isomorphism which is functorial in R. The result follows.
(4) If D is an O k -algebra scheme, the functorial R-algebra maps and h finite locally free, this is none other than the isomorphism Hom k- Finally we prove the adjointness property. We recall from paragraph 2.1.3 (see also Remark 2.1.4) that (O k -Alg) is the category of O k -algebra schemes whose underlying O k -module scheme is of the form V (F ) = Spec S(F ) for some quasi-coherent O Spec(k) -module F , and that (k-Gr) is the category of affine k-group schemes.

Theorem (Adjointness property of the group algebra
Proof. We describe a map in the other direction and we leave to the reader the proof that it is an inverse. Let f : G → D × be a morphism of k-group schemes. We will construct a map of functors f : We know from paragraph 2.1.3 that D = V (F) where F is a k-module, and that the comultiplication . We define f as follows: where i : F → S(F) is the inclusion as the degree 1 piece in the symmetric algebra. The map f is a map of modules, and we only have to check that it respects the multiplication. Let u, v : A → R be module homomorphisms. We have the following commutative diagram: With the notation as in Subsection 2.2, we compute: → D is a map of algebra schemes and this ends the construction.

Remark.
It follows from this result that a smooth vector bundle with action of G is the same as a (smooth) O k [G]-module scheme. Indeed, the endomorphism algebra of such a vector bundle is representable by a linear O k -algebra scheme D. For example, if G is an affine, finite type, differentially flat k-group scheme, then the adjoint action on Lie G makes it an O k [G]-module scheme.

Weil restriction
We keep the notations from the previous sections. In this section, we describe the Weil restriction E = h * G of a k[I]-group scheme G ∈ Gr/k[I] and show how it carries the structure of an object of the category of extensions Ext(I)/k. The necessary notions of differential calculus (tangent bundle, Lie algebra, exponential) are recalled in Appendix A.

Weil restriction of the group algebra
(It also embeds in the algebra O k [h * G ] which however is less interesting in that it does not reflect the Weil restriction structure.) Our aim in this subsection is to give a description of h * O k [I] [G ] suited to the computation of the adjunction map β G .
The starting point is the following definition and lemma. Let A be a k[I]-algebra and R a k-algebra.
for all i ∈ I and x ∈ A. Thenv is uniquely determined by v; in fact it is already determined by the identity v(ix) = iv(x) for any fixed i belonging to a basis of I as a k-module.

Definition. Let
A be a k[I]-algebra and R a k-algebra. We say that a k-linear map v : We denote by Homc k (A, I ⊗ k R) the R-module of I-compatible maps and Homc k (A, I ⊗ k R) → Hom k (A, R), v →v the R-module map that sends v to the unique mapv with the properties above.
Note that since In other words, the map v →v factors through Hom k (A/IA, R).

Lemma. Let A be a k[I]-algebra and R a k-algebra.
( Taking into account that I 2 = 0, this means that v is I-compatible and u =v. (2) The condition f (1) = 1 means thatv(1) = 1, that is v(i) = i for all i ∈ I, and v(1) = 0. The condition of multiplicativity of f means thatv is multiplicative and v is av-derivation, i.e. v(xy) =v(x)v(y) +v(y)v(x).
In the presence of the derivation property, the multiplicativity ofv is automatic (computing v(ixy) in two different ways) as well as the condition v(1) = 0 (setting (1) Let R be a k-algebra and let v, w : (2) For v, w as before let:

Then (O c (G ), +, ) is an associative unitary O k -algebra with multiplicative unit d, and the map
Proof.
(1) The k-linear morphismv ⊗ k w + v ⊗ kw takes the same value iv(a)w(b) on the tensors ia ⊗ b and a ⊗ ib for all i ∈ I, a, b ∈ A. Therefore it vanishes on tensors of the form (a ⊗ b)(i ⊗ 1 − 1 ⊗ i). Since these tensors generate the kernel of the ring map (2) According to (1) the definition of v w makes sense. For the rest of the statement, it is enough to On the other hand, we have: The maps in the brackets are equal, whence Then we have an isomorphism:  (2) Explicit description. We have embeddings of monoids

