Rigid analytic functions on the universal vector extension

Marco Maculan
Institut de Mathématiques de Jussieu, Sorbonne Université, 4 place Jussieu, F-75252 Paris

Let $K$ be a complete non-trivially valued  non-Archimedean field. Given an algebraic group over $K$ on which every regular function is constant, any rigid analytic function is shown to be constant too. It follows that an algebraic group over $K$ is affine if and only if the associated $K$-analytic space is Stein; that is,
rigid analytic embeddings of it in an affine space may always be chosen to be given by algebraic functions. Arguably curiously, the corresponding statement over the complex numbers is false.

§ INTRODUCTION

§.§ Motivation

Throughout this introduction, let $K$ be a complete non-trivially valued non-Archimedean field, e.g. $K = \bbQ_p$, $\bbC_p$, $k((t))$ for a field $k$, etc.

For an algebraic variety  $X$ over $K$, that is, a finite type $K$-scheme, let $X^\an$ denote the $K$-analytic space[In this paper analytic spaces over $K$ are considered in the sense of Berkovich. Nonetheless, the main results are insensible to the choice of the language.] attached to it. The algebraic variety $X$ is said to be Stein if there is a closed embedding of $K$-analytic spaces $i \colon X^\an \to \bbA^{n, \an}_K$ for some integer $n \ge 0$. Of course, affine varieties are Stein.

The frivolous question at the origin of this note is whether the converse holds true. Counterexamples exist for the corresponding affirmation over the complex numbers, and they can be found already in the restrictive class of algebraic groups. Instead, within the same species of varieties, the answer is seen to be positive over the field $K$:

[infra Corollary <ref>]  An algebraic group[Namely, a group $K$-scheme of finite type.] over $K$ is Stein if and only if it is affine.

The interest of such a statement, if any, resides in that it reveals yet another divergence between complex and rigid analysis. Before describing more in detail the content of this article, let me briefly recall what happens over $\bbC$.

§.§.§ Serre's example

As pointed out by Serre, there are non-affine complex algebraic groups that admit a  closed holomorphic embedding in $\bbC^n$ (for some integer $n \ge 0$).

The leading example is the universal vector extension $A^\natural$ of a (non-trivial) complex abelian variety $A$, that is, the moduli space of rank $1$ connections on the dual abelian variety $\check{A}$. The tensor product of line bundles equipped with a connection endows $A^\natural$ with the structure of a complex algebraic group, which sits in the following short exact sequence:

\[ 0 \too \bbV(\omega_{\check{A}}) \too A^\natural \too A \too 0,\]

where the map $A^\natural \to A$ forgets the connection on an algebraically trivial line bundle on $\check{A}$, and the map $\bbV(\omega_{\check{A}}) \to A^\natural$ associates to a differential form $\eta$ on $\check{A}$, the connection on the trivial line bundle $\cO_{\check{A}}$ given by the sum $\rd + \eta$ of the canonical derivation and $\eta$. (The space  $\Gamma(\check{A}, \Omega^1_{\check{A}})$ of global differential forms on $\check{A}$ is identified with the dual $\omega_{\check{A}}$ of the Lie algebra of $\check{A}$.)

On the one hand, the algebraic variety $A^\natural$ is not affine because the quotient of an affine algebraic group by a subgroup is always affine—and $A$ is certainly not so. On the other, a holomorphic connection on a line bundle on $A$ being automatically integrable, the Riemann-Hilbert correspondence yields a biholomorphism

\[ A^{\natural, \an} \stackrel{\sim}{\too} \Hom_{\textup{Groups}}(\pi_1(\check{A}(\bbC), 0), \bbC^\times) \cong (\bbC^\times)^{2g},\]

where $g := \dim A$, and $\pi_1(\check{A}(\bbC), 0)$ is the topological fundamental group of $\check{A}(\bbC)$ with $0$ as base-point. In particular, the complex manifold $A^{\natural, \an}$ admits a closed holomorphic embedding in $\bbC^{2g + 1}$.

The biholomorphism given by the Riemann-Hilbert correspondence admits a quite explicit description, which should hopefully ease the reading of the paper. Let $\exp \colon \Lie A \to A^\an$ be the exponential map. Identify $V:= \Lie A$ with the universal cover of $A(\bbC)$, and $\Lambda := \ker \exp$ with the fundamental group. The pull-back of $A^{\natural, \an}$ to $\Lie A$ splits as an affine bundle because the cohomology group $\rH^1(V^\an, \cO_{V^\an})$ vanishes. More precisely, Hodge theory permits to identify $\omega_{\check{A}}$ with the conjugated complex vector space $\bar{V}$ of $V = \Lie A$. Let $\theta_\Lambda \colon \Lambda \to \bar{V}$ be the inclusion. Then,

\[ A^{\natural, \an} = (V \times \bar{V}) / \Lambda\]

with $\Lambda$ embedded diagonally. The choice of a basis of $\Lambda$ induces biholomorphisms

\[ A^{\natural, \an} \cong (\bbC / \bbZ)^{2g} \cong (\bbC^\times)^{2g}.\]

§.§.§ Neeman's counter-example to Hilbert's fourteenth problem
The difference between affine and Stein varieties is quite subtle. Neeman exhibited a quasi-affine (that is, admitting an open immersion in an affine variety) complex variety which is Stein but not affine.[Note that algebraic functions on $A^\natural$ are all constant (<cit.>), thus $A^\natural$ is not quasi-affine.] Let me recall his construction.

