# Smooth affine group schemes over the dual numbers

Matthieu Romagny and Dajano Tossici

###### Abstract

In this article we provide an equivalence between the category of smooth
affine group schemes over the ring of generalized dual numbers $k[I]$, and the
category of extensions of the form $1\to\operatorname{Lie}(G,I)\to E\to G\to
1$ where $G$ is a smooth affine 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
$\mathbb{O}_{k}$-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 Dieudonné classification for smooth, commutative, unipotent group
schemes over $k[I]$ when $k$ is a perfect field.

  *     * Keywords. Group scheme, deformation, dual numbers, adjoint representation, Weil restriction, Dieudonné classification of unipotent groups

2010 Mathematics Subject Classification. 14L15 (primary); 14D15, 14G17
(secondary)

[Français]

Titre. Schémas en groupes affines et lisses sur les nombres duaux Résumé. Nous
construisons une équivalence entre la catégorie des schémas en groupes affines
et lisses sur l’anneau des nombres duaux généralisés $k[I]$, et la catégorie
des extensions de la forme $1\to\operatorname{Lie}(G,I)\to E\to G\to 1$ où $G$
est un schéma en groupes affine, lisse sur $k$. Ici $k$ est un anneau
commutatif arbitraire et $k[I]=k\oplus I$ avec $I^{2}=0$. L’équivalence est
donnée par la restriction de Weil, et nous construisons un foncteur quasi-
inverse explicite que nous appelons extension de Weil. Ces foncteurs sont
compatibles avec les structures exactes et avec les structures de champs en
$\mathbb{O}_{k}$-modules des deux catégories. Nos constructions s’appuient sur
le schéma en algèbres de groupe d’un schéma en groupes affines, que nous
introduisons et dont nous donnons les propriétés principales. En application,
nous donnons une classification de Dieudonné pour les schémas en groupes
commutatifs, lisses, unipotents sur $k[I]$ lorsque $k$ est un corps parfait.

###### Contents

  1. 1\. Introduction
    1. 1.1. Motivation, results, plan of the article
    2. 1.2. The $\mathbb{O}_{k}$-module stack structure of $\operatorname{Gr}\\!/k[I]$ and $\operatorname{Ext}(I)/k$
    3. 1.3. Weil restriction generalities
  2. 2\. Group algebras of group schemes
    1. 2.1. Linear algebra schemes
    2. 2.2. Group algebra: construction and examples
    3. 2.3. Properties: functoriality and adjointness
  3. 3\. Weil restriction
    1. 3.1. Weil restriction of the group algebra
    2. 3.2. Kernel of the adjunction $\beta\mathrel{\mathop{\mathchar 58\relax}}h^{*}h_{*}\mathscr{G}\to\mathscr{G}$
    3. 3.3. Extension structure of the Weil restriction
  4. 4\. Weil extension
    1. 4.1. Hochschild extensions
    2. 4.2. Kernel of the adjunction, reprise
    3. 4.3. Weil extension functor
  5. 5\. The equivalence of categories
    1. 5.1. Equivariance of rigidifications under Lie algebra translation
    2. 5.2. Proof of the main theorem: equivalence and exactness
    3. 5.3. Proof of the main theorem: isomorphism of $\mathbb{O}_{k}$-module stacks
  6. 6\. Dieudonné theory for unipotent groups over the dual numbers
    1. 6.1. Reminder on Dieudonné theory
    2. 6.2. Dieudonné theory over the dual numbers
  7. Appendix A. Differential calculus on group schemes
    1. A.1. Tangent bundle and Lie algebra
    2. A.2. Exponential and infinitesimal translation
    3. A.3. Deformations of affine group schemes
  8. Appendix B. Module stacks in groupoids

##  1\. Introduction

Throughout this article, we fix a commutative ring $k$ and a free $k$-module
$I$ of finite rank $r\geqslant 1$. We consider the ring of (generalized) dual
numbers $k[I]\mathrel{\mathop{\mathchar 58\relax}}=k\oplus I$ with $I^{2}=0$.
We write $h\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Spec}(k[I])\to\operatorname{Spec}(k)$ the structure
map and $i\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Spec}(k)\to\operatorname{Spec}(k[I])$ the closed
immersion. Also we denote by $\mathbb{O}_{k}$ the
$\operatorname{Spec}(k)$-ring scheme such that if $R$ is a $k$-algebra then
$\mathbb{O}_{k}(R)=R$ with its ring structure.

###  1.1. Motivation, results, plan of the article

#### 1.1.0. Motivation.

The starting point of our work is a relation between deformations and group
extensions. 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
$\mathrm{H}^{2}(BG,\underline{\ell}{}_{G}^{\scriptscriptstyle\vee}\otimes I)$,
see chap. VII, thm 3.2.1 in loc. cit. Here, the coefficients of cohomology are
the derived dual of the equivariant cotangent complex
$\underline{\ell}{}_{G}\in D(BG)$, 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 $\mathbb{V}(I^{\scriptscriptstyle\vee})$. If we assume
moreover that $G$ is smooth, then the augmentation
$\underline{\ell}{}_{G}\to\omega^{1}_{G}$ to the sheaf of invariant
differential 1-forms is a quasi-isomorphism and it follows that
$\underline{\ell}{}_{G}^{\scriptscriptstyle\vee}\simeq\operatorname{Lie}G$.
Since coherent cohomology of $BG$ is isomorphic to group cohomology of $G$,
the cohomology group of interest ends up being
$\mathrm{H}^{2}(G,\operatorname{Lie}(G,I))$ where
$\operatorname{Lie}(G,I)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Lie}G\otimes\mathbb{V}(I^{\scriptscriptstyle\vee})$.
The latter cohomology group is meaningful also in the theory of group
extensions, where it is known to classify isomorphism classes of extensions of
$G$ by $\operatorname{Lie}(G,I)$ viewed as a $G$-module via the adjoint
representation, see [DG70, chap. II, § 3, no 3.3 and III, § 6, no 2.1]. In
this paper, our aim is to give a direct algebro-geometric construction of this
correspondence between deformations and group extensions. Our main result is
that the Weil restriction functor $h_{*}$ provides such a construction.
Thereby, we obtain a categorification of a link that has been available only
as a bijection between sets of isomorphism classes. This improvement is
crucial for a better understanding of $k[I]$-group schemes, since in
applications most groups occur as kernels or quotients of morphisms. We
illustrate this by giving a Dieudonné-type theory for unipotent group schemes.
Natural extensions of our result to more general thickenings of the base, or
to non-smooth group schemes, would have further interesting applications.
Since we wish to show our main results to the reader without further delay, we
postpone the discussion of these applications to 1.1 below.

#### 1.1.1. Results.

Our main result is an equivalence between the category of smooth, affine
$k[I]$-group schemes and a certain category of extensions of $k$-group
schemes. However, for reasons that are discussed below, it is convenient for
us to work with group schemes slightly more general than smooth ones. We say
that a morphism of schemes $X\to S$ is differentially flat if both $X$ and
$\Omega^{1}_{X/S}$ are flat over $S$. Examples of differentially flat group
schemes include smooth group schemes, pullbacks from the spectrum of a field,
Tate-Oort group schemes with parameter $a=0$ in characteristic $p$, flat
vector bundles i.e. $\mathbb{V}(\mathscr{F})$ with $\mathscr{F}$ flat over the
base, and truncated Barsotti-Tate groups of level $n$ over a base where
$p^{n}=0$ [Il85, 2.2.1]. If $\mathscr{G}$ is a $k[I]$-group scheme, we call
rigidification of $\mathscr{G}$ an isomorphism of $k[I]$-schemes
$\sigma\mathrel{\mathop{\mathchar
58\relax}}h^{*}\mathscr{G}_{k}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\mathscr{G}$
that lifts the identity of the scheme
$\mathscr{G}_{k}\mathrel{\mathop{\mathchar 58\relax}}=i^{*}\mathscr{G}$. Such
a map need not be a morphism of group schemes. We say that $\mathscr{G}$ is
rigid if it admits a rigidification. Examples of rigid group schemes include
smooth group schemes, pullbacks from $\operatorname{Spec}(k)$, and groups of
multiplicative type. Finally a deformation of a flat $k$-group scheme $G$ over
$k[I]$ is a pair composed of a flat $k[I]$-group scheme $\mathscr{G}$ and an
isomorphism of $k$-group schemes
$\mathscr{G}_{k}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}G$.
Let $\operatorname{Gr}\\!/k[I]$ be the category of affine, differentially
flat, rigid $k[I]$-group schemes (this includes all smooth affine $k[I]$-group
schemes). The morphisms in this category are the morphisms of $k[I]$-group
schemes. Let $\operatorname{Ext}(I)/k$ be the category of extensions of the
form $1\to\operatorname{Lie}(G,I)\to E\to G\to 1$ where $G$ is an affine
differentially flat $k$-group scheme, and “extension” means that $E\to G$ is
an fpqc torsor under $\operatorname{Lie}(G,I)$. The morphisms in this category
are the commutative diagrams:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Lie}(G,I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{d}\\!\psi}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{1\>}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Lie}(G^{\prime},I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

where $\operatorname{d}\\!\psi=\operatorname{Lie}(\psi)$ is the differential
of $\psi$. Usually such a morphism will be denoted simply
$\varphi\mathrel{\mathop{\mathchar 58\relax}}E\to E^{\prime}$. The categories
$\operatorname{Gr}\\!/k[I]$ and $\operatorname{Ext}(I)/k$ are exact
categories. They are also fpqc stacks over $\operatorname{Spec}(k)$ equipped
with the structure of $\mathbb{O}_{k}$-module stacks fibred in groupoids over
$\operatorname{Gr}/k$. This means that there exist notions of sum and scalar
multiple for objects of $\operatorname{Gr}\\!/k[I]$ and
$\operatorname{Ext}(I)/k$ (for extensions, the sum is the Baer sum); these
structures are described in 1.2. We can now state our main result.

######  Theorem A (see 5)

(1) The Weil restriction functor provides an equivalence:

$h_{*}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gr}\\!/k[I]\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\operatorname{Ext}(I)/k.$

This equivalence commutes with base changes on $\operatorname{Spec}(k)$. (2)
If $1\to\mathscr{G}^{\prime}\to\mathscr{G}\to\mathscr{G}^{\prime\prime}\to 1$
is an exact sequence in $\operatorname{Gr}\\!/k[I]$, then $1\to
h_{*}\mathscr{G}^{\prime}\to h_{*}\mathscr{G}\to
h_{*}\mathscr{G}^{\prime\prime}$ is exact in $\operatorname{Ext}(I)/k$. If
moreover $\mathscr{G}^{\prime}$ is smooth then $1\to
h_{*}\mathscr{G}^{\prime}\to h_{*}\mathscr{G}\to
h_{*}\mathscr{G}^{\prime\prime}\to 1$ 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
$\mathbb{O}_{k}$-module stacks fibred over $\operatorname{Gr}\\!/k$, i.e. it
transforms the addition and scalar multiplication of deformations of a fixed
$G\in\operatorname{Gr}\\!/k$ 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
$\mathscr{G}\in\operatorname{Gr}\\!/k[I]$ has property $P$ if and only if the
$k$-group scheme $E=h_{*}\mathscr{G}$ has $P$.

In order to show that $h_{*}$ is an equivalence, we build a quasi-inverse
$h^{\mbox{\rm\tiny+}}$ 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).

######  Theorem B (see 6.2)

Let $\operatorname{SCU}/k[I]$ be the category of smooth, commutative,
unipotent $k[I]$-group schemes. Let
$\mathbb{D}\mbox{-}I\mbox{-}\operatorname{Mod}$ be the category of
$I$-extensions of smooth erasable Dieudonné modules. Then the Dieudonné
functor $\underline{M}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{CU}/k\longrightarrow\mathbb{D}\mbox{-}\operatorname{Mod}$
induces a contravariant equivalence of categories:

$\mathscr{M}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{SCU}/k[I]\longrightarrow\mathbb{D}\mbox{-}I\mbox{-}\operatorname{Mod}$

that sends $\mathscr{U}$ to the extension
$0\to\underline{M}(\mathscr{U}_{k})\to\underline{M}(h_{*}\mathscr{U}))\to\underline{M}(\operatorname{Lie}(\mathscr{U}_{k},I))\to
0$.

#### 1.1.2. 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 $\mathbb{O}_{k}[G]$ in the
course of proving the main theorem; and the group algebra $\mathbb{O}_{k}[G]$
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 $k[\varepsilon]$ with
$\varepsilon^{2}=0$. The first is that in order to prove that our equivalence
of categories respects the $\mathbb{O}_{k}$-module stack structure, we have to
introduce the ring $k[I]$ with the two-dimensional $k$-module
$I=k\varepsilon+k\varepsilon^{\prime}$. 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.

#### 1.1.3. 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 non-affine 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 $\mu_{p^{n}}$ 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.

#### 1.1.4. Plan of the article.

The present Section 1 ends with material of preliminary nature on the
description of the $\mathbb{O}_{k}$-module stack structure of the categories
$\operatorname{Gr}\\!/k[I]$ and $\operatorname{Ext}(I)/k$ and on Weil
restriction. In Section 2 we introduce group algebra schemes. In Section 3 we
describe the functor $h_{*}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gr}\\!/k[I]\to\operatorname{Ext}(I)/k$, in Section 4
we construct a functor $h^{\mbox{\rm\tiny+}}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Ext}(I)/k\to\operatorname{Gr}\\!/k[I]$, 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.

#### 1.1.5. Acknowledgements.

For various discussions and remarks, we thank Sylvain Brochard, Xavier Caruso,
Brian Conrad, Bernard Le Stum, Brian Osserman, and Tobias Schmidt. We
acknowledge the help of the referee to make the article much more incisive. We
are also grateful to the CIRM in Luminy where part of this research was done.
Finally, the first author would like to thank the executive and administrative
staff of IRMAR and of the Centre Henri Lebesgue ANR-11-LABX-0020-01 for
creating an attractive mathematical environment.

###  1.2. The $\mathbb{O}_{k}$-module stack structure of
$\operatorname{Gr}\\!/k[I]$ and $\operatorname{Ext}(I)/k$

Both categories $\operatorname{Gr}\\!/k[I]$ and $\operatorname{Ext}(I)/k$ are
endowed with the structure of $\mathbb{O}_{k}$-module stacks in groupoids over
$\operatorname{Gr}\\!/k$. The reader who wishes to see the full-fledged
definition is invited to read Appendix 8. In rough terms, once a $k$-group
scheme $G$ is fixed, the $\mathbb{O}_{k}$-module category structure boils down
to an addition law by which one can add deformations (resp. extensions) of
$G$, and an external law by which one can multiply a deformation (resp. an
extension) by scalars of the ring scheme $\mathbb{O}_{k}$. Here is a
description of these structures.

#### 1.2.0. The $\mathbb{O}_{k}$-module stack
$\operatorname{Gr}\\!/k[I]\to\operatorname{Gr}\\!/k$.

Let $G\in\operatorname{Gr}\\!/k$ be fixed. Let
$\mathscr{G}_{1},\mathscr{G}_{2}\in\operatorname{Gr}\\!/k[I]$ with
identifications $i^{*}\mathscr{G}_{1}\simeq G\simeq i^{*}\mathscr{G}_{2}$. The
addition is obtained by a two-step process. First we glue these group schemes
along their common closed subscheme $G$:

$\mathscr{G}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=\mathscr{G}_{1}\amalg_{G}\mathscr{G}_{2}.$

This lives over the scheme
$\operatorname{Spec}(k[I])\times_{\operatorname{Spec}(k)}\operatorname{Spec}(k[I])=\operatorname{Spec}(k[I\oplus
I])$. The properties of gluing along infinitesimal thickenings are studied in
the Stacks Project [SP]. We point out some statements relevant to our
situation: existence of the coproduct in Tag 07RV, a list of properties
preserved by gluing in Tag 07RX, gluing of modules in Tag 08KU and
preservation of flatness of modules in Tag 07RW. It follows from these results
that $\mathscr{G}^{\prime}$ is an object of $\operatorname{Gr}\\!/k[I\oplus
I]$. Then we form the desired sum

$\mathscr{G}_{1}+\mathscr{G}_{2}\mathrel{\mathop{\mathchar
58\relax}}=j^{*}(\mathscr{G}^{\prime})=j^{*}(\mathscr{G}_{1}\amalg_{G}\mathscr{G}_{2})$

by pullback along the closed immersion

$j\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Spec}(k[I])\lhook\joinrel\relbar\joinrel\rightarrow\operatorname{Spec}(k[I\oplus
I])$

induced by the addition morphism $I\oplus I\to I$, $i_{1}\oplus i_{2}\mapsto
i_{1}+i_{2}$. The neutral element for this addition is the group scheme
$h^{*}G$. The scalar multiple

$\lambda\mathscr{G}\mathrel{\mathop{\mathchar
58\relax}}=s_{\lambda}^{*}\mathscr{G}$

is given by rescaling using the scheme map
$s_{\lambda}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Spec}(k[I])\to\operatorname{Spec}(k[I])$ induced by
the $k$-algebra map $k[I]\to k[I]$ which is multiplication by $\lambda$ in
$I$. The axioms of $\mathbb{O}_{k}$-module stacks can be checked but we leave
the details to the courageous reader.