Kernel of the adjunction
, and the isomorphism is given by (x, g) → x. Proof.
(1) If ϕ : G → G is a morphism of pointed schemes, then by functoriality of β the morphism h * h * ϕ takes the kernel of β G into the kernel of β G . If moreover ϕ is a map of group schemes then the restriction of h * h * ϕ to L(G ) also.
(2) In the rest of the proof we use the possibility to compute in the group algebra (O c (G ), +, ), see (3) -first claim. The pullback i * is the restriction to the category of those k[I]-algebras R such that Conversely, any e R -derivation δ : A → I ⊗ k R vanishing on IA gives rise to a k-linear map v : A → I ⊗ k R defined by v := d R + δ and satisfying the properties required to be a point of (i * L(G ))(R). Finally let δ 1 , δ 2 ∈ Lie(G k , I)(R). Since δ 1 , δ 2 vanish on IA, we have d * R + δ * 1 = d * R + δ * 2 = e R and then: All three morphisms e R ⊗ d R + d R ⊗ e R , e R ⊗ δ 2 and δ 1 ⊗ e R factor through A ⊗ k[I] A, so the precomposition with ∆ is distributive for them. Since also d R is the neutral element for the law and e R is the neutral element for the law , we obtain: This shows that the isomorphism We see that u is acting on δ by conjugation in the group algebra. This is the adjoint action, as explained in Proposition A.3.1.
(4) When G = h * G, the adjunction map is the infinitesimal translation as in Proposition A.2.2(3). The R-points of L(G ) are the pairs (x, g) such that exp(x)g = 1 in G(R). This amounts to g = exp(−x) which proves (4).

Extension structure of the Weil restriction
Let G be a k[I]-group scheme. The notion of rigidification for G and the property that G be rigid are defined in 1.1.2. Here are some remarks.
(2) By the infinitesimal lifting criterion, all smooth affine k[I]-group schemes are rigid. By Cartier duality, k[I]-group schemes of multiplicative type are rigid.
(3) If α : G → G is a morphism between rigid k[I]-group schemes, it is not always possible to choose rigidifications for G and G that are compatible in the sense that σ • h * α = α • σ . For instance let I = kε and let α : G a → G a be the morphism defined by α(x) = εx. Then G and G are rigid but since h * α = 0, there do not exist compatible rigidifications.

Lemma. Let G be a k[I]-group scheme such that the restriction homomorphism
Conversely let s be a section of π and σ := β G • h * s.
In particular σ is an affine morphism. Since moreover h * G k is flat, we conclude that σ is an isomorphism, hence a rigidification.

Lemma.
Let G be an affine, differentially flat and rigid k[I]-group scheme. Then β : h * h * G → G is faithfully flat and we have an exact sequence: It follows also that π : h * G → G k is faithfully flat and by the "critère de platitude par fibres" in the nilpotent case ([SP, Tag 06A5]) we deduce that the morphism β is faithfully flat. Finally if G is of finite type over k [I], then the special fibre of L(G ) is the smooth vector group Lie(G k ), hence L(G ) is smooth and so is β.

Example.
Here is an example where the result above fails, for a non-rigid group. Assume k is a field of characteristic p > 0. Let I = kε be free of rank 1. Let G be the kernel of the endomorphism It follows from point (2) in Proposition 3.2.1 that when we restrict to the closed fibre, we obtain an exact sequence: where the G k -action on Lie(G k ) induced by the extension is the adjoint representation. The same reference proves that this extension is functorial in G . More precisely, if u : G → G is a morphism of affine, differentially flat, rigid k[I]-group schemes, then we obtain a morphism between the extensions E = h * G and E = h * G as follows: where ϕ = h * u and ψ = u k = i * u, the restriction of u along i : Spec(k) → Spec(k[I]).
We draw a corollary that will be useful in Section 5.

Corollary. Let Y be an affine, flat, rigid k[I]-scheme and
) and π is the projection onto the first factor, i.e. the map given by reduction modulo I.

Weil extension
In this section, we construct a functor h + called Weil extension which is a quasi-inverse to the functor h * of Weil restriction described in the previous section. The idea behind the construction is that one can recover a k[I]-group scheme G from the extension E = h * G by looking at the target of the adjunction β G : h * E = h * h * G → G . In turn, in order to reconstruct the faithfully flat morphism β G it is enough to know its kernel K. In the case where G is a constant group h * G, which in other words is the case where E is a tangent bundle T(G, I), Proposition 3.2.1(4) hints the correct expression K = {(x, g) ∈ h * E; g = exp(−x)}. The definition of K for general extensions 1 → Lie(G, I) → E → G → 1 where G is an affine, differentially flat k-group scheme, builds on this intuition.

Hochschild extensions
The construction of an extension from a 2-cocycle is well-known; we recall it to set up the notations. Recall from [DG70, chap. II, § 3, no 2] that if G is a k-group functor and M is a k-G-module functor, then a Hochschild extension or simply H-extension of G by M is an exact sequence of group functors such that π has a section (which is not required to be a morphism of groups). From a given section s : G → E, we can produce a unique morphism c : G × G → M such that i(c(g, g )) := s(g)s(g )s(gg ) −1 . This is a 2-cocycle, i.e. it satisfies the identity c(g, g ) + c(gg , g ) = g · c(g , g ) + c(g, g g ).
Note that we may always replace s by the section G → E, g → s(1) −1 s(g) to obtain a section such that s(1) = 1. When this is the case, we have c(g, 1) = c(1, g ) = 0 for all g, g and we say that c is normalized. Conversely, starting from a cocycle c, the functor E c = M × G with multiplication defined by is an H-extension. The map s : G → E c , g → (0, g) is a possible choice of section for π. It follows from the previous comments that we may always change the cocycle into a normalized cocycle.