Let $H$ be an ample line bundle on the abelian variety $A$ and $P$ be the total space of the line bundle $p^\ast H^\vee$ on $A^\natural$ deprived of its zero section, where $p \colon A^\natural \to A$ is the projection. The principal $\Gm$-bundle $P$ is quasi-affine by ampleness of $H$, but not affine: otherwise, by Geometric Invariant Theory (<cit.>), the quotient $P / \Gm = A^\natural$ would be so. On the other hand, a principal $\bbC^\times$-bundle on a Stein space is itself Stein[This is true for principal bundle under a Stein complex Lie group, see <cit.>.], hence so is $P$.

As remarked by Neeman (<cit.>), a quasi-affine complex variety $X$ which is Stein but not affine forces its  $\bbC$-algebra of global sections $\Gamma(X, \cO_X)$ not to be finitely generated. As $P$ is such a variety, it yields a counter-example of Hilbert's fourteenth problem, as extended by Zariski (op.cit. Remark 8.2).

§.§ Main results

§.§.§ Rigid analytic functions on anti-affine groups
Needless to say, in order to prove Theorem <ref> rigid analytic functions on algebraic groups have to be understood.

Employing Brion's nomenclature, an algebraic group $G$ over $K$ is said to be anti-affine if $\Gamma(G, \cO_G) = K$. An anti-affine group is commutative (otherwise the image of its adjoint representation would be a non-trivial affine quotient of $G$), connected and reduced (otherwise $G / G^0_{\red}$ would be a non-trivial finite algebraic group).

[infra Corollary <ref>]  A $K$-analytic function on (the analytification of) an anti-affine algebraic group over $K$ is constant.

When the field $K$ is of positive characteristic, the proof of Theorem <ref> is rather straightforward, for the only anti-affine groups in positive characteristic are (anti-affine) semi-abelian varieties (<cit.>).

For a semi-abelian variety, in arbitrary characteristic, analytic functions are expressed by means of `Fourier expansion' (cf. <ref>) as series of global sections of (translation-invariant) lines bundles on the underlying abelian variety, and the GAGA principle permits to conclude (see Corollary <ref>). Note that these considerations go through also in the complex case.

Given an algebraic group $G$, the $K$-algebra $\Gamma(G, \cO_G)$ is of finite type and the natural structure of Hopf algebra on it endows  $G_\aff := \Spec \Gamma(G, \cO_G)$ with the structure of an algebraic group. The canonical map $\pi\colon G \to G_{\aff}$ is a morphism of algebraic groups.

[infra Theorem <ref>]   For an algebraic group $G$ over $K$, precomposing a $K$-analytic function on $G^\an_{\aff}$ with $\pi^\an$ yields an isomorphism

\[ \Gamma(G^\an_{\aff}, \cO^\an_{G{\aff}}) \stackrel{\sim}{\too}\Gamma(G^\an, \cO^\an_G). \]

Theorem <ref> is an immediate consequence of Theorem <ref>. Indeed, the latter implies that a morphism $f \colon G^\an \to \bbA^{n, \an}_K$ of $K$-analytic spaces factors through $G_{\aff}^\an$. If $f$ happens to be a closed embedding, then $\ker \pi$ is trivial. Also, Theorem <ref> will be easily deduced from <ref> by a descent argument.

§.§.§ Brion's generalization of Neeman's counter-example
Thanks to Chevalley's theorem (<cit.>), a connected and reduced algebraic group $G$  over $K$ is an extension of an abelian variety $A$ by a linear algebraic group $L$:

\[ 0 \too L \too G \stackrel{p}{\too} A \too 0.\]

Let $H$ be a line bundle on the abelian variety $A$ and $P$ the total space of the line bundle $p^\ast H^\vee$ on $G$ deprived of its zero section. If $H$ is ample, then the algebraic variety $P$ is quasi-affine. Moreover, if $G$ is anti-affine and its linear part $L$ is non-trivial, Brion shows that the ring $\Gamma(P, \cO_P)$ is not Noetherian (<cit.>), furnishing a counter-example to Zariski's version of Hilbert's fourteenth problem.[Such an algebraic group $G$ exists because $K$ is not an algebraic extension of $\bbF_p$: the only anti-affine groups on $\bar{\bbF}_p$ are abelian varieties (<cit.>).]