#### 1.2.1. The $\mathbb{O}_{k}$-module stack
$\operatorname{Ext}(I)/k\to\operatorname{Gr}\\!/k$.

Again let $G\in\operatorname{Gr}\\!/k$ be fixed and set
$L\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Lie}(G,I)$. Let
$E_{1},E_{2}\in{\mathcal{E}}xt(G,L)$ be two extensions. Their addition is
given by Baer sum; here again this is a two-step process. Namely, we first
form the fibre product $E^{\prime}=E_{1}\times_{G}E_{2}$ which is an extension
of $G$ by $L\times L$. Then the Baer sum is the pushforward of this extension
along the addition map $+\mathrel{\mathop{\mathchar 58\relax}}L\times L\to L$.
All in all, we have the following diagram which serves as a definition of
$E_{1}+E_{2}$:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\times
L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+}$$\ulcorner$
$\textstyle{E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}+E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1.}$

Explicitly, the underlying group scheme of the Baer sum is given by
$E_{1}+E_{2}=E^{\prime}/M$ where

$M=\ker(L\times L\to L)=\\{(x,-x),x\in L\\}.$

The neutral element for this addition is the trivial extension $E=L\rtimes G$.
Even though this is not emphasized in the literature, the usual proofs of the
fact that the set of extensions endowed with the Baer sum operation is an
abelian group provide explicit associativity and commutativity constraints
proving that ${\mathcal{E}}xt(G,L)$ is a Picard category. The associativity
constraint is obtained by expressing $(E_{1}+E_{2})+E_{3}$ and
$E_{1}+(E_{2}+E_{3})$ as isomorphic quotients of
$E_{1}\times_{G}E_{2}\times_{G}E_{3}$, and the commutativity constraint is
obtained from the flipping morphism in $E_{1}\times_{G}E_{2}$. The scalar
multiplication by $\lambda\in k$ is given by pushforward along the
multiplication-by-$\lambda$ morphism in the module scheme
$L=\operatorname{Lie}(G,I)$. All in all, we have the following diagram which
serves as a definition of $\lambda E$:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\ulcorner$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lambda
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1.}$

Again, the verification of the axioms of an $\mathbb{O}_{k}$-module stack is
tedious but not difficult.

###  1.3. 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
$h\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Spec}(k^{\prime})\to\operatorname{Spec}(k)$ be a
finite, locally free morphism of affine schemes. Let $(\operatorname{Sch}/k)$
be the category of $k$-schemes and $(\operatorname{Fun}/k)$ the category of
functors $(\operatorname{Sch}/k)^{\circ}\to(\operatorname{Sets})$. The Yoneda
functor embeds the former category into the latter. By sending a morphism of
functors $f\mathrel{\mathop{\mathchar
58\relax}}X^{\prime}\to\operatorname{Spec}(k^{\prime})$ to the morphism
$h\circ f\mathrel{\mathop{\mathchar
58\relax}}X^{\prime}\to\operatorname{Spec}(k)$ we obtain a functor
$h_{!}\mathrel{\mathop{\mathchar
58\relax}}(\operatorname{Fun}/k^{\prime})\to(\operatorname{Fun}/k)$. Sometimes
we will refer to $h_{!}X^{\prime}$ as the $k^{\prime}$-functor $X^{\prime}$
viewed as a $k$-functor and the notation $h_{!}$ will be omitted. The pullback
functor $h^{*}\mathrel{\mathop{\mathchar
58\relax}}(\operatorname{Fun}/k)\to(\operatorname{Fun}/k^{\prime})$ is right
adjoint to $h_{!}$, in particular $(h^{*}X)(S^{\prime})=X(h_{!}S^{\prime})$
for all $k^{\prime}$-schemes $S^{\prime}$. The Weil restriction functor
$h_{*}\mathrel{\mathop{\mathchar
58\relax}}(\operatorname{Fun}/k^{\prime})\to(\operatorname{Fun}/k)$ is right
adjoint to $h^{*}$, in particular we have
$(h_{*}X^{\prime})(S)=X^{\prime}(h^{*}S)$ for all $k$-schemes $S$. Thus we
have a sequence of adjoint functors:

$h_{!},h^{*},h_{*}.$

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

$\alpha\mathrel{\mathop{\mathchar 58\relax}}1\longrightarrow
h_{*}h^{*}\quad\mbox{and}\quad\beta\mathrel{\mathop{\mathchar
58\relax}}h^{*}h_{*}\longrightarrow 1$

for the unit and counit of the $(h^{*},h_{*})$-adjunction. If $X$ is a
separated $k$-scheme then $\alpha_{X}\mathrel{\mathop{\mathchar 58\relax}}X\to
h_{*}h^{*}X$ is a closed immersion. If $X^{\prime}$ is a $k^{\prime}$-group
(resp. algebra) functor (resp. scheme), then also $h_{*}X^{\prime}$ is a
$k$-group (resp. algebra) functor (resp. scheme). If moreover
$X^{\prime}\to\operatorname{Spec}(k^{\prime})$ is smooth of relative dimension
$n$, then $h_{*}X^{\prime}\to\operatorname{Spec}(k)$ is smooth of relative
dimension $n[k^{\prime}\mathrel{\mathop{\mathchar 58\relax}}k]$ where
$[k^{\prime}\mathrel{\mathop{\mathchar 58\relax}}k]$ 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.

##  2\. Group algebras of group schemes

Let $G$ be an affine $k$-group scheme. In this subsection, we explain the
construction of the group algebra $\mathbb{O}_{k}[G]$, 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
$\mathbb{O}_{k}[G](k)$ 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. Since we are
not aware of any appearance of the group algebra in the literature, we give a
somewhat detailed treatment, including examples (2.2–2.2), basic properties
(2.3) and the universal property (2.3).

###  2.1. Linear algebra schemes

#### 2.1.0. Vector bundles.

Let $S=\operatorname{Spec}(k)$. As in [EGA$1_{\mbox{\tiny new}}$], we call
vector bundle an $\mathbb{O}_{k}$-module scheme of the form
$\mathbb{V}(\mathscr{F})=\operatorname{Spec}\operatorname{S}(\mathscr{F})$
where $\mathscr{F}$ is a quasi-coherent ${\mathcal{O}}_{S}$-module and
$\operatorname{S}(\mathscr{F})$ is its symmetric algebra. We say smooth vector
bundle if $\mathscr{F}$ is locally free of finite rank. If
$f\mathrel{\mathop{\mathchar 58\relax}}X\to S$ is quasi-compact and quasi-
separated, we set $\mathbb{V}(X/S)\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{V}(f_{*}{\mathcal{O}}_{X})$ so
$\mathbb{V}(X/S)(T)=\operatorname{Hom}_{{\mathcal{O}}_{T}\mbox{-}\operatorname{Mod}}((f_{*}{\mathcal{O}}_{X})_{T},{\mathcal{O}}_{T})$
for all $S$-schemes $T$. There is a canonical $S$-morphism
$\nu_{X}\mathrel{\mathop{\mathchar 58\relax}}X\longrightarrow\mathbb{V}(X/S)$
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
$\nu_{X}$ is a closed immersion because it is induced by the surjective
morphism of algebras $\operatorname{S}(f_{*}{\mathcal{O}}_{X})\longrightarrow
f_{*}{\mathcal{O}}_{X}$ induced by the identity
$f_{*}{\mathcal{O}}_{X}\longrightarrow f_{*}{\mathcal{O}}_{X}$.

#### 2.1.1. Base change and ring scheme.

If $h\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Spec}(k^{\prime})\to\operatorname{Spec}(k)$ is a
morphism of affine schemes, there is a morphism of $k^{\prime}$-ring schemes
$h^{*}\mathbb{O}_{k}\to\mathbb{O}_{k^{\prime}}$ which is the identity on
points, hence an isomorphism of ring schemes. However, whereas the target has
a natural structure of $\mathbb{O}_{k^{\prime}}$-algebra, the source does not.
For this reason, the pullback of module functors or module schemes along $h$
is defined as $M\mapsto
h^{*}M\otimes_{h^{*}\mathbb{O}_{k}}\mathbb{O}_{k^{\prime}}$, as is familiar
for the pullback of modules on ringed spaces. Usually we will write simply
$h^{*}M$. For later use, we give some complements on the case where
$k^{\prime}=k[I]$ is the ring of generalized dual numbers, for some finite
free $k$-module $I$. We will identify $I$ and its dual
$I^{\scriptscriptstyle\vee}=\operatorname{Hom}_{k}(I,k)$ with the coherent
${\mathcal{O}}_{\operatorname{Spec}(k)}$-modules they define, thus we have the
vector bundle $\mathbb{V}(I^{\scriptscriptstyle\vee})$. For each $k$-algebra
$R$, we have $R[I]=R\oplus I\otimes_{k}R$ where $IR\simeq I\otimes_{k}R$ has
square 0. This decomposition functorial in $R$ gives rise to a direct sum
decomposition of $\mathbb{O}_{k}$-module schemes:

$h_{*}\mathbb{O}_{k[I]}=\mathbb{O}_{k}\oplus\mathbb{V}(I^{\scriptscriptstyle\vee}).$

It is natural to use the notation
$\mathbb{O}_{k}[\mathbb{V}(I^{\scriptscriptstyle\vee})]$ for the
$\mathbb{O}_{k}$-algebra scheme on the right-hand side, however we will write
more simply $\mathbb{O}_{k}[I]$. Now we move up on
$\operatorname{Spec}(k[I])$, where there is a morphism of
$\mathbb{O}_{k[I]}$-module schemes
$h^{*}\mathbb{V}(I^{\scriptscriptstyle\vee})\to\mathbb{O}_{k[I]}$ defined for
all $k[I]$-algebras $R$ as the morphism $I\otimes_{k}R\to R$, $i\otimes
x\mapsto i_{R}x$ where $i_{R}\mathrel{\mathop{\mathchar 58\relax}}=i1_{R}$.
According to what we said before, the pullback
$h^{*}\mathbb{V}(I^{\scriptscriptstyle\vee})$ has the module structure such
that a section $a\in\mathbb{O}_{k[I]}$ acts by $a\cdot i\otimes
x\mathrel{\mathop{\mathchar 58\relax}}=i\otimes ax$. The image of
$h^{*}\mathbb{V}(I^{\scriptscriptstyle\vee})\to\mathbb{O}_{k[I]}$ is the ideal
$I\\!\cdot\\!\mathbb{O}_{k[I]}$ defined by
$(I\\!\cdot\\!\mathbb{O}_{k[I]})(R)=IR$ for all $k[I]$-algebras $R$. There is
an exact sequence of $\mathbb{O}_{k[I]}$-module schemes:

$0\longrightarrow
I\\!\cdot\\!\mathbb{O}_{k[I]}\longrightarrow\mathbb{O}_{k[I]}\longrightarrow
i_{*}\mathbb{O}_{k}\longrightarrow 0.$

#### 2.1.2. $\mathbb{O}_{k}$-Algebra schemes.

Here we define a category $(\mathbb{O}_{k}\mbox{-}\operatorname{Alg})$ of
linear $\mathbb{O}_{k}$-algebra schemes and give a summary of elementary
properties. For us an $\mathbb{O}_{k}$-algebra scheme is a $k$-scheme $D$
endowed with two internal composition laws $+,\times\mathrel{\mathop{\mathchar
58\relax}}D\times D\to D$ called addition and multiplication possessing two
neutral sections $0,1\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Spec}(k)\to D$, and an external law
$\cdot\mathrel{\mathop{\mathchar 58\relax}}\mathbb{O}_{k}\times D\to D$, such
that for each $k$-algebra $R$ the tuple $(D(R),+,\times,0,1,\cdot)$ is an
associative unitary $R$-algebra. In particular $(D,+,0)$ is a commutative
group scheme, and $(D(R),\times,1)$ is a (possibly noncommutative) monoid. We
say that $D$ is a linear $\mathbb{O}_{k}$-algebra scheme if its underlying
$\mathbb{O}_{k}$-module scheme is a vector bundle. In this case $\mathscr{F}$
can be recovered as the “dual bundle” sheaf
$\mathscr{F}=\operatorname{Hom}_{\mathbb{O}_{k}\mbox{-}\operatorname{Mod}}(D,\mathbb{O}_{k})$,
the Zariski sheaf over $S$ whose sections over an open $U$ are the morphisms
of ${\mathbb{O}_{k}}_{|U}$-modules $D_{|U}\to{\mathbb{O}_{k}}_{|U}$ (read
[EGA$1_{\mbox{\tiny new}}$, 9.4.9] between the lines). For an affine
$\mathbb{O}_{k}$-algebra scheme in our sense, the comultiplication is a map
$\Delta\mathrel{\mathop{\mathchar
58\relax}}\operatorname{S}(\mathscr{F})\to\operatorname{S}(\mathscr{F})\otimes\operatorname{S}(\mathscr{F})$
and the bilinearity of $m$ implies that this map is induced from a map
$\Delta_{0}\mathrel{\mathop{\mathchar
58\relax}}\mathscr{F}\to\mathscr{F}\otimes\mathscr{F}$. Finally we point out
two constructions on linear $\mathbb{O}_{k}$-algebra schemes. The first is the
tensor product $D\otimes_{\mathbb{O}_{k}}D^{\prime}$, which as a functor is
defined as $R\mapsto D(R)\otimes_{R}D^{\prime}(R)$. If
$D=\mathbb{V}(\mathscr{F})$ and $D^{\prime}=\mathbb{V}(\mathscr{F}^{\prime})$
with $\mathscr{F},\mathscr{F}^{\prime}$ locally free of finite ranks, then
$D\otimes_{\mathbb{O}_{k}}D^{\prime}\simeq\mathbb{V}(\mathscr{F}\otimes\mathscr{F}^{\prime})$.
The second construction is the group of units. We observe that for any linear
$\mathbb{O}_{k}$-algebra scheme $D$, the subfunctor $D^{\times}\subset D$ of
(multiplicative) units is the preimage under the multiplication $D\times D\to
D$ of the unit section $1\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Spec}(k)\to D$ and is therefore representable by an
affine scheme. This gives rise to the group of units functor
$(\mathbb{O}_{k}\mbox{-}\operatorname{Alg})\to(k\mbox{-}\operatorname{Gr})$,
$D\mapsto D^{\times}$ where $(k\mbox{-}\operatorname{Gr})$ is the category of
affine $k$-group schemes.

#### 2.1.3. Remark.

We do not know if an $\mathbb{O}_{k}$-algebra scheme whose underlying scheme
is affine over $S$ is always of the form $\mathbb{V}(\mathscr{F})$.

###  2.2. Group algebra: construction and examples

Let $G=\operatorname{Spec}(A)$ be an affine $k$-group scheme. We write
$(u,v)\mapsto u\star v$ or sometimes simply $(u,v)\mapsto uv$ the
multiplication of $G$. This operation extends to the vector bundle envelope
$\mathbb{V}(G/k)$, as follows. Let $\Delta\mathrel{\mathop{\mathchar
58\relax}}A\to A\otimes_{k}A$ be the comultiplication. For each $k$-algebra
$R$, we have
$\mathbb{V}(G/k)(R)=\operatorname{Hom}_{k\mbox{-}\operatorname{Mod}}(A,R)$. If
$u,v\mathrel{\mathop{\mathchar 58\relax}}A\to R$ are morphisms of $k$-modules,
we consider the composition $u\star v\mathrel{\mathop{\mathchar
58\relax}}=(u\otimes v)\circ\Delta$:

$u\star v\mathrel{\mathop{\mathchar
58\relax}}A\stackrel{{\scriptstyle\Delta}}{{\relbar\joinrel\relbar\joinrel\longrightarrow}}A\otimes_{k}A\stackrel{{\scriptstyle
u\otimes v}}{{\relbar\joinrel\relbar\joinrel\longrightarrow}}R.$

Here the map $u\otimes v\mathrel{\mathop{\mathchar 58\relax}}A\otimes_{k}A\to
R$ is $a\otimes b\mapsto u(a)v(b)$.