Kernel of the adjunction, reprise
In this subsection, we prepare the construction of the kernel of the adjunction map β h + E of the (yet to be produced) Weil extension h + E. The end result is in Proposition 4.3.1 of the next subsection. Note that in spite of the similarity of titles, the viewpoint is different from that of Subsection 3.2.
Let G be an affine k-group scheme, and Lie(G, I) its Lie algebra relative to I, viewed as an affine k-group scheme with the adjoint action of G. To any 2-cocycle c : G × G −→ Lie(G, I) we attach as before an H-extension E c = Lie(G, I) × G with multiplication: Our group E c has a structure of H-extension: The following result is the heart of the construction of the Weil extension functor h + . We point out that among the groups K λ (E c ) introduced here, it is especially K −1 (E c ) that will be relevant in the sequel, as Proposition 3.2.1(4) shows. However, we include the whole family K λ (E c ) since it comes with no extra cost and brings interesting insight, in the sense that it ultimately provides an explicit linear path in the (1) Let G be an affine k-group scheme and let E c be the H-extension constructed out of a normalized 2-cocycle c : G × G −→ Lie(G, I). Let K λ (E c ) ⊂ h * E c be the subfunctor defined by: (2) Let G, G be affine k-group schemes and E c , E c be the H-extensions constructed out of some chosen normalized 2-cocycles c, c . Let f : E c → E c be a morphism of extensions: When the extension E c is clear from context, we write K λ instead of K λ (E c ). We will prove the proposition after a few preliminaries. First of all, for the convenience of the reader, we recall the description of morphisms of extensions, in the abstract group setting for simplicity.

Lemma.
Let α : G → G be a morphism of groups and δ : L → L be a morphism from a G-module to a G -module which is α-equivariant. Let E ∈ Ext(G, L) and E ∈ Ext(G , L ) be two extensions.
(1) There exists a morphism of extensions f : E → E , i.e. a diagram in H 2 (G, L ), and if this condition holds then the set of morphisms is a principal homogeneous space under the set of 1-cocycles Z 1 (G, L ). More precisely, assume that we describe E with a normalized cocycle c : G × G → L so that E L × G with multiplication (x, g) · (x , g ) = (x + g · x + c(g, g ), gg ), and we describe E similarly with a normalized cocycle c . Then all morphisms f : E → E are of the form f (x, g) = (δ(x) + ϕ(g), α(g)) for a unique 1-cochain ϕ : (2) If E, E are two extensions of G by L, then the set of morphisms of extensions E → E is a principal homogeneous space under the group Z 1 (G, L), more precisely all morphisms are of the form f (x, g) = (x + ϕ(g), g) for a unique ϕ ∈ Z 1 (G, L). All of them are isomorphisms.
(3) Assume that the extension is trivial, so that [E] = 0 ∈ H 2 (G, L). Then all group sections G → E of the extension are of the form s(x, g) = (ϕ(g), g) for a unique ϕ ∈ Z 1 (G, L) such that ∂ϕ = c.
Setting x 1 = x, x 2 = 0, g 1 = 1, g 2 = g, and ϕ(g) := u(0, g) we find u(x, g) = δ(x) + ϕ(g) for all x, g. The above identity implies ϕ(g 1 g 2 ) − ϕ(g 1 ) − α(g 1 ) · ϕ(g 2 ) = c (αg 1 , αg 2 ) − δ(c (g 1 , g 2 )). This means that ∂ϕ = c • α − δ • c as claimed in (1). Considering the particular case of morphisms we get (2), and considering the case of morphisms We come back to the extension E c . The lemma tells us that the group Aut ext (E c ) of automorphisms of E c as an extension is isomorphic to the group of 1-cocycles Z 1 (G, Lie(G, I)). Item (2) of Proposition 4.2.1 says in particular that K λ (E c ) is stable under these particular automorphisms. Now we record a few technical properties concerning the exponential and the cocycles. For simplicity we write exp instead of exp G .
The same statements hold with c replaced by λc, for each λ ∈ k.
(2) Write g = exp(x). Since Lie(G, I) = Lie(G) ⊗ V (I ∨ ) we can write x as a sum of tensors y ⊗ i. Working inductively on the number of tensors in the sum, we can assume that x = y ⊗ i. We prove successively that each of the first four terms equals exp(c (g , g )).
c. The cocycle identity with g and g exchanged reads c(g , g) + c(g g, g ) = Ad(g )c(g, g ) + c(g , gg ). We deduce exp(c(g g, g )) = exp(c(g , gg )). We conclude with b.
d. Again this follows from the fact that g g(g ) −1 is an exponential.
This proves that the product (x, g) · (x , g ) is a point of K. Using the same arguments we prove that the inverse (x, g) −1 = (− Ad(g −1 )x − c(g −1 , g), g −1 ) is a point of K. Hence K is a subgroup scheme.
Second, let us prove that K is stable by inner automorphisms. Let (x, g) and (x , g ) be R-valued points of h * E c and K respectively. We must prove that (x , g ) := (x, g) · (x , g ) · (x, g) −1 lies in K. Writing x as a sum of tensors x s = y s ⊗ i s and setting g s = exp(λx s ), we have (x s , g s ) ∈ K(R) and since K is a subgroup scheme, it is enough to prove that (x, g) · (x s , g s ) · (x, g) −1 lies in K. In other words, we may and do assume in the sequel that x = y ⊗ i. We first consider (x 1 , g 1 ) := (x , g ) · (x, g) −1 . Using the fact that g = exp(λx ) and Proposition A.2.2(4), we find where b ∈ I ·Lie(G, I)(R) is a certain bracket, and hence: Now (x , g ) = (x, g) · (x 1 , g 1 ) = (x + Ad(g)x 1 + c(g, g 1 ), gg 1 ) and our task is to check that exp λx + λ Ad(g)x 1 + λc(g, g 1 ) = gg 1 .
When f is an isomorphism, applying the statement to f −1 , we find (h * f )(K) = K .