[infra Theorem <ref>] With the notation above,

\[ G \text{ is affine} \iff G \text{ is Stein} \iff P \text{ is affine} \iff P \text{ is Stein}. \]

In particular, if $G$ is anti-affine and non-trivial, then $P$ is not Stein, showing that the equivalence between the notions of Stein and affine varieties hold even in more intricate examples.

§.§.§ The critical case
According to Remark <ref>, Theorem <ref> is most interesting (and strikes the difference with the complex framework) when $G$ is a vector extension of an abelian variety $A$ and $K$ is of characteristic $0$. As recalled below, all such $G$ are obtained as the push-out of the universal vector extension $A^\natural$ along a linear map.  Besides, the techniques employed go through when $A^\an$ is replaced by an abeloid variety (the rigid analytic analogue of a complex torus).

For these two reasons, in the remainder of this introduction, I will focus on the proof the following result:

[infra Theorem <ref>]  Suppose $K$ of characteristic $0$. Let $A$ be an abeloid variety over $K$ and $A^\natural$ the universal vector extension of $A$. Then,

\[ \Gamma(A^\natural, \cO_{A^\natural}) = K.\]

§.§ Definition of the universal vector extension

§.§.§ Moduli space of connections
The first problem to tackle is settling solid foundations for the universal vector extension in algebraic, formal, and rigid geometry. Not to abuse of reader's patience, I leaned toward a complete treatment exclusively in the scheme-theoretic framework, but an effort has been made to present arguments that can, abusing of reader's expertise, be translated right away in the other contexts.

Truth to be told, the universal vector extension is not introduced as such, but rather as the moduli space of rank $1$ connections. Stricto sensu, the universal vector extension $\rE(A)$ of an abelian scheme $\alpha \colon A \to S$ is a vector extension

\[ 0 \too \bbV((\rR^1 \alpha_\ast \cO_A)^\vee) \too \rE(A) \too A \too 0\]

from which any other vector extension is obtained by push-out along a linear map (<cit.>).

Instead, the moduli space of rank $1$ connections on $\check{A}$ is the $S$-scheme $A^\natural$ representing the functor associating to an $S$-scheme $S'$ the isomorphism classes of couples $(L, \nabla)$ made of a (fiberwise) translation-invariant line bundle $L$ on $\check{A} \times_S S'$ and a connection $\nabla$ on $L$. Tensor product of line bundles endowed with a connection defines a group $S$-scheme structure on $A^\natural$, giving rise to a short exact sequence of commutative group $S$-schemes

\[ 0 \too \bbV(\omega_{\check{A}}) \too A^\natural \too A \too 0,\]

where $\omega_{\check{A}}$ is the dual of the Lie algebra of $\check{A}$ and the arrow $A^\natural \to A$ forgets the connection (by biduality, the abelian scheme $A$ parametrizes translation-invariant line bundles on $\check{A}$).

To do `explicit' computations, it is worth having a concrete construction of $A^\natural$ at hand. The one presented here is inspired from <cit.>, even though the Hodge-theoretical reasonings therein had to be circumvented. Let $\cL$ be the Poincaré bundle on $A \times_S \check{A}$ and

\begin{equation} \tag{$\At(\cL)$} 0 \too q^\ast \Omega^1_{\check{A}/S} \too \At (\cL) \too \cO_{A\times \check{A}} \too 0
\end{equation}

the Atiyah extension of $\cL$ (relative to the first projection $q \colon A \times_S \check{A} \to A$). Note that other conventions for the Atiyah extension exist in the literature; for instance, in op.cit., the dual one is adopted.

For a morphism of schemes $x \colon S' \to A$, the pull-back to $\check{A}_{S'}$ of the short exact sequence $(\At(\cL))$  is the Atiyah extension $\At(\cL_x)$  of the line bundle
\[\cL_x := (x \times \id_{\check{A}})^\ast \cL\]
on $A_{S'}$. Splittings of $\At(\cL_x)$ correspond to connections on the line bundle $\cL_x$. Mazur and Messing prove that such splittings exist as soon as the scheme $S'$ is affine (see <cit.> or Proposition <ref>). It follows that the sequence of $\cO_{A}$-modules

\[ 0 \too \alpha^\ast \omega_{\check{A}} \too q_\ast \At(\cL) \too \cO_{A} \too 0,\]

obtained by pushing-forward $(\At(\cL))$ along the map $q$, is short exact.  Upon setting $\cU_{A} := q_\ast \At(\cL)$, the affine bundle

\[ A^\natural := \bbP(\cU_{A}) \smallsetminus \bbP(\alpha^\ast \omega_{\check{A}})\]

over $A$, parametrizing splittings of the above short exact sequence, is seen to be the moduli space of rank $1$ connections on $\check{A}$ almost by definition (Theorem <ref>).