#### 2.2.0. Definition.

The group algebra of the $k$-group scheme $G$:

$\mathbb{O}_{k}[G]\mathrel{\mathop{\mathchar
58\relax}}=(\mathbb{V}(G/k),+,\star),$

is the vector bundle $\mathbb{V}(G/k)$ endowed with the product just defined.
We write $\nu_{G}\mathrel{\mathop{\mathchar
58\relax}}G\hookrightarrow\mathbb{O}_{k}[G]$ for the closed immersion as in
paragraph 2.1.

We check below that $\mathbb{O}_{k}[G]$ is a linear $\mathbb{O}_{k}$-algebra
scheme. Apart from $G(R)$, there is another noteworthy subset inside
$\mathbb{O}_{k}[G](R)$, namely the set $\operatorname{Der}_{G}(R)$ of
$k$-module maps $d\mathrel{\mathop{\mathchar 58\relax}}A\to R$ which are
$u$-derivations for some $k$-algebra map $u\mathrel{\mathop{\mathchar
58\relax}}A\to R$ (which need not be determined by $d$); a more accurate
notation would be $\operatorname{Der}({\mathcal{O}}_{G},\mathbb{O}_{k})(R)$
but we favour lightness. Here are the first basic properties coming out of the
construction.

#### 2.2.1. Proposition.

Let $G$ be an affine $k$-group scheme. Let $\mathbb{O}_{k}[G]$ and
$\operatorname{Der}_{G}$ be as described above.

  * (1)

The tuple $\mathbb{O}_{k}[G]\mathrel{\mathop{\mathchar
58\relax}}=(\mathbb{V}(G/k),+,\star)$ is a linear $\mathbb{O}_{k}$-algebra
scheme.

  * (2)

As a $k$-scheme, $\mathbb{O}_{k}[G]$ is $k$-flat (resp. has $k$-projective
function ring) iff $G$ has the same property.

  * (3)

The composition
$G\hookrightarrow\mathbb{O}_{k}[G]^{\times}\hookrightarrow\mathbb{O}_{k}[G]$
is a closed embedding, hence also
$G\hookrightarrow\mathbb{O}_{k}[G]^{\times}$.

  * (4)

The subfunctor $\operatorname{Der}_{G}\subset\mathbb{O}_{k}[G]$ is stable by
multiplication by $G$ on the left and on the right, so it acquires left and
right $G$-actions. More precisely, if $u,v\mathrel{\mathop{\mathchar
58\relax}}A\to R$ are maps of algebras, $d\mathrel{\mathop{\mathchar
58\relax}}A\to R$ is a $u$-derivation and
$d^{\prime}\mathrel{\mathop{\mathchar 58\relax}}A\to R$ is a $v$-derivation,
then $d\star v$ and $u\star d^{\prime}$ are $(u\star v)$-derivations.

  * Proof.

Omitted. $\Box$

#### 2.2.2. Remark on Hopf algebra structure.

If $G$ is a finite, locally free commutative $k$-group scheme, then the
multiplication of its function ring induces a comultiplication on
$\mathbb{O}_{k}[G]$ making it into a Hopf $\mathbb{O}_{k}$-algebra scheme.
Moreover the $k$-algebra $\mathbb{O}_{k}[G](k)$ is the ring of functions of
the Cartier dual $G^{\scriptscriptstyle\vee}$. This Hopf algebra structure
highlights Examples 2.2 and 2.2 below.

#### 2.2.3. Example 1: finite locally free groups.

If $G$ is a finite locally free $k$-group scheme, the algebra
$\mathbb{O}_{k}[G]$ is a smooth vector bundle of rank
$[G\mathrel{\mathop{\mathchar 58\relax}}\operatorname{Spec}(k)]$ and its group
of units is the complement in $\mathbb{O}_{k}[G]$ of the Cartier divisor equal
to the zero locus of the determinant of the left regular representation:

$\mathbb{O}_{k}[G]\stackrel{{\scriptstyle\mathbb{L}}}{{\lhook\joinrel\relbar\joinrel\rightarrow}}\operatorname{End}_{\mathbb{O}_{k}\mbox{-}\operatorname{Mod}}(\mathbb{O}_{k}[G])\stackrel{{\scriptstyle\det}}{{\relbar\joinrel\relbar\joinrel\longrightarrow}}\mathbb{A}^{1}.$

If moreover $G$ is the finite constant $k$-group scheme defined by a finite
abstract group $\Gamma$, then $\mathbb{O}_{k}[G]$ is isomorphic to the algebra
scheme defined by the abstract group algebra $k[\Gamma]$, that is to say
$\mathbb{O}_{k}[G](R)\simeq R[\Gamma]$ functorially in $R$.

#### 2.2.4. Example 2: the additive group.

Let $G=\mathbb{G}_{a}=\operatorname{Spec}(k[x])$. For a $k$-algebra $R$, let

$R\langle\\!\langle t\rangle\\!\rangle=R[[t,t^{[2]},t^{[3]},\dots]]$

be the $R$-algebra of divided power formal power series in one variable $t$,
with $t^{[i]}t^{[j]}={i+j\choose i}t^{[i+j]}$ for all $i,j\geqslant 0$.
Setting $\mathbb{O}\langle\\!\langle t\rangle\\!\rangle(R)=R\langle\\!\langle
t\rangle\\!\rangle$, we have an isomorphism
$\mathbb{O}_{k}[G]\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\mathbb{O}\langle\\!\langle
t\rangle\\!\rangle$ given by:

$\mathbb{O}_{k}[G](R)\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}R\langle\\!\langle
t\rangle\\!\rangle\quad,\quad(k[x]\stackrel{{\scriptstyle
u}}{{\longrightarrow}}R)\longmapsto\sum_{i\geqslant 0}u(x^{i})t^{i}.$

If $k$ is a ring of characteristic $p>0$, let $H=\alpha_{p}$ be the kernel of
Frobenius. The algebra $\mathbb{O}_{k}[H](R)$ is identified with the
$R$-subalgebra of $R\langle\\!\langle t\rangle\\!\rangle$ generated by $t$,
which is isomorphic to $R[t]/(t^{p})$ because $t^{p}=pt^{[p]}=0$.

#### 2.2.5. Example 3: the multiplicative group.

Let $G=\mathbb{G}_{m}=\operatorname{Spec}(k[x,1/x])$. Let $R^{\mathbb{Z}}$ be
the product algebra, whose elements are sequences with componentwise addition
and multiplication. We have an isomorphism
$\mathbb{O}_{k}[G]\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\prod_{i\in\mathbb{Z}}\mathbb{O}_{k}$
given by:

$\mathbb{O}_{k}[G](R)\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}R^{\mathbb{Z}}\quad,\quad(k[x]\stackrel{{\scriptstyle
u}}{{\longrightarrow}}R)\longmapsto\\{u(x^{i})\\}_{i\in\mathbb{Z}}.$

More generally, for any torus $T$ with character group $X(T)$, we have
$\mathbb{O}_{k}[T]\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\prod_{i\in
X(T)}\mathbb{O}_{k}$. Let $H=\mu_{n}$ be the subgroup of $n$-th roots of
unity. The algebra $\mathbb{O}_{k}[H](R)$ is identified with the
$R$-subalgebra of $\mathbb{Z}/n\mathbb{Z}$-invariants
$\mathbb{O}_{k}[G](R)^{\mathbb{Z}/n\mathbb{Z}}$, composed of sequences
$\\{r_{i}\\}_{i\in\mathbb{Z}}$ such that $r_{i+nj}=r_{i}$ for all
$i,j\in\mathbb{Z}$.

###  2.3. Properties: functoriality and adjointness

Here are some functoriality properties of the group algebra.

#### 2.3.0. Proposition.

Let $G$ be an affine $k$-group scheme. The formation of the group algebra
$\mathbb{O}_{k}[G]$:

  * (1)

is functorial in $G$ and faithful;

  * (2)

commutes with base change $k^{\prime}/k$;

  * (3)

is compatible with products: there is a canonical isomorphism
$\mathbb{O}_{k}[G]\otimes_{\mathbb{O}_{k}}\mathbb{O}_{k}[H]\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\mathbb{O}_{k}[G\times
H]$;

  * (4)

is compatible with Weil restriction: if $h\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Spec}(k^{\prime})\to\operatorname{Spec}(k)$ is a
finite locally free morphism of schemes, there is an isomorphism of
$\mathbb{O}_{k}$-algebra schemes
$\mathbb{O}_{k}[G]\otimes_{\mathbb{O}_{k}}h_{*}\mathbb{O}_{k^{\prime}}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}h_{*}h^{*}\mathbb{O}_{k}[G]$.

  * Proof.

(1) Any map of affine $k$-group schemes $G=\operatorname{Spec}(A)\to
H=\operatorname{Spec}(B)$ gives rise to a map of $k$-Hopf algebras $B\to A$
and then by precomposition to a map of $R$-algebras
$\mathbb{O}_{k}[G](R)\to\mathbb{O}_{k}[H](R)$ which is functorial in $R$.
Faithfulness follows from the fact that $G\hookrightarrow\mathbb{O}_{k}[G]$ is
a closed immersion. (2) The isomorphism
$\operatorname{Hom}_{k^{\prime}\mbox{-}\operatorname{Mod}}(A\otimes_{k}k^{\prime},R^{\prime})\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\operatorname{Hom}_{k\mbox{-}\operatorname{Mod}}(A,R^{\prime})$,
functorial in the $k^{\prime}$-algebra $R^{\prime}$, gives an isomorphism
$\mathbb{O}_{k^{\prime}}[G_{k^{\prime}}]\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\mathbb{O}_{k}[G]\times_{\operatorname{Spec}(k)}\operatorname{Spec}(k^{\prime})$.
(3) Write $G=\operatorname{Spec}(A)$ and $H=\operatorname{Spec}(B)$. To a pair
of $k$-module maps $u\mathrel{\mathop{\mathchar 58\relax}}A\to R$ and
$v\mathrel{\mathop{\mathchar 58\relax}}B\to R$ we attach the map $u\otimes
v\mathrel{\mathop{\mathchar 58\relax}}A\otimes_{k}B\to R$, $a\otimes b\mapsto
u(a)v(b)$. This defines an isomorphism

$\operatorname{Hom}_{k\mbox{-}\operatorname{Mod}}(A,R)\otimes_{R}\operatorname{Hom}_{k\mbox{-}\operatorname{Mod}}(B,R)\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\operatorname{Hom}_{k\mbox{-}\operatorname{Mod}}(A\otimes_{k}B,R)$

which is functorial in $R$. The result follows. (4) If $D$ is an
$\mathbb{O}_{k}$-algebra scheme, the functorial $R$-algebra maps

$\begin{array}[]{rcl}D(R)\otimes_{R}(R\otimes_{k}k^{\prime})&\longrightarrow&D(R\otimes_{k}k^{\prime})\\\
d\otimes r^{\prime}&\longmapsto&r^{\prime}d\end{array}$

fit together to give a morphism
$D\otimes_{\mathbb{O}_{k}}h_{*}\mathbb{O}_{k^{\prime}}\to h_{*}h^{*}D$. In
case $D=\mathbb{O}_{k}[G]$ and $h$ finite locally free, this is none other
than the isomorphism
$\operatorname{Hom}_{k\mbox{-}\operatorname{Mod}}(A,R)\otimes_{k}k^{\prime}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\operatorname{Hom}_{k\mbox{-}\operatorname{Mod}}(A,R\otimes_{k}k^{\prime})$.
$\Box$

Finally we prove the adjointness property. We recall from paragraph 2.1 (see
also Remark 2.1) that $(\mathbb{O}_{k}\mbox{-}\operatorname{Alg})$ is the
category of $\mathbb{O}_{k}$-algebra schemes whose underlying
$\mathbb{O}_{k}$-module scheme is of the form
$\mathbb{V}(\mathscr{F})=\operatorname{Spec}\operatorname{S}(\mathscr{F})$ for
some quasi-coherent ${\mathcal{O}}_{\operatorname{Spec}(k)}$-module
$\mathscr{F}$, and that $(k\mbox{-}\operatorname{Gr})$ is the category of
affine $k$-group schemes.

#### 2.3.1. Theorem (Adjointness property of the group algebra).

The group algebra functor is left adjoint to the group of units functor. In
other words, for all affine $k$-group schemes $G$ and all linear
$\mathbb{O}_{k}$-algebra schemes $D$, the map that sends a morphism of algebra
schemes $\mathbb{O}_{k}[G]\to D$ to the composition
$G\subset\mathbb{O}_{k}[G]^{\times}\to D^{\times}$ gives a bifunctorial
bijection:

$\operatorname{Hom}_{\mathbb{O}_{k}\mbox{-}\operatorname{Alg}}(\mathbb{O}_{k}[G],D)\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\operatorname{Hom}_{k\mbox{-}\operatorname{Gr}}(G,D^{\times}).$

  * Proof.

We describe a map in the other direction and we leave to the reader the proof
that it is an inverse. Let $f\mathrel{\mathop{\mathchar 58\relax}}G\to
D^{\times}$ be a morphism of $k$-group schemes. We will construct a map of
functors $f^{\prime}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{O}_{k}[G]\to
D$. We know from paragraph 2.1 that $D=\mathbb{V}(F)$ where $F$ is a
$k$-module, and that the comultiplication
$\Delta_{D}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{S}(F)\to\operatorname{S}(F)\otimes\operatorname{S}(F)$
is induced by a morphism $\Delta_{0}\mathrel{\mathop{\mathchar 58\relax}}F\to
F\otimes F$. Consider the composition $G\to D^{\times}\subset D$ and let
$g\mathrel{\mathop{\mathchar 58\relax}}\operatorname{S}(F)\to A$ be the
corresponding map of algebras. For each $k$-algebra $R$ we have the equalities
$\mathbb{O}_{k}[G](R)=\operatorname{Hom}_{k\mbox{-}\operatorname{Mod}}(A,R)$
and $D(R)=\operatorname{Hom}_{k\mbox{-}\operatorname{Mod}}(F,R)$. We define
$f^{\prime}$ as follows:

$\begin{array}[]{rcl}\mathbb{O}_{k}[G](R)&\longrightarrow&D(R)\\\
(u\mathrel{\mathop{\mathchar 58\relax}}A\to R)&\longmapsto&(u\circ g\circ
i\mathrel{\mathop{\mathchar 58\relax}}F\to R)\end{array}$

where $i\mathrel{\mathop{\mathchar
58\relax}}F\hookrightarrow\operatorname{S}(F)$ is the inclusion as the degree
1 piece in the symmetric algebra. The map $f^{\prime}$ is a map of modules,
and we only have to check that it respects the multiplication. Let
$u,v\mathrel{\mathop{\mathchar 58\relax}}A\to R$ be module homomorphisms. We
have the following commutative diagram:

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta_{G}}$$\textstyle{A\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u\otimes
v}$$\textstyle{R}$$\textstyle{\operatorname{S}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta_{D}}$$\scriptstyle{g}$$\textstyle{\operatorname{S}(F)\otimes\operatorname{S}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g\otimes
g}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta_{0}}$$\scriptstyle{i}$$\textstyle{F\otimes
F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i\otimes
i}$

With the $\star$ notation as in Subsection 2.2, we compute:

$\displaystyle(u\star_{G}v)\circ g\circ i$ $\displaystyle=(u\otimes
v)\circ\Delta_{G}\circ g\circ i$ $\displaystyle=(ugi\otimes
vgi)\circ\Delta_{0}$ $\displaystyle=ugi\star_{D}vgi.$

Thus $f^{\prime}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{O}_{k}[G]\to D$
is a map of algebra schemes and this ends the construction. $\Box$

#### 2.3.2. Remark.

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

##  3\. Weil restriction

We keep the notations from the previous sections. In this section, we describe
the Weil restriction $E=h_{*}\mathscr{G}$ of a $k[I]$-group scheme
$\mathscr{G}\in\operatorname{Gr}\\!/k[I]$ and show how it carries the
structure of an object of the category of extensions
$\operatorname{Ext}(I)/k$. The necessary notions of differential calculus
(tangent bundle, Lie algebra, exponential) are recalled in Appendix 7.