Weil extension functor
Now let G be an affine and differentially flat k-group scheme. Thus G as well as the adjoint representation Lie(G, I) are k-flat. We consider an arbitrary extension: Then E → G is an fpqc torsor under Lie(G, I). It has a cohomology class in H 1 (G, Lie(G, I)) which vanishes, being quasi-coherent cohomology of an affine scheme. It follows that π has a section s : G → E, and the extension becomes an H-extension. We may and do replace s by s(1) −1 · s in order to ensure that s(1) = 1. (1) The map τ s : E c → E, (x, g) → i(x)s(g) is an isomorphism of extensions.
(2) The closed normal subgroup scheme K λ (E) := (h * τ s )(K λ (E c )) ⊂ h * E does not depend on the choice of s.
(3) For all morphisms f : If the extension E is clear from context, we write K λ instead of K λ (E). Note that if E is the trivial extension and s = α, the map τ α is the map G defined in paragraph A.1.

Proof. (1) follows from the constructions of c and E c .
For the proof of (2) and (3) we will rely on the following basic remark. Let f : E → E be a morphism in Ext(I)/k. Let τ s : E c → E and τ s : E c → E be the isomorphisms associated to choices of sections s, s preserving 1 and corresponding normalized cocycles c, c . Let K λ,s (E) := (h * τ s )(K λ (E c )) ⊂ h * E and similarly K λ,s (E ) := (h * τ s )(K λ (E c )) ⊂ h * E . We have a morphism of extensions: According to Proposition 4.2.1(2) we have (h * ρ)(K λ (E c )) ⊂ K λ (E c ). It follows that: When f is an isomorphism, applying the statement to f −1 gives equality.
(2) Applying the basic remark to E = E and f = id : E → E proves that K λ,s (E) = K λ,s (E), that is, the subgroup K λ,s (E) does not depend on the choice of s. Since τ s is an isomorphism of groups, the fact that K λ (E) is a closed normal subgroup scheme follows from Proposition 4.2.1(1).
(3) Applying the basic remark to a general f gives the statement.

Definition.
We call Weil extension the quotient h + E := h * E/K −1 .

Lemma. Weil extension is a functor Ext(I)/k → Gr/k[I].
Proof. The k[I]-group scheme G := h * E/K −1 is affine and flat. Let s : G → E be a section of E → G such that s(1) = 1. By pullback, this induces a morphism h * G → h * E → G which is the identity on the special fibre, hence an isomorphism, hence a rigidification. This proves that the functor of the statement is well-defined on objets. Proposition 4.3.1 proves that the functor is well-defined on morphisms.

The equivalence of categories
This section is devoted to the proof of Theorem A, which we recall below for ease of reading. The plan is as follows. In Subsection 5.1 we prove a preliminary result used in the proof of (1). In Subsection 5.2 we prove (1), (2), (4). Finally in Subsection 5.3 we prove (3).