§.§.§ Comparison with the universal vector extension
By universal property of the universal vector extension, the moduli space of rank $1$ connections on $A^\natural$ is the push-out of $\rE(A)$ along a uniquely determined homomorphism of $\cO_S$-modules $\phi_A \colon (\rR^1 \alpha_\ast \cO_A)^\vee \to \omega_{\check{A}}$:

\[
\begin{tikzcd}
0 \ar[r] & \bbV((\rR^1 \alpha_\ast \cO_A)^\vee) \ar[d, "\phi_A"] \ar[r] & \rE(A) \ar[d] \ar[r] &A \ar[d, equal] \ar[r] & 0\ \\
0 \ar[r] &  \bbV(\omega_{\check{A}}) \ar[r] & A^\natural \ar[r] & A \ar[r] & 0.
\end{tikzcd}
\]

The homomorphism $\phi_A$ can also be seen as a section of the vector bundle $\rR^1 \alpha_\ast \alpha^\ast \omega_{\check{A}}$ on $S$, and as such is the isomorphism class of the extension $\cU_A$. In order to recognize $\phi_A$, recall the canonical isomorphism (<cit.>)

\[ \Phi_A \colon \Lie \check{A} \stackrel{\sim}{\too}  \rR^1 \alpha_\ast \cO_A.\]

[infra Proposition <ref>] With the notation above, by means of the isomorphism $\rR^1 \alpha_\ast \alpha^\ast \omega_{\check{A}} \cong \cHom(\Lie \check{A}, \rR^1 \alpha_\ast \cO_A)$, the following equality holds:

\[ \Phi_A = \phi_A.\]

I could not pin down in the literature a proof for this, perhaps because of its what-else-could-it-be nature. Of course, Mazur and Messing prove that $\phi_A$ is an isomorphism and is functorial in $A$ (<cit.>) but its explicit form  is missing.
This gap was already pointed out by Crew (<cit.>), but in the description he proposes (op.cit., Theorem 2.7) the isomorphism $\Phi_{A}$ does not explicitly enter into play.

By means of the preceding isomorphism, the moduli space of rank $1$ connections $A^\natural$ and the universal vector extension $\rE(A)$ are henceforth identified.

§.§ Algebraic functions on the universal vector extension

§.§.§ Finite vector bundles
A vector bundle $V$ on the abelian scheme $\alpha \colon A \to S$ is said to be finite if there is an integer $m \ge 1$ such that $[m]^\ast V$ is `constant', that is, it is isomorphic to $\alpha^\ast E$ for some vector bundle $E$ on $S$.

The multiplication-by-$m$ map $[m] \colon A \to A$ makes $A$ into an fppf principal $A[m]$-bundle over itself, which is étale as soon as the integer $m$ is invertible on $S$ (here, as usual, $A[m]$ stays for the $m$-torsion of the abelian scheme $A$). The finite flat group $S$-scheme $A[m]$ acts naturally on the vector bundle $[m]^\ast V$ (in an equivariant fashion with respect to its translation operation on $A$), thus its push-forward $\alpha_\ast [m]^\ast V$ onto $S$ inherits a linear action of $A[m]$. Identifying the latter with the fiber $e^\ast V$ of $V$ at the zero section $e$ of $A$ gives rise to a representation

\[ \rho_{V, m} \colon A[m] \too \GL(e^\ast V).\]

By faithfully flat descent, the previous construction furnishes an equivalence (compatible with tensor operations) between the full subcategory of vector bundles $V$ on $A$ such that $[m]^\ast V$ is constant and that of representation of finite flat group $S$-scheme $A[m]$.

While dealing with formal functions, it will be useful to make the representation $\rho_{V, m}$ independent of $m$. In order to do so, consider the Tate group scheme

\[ \rT A = \projlim_{m \ge 1} A[m],\]

the projective limit being taken in the sense of schemes (which is well-defined as all transition maps are affine). The composite morphism

\[ \rho_{V} \colon \rT A \too A[m] \stackrel{\rho_{V, m}}{\too} \GL(e^\ast V)\]

is then seen to be independent of the chosen $m$. Note that when the scheme $S$ is the spectrum of an algebraically closed field of characteristic $0$, the Tate group scheme coincides with the Tate module of $A$; however, in positive characteristic $p$, the nonreduceness of $A[p^i]$ radiates onto $\rT A$. As for the Tate module, the Tate group scheme $\rT A$ decomposes as the product

\[ \rT A \cong \prod_{p \textup{ prime}} \rT_p A\]

of $p$-adic Tate group schemes $\rT_p A = \projlim_{i \ge 1} A[p^i]$.

The most interesting example here is $V = \cU_A$. In characteristic $0$, the unipotent bundle $\cU_A$ is not finite; but, when an integer $m$ vanishes vanishing on $S$, the vector bundle $[m]^\ast \cU_A$ is isomorphic to $\alpha^\ast \omega_{\check{A}} \oplus \cO_A$, because pulling-back by $[m]$ acts on $\rR^1 \alpha_\ast \cO_A$  as the multiplication by $m = 0$.