###  3.1. Weil restriction of the group algebra

If $\mathscr{G}$ is an affine $k[I]$-group scheme, its Weil restriction
$h_{*}\mathscr{G}$ embeds in the pushforward algebra
$h_{*}\mathbb{O}_{k[I]}[\mathscr{G}]$. (It also embeds in the algebra
$\mathbb{O}_{k}[h_{*}\mathscr{G}]$ 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 $h_{*}\mathbb{O}_{k[I]}[\mathscr{G}]$
suited to the computation of the adjunction map $\beta_{\mathscr{G}}$. The
starting point is the following definition and lemma. Let $A$ be a
$k[I]$-algebra and $R$ a $k$-algebra. Let $v\mathrel{\mathop{\mathchar
58\relax}}A\to I\otimes_{k}R$ be a $k$-linear map such that there is a
$k$-linear map ${\bar{v}}\mathrel{\mathop{\mathchar 58\relax}}A\to R$
satisfying $v(ix)=i\bar{v}(x)$ for all $i\in I$ and $x\in A$. Then ${\bar{v}}$
is uniquely determined by $v$; in fact it is already determined by the
identity $v(ix)=i{\bar{v}}(x)$ for any fixed $i$ belonging to a basis of $I$
as a $k$-module.

#### 3.1.0. Definition.

Let $A$ be a $k[I]$-algebra and $R$ a $k$-algebra. We say that a $k$-linear
map $v\mathrel{\mathop{\mathchar 58\relax}}A\to I\otimes_{k}R$ is
$I$-compatible if there is a $k$-linear map
${\bar{v}}\mathrel{\mathop{\mathchar 58\relax}}A\to R$ such that
$v(ix)=i{\bar{v}}(x)$ for all $i\in I$ and $x\in A$. We denote by
$\operatorname{Homc}_{k}(A,I\otimes_{k}R)$ the $R$-module of $I$-compatible
maps and
$\operatorname{Homc}_{k}(A,I\otimes_{k}R)\to\operatorname{Hom}_{k}(A,R)$,
$v\mapsto{\bar{v}}$ the $R$-module map that sends $v$ to the unique map
${\bar{v}}$ with the properties above.

Note that since $I^{2}=0$ in $R[I]$, if $v$ is $I$-compatible then ${\bar{v}}$
vanishes on $IA$. In other words, the map $v\mapsto{\bar{v}}$ factors through
$\operatorname{Hom}_{k}(A/IA,R)$.

#### 3.1.1. Lemma.

Let $A$ be a $k[I]$-algebra and $R$ a $k$-algebra.

  * (1)

Each morphism of $k[I]$-modules $f\mathrel{\mathop{\mathchar 58\relax}}A\to
R[I]$ is of the form $f={\bar{v}}+v$ for a unique $I$-compatible $k$-linear
map $v\mathrel{\mathop{\mathchar 58\relax}}A\to I\otimes_{k}R$, and
conversely.

  * (2)

Each morphism of $k[I]$-algebras $f\mathrel{\mathop{\mathchar 58\relax}}A\to
R[I]$ is of the form $f={\bar{v}}+v$ as above with $v$ satisfying moreover
$v(i)=i$ for all $i\in I$ and $v(xy)={\bar{v}}(x)v(y)+{\bar{v}}(y)v(x)$ for
all $x,y\in A$, and conversely. In particular
${\bar{v}}\mathrel{\mathop{\mathchar 58\relax}}A\to R$ is a $k$-algebra
homomorphism and $v\mathrel{\mathop{\mathchar 58\relax}}A\to I\otimes_{k}R$ is
a ${\bar{v}}$-derivation.

  * Proof.

(1) Using the decomposition $R[I]=R\oplus I\otimes_{k}R$, we can write
$f(x)=u(x)+v(x)$ for some unique $k$-linear maps $u\mathrel{\mathop{\mathchar
58\relax}}A\to R$ and $v\mathrel{\mathop{\mathchar 58\relax}}A\to
I\otimes_{k}R$. Then $f$ is $k[I]$-linear if and only if $f(ix)=if(x)$ for all
$i\in I$ and $x\in A$. Taking into account that $I^{2}=0$, this means that $v$
is $I$-compatible and $u={\bar{v}}$. (2) The condition $f(1)=1$ means that
${\bar{v}}(1)=1$, that is $v(i)=i$ for all $i\in I$, and $v(1)=0$. The
condition of multiplicativity of $f$ means that ${\bar{v}}$ is multiplicative
and $v$ is a ${\bar{v}}$-derivation, i.e.
$v(xy)={\bar{v}}(x)v(y)+{\bar{v}}(y)v(x)$. In the presence of the derivation
property, the multiplicativity of ${\bar{v}}$ is automatic (computing $v(ixy)$
in two different ways) as well as the condition $v(1)=0$ (setting $x=y=1$).
Conversely if $v$ is $I$-compatible with $v(i)=i$ and
$v(xy)={\bar{v}}(x)v(y)+{\bar{v}}(y)v(x)$, one sees that ${\bar{v}}$ is a
morphism of rings and $f={\bar{v}}+v$ is a morphism of $k[I]$-algebras. $\Box$

Now let $\mathscr{G}$ be an affine $k[I]$-group scheme. Lemma 3.1 shows that
the Weil restriction $h_{*}\mathbb{V}(\mathscr{G}/k[I])$ can be described in
terms of the scheme of $I$-compatible maps, defined as a functor on
$k$-algebras by:

$\mathbb{O}_{\mathrm{c}}(\mathscr{G})(R)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Homc}_{k}(A,I\otimes_{k}R).$

We know that $h_{*}\mathbb{V}(\mathscr{G}/k[I])$ supports the algebra scheme
structure $h_{*}\mathbb{O}_{k[I]}[\mathscr{G}]$, and we will now identify the
multiplication induced on $\mathbb{O}_{\mathrm{c}}(\mathscr{G})$ by means of
this isomorphism.

#### 3.1.2. Proposition.

Let $\mathscr{G}=\operatorname{Spec}(A)$ be an affine $k[I]$-group scheme with
comultiplication $\Delta\mathrel{\mathop{\mathchar 58\relax}}A\to
A\otimes_{k[I]}A$ and counit $e\mathrel{\mathop{\mathchar 58\relax}}A\to
k[I]$, with $e={\bar{d}}+d$ for a unique $I$-compatible $k$-linear map
$d\mathrel{\mathop{\mathchar 58\relax}}A\to I$.

  * (1)

Let $R$ be a $k$-algebra and let $v,w\mathrel{\mathop{\mathchar 58\relax}}A\to
I\otimes_{k}R$ be two $I$-compatible $k$-linear maps. Then the morphism
${\bar{v}}\otimes_{k}w+v\otimes_{k}{\bar{w}}\mathrel{\mathop{\mathchar
58\relax}}A\otimes_{k}A\to R$ factors through a well-defined $k$-linear
morphism

${\bar{v}}\otimes_{k}w+v\otimes_{k}{\bar{w}}\mathrel{\mathop{\mathchar
58\relax}}A\otimes_{k[I]}A\to R.$

  * (2)

For $v,w$ as before let:

$v\diamond w\mathrel{\mathop{\mathchar
58\relax}}=({\bar{v}}\otimes_{k}w+v\otimes_{k}{\bar{w}})\circ\Delta.$

Then $(\mathbb{O}_{\mathrm{c}}(\mathscr{G}),+,\diamond)$ is an associative
unitary $\mathbb{O}_{k}$-algebra with multiplicative unit $d$, and the map

$\theta_{\mathscr{G}}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{O}_{\mathrm{c}}(\mathscr{G})\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}h_{*}\mathbb{O}_{k[I]}[\mathscr{G}],\quad
v\longmapsto{\bar{v}}+v$

is an isomorphism of associative unitary $\mathbb{O}_{k}$-algebras.

  * Proof.

(1) The $k$-linear morphism ${\bar{v}}\otimes_{k}w+v\otimes_{k}{\bar{w}}$
takes the same value $i{\bar{v}}(a){\bar{w}}(b)$ on the tensors $ia\otimes b$
and $a\otimes ib$ for all $i\in I$, $a,b\in A$. Therefore it vanishes on
tensors of the form $(a\otimes b)(i\otimes 1-1\otimes i)$. Since these tensors
generate the kernel of the ring map $A\otimes_{k}A\to A\otimes_{k[I]}A$, we
obtain an induced morphism
${\bar{v}}\otimes_{k}w+v\otimes_{k}{\bar{w}}\mathrel{\mathop{\mathchar
58\relax}}A\otimes_{k[I]}A\to R$. (2) According to (1) the definition of
$v\diamond w$ makes sense. For the rest of the statement, it is enough to
prove that $\theta_{\mathscr{G}}(v\diamond
w)=\theta_{\mathscr{G}}(v)\star\theta_{\mathscr{G}}(w)$ because if this is the
case then all the known properties of the product $\star$ in
$h_{*}\mathbb{O}_{k[I]}[\mathscr{G}]$ are transferred to $\diamond$ by the
isomorphism $\theta_{\mathscr{G}}$. On one hand, using the expression
$v\diamond w=({\bar{v}}\otimes_{k}w+v\otimes_{k}{\bar{w}})\circ\Delta$ and the
fact that $\Delta$ is $k[I]$-linear, we find $\overline{v\diamond
w}=({\bar{v}}\otimes_{k}{\bar{w}})\circ\Delta$ hence:

$\displaystyle\theta_{\mathscr{G}}(v\diamond w)$
$\displaystyle=\big{(}{\bar{v}}\otimes_{k}{\bar{w}}\big{)}\circ\Delta+\big{(}{\bar{v}}\otimes_{k}w+v\otimes_{k}{\bar{w}}\big{)}\circ\Delta$
$\displaystyle=\big{[}{\bar{v}}\otimes_{k}{\bar{w}}+({\bar{v}}\otimes_{k}w+v\otimes_{k}{\bar{w}})\big{]}\circ\Delta.$

On the other hand, we have:

$\theta_{\mathscr{G}}(v)\star\theta_{\mathscr{G}}(w)=\big{[}({\bar{v}}+v)\otimes_{k}({\bar{w}}+w)\big{]}\circ\Delta.$

The maps in the brackets are equal, whence $\theta_{\mathscr{G}}(v\diamond
w)=\theta_{\mathscr{G}}(v)\star\theta_{\mathscr{G}}(w)$ as desired. $\Box$

#### 3.1.3. Remark.

If $(\varepsilon_{1},\dots,\varepsilon_{r})$ is a basis of $I$ we have a
concrete description as follows. A $k$-linear map $v\mathrel{\mathop{\mathchar
58\relax}}A\to I\otimes_{k}R$ can be written
$v=\varepsilon_{1}v_{1}+\dots+\varepsilon_{r}v_{r}$ for some maps
$v_{j}\mathrel{\mathop{\mathchar 58\relax}}A\to R$. Then $v$ is $I$-compatible
if and only if $v_{j}\varepsilon_{i}=\delta_{i,j}{\bar{v}}$ for all $i,j$. If
this is the case, $v_{j}$ induces a $k$-linear morphism
$A\otimes_{k[I]}k[\varepsilon_{j}]\to R$ and ${\bar{v}}=v_{j}\varepsilon_{j}$
for each $j$. Now write
$\mathscr{G}_{j}=\mathscr{G}\otimes_{k[I]}k[\varepsilon_{j}]$ and
$h_{j}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Spec}(k[\varepsilon_{j}])\to\operatorname{Spec}(k)$
the structure map. Also let $\mathscr{G}_{k}=\mathscr{G}\otimes_{k[I]}k$ so we
have maps
$\mathbb{V}(h_{j,!}\mathscr{G}_{j}/k)\to\mathbb{V}(\mathscr{G}_{k}/k)$,
$v_{j}\mapsto{\bar{v}}\mathrel{\mathop{\mathchar
58\relax}}=v_{j}\varepsilon_{j}$. Then we have an isomorphism:

$\mathbb{V}(h_{1,!}\mathscr{G}_{1}/k)\underset{\mathbb{V}(\mathscr{G}_{k}/k)}{\times}\dots\underset{\mathbb{V}(\mathscr{G}_{k}/k)}{\times}\mathbb{V}(h_{r,!}\mathscr{G}_{r}/k)\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\mathbb{O}_{\mathrm{c}}(\mathscr{G})$

given by $(v_{1},\dots,v_{r})\mapsto
v=\varepsilon_{1}v_{1}+\dots+\varepsilon_{r}v_{r}$.

###  3.2. Kernel of the adjunction $\beta\mathrel{\mathop{\mathchar
58\relax}}h^{*}h_{*}\mathscr{G}\to\mathscr{G}$

Again let $\mathscr{G}=\operatorname{Spec}(A)$ be an affine $k[I]$-group
scheme. Denote by $\Delta\mathrel{\mathop{\mathchar 58\relax}}A\to
A\otimes_{k[I]}A$ the comultiplication and $e\mathrel{\mathop{\mathchar
58\relax}}A\to k[I]$ the counit, of the form $e={\bar{d}}+d$ for a unique
$I$-compatible $k$-linear map $d\mathrel{\mathop{\mathchar 58\relax}}A\to I$.
The purpose of this subsection is to prove the following proposition.

#### 3.2.0. Proposition.

Let $\mathscr{G}$ be an affine $k[I]$-group scheme and
$\mathscr{G}_{k}=i^{*}\mathscr{G}$. Let
$\beta=\beta_{\mathscr{G}}\mathrel{\mathop{\mathchar
58\relax}}h^{*}h_{*}\mathscr{G}\to\mathscr{G}$ be the adjunction, and
$\mathrm{L}(\mathscr{G})\mathrel{\mathop{\mathchar 58\relax}}=\ker(\beta)$.

  * (1)

Functoriality. The formation of $\mathrm{L}(\mathscr{G})$ is functorial for
morphisms of pointed $k[I]$-schemes, and for morphisms of $k[I]$-group
schemes.

  * (2)

Explicit description. We have embeddings of monoids
$\mathrm{L}(\mathscr{G})\subset
h^{*}h_{*}\mathscr{G}\subset(h^{*}\mathbb{O}_{\mathrm{c}}(\mathscr{G}),\diamond)$
under which, functorially in the $k[I]$-algebra $R$:

$(h^{*}h_{*}\mathscr{G})(R)=\left\\{v\in(h^{*}\mathbb{O}_{\mathrm{c}}(\mathscr{G}))(R)\
\left|\ \begin{array}[]{l}v(i)=i\mbox{ for all }i\in I\\\
v(xy)={\bar{v}}(x)v(y)+{\bar{v}}(y)v(x)\mbox{ for all }x,y\in
A\end{array}\right.\right\\}$

and if $v_{R}\mathrel{\mathop{\mathchar 58\relax}}A\to R$ denotes the
composition $A\stackrel{{\scriptstyle
v}}{{\longrightarrow}}I\otimes_{k}R\stackrel{{\scriptstyle i\otimes r\mapsto
ir}}{{\relbar\joinrel\relbar\joinrel\longrightarrow}}R$:

$\mathrm{L}(\mathscr{G})(R)=\big{\\{}v\in(h^{*}h_{*}\mathscr{G})(R)\
\mathrel{\mathop{\mathchar 58\relax}}\ {\bar{v}}+v_{R}=e_{R}\big{\\}}.$

  * (3)

Special fibre. There is a functorial isomorphism
$i^{*}\mathrm{L}(\mathscr{G})\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\operatorname{Lie}(\mathscr{G}_{k},I)$,
$v\mapsto v-d$. Under this isomorphism, the action by conjugation of
$h_{*}\mathscr{G}=i^{*}h^{*}h_{*}\mathscr{G}$ on
$i^{*}\mathrm{L}(\mathscr{G})$ is given by the morphism:

$h_{*}\mathscr{G}\stackrel{{\scriptstyle
i^{*}\beta}}{{\relbar\joinrel\relbar\joinrel\longrightarrow}}\mathscr{G}_{k}\stackrel{{\scriptstyle\operatorname{Ad}}}{{\relbar\joinrel\relbar\joinrel\longrightarrow}}\operatorname{GL}(\operatorname{Lie}(\mathscr{G}_{k},I)).$

  * (4)

Case of trivial deformation groups. If $\mathscr{G}=h^{*}G$ for some affine
$k$-group scheme $G$, there is a canonical and functorial isomorphism
$\mathrm{L}(\mathscr{G})\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}h^{*}\operatorname{Lie}(G,I)$.
More precisely, let $\exp_{G,I}\mathrel{\mathop{\mathchar
58\relax}}h^{*}\operatorname{Lie}(G,I)\to h^{*}G$ be the exponential morphism
as defined in A.2. Then under the isomorphism (see A.1)

$\varrho_{G}\mathrel{\mathop{\mathchar 58\relax}}\operatorname{Lie}(G,I)\times
G\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\operatorname{T}(G,I),$

the subgroup $\mathrm{L}(\mathscr{G})\subset h^{*}\operatorname{T}(G,I)$ has
for points the pairs $(x,g)\in h^{*}\operatorname{Lie}(G,I)\times h^{*}G$ such
that $g=\exp(-x)$, and the isomorphism is given by $(x,g)\mapsto x$.