Theorem. (1) The Weil restriction/extension functors provide quasi-inverse equivalences:
These equivalences commute with base changes Spec(k ) → Spec(k). ( In particular, h * is an exact equivalence between the subcategories of smooth objects endowed with their natural exact structure. (3) The equivalence h * is a morphism of O k -module stacks fibred over Gr/k, i.e. it transforms the addition and scalar multiplication of deformations of a fixed G ∈ Gr/k into the Baer sum and scalar multiplication of extensions of G by Lie(G, I).
(4) Let P be one of the properties: of finite type, smooth, connected, unipotent, split unipotent, solvable, commutative. Then G ∈ Gr/k[I] has the property P if and only if the k-group scheme E = h * G has P .

Equivariance of rigidifications under Lie algebra translation
Let G be an affine, differentially flat, rigid k[I]-group scheme. Let σ : h * G k ∼ − − → G be a rigidification such that σ (1) = 1. We consider the morphism of k-schemes: This is not a morphism of group schemes, because source and target are not isomorphic groups in general. However, it satisfies an important equivariance property. To state it, note that source and target are extensions of G k by Lie(G k , I); in particular both carry an action of Lie(G k , I) by left translation.

Proposition. With notation as above, the morphism of k-schemes
when the base is clear from context. Consider the extension of σ to the group algebras: Note that by compatibility of O[−] with base change and Weil restriction (see Proposition 2.3.1, (2)-(4)), we We obtain a commutative diagram: To prove this, we introduce another copy J = I of our square-zero ideal as follows: = (h * σ )(y) + x 1 y by the Claim above, = (h * σ )(y) + x 2 y by Corollary 3.3.5 since 1 = 2 modulo I, = (h * σ )(y) + x 2 (h * σ )(y) by Corollary 3.3.5 since σ = id modulo I, This proves that h * σ is Lie(G k , I)-equivariant.

Proof of the main theorem
We fix a rigidification σ : h * G k ∼ − − → G such that σ (1) = 1. We know from Proposition 5.1.1 that the map If we use the letter γ to denote the inclusions of Lie(G k , I) into the relevant extensions, this can be written: Restricting to y in the image of α = α G k : G k → h * h * G k , so τ α = G k , we obtain: Using functoriality of β and the fact that σ (1) = 1, we build a commutative diagram: Here the horizontal maps are morphisms of groups and the vertical maps are not morphisms of groups (not even the leftmost map id : h * F → h * E c ). Note also that F is E 0 = T(G k , I), that is, the extension E c with the zero cocycle c = 0. Now we consider K −1 (E) as defined in Proposition 4.3.1. According to Proposition 3.2.1(4), the group On one hand the identity takes K −1 (E 0 ) to K −1 (E c ), and on the other hand the map h * σ takes ker(β h * G k ) onto ker(β G ) since it takes 1 to 1. By commutativity of the left-hand square, we find K −1 (E) = ker(β G ) and thefore β G induces an isomorphism h + E = h * E/K −1 (E) G which is visibly functorial.
Secondly, we prove that h * • h + is isomorphic to the identity. Let 1 → Lie(G, I) → E → G → 1 be an extension. We fix a section s : G → E such that s(1) = 1 and we let c be the normalized cocycle defined by s. Let K −1 = K −1 (E) ⊂ h * E be the closed normal subgroup defined in Proposition 4.3.1, and let G := h + E = h * E/K −1 with quotient map π : h * E → G . Define σ = π•h * s : h * G → G . Since i * K −1 = Lie(G, I) as a subgroup of E, we see that i * G G and i * σ is the identity of G. Since G is k-flat, it follows that σ is an isomorphism. From the construction of K −1 , we see that after we compose with the isomorphisms the flat surjection π : h * E → G is identified with the counit of the adjunction: We apply h * and obtain the commutative diagram: Since the top row is the identity, we see that the bottom row is an isomorphism, i.e. E ∼ − − → h * G . Again, it is clear that this isomorphism is functorial.
Finally, we consider the commutation with base changes. For Weil restriction, this is a standard fact. For Weil extension, this follows from base change commutation for pullbacks and for quotients by flat subgroups.
This proves the claim. 5.0.1(4). If G is of finite type, or smooth, or connected, or unipotent, or split unipotent, or solvable, then G = i * G as well as Lie(G, I) have the same property. It follows that E = h * G has the property. Moreover, if G is commutative then E also. Conversely, if E is of finite type, or smooth, or connected, or unipotent, or split unipotent, or solvable, or commutative, then h * E has the same property. Therefore the quotient h + E := h * E/K −1 has the same property.