To compute $\rho_{\cU_A, m}$, recall that the universal vector hull $\theta_{A[m]} \colon A[m] \to \bbV(\omega_{\check{A}[m]})$ of the finite flat group $S$-scheme $A[m]$ is the group $S$-schemes morphism with the following universal property: given a vector bundle $E$ on $S$ and a group $S$-scheme morphism $f \colon A[m] \to \bbV(E)$, there is a unique homomorphism of $\cO_S$-modules $\phi \colon \omega_{\check{A}[m]} \to E$ such that $f = \phi \circ \theta_{A[m]}$.

[infra Theorem <ref>]  Suppose $m = 0$ on $S$. With the notation above, by means of a splitting $e^\ast \cU_A \cong \omega_{\check{A}} \oplus \cO_S$,

\[
\rho_{\cU_A, m} =
\begin{pmatrix}
\id & \theta_{A[m]} \\
0 & 1
\end{pmatrix},
\]

where $\omega_{\check{A}}$ is identified with $\omega_{\check{A}[m]}$ thanks to the vanishing of $m$.

§.§.§ Computing algebraic functions
The relation with the problem of determining functions on the universal vector extension is readily elucidated. Start with recalling the following result of Coleman:

[<cit.>]  If the scheme $S$ is flat over $\bbZ$, then

\[ \Gamma(A^\natural, \cO_{A^\natural}) = \Gamma(S, \cO_S). \]

Under the stronger hypothesis of $S$ being of characteristic $0$, a generality anyway sufficient for our ultimate goal, the previous statement was also obtained by Laumon (<cit.>). This leaves to treat the case where an integer $m$ vanishes on $S$.

For, the projection $s \colon \cU_A \to \cO_A$ in the datum of $\cU_A$ as an extension can be seen as a global section of the dual vector bundle $\cU_A^\vee$. Then,

\[ \Gamma(A^\natural, \cO_{A^\natural}) = \injlim_{d \ge 0} \Gamma(A, \Sym^d \cU_A^\vee),\]

the transition maps in the direct limit being the multiplication by $s$. Now, global sections of the vector bundle $\Sym^d \cU_A^\vee$ are identified, by the arguments above, to those of $e^\ast \Sym^d \cU_A^\vee$ invariant under the linear action of $A[m]$. This leads to the equality

\[ \Gamma(A^\natural, \cO_{A^\natural}) \cong \injlim_{d \ge 0} \Gamma(S, e^\ast \Sym^d \cU_A^\vee)^{A[m]}.\]

Upon choosing a splitting $e^\ast \cU_A \cong \omega_{\check{A}} \oplus \cO_S$, the preceding identity can be reformulated as follows:

[infra Theorem <ref>]  Suppose $m = 0$ on $S$. Then, with the notation above,

\[ \Gamma(A^\natural, \cO_{A^\natural})\cong \Gamma(\bbV(\omega_{\check{A}}),\cO_{\bbV(\omega_{\check{A}})})^{A[m]},\]

the right-hand side being the set of functions on $\bbV(\omega_{\check{A}})$ invariant the transformation $x \mapsto x + \theta_{A[m]}$ on $\bbV(\omega_{\check{A}})$.

Suppose that $S$ is the spectrum of an algebraically closed field $k$ of characteristic $p > 0$ and let $E$ be an elliptic curve over $k$.

If the elliptic curve $E$ is ordinary, then $E[p] \cong \mu_p \times \bbZ/p\bbZ$.  The universal vector hull kills $\mu_p$ and injects $\bbZ/p\bbZ$ into $\bbA^1_k$ by means of the inclusion $\bbZ/p\bbZ \subseteq k$. In particular,

\[ \Gamma(E^\natural, \cO_{E^\natural}) \cong k[x]^{E[p]} = k[x^p - x],\]

where $x$ is the coordinate function on $\bbA^1_k$.

In the supersingular case, the finite group scheme $E[p]$ is an extension

\[ 0 \too \alpha_p \too E[p] \too \alpha_p \too 0\]

of $\alpha_p = \Spec k[x] / (x^p)$ by itself. The universal vector hull kills the copy of $\alpha_p$ on the left and injects the one on the right in $\bbA^1_k$ by its natural inclusion. Therefore,

\[ \Gamma(E^\natural, \cO_{E^\natural}) \cong k[x]^{E[p]} = k[x^p].\]

§.§ Formal functions on the universal vector extension

Let $K$ be a nontrivally valued complete non-Archimedean field of characteristic $0$,  $R$ its ring of integers, and $\varpi \in R$ a non-zero topologically nilpotent element. For a formal scheme $X$ over $R$, let $X_n$ the closed subscheme over $R_n := R /(\varpi^n)$ defined by the equation $\varpi^n = 0$.