  * Proof.

(1) If $\varphi\mathrel{\mathop{\mathchar
58\relax}}\mathscr{G}\to\mathscr{G}^{\prime}$ is a morphism of pointed
schemes, then by functoriality of $\beta$ the morphism $h_{*}h^{*}\varphi$
takes the kernel of $\beta_{\mathscr{G}}$ into the kernel of
$\beta_{\mathscr{G}^{\prime}}$. If moreover $\varphi$ is a map of group
schemes then the restriction of $h_{*}h^{*}\varphi$ to
$\mathrm{L}(\mathscr{G})$ also. (2) In the rest of the proof we use the
possibility to compute in the group algebra
$(\mathbb{O}_{\mathrm{c}}(\mathscr{G}),+,\diamond)$, see Proposition 3.1. The
description of $h^{*}h_{*}\mathscr{G}$ as a submonoid of the multiplicative
monoid of $\mathbb{O}_{\mathrm{c}}(\mathscr{G})$ is copied from Lemma 3.1. The
description of $\mathrm{L}(\mathscr{G})$ follows from the fact that for
$f\in\mathscr{G}(R[I])$, $f={\bar{v}}+v\mathrel{\mathop{\mathchar
58\relax}}A\to R[I]$, the image $\beta(f)\in\mathscr{G}(R)$,
$\beta(f)\mathrel{\mathop{\mathchar 58\relax}}A\to R$ is the map
${\bar{v}}+v_{R}$. (3) – first claim. The pullback $i^{*}$ is the restriction
to the category of those $k[I]$-algebras $R$ such that $IR=0$. For such an
$R$, an element $v\in(i^{*}\mathrm{L}(\mathscr{G}))(R)$ is a $k$-linear map
$v\mathrel{\mathop{\mathchar 58\relax}}A\to R$ such that $v(i)=i$,
$v(xy)={\bar{v}}(x)v(y)+{\bar{v}}(y)v(x)$ and ${\bar{v}}=e_{R}$. In particular
we see that $v$ is an $e_{R}$-derivation. Since also $d_{R}$ is an
$e_{R}$-derivation with $d_{R}(i)=i$ for all $i\in I$, the difference
$\delta\mathrel{\mathop{\mathchar 58\relax}}=v-d_{R}$ induces an
$e_{R}$-derivation $\delta\mathrel{\mathop{\mathchar 58\relax}}A\to
I\otimes_{k}R$ vanishing on $IA$, i.e. an $R$-point of
$\operatorname{Lie}(\mathscr{G}_{k},I)$. Conversely, any $e_{R}$-derivation
$\delta\mathrel{\mathop{\mathchar 58\relax}}A\to I\otimes_{k}R$ vanishing on
$IA$ gives rise to a $k$-linear map $v\mathrel{\mathop{\mathchar
58\relax}}A\to I\otimes_{k}R$ defined by $v\mathrel{\mathop{\mathchar
58\relax}}=d_{R}+\delta$ and satisfying the properties required to be a point
of $(i^{*}\mathrm{L}(\mathscr{G}))(R)$. Finally let
$\delta_{1},\delta_{2}\in\operatorname{Lie}(\mathscr{G}_{k},I)(R)$. Since
$\delta_{1},\delta_{2}$ vanish on $IA$, we have
$d_{R}^{*}+\delta_{1}^{*}=d_{R}^{*}+\delta_{2}^{*}=e_{R}$ and then:

$\displaystyle(d_{R}+\delta_{1})\diamond(d_{R}+\delta_{2})$
$\displaystyle=\big{[}e_{R}\otimes(d_{R}+\delta_{2})+(d_{R}+\delta_{1})\otimes
e_{R}\big{]}\circ\Delta$ $\displaystyle=\big{[}(e_{R}\otimes
d_{R}+d_{R}\otimes e_{R})+(e_{R}\otimes\delta_{2}+\delta_{1}\otimes
e_{R})\big{]}\circ\Delta.$

All three morphisms $e_{R}\otimes d_{R}+d_{R}\otimes e_{R}$,
$e_{R}\otimes\delta_{2}$ and $\delta_{1}\otimes e_{R}$ factor through
$A\otimes_{k[I]}A$, so the precomposition with $\Delta$ is distributive for
them. Since also $d_{R}$ is the neutral element for the law $\diamond$ and
$e_{R}$ is the neutral element for the law $\star$, we obtain:

$\displaystyle(d_{R}+\delta_{1})\diamond(d_{R}+\delta_{2})$
$\displaystyle=\big{[}e_{R}\otimes d_{R}+d_{R}\otimes
e_{R}\big{]}\circ\Delta+\big{[}e_{R}\otimes\delta_{2}\big{]}\circ\Delta+\big{[}\delta_{1}\otimes
e_{R}\big{]}\circ\Delta$ $\displaystyle=d_{R}+\delta_{1}+\delta_{2}.$

This shows that the isomorphism
$i^{*}\mathrm{L}(\mathscr{G})\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\operatorname{Lie}(\mathscr{G}_{k})$,
$v\mapsto v-d_{R}$ is a morphism of groups. (3) – second claim. Let $R$ be a
$k[I]$-algebra such that $IR=0$. The action of an element
$u\in(h^{*}h_{*}\mathscr{G})(R)$ on an element
$v=d_{R}+\delta\in\mathrm{L}(\mathscr{G})(R)$ by conjugation can be computed
in the algebra $h^{*}\mathbb{O}_{\mathrm{c}}(\mathscr{G})$:

$u\diamond v\diamond u^{-1}=u\diamond(d_{R}+\delta)\diamond u^{-1}=u\diamond
d_{R}\diamond u^{-1}+u\diamond\delta\diamond
u^{-1}=d_{R}+u\diamond\delta\diamond u^{-1}.$

We see that $u$ is acting on $\delta$ by conjugation in the group algebra.
This is the adjoint action, as explained in Proposition A.3. (4) When
$\mathscr{G}=h^{*}G$, the adjunction map is the infinitesimal translation as
in Proposition A.2(3). The $R$-points of $\mathrm{L}(\mathscr{G})$ are the
pairs $(x,g)$ such that $\exp(x)g=1$ in $G(R)$. This amounts to $g=\exp(-x)$
which proves (4). $\Box$

###  3.3. Extension structure of the Weil restriction

Let $\mathscr{G}$ be a $k[I]$-group scheme. The notion of rigidification for
$\mathscr{G}$ and the property that $\mathscr{G}$ be rigid are defined in 1.1.
Here are some remarks.

  * (1)

If $\sigma\mathrel{\mathop{\mathchar
58\relax}}h^{*}\mathscr{G}_{k}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\mathscr{G}$
is a rigidification, then $\sigma(1)^{-1}\cdot\sigma$ is another. Therefore if
there exists a rigidification $\sigma$, we may always assume moreover that
$\sigma(1)=1$.

  * (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 $\alpha\mathrel{\mathop{\mathchar
58\relax}}\mathscr{G}\to\mathscr{G}^{\prime}$ is a morphism between rigid
$k[I]$-group schemes, it is not always possible to choose rigidifications for
$\mathscr{G}$ and $\mathscr{G}^{\prime}$ that are compatible in the sense that
$\sigma^{\prime}\circ h^{*}\alpha=\alpha\circ\sigma$. For instance let
$I=k\varepsilon$ and let $\alpha\mathrel{\mathop{\mathchar
58\relax}}\mathbb{G}_{a}\to\mathbb{G}_{a}$ be the morphism defined by
$\alpha(x)=\varepsilon x$. Then $\mathscr{G}$ and $\mathscr{G}^{\prime}$ are
rigid but since $h^{*}\alpha=0$, there do not exist compatible
rigidifications.

#### 3.3.0. Lemma.

Let $\mathscr{G}$ be a $k[I]$-group scheme such that the restriction
homomorphism $\mathscr{G}_{k}=i^{*}\mathscr{G}$ is $k$-flat. Let
$\pi\mathrel{\mathop{\mathchar
58\relax}}=i^{*}\beta_{\mathscr{G}}\mathrel{\mathop{\mathchar
58\relax}}h_{*}\mathscr{G}\to\mathscr{G}_{k}$. Then the adjunction
$\operatorname{Hom}_{\operatorname{Sch}/k[I]}(h^{*}\mathscr{G}_{k},\mathscr{G})=\operatorname{Hom}_{\operatorname{Sch}/k}(\mathscr{G}_{k},h_{*}\mathscr{G})$
induces a bijection between rigidifications of $\mathscr{G}$ and sections of
$\pi$.

  * Proof.

Let $\sigma\mathrel{\mathop{\mathchar
58\relax}}h^{*}\mathscr{G}_{k}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\mathscr{G}$
be a rigidification and $s\mathrel{\mathop{\mathchar
58\relax}}=h_{*}\sigma\circ\alpha_{\mathscr{G}_{k}}$. Then we have
$\sigma=\beta_{\mathscr{G}}\circ h^{*}s$ and by applying $i^{*}$ we find
$\operatorname{id}_{\mathscr{G}_{k}}=i^{*}\sigma=\pi\circ s$ hence $s$ is a
section of $\pi$. Conversely let $s$ be a section of $\pi$ and
$\sigma\mathrel{\mathop{\mathchar 58\relax}}=\beta_{\mathscr{G}}\circ h^{*}s$.
Then $i^{*}\sigma=i^{*}\beta_{\mathscr{G}}\circ i^{*}h^{*}s=\pi\circ
s=\operatorname{id}_{\mathscr{G}_{k}}$ hence $\sigma$ lifts the identity. In
particular $\sigma$ is an affine morphism. Since moreover
$h^{*}\mathscr{G}_{k}$ is flat, we conclude that $\sigma$ is an isomorphism,
hence a rigidification. $\Box$

#### 3.3.1. Lemma.

Let $\mathscr{G}$ be an affine, differentially flat and rigid $k[I]$-group
scheme. Then $\beta\mathrel{\mathop{\mathchar
58\relax}}h^{*}h_{*}\mathscr{G}\to\mathscr{G}$ is faithfully flat and we have
an exact sequence:

$1\longrightarrow\mathrm{L}(\mathscr{G})\longrightarrow
h^{*}h_{*}\mathscr{G}\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}\mathscr{G}\longrightarrow
1.$

If $\mathscr{G}$ is of finite type over $k[I]$, then $\beta$ is smooth.

  * Proof.

Again we put $\pi=i^{*}\beta\mathrel{\mathop{\mathchar
58\relax}}h_{*}\mathscr{G}\to\mathscr{G}_{k}$. Let
$\sigma\mathrel{\mathop{\mathchar
58\relax}}h^{*}\mathscr{G}_{k}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\mathscr{G}$
be a rigidification and $s\mathrel{\mathop{\mathchar
58\relax}}\mathscr{G}_{k}\to h_{*}\mathscr{G}$ the corresponding section of
$\pi$, see Lemma 3.3. By Proposition 3.2(3) the kernel of $\pi$ is
$\operatorname{Lie}(\mathscr{G}_{k})$ which is $k$-flat. The section $s$
provides an isomorphism of $k$-schemes
$\operatorname{Lie}(\mathscr{G}_{k})\times\mathscr{G}_{k}\simeq
h_{*}\mathscr{G}$ which shows that $h_{*}\mathscr{G}$ is $k$-flat so that
$h^{*}h_{*}\mathscr{G}$ is $k[I]$-flat. It follows also that
$\pi\mathrel{\mathop{\mathchar 58\relax}}h_{*}\mathscr{G}\to\mathscr{G}_{k}$
is faithfully flat and by the “critère de platitude par fibres” in the
nilpotent case ([SP, Tag 06A5]) we deduce that the morphism $\beta$ is
faithfully flat. Finally if $\mathscr{G}$ is of finite type over $k[I]$, then
the special fibre of $\mathrm{L}(\mathscr{G})$ is the smooth vector group
$\operatorname{Lie}(\mathscr{G}_{k})$, hence $\mathrm{L}(\mathscr{G})$ is
smooth and so is $\beta$. $\Box$

#### 3.3.2. 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 $I=k\varepsilon$ be free of rank
1. Let $\mathscr{G}$ be the kernel of the endomorphism
$\mathbb{G}_{a}\to\mathbb{G}_{a}$, $x\mapsto x^{p}-\varepsilon x$. Then
$(h^{*}h_{*}\mathscr{G})(R)$ is the set of elements $a\oplus b\varepsilon\in
R\oplus R\varepsilon$ such that $(a\oplus
b\varepsilon)^{p}=\varepsilon(a\oplus b\varepsilon)$. This equation is
equivalent to $a^{p}=\varepsilon a$, hence $a=0$. Thus $(a,b)\mapsto b$ is an
isomorphism
$h^{*}h_{*}\mathscr{G}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\mathbb{G}_{a,k[\varepsilon]}$.
The map $(h^{*}h_{*}\mathscr{G})(R)\to\mathscr{G}(R)$ sends $b$ to
$b\varepsilon_{R}$. In other words, if we let
$K_{\varepsilon}\simeq\operatorname{Spec}(k[\varepsilon][x]/(\varepsilon x))$
denote the kernel of $\varepsilon\mathrel{\mathop{\mathchar
58\relax}}\mathbb{G}_{a,k[\varepsilon]}\to\mathbb{G}_{a,k[\varepsilon]}$, then
the sequence of the lemma is $1\to
K_{\varepsilon}\to\mathbb{G}_{a,k[\varepsilon]}\stackrel{{\scriptstyle\varepsilon}}{{\longrightarrow}}\mathscr{G}\to
1$. Here, the map $\varepsilon\mathrel{\mathop{\mathchar
58\relax}}\mathbb{G}_{a,k[\varepsilon]}\to\mathscr{G}$ is not flat so this is
not an exact sequence of flat group schemes. $\square$

#### 3.3.3. Proposition.

For each $\mathscr{G}\in\operatorname{Gr}\\!/k[I]$, the restriction via
$i^{*}$ of the exact sequence of Lemma 3.3 gives $E\mathrel{\mathop{\mathchar
58\relax}}=h_{*}\mathscr{G}$ the structure of an object of
$\operatorname{Ext}(I)/k$. Hence Weil restriction gives a functor:

$h_{*}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gr}\\!/k[I]\to\operatorname{Ext}(I)/k.$

  * Proof.

From Lemma 3.3 we have an exact sequence $1\to\mathrm{L}(\mathscr{G})\to
h^{*}h_{*}\mathscr{G}\to\mathscr{G}\to 1$. It follows from point (2) in
Proposition 3.2 that when we restrict to the closed fibre, we obtain an exact
sequence:

$1\longrightarrow\operatorname{Lie}(\mathscr{G}_{k},I)\longrightarrow
E\longrightarrow\mathscr{G}_{k}\longrightarrow 1$

where the $\mathscr{G}_{k}$-action on $\operatorname{Lie}(\mathscr{G}_{k})$
induced by the extension is the adjoint representation. The same reference
proves that this extension is functorial in $\mathscr{G}$. More precisely, if
$u\mathrel{\mathop{\mathchar 58\relax}}\mathscr{G}\to\mathscr{G}^{\prime}$ is
a morphism of affine, differentially flat, rigid $k[I]$-group schemes, then we
obtain a morphism between the extensions $E=h_{*}\mathscr{G}$ and
$E^{\prime}=h_{*}\mathscr{G}^{\prime}$ as follows:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Lie}(\mathscr{G}_{k},I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{d}\\!\psi}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\mathscr{G}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Lie}(\mathscr{G}^{\prime}_{k},I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{G}^{\prime}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

where $\varphi=h_{*}u$ and $\psi=u_{k}=i^{*}u$, the restriction of $u$ along
$i\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Spec}(k)\hookrightarrow\operatorname{Spec}(k[I])$.
$\Box$

We draw a corollary that will be useful in Section 5.