Proof of the main theorem: isomorphism of O k -module stacks
In this paragraph, we prove 5.0.1(3), i.e. that the Weil restriction functor h * : Gr/k[I] → Ext(I)/k exchanges the addition and the scalar multiplication on both sides. Before we start, we point out that these properties will imply that the image of a trivial deformation group scheme G = h * G under Weil restriction is the tangent bundle (i.e. trivial) extension T(G, I), a fact which can be shown directly using Proposition 3. 2.1(4).
We work in the fibre category over a fixed G ∈ Gr/k and we set L := Lie(G, I). Let G 1 , G 2 ∈ Gr/k[I] with identifications i * G 1 G i * G 2 . For clarity, we introduce three copies I 1 = I 2 = I of the same finite free k-module. For c = 1, 2 we have obvious maps:

Lemma.
The morphism ξ is an isomorphism and it induces an isomorphism of extensions on the bottom row of the following commutative square: There is a morphism of algebra schemes ξ : contructed in the same way as ξ. It order to describe ξ we can express the Weil restrictions in terms of I-compatible maps as in Lemma 3.1.2. For a k-algebra R, we have: is v modulo I 2 (resp. I 1 ). This is a bijection whose inverse sends a pair a 2 ) → v 1 (a 1 ) + v 2 (a 2 ). The morphism ξ is the bijection obtained by restriction of ξ to the subsets of algebra maps as in Lemma 3.1.2(2). Namely, an algebra map is of the form In order to describe ω note that (h * (G 1 + G 2 ))( Thus ω(R) is surjective because R[I 1 ⊕ I 2 ] −→ R[I] has R-algebra sections, i.e. ω is a surjection of functors. Its kernel is the set of maps f =v + v 1 + v 2 such that v 1 + v 2 = d 1 + d 2 : the counits of the Hopf algebras. After translation by the derivations as indicated by Proposition 3.2.1(3), on the side of extensions the kernel is ker(+ : L × L → L), giving rise to a quotient isomorphic to the Baer sum extension h 1, * G 1 + h 2, * G 2 .
It remains to prove that h * : Gr/k[I] → Ext(I)/k exchanges the scalar multiplication on both sides. Let G ∈ Gr/k[I] with an identification i * G G. We will reduce to a situation similar as that of Lemma 5.3.1 thanks to the following trick. It follows that: Set G 1 = G and G 2 = h * G. Recall that L = Lie(G, I). The Weil restrictions are E := E 1 = h * G and the trivial extension E 2 = h * h * G = L G. As in Lemma 5.3.1, there are morphisms ξ : * G → E 1 × G E 2 = E × L.

Lemma. The morphism ξ is an isomorphism which induces an isomorphism of extensions on the bottom
row of the following commutative square: Proof. The proof is the same as that of Lemma 5.3.1 except that in the final step we use the map I ⊕ I → I, i 1 ⊕ i 2 → λi 1 + i 2 . Again this morphism is surjective and on the side of extensions, the kernel corresponds to the kernel of L × L → L, (v 1 , v 2 ) → λv 1 + v 2 . The quotient of E × L by this kernel is exactly the extension λE, the pushout of the diagram: L This finishes the proof.

Dieudonné theory for unipotent groups over the dual numbers
In this section, as an application of Theorem A, we give a classification of smooth, unipotent group schemes over the dual numbers of a perfect field k, in terms of extensions of Dieudonné modules. So throughout the section, the ring k is a perfect field of characteristic p > 0.

Reminder on Dieudonné theory
We denote by W the Witt ring scheme over k and F, V its Frobenius and Verschiebung endomorphisms. For all n 1, we write W n := W / V n W the ring scheme of Witt vectors of length n. We use the same notation also for these operators over the R-points, with R a k-algebra. We also define V : W n −→ W n+1 as the morphism induced on W n by the composition where π n+1 is the natural projection.
The Dieudonné ring D is the W (k)-algebra generated by two variables F and V with the relations: for varying x ∈ W (k). A Dieudonné module is a left D-module. A Dieudonné module M is called erasable if for any m ∈ M there exists a positive integer n such that V n m = 0.
Let R be a k-algebra. Then, for any n 1, we make W n (R) a left D-module with the rules: for all u ∈ W n (R) and x ∈ W (k). The twist in the latter definition is designed to make V : W n (R) → W n+1 (R) a morphism of D-modules, see Demazure and Gabriel [DG70, chap. V, § 1, no 3.3]. All of this is functorial in R and gives W n a structure of D-module scheme. In particular, End k (W n ) is a D-module. According to [DG70, chap. V, § 1, no 3.4] the morphism D → End k (W n ) induces an isomorphism of D-modules: If U is a commutative, unipotent k-group scheme, the set Hom k (U , W n ) is a Dieudonné module with its structure given by postcomposition, i.e. for any f : U → W n : We define the Dieudonné module of U as: where the transition maps of the inductive system are induced by V : W n → W n+1 . Since Hom k (U , W n ) is killed by V n and M(U ) is a union of these subgroups, we see that M(U ) is erasable. If M is a Dieudonné module, we define its Frobenius twist M (p) as the module with underlying group M (p) = M and D-module structure given by:

Dieudonné theory over the dual numbers
Before stating our Dieudonné classification, we need to define the notions of Lie algebra and smoothness of Dieudonné modules. We let D-Mod e ⊂ D-Mod be the subcategory of erasable D-modules. We have an exact sequence: We deduce isomorphisms Lie U ∼ − − → Lie U and M = M/F M. In the sequel set L U := (Lie U )(k), a k-vector space. Since U is a finite commutative group scheme, according to Fontaine [Fo77,chap. III,4.2] there is a canonical isomorphism: We deduce a composed isomorphism η U as follows: By tensoring η U with k[F ], we find (1) L is right exact and commutes with filtering inductive limits; (2) L (D/DV n ) = k[F ] n for all n 1; (3) L : • V to the endomorphism (a 0 , a 1 , . . . , a n−1 ) → (0, a 0 , a 1 , . . . , a n−2 ); Proof. Uniqueness. The key is the fact that D/DV n is a projective generator of the full subcategory C n := (D-Mod e ) V n =0 of objects killed by V n . More precisely, since any erasable D-module is a filtering union of its submodules of finite type, property (1) implies that L is determined by its restriction to the subcategory of modules of finite type. Any finite type module M is killed by V n for some n 1. Since D/DV n is noetherian, for any M ∈ C n there exist r, s and an exact sequence: and for any morphism f : M → M in C n there is a commutative diagram: We come to the notion of smoothness. It is known that a k-group scheme of finite type U is smooth if and only if its relative Frobenius F U /k : U → U (p) is an epimorphism of k-group schemes. This motivates the following definition. Proof. It suffices to put together Theorem A and Theorem 6.1.1. In little more detail, let U be a smooth, commutative, unipotent group scheme over the ring of dual numbers k[I], and let U = U k be its special fibre. By Theorem A this datum is equivalent to an extension This is an exact sequence of functors, hence also an exact sequence of group schemes (i.e. of fppf sheaves). Let m be the multiplication of T(G, I). The splitting gives rise to an isomorphism of k-schemes: that is, any point of T(G, I) may be written uniquely as a product γ G (x)·α G (g) for some points x ∈ Lie(G, I) and g ∈ G. We will sometimes write briefly (x, g) = γ G (x) · α G (g) to denote this point of T(G, I). The conjugation action of G on Lie(G, I) related to the extension structure is given by the adjoint action, thus the group structure of T(G, I) can be described by: The dependence of T(G, I) and Lie(G, I) on I can be described further. If I = kε so that k[I] = k[ε] with ε 2 = 0, we write simply TG = T(G, kε) and Lie G = Lie(G, kε) and we call them the tangent bundle and the Lie algebra of G. Let u : G → Spec(k) be the structure map. For general I, the isomorphisms Similarly, the isomorphisms With G acting trivially on V (I ∨ ), this isomorphism is G-equivariant. For simplicity of notation, we will write d as an equality: Lie(G, I ⊗ k J) = Lie(Lie(G, I), J). This will not cause any ambiguity. If I = J = kε this means simply that Lie G = Lie(Lie G).

A.2. Exponential and infinitesimal translation
Demazure and Gabriel in [DG70] use an exponential notation which is flexible enough to coincide in some places with the morphism exp G as we define it below (loc. cit. chap. II, § 4, 3.7) and in other places with the morphism γ G (loc. cit. chap. II, § 4, 4.2). The drawback of flexibility is a little loss of precision. We introduce the exponential in a somehow more formal way, as an actual morphism between functors.
A.2.1. Definition. The exponential of a k-group scheme G is the composition: When I is clear from context, and also when I = kε, we write exp G instead of exp G,I . The following proposition collects some elementary properties of the exponential.
A.2.2. Proposition. The exponential exp G,I of a k-group scheme G has the following properties.
(1. Functoriality) For all morphisms of group functors f : G → G we have a commutative square: (2. Equivariance) The map exp G,I is h * G-equivariant for the adjoint action on h * Lie(G, I) and the conjugation action of h * G on itself.   have equal kernels, thus I ·h * Lie(G, I) ⊂ ker(exp G,I ). In particular, in case I = kε, the kernel of the morphism exp G : h * Lie G → h * G is equal to the kernel of the multiplication-by-ε map in h * Lie G.
We finish this subsection with a corollary of the computation of the exponential of a Lie algebra.
The first two rows are split exact sequences of group schemes. In the last row we have written the points of  Proof.
(1) By Proposition 2.3.1(4), we have an isomorphism of O k -algebra schemes: Under this identification, the map T This is the central vertical map in the pictured diagram. The rest is clear.
(2) Using the inclusions of multiplicative monoids α G : G → TG and T ν G : TG → O k [G][ε], we can view the conjugation action by G inside the tangent bundle or inside the tangent group algebra, as we wish. The result follows.
(3) In order to compute ad we differentiate and hence work in O k [G][ε, ε ]. That is, the Lie algebra embedded by y → 1 + εy is acted upon by the Lie algebra embedded by x → 1 + ε x, via conjugation in the ambient O k [G][ε, ε ]. With these notations, the identification End(Lie G) ∼ − − → Lie(GL(Lie G)) goes by f → 1 + ε f . All in all, the outcome is that ad(x) is determined by the condition that for all y we have: (1 + ε x)(1 + εy)(1 − ε x) = 1 + ε (id +ε ad(x))(y) .
Since the left-hand side is equal to 1 + εy + εε (xy − yx), this proves our claim.