§.§.§ Setup
Let $A$ be a formal abelian scheme over $R$ and $A^\natural$ its universal vector extension. By definition, formal functions on $A^\natural$ are limit of functions on the successive thickenings,

\[ \Gamma(A^\natural, \cO_{A^\natural}) = \projlim_{n \ge 1} \Gamma(A^\natural_n, \cO_{A^\natural_n}).\]

[infra Theorem <ref>]  With the notation above,

\[ \Gamma(A^\natural, \cO_{A^\natural}) = R.\]

For $n \ge 1$, it is clear by construction that $A^\natural_n$ is the universal vector extension of abelian scheme $A_n$ over $R_n$. It follows that, if $K$ is of residue characteristic $0$, then the previous statement is obtained applying Theorem <ref> to $A^\natural_n$ for each $n \ge1$. Thus, from now on, the field $K$ is supposed to be $p$-adic and the topologically nilpotent element $\varpi$ to be $p$.

§.§.§ Almost finite vector bundles
Similarly to what have been done in the algebraic context, a vector bundle $V$ on the formal abelian scheme $A$ is said to be almost finite if, for each $n \ge 1$, the restriction $V_{\rvert A_n}$ to $A_n$ is finite. If this is the case, for $n \ge 1$, let

\[ \rho_V^{(n)} \colon \rT A_n \too \GL(e_n^\ast V)\]

be the associated representation of the Tate group scheme of $A_n$, where $e_n$ is neutral element of $A_n$. The construction of the Tate group scheme and of the above representations being compatible with base change, they give rise to a morphism of formal schemes in groups over $R$,

\[ \rho_V \colon \rT A \too \GL(e^\ast V),\]

where $\rT A$ is the formal scheme $\injlim_{n \ge 1} \rT A_n$.

§.§.§ The Hodge-Tate decomposition
Consider the universal vector hull of the $p$-adic Tate group scheme $\theta_{\rT_p A} \colon \rT_p A \to \bbV(\omega_{\check{A}})$ defined as the limit, for $n \ge 1$, of the composite morphisms

\[ \rT_p A_n \too A_n[p^n] \stackrel{\theta_{n}}{\too} \bbV(\omega_{\check{A}_n}),\]

where $\rT_p A = \injlim_{n \ge 1} \rT_p A_n$ and $\theta_n$ is the universal vector hull of the finite flat group $R_n$-scheme $A_n[p^n]$ and $\omega_{\check{A}_n[p^n]}$ is identified with $\omega_{\check{A}_n}$ because $p^n = 0$ in $R_n$.

Suppose that the field $K$ is algebraically closed (which can always be done, as constancy of functions can be tested after having extended scalars). Then, following Crew (<cit.>), Coleman (<cit.>, <cit.>) and Faltings (<cit.>), the $K$-linear map

\[ \rT_p A(R) \otimes_{\bbZ_p} K \too \omega_{\check{A}} \otimes_{R} K \]

is surjective, the dual one

\[ \Lie \check{A} \otimes_{R} K = \rH^1(A, \cO_A) \otimes_R K \too \rH^1_{\et}(A, \bbQ) = \Hom_{\bbZ_p}(\rT_p A(R), K)  \]

being the $K$-linear map involved in the Hodge-Tate decomposition; see also <cit.> and Proposition <ref>.

Up to choosing a splitting $e^\ast \cU_A \cong \omega_{\check{A}} \oplus \cO_S$, Theorem <ref> applied to each successive thickening implies that the representation $\rho_{\cU_A} \colon \rT_p A \to \GL(e^\ast \cU_A)$ is

\[ \rho_{\cU_A} =
\begin{pmatrix} \id & \theta_{\rT_p A} \\
0 & 1
\end{pmatrix}.
\]

This point of view à la Narasimhan-Seshadri on the Hodge-Tate decomposition is borrowed from Deninger-Werner (<cit.>).

§.§.§ Computing functions
The advantage of employing with the Tate group scheme instead of the more classical Tate module as in <cit.> is palpable when it gets to determine the formal functions on $A^\natural$.

Indeed, it is not clear whether functions on $A^\natural_n$ can be computed as those on $\bbV(\omega_{\check{A}_n})$ left invariant under the action of the (abstract) group $\rT_p A(R_n)$. (Other instances of the considerable complications popping up by working only with the Tate module can be found in <cit.>.)

Instead, by making use of the Tate group scheme and Theorem <ref> on all successive thickenings, formal functions on $A^\natural$ are seen to be the formal functions on $\bbV(\omega_{\check{A}})$ invariant under the action of the $p$-adic Tate group scheme $\rT_p A$ given by

\[ x \longmapsto x + \theta_{\rT_p A}.\]

By the Hodge-Tate decomposition (in the form above), the image of the $\bbZ_p$-module $\rT_p A(R)$ in the $K$-vector space $\omega_{\check{A}} \otimes_R K$ is a lattice. The accumulation at $0$ of lattices in $p$-adic vector spaces forces a translation-invariant function to be constant (Lemma <ref>), permitting to conclude the proof of Theorem <ref>.