#### 3.3.4. Corollary.

Let $Y$ be an affine, flat, rigid $k[I]$-scheme and
$\mathbb{V}(Y)=\mathbb{V}(Y/k[I])$, $\mathbb{V}(Y_{k})=\mathbb{V}(Y_{k}/k)$
the vector bundle envelopes. Then we have a split exact sequence of flat
$k$-group schemes:

$0\longrightarrow\mathbb{V}(Y_{k})\otimes_{\mathbb{O}_{k}}\mathbb{V}(I^{{\scriptscriptstyle\vee}})\longrightarrow
h_{*}\mathbb{V}(Y)\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\mathbb{V}(Y_{k})\longrightarrow
0.$

Moreover $h_{*}\mathbb{V}(Y)$ is flat as an $\mathbb{O}_{k}[I]$-module scheme
and the surjection $\pi$ is isomorphic to the map given by reduction modulo
$I$.

  * Proof.

Recall from paragraph 2.1 that $\mathbb{O}_{k}[I]\mathrel{\mathop{\mathchar
58\relax}}=h_{*}\mathbb{O}_{k[I]}$. By the assumptions on $Y$, the vector
bundle $\mathbb{V}(Y)$ is affine, differentially flat and rigid over $k[I]$.
Thus Proposition 3.3 yields the displayed exact sequence. Using a
rigidification for $Y$ and base change for the functor $\mathbb{V}$, we have
$h_{*}\mathbb{V}(Y)\simeq h_{*}\mathbb{V}(h^{*}Y_{k})\simeq
h_{*}h^{*}\mathbb{V}(Y_{k})$. Like in Proposition 2.3(4), we have an
isomorphism of $\mathbb{O}_{k}$-modules
$\mathbb{V}(Y_{k})\otimes_{\mathbb{O}_{k}}\mathbb{O}_{k}[I]\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}h_{*}h^{*}\mathbb{V}(Y_{k})$
defined by $x\otimes a\mapsto ax$. Given that $\mathbb{V}(Y_{k})$ is
$\mathbb{O}_{k}$-flat, this proves that $h_{*}\mathbb{V}(Y)$ is
$\mathbb{O}_{k}[I]$-flat. Even more, we have an isomorphism
$h_{*}\mathbb{V}(Y)\simeq\mathbb{V}(Y_{k})\oplus(\mathbb{V}(Y_{k})\otimes_{\mathbb{O}_{k}}\mathbb{V}(I^{{\scriptscriptstyle\vee}}))$
and $\pi$ is the projection onto the first factor, i.e. the map given by
reduction modulo $I$. $\Box$

##  4\. Weil extension

In this section, we construct a functor $h^{\mbox{\tiny+}}$ 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 $\mathscr{G}$ from the extension
$E=h_{*}\mathscr{G}$ by looking at the target of the adjunction
$\beta_{\mathscr{G}}\mathrel{\mathop{\mathchar
58\relax}}h^{*}E=h^{*}h_{*}\mathscr{G}\to\mathscr{G}$. In turn, in order to
reconstruct the faithfully flat morphism $\beta_{\mathscr{G}}$ it is enough to
know its kernel $K$. In the case where $\mathscr{G}$ is a constant group
$h^{*}G$, which in other words is the case where $E$ is a tangent bundle
$\operatorname{T}(G,I)$, Proposition 3.2(4) hints the correct expression
$K=\\{(x,g)\in h^{*}E;\ g=\exp(-x)\\}$. The definition of $K$ for general
extensions $1\to\operatorname{Lie}(G,I)\to E\to G\to 1$ where $G$ is an
affine, differentially flat $k$-group scheme, builds on this intuition.

###  4.1. 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

$1\longrightarrow M\stackrel{{\scriptstyle
i}}{{\longrightarrow}}E\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}G$

such that $\pi$ has a section (which is not required to be a morphism of
groups). From a given section $s\mathrel{\mathop{\mathchar 58\relax}}G\to E$,
we can produce a unique morphism $c\mathrel{\mathop{\mathchar
58\relax}}G\times G\to M$ such that
$i(c(g,g^{\prime}))\mathrel{\mathop{\mathchar
58\relax}}=s(g)s(g^{\prime})s(gg^{\prime})^{-1}$. This is a 2-cocycle, i.e. it
satisfies the identity

$c(g,g^{\prime})+c(gg^{\prime},g^{\prime\prime})=g\cdot
c(g^{\prime},g^{\prime\prime})+c(g,g^{\prime}g^{\prime\prime}).$

Note that we may always replace $s$ by the section $G\to E$, $g\mapsto
s(1)^{-1}s(g)$ to obtain a section such that $s(1)=1$. When this is the case,
we have $c(g,1)=c(1,g^{\prime})=0$ for all $g,g^{\prime}$ and we say that $c$
is normalized. Conversely, starting from a cocycle $c$, the functor
$E_{c}=M\times G$ with multiplication defined by

$(x,g)\cdot(x^{\prime},g^{\prime})\mathrel{\mathop{\mathchar
58\relax}}=\big{(}x+g\cdot x^{\prime}+c(g,g^{\prime}),gg^{\prime}\big{)}$

is an $H$-extension. The map $s\mathrel{\mathop{\mathchar 58\relax}}G\to
E_{c}$, $g\mapsto(0,g)$ is a possible choice of section for $\pi$. It follows
from the previous comments that we may always change the cocycle into a
normalized cocycle.

###  4.2. Kernel of the adjunction, reprise

In this subsection, we prepare the construction of the kernel of the
adjunction map $\beta_{h^{\mbox{\tiny+}}E}$ of the (yet to be produced) Weil
extension $h^{\mbox{\tiny+}}E$. The end result is in Proposition 4.3 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 $\operatorname{Lie}(G,I)$ its Lie algebra relative to $I$, viewed
as an affine $k$-group scheme with the adjoint action of $G$. To any 2-cocycle
$c\mathrel{\mathop{\mathchar 58\relax}}G\times
G\longrightarrow\operatorname{Lie}(G,I)$ we attach as before an $H$-extension
$E_{c}=\operatorname{Lie}(G,I)\times G$ with multiplication:

$(x,g)\cdot(x^{\prime},g^{\prime})\mathrel{\mathop{\mathchar
58\relax}}=\big{(}x+\operatorname{Ad}(g)x^{\prime}+c(g,g^{\prime}),gg^{\prime}\big{)}.$

Our group $E_{c}$ has a structure of $H$-extension:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Lie}(G,I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\mapsto(x,1)}$$\textstyle{E_{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(x,g)\mapsto
g}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1.}$

The following result is the heart of the construction of the Weil extension
functor $h^{\mbox{\tiny+}}$. We point out that among the groups
$K_{\lambda}(E_{c})$ introduced here, it is especially $K_{-1}(E_{c})$ that
will be relevant in the sequel, as Proposition 3.2(4) shows. However, we
include the whole family $K_{\lambda}(E_{c})$ since it comes with no extra
cost and brings interesting insight, in the sense that it ultimately provides
an explicit linear path in the $\mathbb{O}_{k}$-module stack
$\operatorname{Gr}/k[I]$.

#### 4.2.0. Proposition.

Let $h\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Spec}(k[I])\to\operatorname{Spec}(k)$ be the
$k$-scheme of dual numbers. Fix $\lambda\in k$.

  * (1)

Let $G$ be an affine $k$-group scheme and let $E_{c}$ be the $H$-extension
constructed out of a normalized 2-cocycle $c\mathrel{\mathop{\mathchar
58\relax}}G\times G\longrightarrow\operatorname{Lie}(G,I)$. Let
$K_{\lambda}(E_{c})\subset h^{*}E_{c}$ be the subfunctor defined by:

$K_{\lambda}(E_{c})=\\{(x,g)\in h^{*}E_{c}\,;\,g=\exp(\lambda x)\\}.$

Then $K_{\lambda}(E_{c})$ is a closed normal sub-$k[I]$-group scheme of
$h^{*}E_{c}$.

  * (2)

Let $G,G^{\prime}$ be affine $k$-group schemes and $E_{c},E_{c^{\prime}}$ be
the $H$-extensions constructed out of some chosen normalized 2-cocycles
$c,c^{\prime}$. Let $f\mathrel{\mathop{\mathchar 58\relax}}E_{c}\to
E_{c^{\prime}}$ be a morphism of extensions:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Lie}(G,I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Lie}(\alpha)}$$\textstyle{E_{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Lie}(G^{\prime},I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{c^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

Then $(h^{*}f)(K_{\lambda}(E_{c}))\subset K_{\lambda}(E_{c^{\prime}})$, with
equality if $f$ is an isomorphism.

When the extension $E_{c}$ is clear from context, we write $K_{\lambda}$
instead of $K_{\lambda}(E_{c})$. 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.

#### 4.2.1. Lemma.

Let $\alpha\mathrel{\mathop{\mathchar 58\relax}}G\to G^{\prime}$ be a morphism
of groups and $\delta\mathrel{\mathop{\mathchar 58\relax}}L\to L^{\prime}$ be
a morphism from a $G$-module to a $G^{\prime}$-module which is
$\alpha$-equivariant. Let $E\in{\mathcal{E}}xt(G,L)$ and
$E^{\prime}\in{\mathcal{E}}xt(G^{\prime},L^{\prime})$ be two extensions.

  * (1)

There exists a morphism of extensions $f\mathrel{\mathop{\mathchar
58\relax}}E\to E^{\prime}$, i.e. a diagram

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\scriptstyle{f}$
$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1,}$

if and only if $\alpha^{*}[E^{\prime}]=\delta_{*}[E]$ in
$\mathrm{H}^{2}(G,L^{\prime})$, and if this condition holds then the set of
morphisms is a principal homogeneous space under the set of 1-cocycles
$\mathrm{Z}^{1}(G,L^{\prime})$. More precisely, assume that we describe $E$
with a normalized cocycle $c\mathrel{\mathop{\mathchar 58\relax}}G\times G\to
L$ so that $E\simeq L\times G$ with multiplication
$(x,g)\cdot(x^{\prime},g^{\prime})=(x+g\cdot
x^{\prime}+c(g,g^{\prime}),gg^{\prime})$, and we describe $E^{\prime}$
similarly with a normalized cocycle $c^{\prime}$. Then all morphisms
$f\mathrel{\mathop{\mathchar 58\relax}}E\to E^{\prime}$ are of the form
$f(x,g)=(\delta(x)+\varphi(g),\alpha(g))$ for a unique 1-cochain
$\varphi\mathrel{\mathop{\mathchar 58\relax}}G\to L^{\prime}$ such that
$\partial\varphi=c^{\prime}\circ\alpha-\delta\circ c$.

  * (2)

If $E,E^{\prime}$ are two extensions of $G$ by $L$, then the set of morphisms
of extensions $E\to E^{\prime}$ is a principal homogeneous space under the
group $\mathrm{Z}^{1}(G,L)$, more precisely all morphisms are of the form
$f(x,g)=(x+\varphi(g),g)$ for a unique $\varphi\in\mathrm{Z}^{1}(G,L)$. All of
them are isomorphisms.

  * (3)

Assume that the extension is trivial, so that $[E]=0\in\mathrm{H}^{2}(G,L)$.
Then all group sections $G\to E$ of the extension are of the form
$s(x,g)=(\varphi(g),g)$ for a unique $\varphi\in\mathrm{Z}^{1}(G,L)$ such that
$\partial\varphi=c$.

  * Proof.

Any morphism of extensions can be written as $f\mathrel{\mathop{\mathchar
58\relax}}L\times G\to L^{\prime}\times G^{\prime}$,
$(x,g)\mapsto(u(x,g),\alpha(g))$ with $u(x,1)=\delta(x)$. The property that
$f$ is a morphism of groups translates into the identity:

$u(x_{1}+g_{1}\cdot
x_{2}+c(g_{1},g_{2}),g_{1}g_{2})=u(x_{1},g_{1})+\alpha(g_{1})\cdot
u(x_{2},g_{2})+c^{\prime}(\alpha g_{1},\alpha g_{2}).$

Setting $x_{1}=x$, $x_{2}=0$, $g_{1}=1$, $g_{2}=g$, and
$\varphi(g)\mathrel{\mathop{\mathchar 58\relax}}=u(0,g)$ we find
$u(x,g)=\delta(x)+\varphi(g)$ for all $x,g$. The above identity implies
$\varphi(g_{1}g_{2})-\varphi(g_{1})-\alpha(g_{1})\cdot\varphi(g_{2})=c^{\prime}(\alpha
g_{1},\alpha g_{2})-\delta(c(g_{1},g_{2}))$. This means that
$\partial\varphi=c^{\prime}\circ\alpha-\delta\circ c$ as claimed in (1).
Considering the particular case of morphisms

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

we get (2), and considering the case of morphisms

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$
$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

we get (3). $\Box$

We come back to the extension $E_{c}$. The lemma tells us that the group
$\operatorname{Aut}_{\operatorname{ext}}(E_{c})$ of automorphisms of $E_{c}$
as an extension is isomorphic to the group of 1-cocycles
$\mathrm{Z}^{1}(G,\operatorname{Lie}(G,I))$. Item (2) of Proposition 4.2 says
in particular that $K_{\lambda}(E_{c})$ 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
$\exp_{G}$.

#### 4.2.2. Lemma.

Let $G$ be an affine $k$-group scheme and $c\mathrel{\mathop{\mathchar
58\relax}}G\times G\longrightarrow\operatorname{Lie}(G,I)$ a normalized
2-cocycle. Let $\exp\mathrel{\mathop{\mathchar
58\relax}}h^{*}\operatorname{Lie}(G,I)\to h^{*}G$ be the exponential morphism
as defined in Subsection A.2. Let $R$ be a $k[I]$-algebra and
$g,g^{\prime},g^{\prime\prime}\in G(R)$. Assume that $g$ is an exponential
i.e. an element in the image of $\exp$. Then, we have:

  * (1)

$\exp(c(g,g^{\prime}))=\exp(c(g^{\prime},g))=1$,

  * (2)

$\exp(c(gg^{\prime},g^{\prime\prime}))=\exp(c(g^{\prime}g,g^{\prime\prime}))=\exp(c(g^{\prime},gg^{\prime\prime}))=\exp(c(g^{\prime},g^{\prime\prime}g))=\exp(c(g^{\prime},g^{\prime\prime}))$.

The same statements hold with $c$ replaced by $\lambda c$, for each
$\lambda\in k$.

  * Proof.

(1) Apply Lemma A.2 to the morphism of pointed schemes
$\phi=c(-,g^{\prime})\mathrel{\mathop{\mathchar
58\relax}}G\otimes_{k}R\to(\operatorname{Lie}(G,I))\otimes_{k}R$ to obtain
$\exp(c(g,g^{\prime}))=1$. Similarly $\exp(c(g^{\prime},g))=1$. (2) Write
$g=\exp(x)$. Since
$\operatorname{Lie}(G,I)=\operatorname{Lie}(G)\otimes\mathbb{V}(I^{\scriptscriptstyle\vee})$
we can write $x$ as a sum of tensors $y\otimes i$. Working inductively on the
number of tensors in the sum, we can assume that $x=y\otimes i$. We prove
successively that each of the first four terms equals
$\exp(c(g^{\prime},g^{\prime\prime}))$.

    1. a.

The cocycle identity
$c(g,g^{\prime})\,+\,c(gg^{\prime},g^{\prime\prime})\,=\,\operatorname{Ad}(g)\,c(g^{\prime},g^{\prime\prime})\,+\,c(g,g^{\prime}g^{\prime\prime})$
together with (1) imply
$\exp(c(gg^{\prime},g^{\prime\prime}))=\exp(\operatorname{Ad}(g)c(g^{\prime},g^{\prime\prime}))$.
Since $g$ is an exponential, according to Proposition A.2(4) its adjoint
action is given by
$\operatorname{Ad}(g)c(g^{\prime},g^{\prime\prime})=c(g^{\prime},g^{\prime\prime})+i[x,c(g^{\prime},g^{\prime\prime})]$.
Since multiples of $i$ lie in the kernel of $\exp$, see Proposition A.2(6), we
deduce
$\exp(\operatorname{Ad}(g)c(g^{\prime},g^{\prime\prime}))=\exp(c(g^{\prime},g^{\prime\prime}))$.

    2. b.

Since $\gamma\mathrel{\mathop{\mathchar
58\relax}}=g^{\prime}g(g^{\prime})^{-1}$ is an exponential,
$\exp(c(g^{\prime}g,g^{\prime\prime}))=\exp(c(\gamma
g^{\prime},g^{\prime\prime}))=\exp(c(g^{\prime},g^{\prime\prime}))$ by a.

    3. c.