A.3.3. Proposition.
Let k be a base ring and let G be an affine k-group scheme.
(1) The set of k[I]-group scheme structures on the scheme h * G that lift the k-group scheme structure of G is in bijection with the set of 2-cocycles c : G × G → Lie G.
(2) The set of isomorphism classes of rigid deformations of G over k[I] is in bijection with H 2 (G, Lie(G, I)), the second group cohomology of G with coefficients in the adjoint representation Lie(G, I) Lie G ⊗ V (I ∨ ).

Proof.
(1) We want to deform the multiplication m : G × G −→ G, (u, v) → uv into a multiplication m : h * G × h * G → h * G which by adjunction we can view as a map: B.2. Definition. Let (P 1 , +) and (P 2 , +) be categories endowed with bifunctors. Let F : P 1 → P 2 be a functor and ϕ F,x,y : F(x + y) ∼ − − → F(x) + F(y) an isomorphism of functors.

B.3. Definition.
A Picard category is a quadruple (P , +, a, c) composed of a nonempty groupoid P , a bifunctor + : P × P → P with compatible associativity and commutativity constraints a and c, such that for each x ∈ P the functor P → P , y → x + y is an equivalence.
Any Picard category P has a neutral element 0 which is unique up to a unique isomorphism ([SGA4.3], Exp. XVIII, 1.4.4). Moreover, for each x, y ∈ P the set of morphisms Hom(x, y) is either empty or a torsor under the group G := Aut(0). More precisely, the functors +x : P → P and x+ : P → P induce the same bijection G → Aut(x), ϕ → ϕ + id x . Viewing this bijection as an identification, the set Hom(x, y) with its right Aut(x)-action and left Aut(y)-action becomes a pseudo-G-bitorsor, i.e. it is either empty or a G-bitorsor. B.4. Definition. Let P 1 , P 2 be Picard categories.
(1) An additive functor is a pair (F, ϕ F ) where F : P 1 → P 2 is a functor and ϕ F,x,y : F(x + y) ∼ − − → F(x) + F(y) is an isomorphism of functors that is compatible with associativity and commutativity constraints.
(2) Let F, G : P 1 → P 2 be additive functors. A morphism of additive functors is a morphism of functors u : F → G such that the following diagram is commutative: We emphasize that since a Picard category is a groupoid (that is, all its morphisms are isomorphisms), all morphisms of additive functors u : F → G are isomorphisms.
The category of additive functors Hom(P 1 , P 2 ) is itself a Picard category ([SGA4.3], Exp. XVIII, 1.4.7). Additive functors can be composed and the identity functors behave as neutral elements. In the particular case where P 1 = P 2 = P , along with its addition law, the Picard category End(P ) = Hom(P , P ) enjoys an internal multiplication given by composition. Note that in this case multiplication is strictly associative, because so is composition of functors in categories.
In fact End(P ) is a ring category, but in order to introduce module groupoids, we do not actually need to define what is such a thing. B.5. Definition. Let Λ be a commutative ring. A Λ-module groupoid is a Picard category P endowed with a functor F = (F, ϕ F , ψ F ) : Λ → End(P ) called scalar multiplication such that: (1) (F, ϕ F ) is an additive functor.
(3) F is compatible with the distributivity of multiplication over addition: commutes.
B.6. Definition. Let S be a site. Let Λ be a sheaf of commutative rings on S. A Λ-module stack (in groupoids) over S is a stack in groupoids P over S endowed with (1) a functor + : P × P → P , (2) isomorphisms of functors a x,y,z : (x + y) + z ∼ − − → x + (y + z) and c x,y : x + y ∼ − − → y + x, (3) a functor F = (F, ϕ F , ψ F ) : Λ → End(P ), such that for each U ∈ S the fibre category P (U ) is a Λ(U )-module groupoid.
There is an obvious corresponding relative notion of Λ-module stack (in groupoids) over a given S-stack Q, namely, it is a morphism of stacks P → Q that makes P a stack fibred in groupoids over Q, with an addition functor + : P × Q P → P etc. It is the relative notion that is useful in the paper.