The difference with the complex picture is rather clear here: given a free abelian group $\Lambda$ of finite rank, there are plenty of nonconstant translation-invariant holomorphic functions on the complex vector space $\Lambda \otimes_{\bbZ} \bbC$, e.g. the exponential.

§.§ Uniformization of the universal vector extension

Let $A$ be an abeloid variety over $K$, namely a proper, smooth, and connected $K$-analytic space in groups. A theorem of Lütkebohmert states that the functor associating to a $K$-analytic space $S$ the set of isomorphism classes of (fiberwise) translation-invariant line bundle on $A_S$ is represented by an abeloid variety $\check{A}$ (<cit.>), the dual of $A$. By the procedure explained in the algebraic context, this permits to define the universal vector extension $A^\natural$ of $A$.

§.§.§ Tate-Raynaud's uniformization
As already argued, the constancy of a function may be tested after extending scalars. Upon replacing $K$ by a finite extension, the abeloid variety $A$ may be therefore supposed to be of the following form (op.cit., Theorem 7.6.1).

Let $B$ be Raynaud's generic fiber of a formal abelian scheme $\cB$ over $R$, $\check{\Lambda}$ a free abelian group of finite rank, $\check{c} \colon \check{\Lambda} \to \check{B}(K)$ a group homomorphism, where $\check{B}$ is Raynaud's generic fiber of the dual formal abelian scheme $\check{\cB}$. This datum defines an extension $E$ of the abeloid variety $B$ by the split torus $T$ over $K$ with group of character $\check{\Lambda}$.

Let $\Lambda$ be a free abelian group of finite rank and $i \colon \Lambda \to E(K)$ an injective group homomorphism whose image is closed and discrete. Then, the abeloid variety $A$ may be supposed to be $A = E / i(\Lambda)$, granted that $E / i(\Lambda)$ is a proper $K$-analytic space. The notation is resumed here:

\[
\begin{tikzcd}
& \Lambda \ar[d, "i"] \ar[dr, "c"] & \\
T \ar[r] & E \ar[r, "p"] \ar[d, "u"] & B \\
& A
\end{tikzcd}
\]

Within the framework of Berkovich spaces, the topological space underlying $A$ is locally path connected and locally contractible, and the one underlying $E$ is contractible (a consequence of the good reduction of $B$). Moreover, the projection $u \colon E \to A$ is seen to be a topological universal cover and $\Lambda$ the topological fundamental group of $A$.

The dual abeloid variety $\check{A}$ admits then a similar description. By biduality, the composition $c \colon \Lambda \to B(K)$ of $i$ with the projection $p \colon E \to B$ defines an extension of the abeloid variety $\check{B}$ by the split torus with group of characters $\Lambda$. Moreover, by symmetry of the Poincaré bundle, the injection $i$ induces an injective group homomorphism $\check{\imath} \colon \check{\Lambda} \to \check{E}$ whose image is closed and discrete, and $\check{A} = \check{E} / \check{\imath}(\check{\Lambda})$.

The reader might find the notation awkward enough, as the abelian group $\check{\Lambda}$ is not the dual of $\Lambda$, but rather the Cartier dual of $T$; and similarly, the torus $\check{T}$ is the Cartier dual of $\Lambda$, not of $T$. However, I felt that a notation faithful to duality would have been even murkier.

§.§.§ The universal cover of the universal vector extension
According to <cit.>, the pull-back to $E \times \check{E}$ of the Poincaré bundles $\cL_A$ on $A \times \check{A}$ and $\cL_B$ on $B \times \check{B}$ are isomorphic. Coupled with the explicit construction of the universal vector extension, this yields a canonical isomorphism of $K$-analytic groups

\[ A^\natural \times_A E \cong E^\natural,\]

where $E^\natural$ is the push-out of $B^\natural \times_B E$ along the $K$-linear map $\omega_{\check{B}} \to \omega_{\check{E}}$ given by pull-back of $1$-forms along the projection $\check{E} \to \check{B}$ (cf. <ref>).

Furthermore, according to Proposition <ref>, the natural action of $\Lambda$  induces an injective group homomorphism $i^\natural \colon  \Lambda \to E^\natural$ with closed and discrete image, whose composition with $E^\natural \to E$ is $i$, and such that

\[ E^\natural/i^\natural(\Lambda) = A^\natural.\]

Besides, the commutative and exact diagram of $K$-analytic groups

\[
\begin{tikzcd}
& 0 \ar[d] & 0 \ar[d] & 0 \ar[d] & \\
0 \ar[r] & \bbV(\omega_{\check{B}}) \ar[d, "\rd \check{p}"] \ar[r] & B^\natural \times_B E \ar[d] \ar[r] & E \ar[d, equal] \ar[r] & 0 \\
0 \ar[r] & \bbV(\omega_{\check{E}}) \ar[d] \ar[r] & E^\natural \ar[d] \ar[r] & E \ar[d] \ar[r] & 0 \\
0 \ar[r] & \bbV(\omega_{\check{T}}) \ar[d] \ar[r, "\sim"] & E^\natural / (B^\natural \times_B E) \ar[d] \ar[r] & 0 \\
& 0 & 0
\end{tikzcd}
\]

given by definition of $E^\natural$ as a push-out, allows to define a projection

\[ \pr_u \colon E^\natural \too \bbV(\omega_{\check{T}}).\]