The cocycle identity with $g$ and $g^{\prime}$ exchanged reads
$c(g^{\prime},g)+c(g^{\prime}g,g^{\prime\prime})=\operatorname{Ad}(g^{\prime})c(g,g^{\prime\prime})+c(g^{\prime},gg^{\prime\prime})$.
We deduce
$\exp(c(g^{\prime}g,g^{\prime\prime}))=\exp(c(g^{\prime},gg^{\prime\prime}))$.
We conclude with b.

    4. d.

Again this follows from the fact that
$g^{\prime\prime}g(g^{\prime\prime})^{-1}$ is an exponential.

The final claim of the lemma holds because $\lambda c$ is again a normalized
cocycle. $\Box$

  * Proof of Proposition 4.2.

Let us write $K=K_{\lambda}(E_{c})$ for simplicity. Obviously $K$ is a closed
subfunctor of $h^{*}E_{c}$ which is isomorphic to
$h^{*}\operatorname{Lie}(G,I)$ as a $k[I]$-scheme. For the verification of
points (1) and (2) we let $R$ be an arbitrary $k[I]$-algebra. (1) First, let
us prove that $K$ is a subgroup scheme. Let $(x,g)$ and
$(x^{\prime},g^{\prime})$ be two $R$-valued points of $K$ so we have
$g=\exp(\lambda x)$ and $g^{\prime}=\exp(\lambda x^{\prime})$. On one hand,
using Proposition A.2(4) we see that
$\operatorname{Ad}(g)x^{\prime}=x^{\prime}+\varepsilon[x,x^{\prime}]$ and by
Proposition A.2(6) we deduce
$\exp(\lambda\operatorname{Ad}(g)x^{\prime})=\exp(\lambda x^{\prime})$. On the
other hand, by Lemma 4.2(1) we have $\exp(\lambda c(g,g^{\prime}))=1$. Putting
all this together we get:

$\displaystyle\exp\big{(}\lambda
x+\lambda\operatorname{Ad}(g)x^{\prime}+\lambda c(g,g^{\prime})\big{)}$
$\displaystyle=\exp(\lambda
x)\,\exp(\lambda\operatorname{Ad}(g)x^{\prime})\,\exp(\lambda
c(g,g^{\prime}))$ $\displaystyle=\exp(\lambda x)\,\exp(\lambda x^{\prime})$
$\displaystyle=gg^{\prime}.$

This proves that the product $(x,g)\cdot(x^{\prime},g^{\prime})$ is a point of
$K$. Using the same arguments we prove that the inverse
$(x,g)^{-1}=(-\operatorname{Ad}(g^{-1})x-c(g^{-1},g),g^{-1})$ 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 $(x^{\prime},g^{\prime})$ be
$R$-valued points of $h^{*}E_{c}$ and $K$ respectively. We must prove that
$(x^{\prime\prime},g^{\prime\prime})\mathrel{\mathop{\mathchar
58\relax}}=(x,g)\cdot(x^{\prime},g^{\prime})\cdot(x,g)^{-1}$ lies in $K$.
Writing $x^{\prime}$ as a sum of tensors $x_{s}=y_{s}\otimes i_{s}$ and
setting $g_{s}=\exp(\lambda x_{s})$, we have $(x_{s},g_{s})\in K(R)$ and since
$K$ is a subgroup scheme, it is enough to prove that
$(x,g)\cdot(x_{s},g_{s})\cdot(x,g)^{-1}$ lies in $K$. In other words, we may
and do assume in the sequel that $x^{\prime}=y\otimes i$. We first consider
$(x_{1},g_{1})\mathrel{\mathop{\mathchar
58\relax}}=(x^{\prime},g^{\prime})\cdot(x,g)^{-1}$. Using the fact that
$g^{\prime}=\exp(\lambda x^{\prime})$ and Proposition A.2(4), we find

$\operatorname{Ad}(g^{\prime})\big{(}-\operatorname{Ad}(g^{-1})x-c(g^{-1},g)\big{)}=-\operatorname{Ad}(g^{-1})x-c(g^{-1},g)+b$

where $b\in I\\!\cdot\\!\operatorname{Lie}(G,I)(R)$ is a certain bracket, and
hence:

$\displaystyle(x_{1},g_{1})$
$\displaystyle=(x^{\prime},g^{\prime})\cdot(-\operatorname{Ad}(g^{-1})x-c(g^{-1},g),g^{-1})$
$\displaystyle=\big{(}x^{\prime}-\operatorname{Ad}(g^{-1})x-c(g^{-1},g)+b+c(g^{\prime},g^{-1}),g^{\prime}g^{-1}\big{)}.$

Now
$(x^{\prime\prime},g^{\prime\prime})=(x,g)\cdot(x_{1},g_{1})=(x+\operatorname{Ad}(g)x_{1}+c(g,g_{1}),gg_{1})$
and our task is to check that

$\exp\big{(}\lambda x+\lambda\operatorname{Ad}(g)x_{1}+\lambda
c(g,g_{1})\big{)}=gg_{1}.$

We note the following:

    1. a.

We have: $\lambda
x+\lambda\operatorname{Ad}(g)x_{1}=\lambda\operatorname{Ad}(g)x^{\prime}-\lambda\operatorname{Ad}(g)c(g^{-1},g)+b+\lambda\operatorname{Ad}(g)c(g^{\prime},g^{-1})$.
By Proposition A.2(6), the term $b$ will disappear upon taking exponentials,
so we may disregard it. Similarly, by Lemma 4.2(1) the exponential of
$\lambda\operatorname{Ad}(g)c(g^{\prime},g^{-1})$ equals 1. Also, using the
cocycle relation we see that $\operatorname{Ad}(g)c(g^{-1},g)=c(g,g^{-1})$.
Hence:

$\exp(\lambda
x+\lambda\operatorname{Ad}(g)x_{1})=\exp\big{(}\lambda\operatorname{Ad}(g)x^{\prime}-\lambda
c(g,g^{-1})\big{)}.$

    2. b.

By Lemma 4.2(2) we have $\exp(\lambda c(g,g_{1}))=\exp(\lambda
c(g,g^{\prime}g^{-1}))=\exp(\lambda c(g,g^{-1}))$.

    3. c.

Using Proposition A.2(3) we have $gg_{1}=gg^{\prime}g^{-1}=g\exp(\lambda
x^{\prime})g^{-1}=\exp(\lambda\operatorname{Ad}(g)x^{\prime})$.

Putting a-b-c together we get $\exp\big{(}\lambda
x+\lambda\operatorname{Ad}(g)x_{1}+\lambda
c(g,g_{1})\big{)}=\exp(\lambda\operatorname{Ad}(g)x^{\prime})=gg_{1}$ as
desired. (2) Let us write $K=K_{\lambda}(E_{c})$ and
$K^{\prime}=K_{\lambda}(E_{c^{\prime}})$ for simplicity. According to Lemma
4.2, any morphism of extensions $f\mathrel{\mathop{\mathchar
58\relax}}E_{c}\to E_{c^{\prime}}$ is of the form
$f(x,g)=(\operatorname{Lie}(\alpha)(x)+\varphi(g),\alpha(g))$ for a unique
$\varphi\mathrel{\mathop{\mathchar 58\relax}}G\to\operatorname{Lie}G^{\prime}$
satisfying
$\varphi(gg^{\prime})-\varphi(g)-\operatorname{Ad}(g)\varphi(g^{\prime})=c^{\prime}(\alpha
g,\alpha g^{\prime})-\operatorname{Lie}(\alpha)(c(g,g^{\prime}))$. Setting
$g=g^{\prime}=1$ we see that such a $\varphi$, hence also $\lambda\varphi$, is
a map of pointed schemes. This being said, if $(x,g)$ is an $R$-valued point
of $K$, the following computation shows that $f(x,g)$ is a point of
$K^{\prime}$:

$\displaystyle\exp_{G^{\prime}}(\lambda\operatorname{Lie}(\alpha)(x)+\lambda\varphi(g))$
$\displaystyle=\exp_{G^{\prime}}(\lambda\operatorname{Lie}(\alpha)(x))\exp_{G^{\prime}}(\lambda\varphi(g))$
$\displaystyle=\exp_{G^{\prime}}(\lambda\operatorname{Lie}(\alpha)(x))\
\mbox{by Lemma\leavevmode\nobreak\ \ref{lemma:cocycle vanishes with eps},}$
$\displaystyle=\exp_{G^{\prime}}(\operatorname{Lie}(\alpha)(\lambda x))$
$\displaystyle=\alpha(\exp_{G}(\lambda x))\ \mbox{by functoriality of
$\exp$,}$ $\displaystyle=\alpha(g)\ \mbox{because $g=\exp_{G}(\lambda x)$}.$

When $f$ is an isomorphism, applying the statement to $f^{-1}$, we find
$(h^{*}f)(K)=K^{\prime}$. $\Box$

###  4.3. Weil extension functor

Now let $G$ be an affine and differentially flat $k$-group scheme. Thus $G$ as
well as the adjoint representation $\operatorname{Lie}(G,I)$ are $k$-flat. We
consider an arbitrary extension:

$1\longrightarrow\operatorname{Lie}(G,I)\stackrel{{\scriptstyle
i}}{{\longrightarrow}}E\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}G\longrightarrow
1.$

Then $E\to G$ is an fpqc torsor under $\operatorname{Lie}(G,I)$. It has a
cohomology class in $\mathrm{H}^{1}(G,\operatorname{Lie}(G,I))$ which
vanishes, being quasi-coherent cohomology of an affine scheme. It follows that
$\pi$ has a section $s\mathrel{\mathop{\mathchar 58\relax}}G\to E$, and the
extension becomes an $H$-extension. We may and do replace $s$ by
$s(1)^{-1}\cdot s$ in order to ensure that $s(1)=1$. From $s$ we build a
normalized cocycle $c\mathrel{\mathop{\mathchar 58\relax}}G\times
G\longrightarrow\operatorname{Lie}(G,I)$ as follows:

$i(c(g,g^{\prime}))\mathrel{\mathop{\mathchar
58\relax}}=s(g)s(g^{\prime})s(gg^{\prime})^{-1}.$

These data give rise to the group $E_{c}$ as defined in Subsection 4.2.

#### 4.3.0. Proposition.

Let $1\to\operatorname{Lie}(G,I)\to E\to G\to 1$ be an object of the category
$\operatorname{Ext}(I)/k$. Let $s\mathrel{\mathop{\mathchar 58\relax}}G\to E$,
with $s(1)=1$, be as chosen above, and $c$ the normalized cocycle derived from
it. Let $\lambda\in k$.

  * (1)

The map $\tau_{s}\mathrel{\mathop{\mathchar 58\relax}}E_{c}\to E$,
$(x,g)\mapsto i(x)s(g)$ is an isomorphism of extensions.

  * (2)

The closed normal subgroup scheme $K_{\lambda}(E)\mathrel{\mathop{\mathchar
58\relax}}=(h^{*}\tau_{s})(K_{\lambda}(E_{c}))\subset h^{*}E$ does not depend
on the choice of $s$.

  * (3)

For all morphisms $f\mathrel{\mathop{\mathchar 58\relax}}E\to E^{\prime}$ in
$\operatorname{Ext}(I)/k$ we have $(h^{*}f)(K_{\lambda}(E))\subset
K_{\lambda}(E^{\prime})$, with equality if $f$ is an isomorphism.

If the extension $E$ is clear from context, we write $K_{\lambda}$ instead of
$K_{\lambda}(E)$. Note that if $E$ is the trivial extension and $s=\alpha$,
the map $\tau_{\alpha}$ is the map $\varrho_{G}$ 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
$f\mathrel{\mathop{\mathchar 58\relax}}E\to E^{\prime}$ be a morphism in
$\operatorname{Ext}(I)/k$. Let $\tau_{s}\mathrel{\mathop{\mathchar
58\relax}}E_{c}\to E$ and $\tau_{s^{\prime}}\mathrel{\mathop{\mathchar
58\relax}}E_{c^{\prime}}\to E^{\prime}$ be the isomorphisms associated to
choices of sections $s,s^{\prime}$ preserving $1$ and corresponding normalized
cocycles $c,c^{\prime}$. Let $K_{\lambda,s}(E)\mathrel{\mathop{\mathchar
58\relax}}=(h^{*}\tau_{s})(K_{\lambda}(E_{c}))\subset h^{*}E$ and similarly
$K_{\lambda,s^{\prime}}(E^{\prime})\mathrel{\mathop{\mathchar
58\relax}}=(h^{*}\tau_{s^{\prime}})(K_{\lambda}(E_{c^{\prime}}))\subset
h^{*}E^{\prime}$. We have a morphism of extensions:

$\rho=\tau_{s^{\prime}}^{-1}\circ f\circ\tau_{s}\mathrel{\mathop{\mathchar
58\relax}}E_{c}\longrightarrow E_{c^{\prime}}.$

According to Proposition 4.2(2) we have
$(h^{*}\rho)(K_{\lambda}(E_{c}))\subset K_{\lambda}(E_{c^{\prime}})$. It
follows that:

$(h^{*}f)(K_{\lambda,s}(E))=(h^{*}f)((h^{*}\tau_{s})(K_{\lambda}(E_{c})))=(h^{*}\tau_{s^{\prime}})((h^{*}\rho)(K_{\lambda}(E_{c})))\subset(h^{*}\tau_{s^{\prime}})(K_{\lambda}(E_{c^{\prime}}))=K_{\lambda,s^{\prime}}(E^{\prime}).$

When $f$ is an isomorphism, applying the statement to $f^{-1}$ gives equality.
(2) Applying the basic remark to $E=E^{\prime}$ and
$f=\operatorname{id}\mathrel{\mathop{\mathchar 58\relax}}E\to E$ proves that
$K_{\lambda,s}(E)=K_{\lambda,s^{\prime}}(E)$, that is, the subgroup
$K_{\lambda,s}(E)$ does not depend on the choice of $s$. Since $\tau_{s}$ is
an isomorphism of groups, the fact that $K_{\lambda}(E)$ is a closed normal
subgroup scheme follows from Proposition 4.2(1). (3) Applying the basic remark
to a general $f$ gives the statement. $\Box$

For an extension $1\to\operatorname{Lie}(G)\to E\to G\to 1$, we let
$K_{\lambda}\mathrel{\mathop{\mathchar 58\relax}}=K_{\lambda}(E)$ be the
normal subgroup defined in the proposition. Point (4) in Proposition 3.2 gives
motivation to consider $K_{-1}$. The fpqc quotient sheaf
$h^{\mbox{\tiny+}}E\mathrel{\mathop{\mathchar 58\relax}}=h^{*}E/K_{-1}$ is
representable by an affine flat $k[I]$-scheme (see Perrin [Per76, Cor. 0.2]).

#### 4.3.1. Definition.

We call Weil extension the quotient
$h^{\mbox{\tiny+}}E\mathrel{\mathop{\mathchar 58\relax}}=h^{*}E/K_{-1}$.

#### 4.3.2. Lemma.

Weil extension is a functor
$\operatorname{Ext}(I)/k\to\operatorname{Gr}\\!/k[I]$.

  * Proof.

The $k[I]$-group scheme $G\mathrel{\mathop{\mathchar 58\relax}}=h^{*}E/K_{-1}$
is affine and flat. Let $s\mathrel{\mathop{\mathchar 58\relax}}G\to E$ be a
section of $E\to G$ such that $s(1)=1$. By pullback, this induces a morphism
$h^{*}G\to h^{*}E\to G$ 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 proves that the functor
is well-defined on morphisms. $\Box$

##  5\. 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).

#### 5.0.0. Theorem.

(1) The Weil restriction/extension functors provide quasi-inverse
equivalences:

$\textstyle{\operatorname{Gr}\\!/k[I]\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{*}}$$\textstyle{\
\operatorname{Ext}(I)/k\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{h^{\mbox{\rm\tiny+}}}$

These equivalences commute with base changes
$\operatorname{Spec}(k^{\prime})\to\operatorname{Spec}(k)$. (2) If
$1\to\mathscr{G}^{\prime}\to\mathscr{G}\to\mathscr{G}^{\prime\prime}\to 1$ is
an exact sequence in $\operatorname{Gr}\\!/k[I]$, then $1\to
h_{*}\mathscr{G}^{\prime}\to h_{*}\mathscr{G}\to
h_{*}\mathscr{G}^{\prime\prime}$ is exact in $\operatorname{Ext}(I)/k$. If
moreover $\mathscr{G}^{\prime}$ is smooth then $1\to
h_{*}\mathscr{G}^{\prime}\to h_{*}\mathscr{G}\to
h_{*}\mathscr{G}^{\prime\prime}\to 1$ 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
$\mathbb{O}_{k}$-module stacks fibred over $\operatorname{Gr}\\!/k$, i.e. it
transforms the addition and scalar multiplication of deformations of a fixed
$G\in\operatorname{Gr}\\!/k$ into the Baer sum and scalar multiplication of
extensions of $G$ by $\operatorname{Lie}(G,I)$. (4) Let $P$ be one of the
properties: of finite type, smooth, connected, unipotent, split unipotent,
solvable, commutative. Then $\mathscr{G}\in\operatorname{Gr}\\!/k[I]$ has the
property $P$ if and only if the $k$-group scheme $E=h_{*}\mathscr{G}$ has $P$.

###  5.1. Equivariance of rigidifications under Lie algebra translation

Let $\mathscr{G}$ be an affine, differentially flat, rigid $k[I]$-group
scheme. Let $\sigma\mathrel{\mathop{\mathchar
58\relax}}h^{*}\mathscr{G}_{k}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\mathscr{G}$
be a rigidification such that $\sigma(1)=1$. We consider the morphism of
$k$-schemes:

$h_{*}\sigma\mathrel{\mathop{\mathchar
58\relax}}h_{*}h^{*}\mathscr{G}_{k}\longrightarrow h_{*}\mathscr{G}.$

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
$\mathscr{G}_{k}$ by $\operatorname{Lie}(\mathscr{G}_{k},I)$; in particular
both carry an action of $\operatorname{Lie}(\mathscr{G}_{k},I)$ by left
translation.

#### 5.1.0. Proposition.

With notation as above, the morphism of $k$-schemes

$h_{*}\sigma\mathrel{\mathop{\mathchar
58\relax}}h_{*}h^{*}\mathscr{G}_{k}\longrightarrow h_{*}\mathscr{G}$

is $\operatorname{Lie}(\mathscr{G}_{k},I)$-equivariant.

  * Proof.

We write simply $\mathbb{O}$ instead of $\mathbb{O}_{k}$ or
$\mathbb{O}_{k[I]}$ when the base is clear from context. Consider the
extension of $\sigma$ to the group algebras:

$\sigma^{\prime}\mathrel{\mathop{\mathchar 58\relax}}=h_{*}\mathbb{O}[\sigma]\
\mathrel{\mathop{\mathchar 58\relax}}h_{*}\mathbb{O}[h^{*}\mathscr{G}_{k}]\to
h_{*}\mathbb{O}[\mathscr{G}].$

Note that by compatibility of $\mathbb{O}[-]$ with base change and Weil
restriction (see Proposition 2.3, (2)-(4)), we have
$h_{*}\mathbb{O}[h^{*}\mathscr{G}_{k}]\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}h_{*}h^{*}\mathbb{O}[\mathscr{G}_{k}]\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\mathbb{O}[\mathscr{G}_{k}][I]$.
We obtain a commutative diagram:

$\textstyle{\underset{\
}{h_{*}h^{*}\mathscr{G}_{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{*}\sigma}$$\textstyle{\underset{\
}{h_{*}\mathscr{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overset{\
}{\mathbb{O}[\mathscr{G}_{k}][I]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{\prime}}$$\textstyle{\overset{\
}{h_{*}\mathbb{O}[\mathscr{G}]}.}$

We identify $I\cdot\mathbb{O}[\mathscr{G}_{k}]\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{O}[\mathscr{G}_{k}]\otimes_{\mathbb{O}_{k}}\mathbb{V}(I^{{\scriptscriptstyle\vee}})$
as the ideal generated by $I$ inside both algebras
$\mathbb{O}[\mathscr{G}_{k}][I]$ and $h_{*}\mathbb{O}[\mathscr{G}]$, see
Corollary 3.3. With this convention we formulate:

Claim. We have $\sigma^{\prime}(y+x)=\sigma^{\prime}(y)+x$ for all points
$y\in\mathbb{O}[\mathscr{G}_{k}][I]$ and $x\in
I\cdot\mathbb{O}[\mathscr{G}_{k}]$.

To prove this, we introduce another copy $J=I$ of our square-zero ideal as
follows:

$\textstyle{h_{*}h^{*}h_{*}h^{*}\mathscr{G}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{*}h^{*}h_{*}\sigma}$$\textstyle{h_{*}h^{*}h_{*}\mathscr{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{O}[\mathscr{G}_{k}][I][J]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{\prime\prime}}$$\textstyle{h_{*}\mathbb{O}[\mathscr{G}][J]}$

where we have set $\sigma^{\prime\prime}=h_{*}h^{*}\sigma^{\prime}$ for
brevity. Let $s,t\mathrel{\mathop{\mathchar
58\relax}}\mathbb{O}[\mathscr{G}_{k}][I]\times
J\cdot\mathbb{O}[\mathscr{G}_{k}][I][J]\to h_{*}\mathbb{O}[\mathscr{G}][J]$ be
defined by

$s(y,x)=\sigma^{\prime\prime}(y+x)\quad\mbox{and}\quad
t(y,x)=\sigma^{\prime\prime}(y)+x.$

Then $s$ and $t$ are equal modulo $I$ because of the fact that $\sigma$ is the
identity modulo $I$. Clearly they are also equal modulo $J$. Since
$h_{*}\mathbb{O}[\mathscr{G}][J]$ is flat as an $\mathbb{O}_{k}[I][J]$-module,
we deduce that $s-t$ takes its values in the ideal $IJ\cdot
h_{*}\mathbb{O}[\mathscr{G}][I]$. Setting $J=I$, that is composing with the
morphism $h_{*}\mathbb{O}[\mathscr{G}][J]\to h_{*}\mathbb{O}[\mathscr{G}]$
that takes a section of $J$ to the corresponding section of $I$, we obtain the
claim.

We now conclude the proof of the proposition. We write $\star_{1}$ and
$\star_{2}$ the multiplications of $h_{*}h^{*}\mathscr{G}_{k}$ and
$h_{*}\mathscr{G}$ respectively, extended to
$h_{*}\mathbb{O}[h^{*}\mathscr{G}_{k}]$ and $h_{*}\mathbb{O}[\mathscr{G}]$. It
is enough to show more generally that

$h_{*}\sigma\mathrel{\mathop{\mathchar
58\relax}}h_{*}\mathbb{O}[h^{*}\mathscr{G}_{k}]\longrightarrow
h_{*}\mathbb{O}[\mathscr{G}]$

is $(1+I\cdot\mathbb{O}[\mathscr{G}_{k}])$-equivariant. We take points $x\in
I\cdot\mathbb{O}[\mathscr{G}_{k}]$ and $y\in
h_{*}\mathbb{O}[h^{*}\mathscr{G}_{k}]$ and compute:

$\begin{array}[]{rll}(h_{*}\sigma)((1+x)\star_{1}y)&=(h_{*}\sigma)(y+x\star_{1}y)\\\
&=(h_{*}\sigma)(y)+x\star_{1}y\ \mbox{ by the Claim above,}\\\
&=(h_{*}\sigma)(y)+x\star_{2}y\ \mbox{ by Corollary\leavevmode\nobreak\
\ref{coro1} since $\star_{1}=\star_{2}$ modulo $I$,}\\\
&=(h_{*}\sigma)(y)+x\star_{2}(h_{*}\sigma)(y)\ \mbox{ by
Corollary\leavevmode\nobreak\ \ref{coro1} since $\sigma=\operatorname{id}$
modulo $I$,}\\\ &=(1+x)\star_{2}(h_{*}\sigma)(y).\end{array}$

This proves that $h_{*}\sigma$ is
$\operatorname{Lie}(\mathscr{G}_{k},I)$-equivariant. $\Box$

###  5.2. Proof of the main theorem: equivalence and exactness

#### 5.2.0. Proof of 5(1).

We shall prove that the functors

$\textstyle{\operatorname{Gr}\\!/k[I]\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{*}}$$\textstyle{\
\operatorname{Ext}(I)/k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\mbox{\rm\tiny+}}}$

provide quasi-inverse equivalences that commute with base changes
$\operatorname{Spec}(k^{\prime})\to\operatorname{Spec}(k)$. Firstly, we prove
that $h^{\mbox{\rm\tiny+}}\circ h_{*}$ is isomorphic to the identity. Let
$\mathscr{G}\to\operatorname{Spec}(k[I])$ be an affine, differentially flat,
rigid $k[I]$-group scheme. Let $E=h_{*}\mathscr{G}$ be the associated
extension:

$1\longrightarrow\operatorname{Lie}(\mathscr{G}_{k},I)\longrightarrow
E\longrightarrow\mathscr{G}_{k}\longrightarrow 1.$

We fix a rigidification $\sigma\mathrel{\mathop{\mathchar
58\relax}}h^{*}\mathscr{G}_{k}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\mathscr{G}$
such that $\sigma(1)=1$. We know from Proposition 5.1 that the map
$h_{*}\sigma\mathrel{\mathop{\mathchar
58\relax}}h_{*}h^{*}\mathscr{G}_{k}\longrightarrow h_{*}\mathscr{G}$ is
$\operatorname{Lie}(\mathscr{G}_{k},I)$-equivariant. If we use the letter
$\gamma$ to denote the inclusions of $\operatorname{Lie}(\mathscr{G}_{k},I)$
into the relevant extensions, this can be written:

$(h_{*}\sigma)(\gamma_{h_{*}h^{*}\mathscr{G}_{k}}(x)\cdot
y)=\gamma_{h_{*}\mathscr{G}}(x)\cdot(h_{*}\sigma)(y),\quad\mbox{all
}x\in\operatorname{Lie}(\mathscr{G}_{k},I),\ y\in h_{*}h^{*}\mathscr{G}_{k}.$

Restricting to $y$ in the image of
$\alpha=\alpha_{\mathscr{G}_{k}}\mathrel{\mathop{\mathchar
58\relax}}\mathscr{G}_{k}\hookrightarrow h_{*}h^{*}\mathscr{G}_{k}$, so
$\tau_{\alpha}=\varrho_{\mathscr{G}_{k}}$, we obtain:

$h_{*}\sigma\circ\tau_{\alpha}=\tau_{s}.$

Using functoriality of $\beta$ and the fact that $\sigma(1)=1$, we build a
commutative diagram:

$\textstyle{h^{*}F=h^{*}(\operatorname{Lie}(\mathscr{G}_{k},I)\times\mathscr{G}_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{*}\tau_{\alpha}}$$\scriptstyle{\operatorname{id}}$$\textstyle{h^{*}h_{*}h^{*}\mathscr{G}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{h^{*}\mathscr{G}_{k}}}$$\scriptstyle{h^{*}h_{*}\sigma}$$\textstyle{h^{*}\mathscr{G}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{1}$$\textstyle{h^{*}E_{c}=h^{*}(\operatorname{Lie}(\mathscr{G}_{k},I)\times\mathscr{G}_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{*}\tau_{s}}$$\textstyle{h^{*}h_{*}\mathscr{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{\mathscr{G}}}$$\textstyle{\mathscr{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

Here the horizontal maps are morphisms of groups and the vertical maps are not
morphisms of groups (not even the leftmost map
$\operatorname{id}\mathrel{\mathop{\mathchar 58\relax}}h^{*}F\to h^{*}E_{c}$).
Note also that $F$ is $E_{0}=\operatorname{T}(\mathscr{G}_{k},I)$, that is,
the extension $E_{c}$ with the zero cocycle $c=0$. Now we consider $K_{-1}(E)$
as defined in Proposition 4.3. According to Proposition 3.2(4), the group
$K_{-1}(E_{0})\subset E_{0}$ is the kernel of
$\beta_{h^{*}\mathscr{G}_{k}}\circ h^{*}\tau_{\alpha}$. On one hand the
identity takes $K_{-1}(E_{0})$ to $K_{-1}(E_{c})$, and on the other hand the
map $h_{*}\sigma$ takes $\ker(\beta_{h^{*}\mathscr{G}_{k}})$ onto
$\ker(\beta_{\mathscr{G}})$ since it takes 1 to 1. By commutativity of the
left-hand square, we find $K_{-1}(E)=\ker(\beta_{\mathscr{G}})$ and thefore
$\beta_{\mathscr{G}}$ induces an isomorphism
$h^{\mbox{\rm\tiny+}}E=h^{*}E/K_{-1}(E)\simeq\mathscr{G}$ which is visibly
functorial. Secondly, we prove that $h_{*}\circ h^{\mbox{\rm\tiny+}}$ is
isomorphic to the identity. Let $1\to\operatorname{Lie}(G,I)\to E\to G\to 1$
be an extension. We fix a section $s\mathrel{\mathop{\mathchar 58\relax}}G\to
E$ such that $s(1)=1$ and we let $c$ be the normalized cocycle defined by $s$.
Let $K_{-1}=K_{-1}(E)\subset h^{*}E$ be the closed normal subgroup defined in
Proposition 4.3, and let $\mathscr{G}\mathrel{\mathop{\mathchar
58\relax}}=h^{\mbox{\rm\tiny+}}E=h^{*}E/K_{-1}$ with quotient map
$\pi\mathrel{\mathop{\mathchar 58\relax}}h^{*}E\to\mathscr{G}$. Define
$\sigma=\pi\circ h^{*}s\mathrel{\mathop{\mathchar
58\relax}}h^{*}G\to\mathscr{G}$. Since $i^{*}K_{-1}=\operatorname{Lie}(G,I)$
as a subgroup of $E$, we see that $i^{*}\mathscr{G}\simeq G$ and $i^{*}\sigma$
is the identity of $G$. Since $G$ is $k$-flat, it follows that $\sigma$ is an
isomorphism. From the construction of $K_{-1}$, we see that after we compose
with the isomorphisms

$h^{*}\tau_{s}\mathrel{\mathop{\mathchar
58\relax}}h^{*}h_{*}h^{*}G\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}h^{*}E\quad\mbox{and}\quad\sigma\mathrel{\mathop{\mathchar
58\relax}}h^{*}G\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}\mathscr{G},$

the flat surjection $\pi\mathrel{\mathop{\mathchar
58\relax}}h^{*}E\to\mathscr{G}$ is identified with the counit of the
adjunction:

$\beta_{h^{*}G}\mathrel{\mathop{\mathchar
58\relax}}h^{*}h_{*}h^{*}G\longrightarrow h^{*}G.$

We apply $h_{*}$ and obtain the commutative diagram:

$\textstyle{h_{*}h^{*}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{h_{*}h^{*}G}}$
$\sim$
$\scriptstyle{\tau_{s}}$$\textstyle{h_{*}h^{*}h_{*}h^{*}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{*}\beta_{h^{*}G}}$
$\sim$
$\scriptstyle{h_{*}h^{*}\tau_{s}}$$\textstyle{h_{*}h^{*}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\sim$
$\scriptstyle{h_{*}\sigma}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{E}}$$\textstyle{h_{*}h^{*}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{*}\pi}$$\textstyle{h_{*}\mathscr{G}}$

Since the top row is the identity, we see that the bottom row is an
isomorphism, i.e.
$E\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\scriptstyle\sim$}}\,}h_{*}\mathscr{G}$.
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.

#### 5.2.1. Proof of 5(2).

Let $1\to\mathscr{G}^{\prime}\to\mathscr{G}\to\mathscr{G}^{\prime\prime}\to 1$
be an exact sequence in $\operatorname{Gr}\\!/k[I]$. Then the exact sequences
with solid arrows are exact:

$\begin{array}[]{c}1\longrightarrow\mathscr{G}^{\prime}_{k}\longrightarrow\mathscr{G}_{k}\longrightarrow\mathscr{G}^{\prime\prime}_{k}\longrightarrow
1\\\ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.05946pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\kern
5.50002pt\vrule
width=0.0pt,depth=0.0pt,height=0.0pt}}\ignorespaces{\hbox{\kern-6.19835pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\xyatipfont\char
111\relax\xybtipfont\char 111\relax}}}}}{\hbox{\kern-5.92056pt\vrule
width=-0.1389pt,depth=0.0pt,height=0.0pt}}{\hbox{\kern-6.05946pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Lie}(\mathscr{G}^{\prime}_{k},I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\kern
40.20276pt\vrule
width=0.0pt,depth=0.0pt,height=0.0pt}}\ignorespaces{\hbox{\kern
10.09746pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\xyatipfont\char 111\relax\xybtipfont\char
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Moreover, if $\mathscr{G}^{\prime}$ is smooth then the second sequence is
exact also if we include the dotted arrow. By an easy diagram chase, we find
that the commutative diagram below has exact rows (again including dotted
arrows if $\mathscr{G}^{\prime}$ is smooth):