Recalling that $\check{T}$ is by definition the split torus with group of characters $\Lambda$, define the universal vector hull $\theta_{\Lambda} \colon \Lambda \to \bbV(\omega_{\check{T}})$ of $\Lambda$ to be the map associating to $\chi \in \Lambda$, seen as a character $\chi \colon \check{T} \to \Gm$, the differential form $\chi^\ast \tfrac{\rd z}{z}$, where $z$ is the coordinate function on $\Gm$.

[infra  Theorem <ref>]  With the notation above,

\[ \pr_u \circ i^\natural = \theta_{\Lambda}. \]

When $A$ has totally degenerate semi-stable reduction, that is, $B = 0$ and thus $E = T$, the abeloid variety $A$ is the quotient $T / i(\Lambda)$ and, by Theorem <ref>, its universal vector extension is

\[ A^\natural = (T \times \bbV(\omega_{\check{T}})) / \{ (i(\chi), \theta_{\Lambda}(\chi)) : \chi \in \Lambda\}.\]

§.§ Rigid analytic functions on the universal vector extension
Keep the notation introduced in the previous paragraph and—Good riddance!—let us approach the proof of Theorem <ref>.

§.§.§ Good reduction
When the abeloid variety $A$ has good reduction, that is, it Raynaud's the generic fiber of a formal abelian scheme $\cA$ over $R$, Theorem <ref> is an easy consequence of Theorem <ref>.

Indeed, Raynaud's generic fiber $\cA_\eta^\natural$ of the formal scheme $\cA^\natural$ is a compact analytic domain in $A^\natural$ dense for the (analytic) Zariski topology of the latter. Each analytic function on $\cA^\natural_\eta$ being constant by Theorem <ref>, the same holds for $A^\natural$.

§.§.§ Totally degenerate reduction
In the diametrically opposite case, when $B = 0$ and $E = T$, according to Example <ref>, analytic functions on $A^\natural$ are identified with analytic functions $f$ on $T \times \bbV(\omega_{\check{T}})$ such that, for $\chi \in \Lambda$,

\[ f(t, x) = f(i(\chi)t, x + \theta_\Lambda(\chi)).\]

It is an easy exercise proving that such functions are necessarily constant (cf. Lemma <ref>), a phenomenon for which accumulation of lattices in non-Archimedean vector spaces ought to be held accountable, again (cf. Remark <ref>).

§.§.§ Intermediate reduction
In the intermediate case, that is, when neither $T$ nor $B$ is trivial, analytic functions on $A^\natural$ are identified with $\Lambda$-invariant analytic functions on $E^\natural$. By design, $E^\natural$ is a principal $T$-bundle over the vector extension $G$ of $B$ obtained as the push-out of the universal vector extension $B^\natural$ along the $K$-linear map $\rd \check{p} \colon \omega_{\check{B}} \to \omega_{\check{E}}$. Therefore, analytic functions on $E^\natural$ are expanded in Fourier series of global sections of line bundles on $X$. These line bundles, by definition of $E$, are pull-back along the projection $\pi \colon X \to B$ of translation-invariant line bundles on $B$.

Alas, this demands to start all over again and study not only global sections on vector extensions of the trivial line bundle, but of (the pull-back of) any (translation-invariant) line bundle. This being done, a careful analysis of the action of $\Lambda$ on these global sections permits to conclude, once again by the afore-mentioned accumulation of lattices (Lemma <ref>).

§.§ Organization of the paper
The introduction left apart, this paper has eight sections.

In the first one, applications to algebraic groups are presented. The second is a collection of some background material in algebraic geometry, for reader's sake. In the third, the moduli space of rank $1$ connections on an abelian scheme is constructed and compared with the universal vector extension. The fourth one  concerns finite vector bundles on abelian schemes and algebraic functions on vector extensions. In the fifth section, the case of formal schemes is treated and therein is found the material on almost finite vector bundles, the Hodge-Tate decomposition and their relation to formal functions on vector extensions. The sixth section gathers some results on toric bundles, including Fourier expansion. The seventh is a brief reminder on Tate-Raynaud's uniformization. In the last one, the topological universal cover of vector extensions of an abeloid variety is studied, leading to the determination of rigid-analytic functions on vector extensions.

§.§.§ Acknowledgements
I want to thank G. Ancona, J.-B. Bost, A. Chambert-Loir, A. Ducros, M. Florence, G. Freixas, J. Fresán, J. Poineau for the interest they showed towards this paper and the discussions I shared with them. Also, I want to thank G. Ancona, J. Fresán and J. Poineau for their thorough reading of a first draft of this paper and their comments that permitted to improve the presentation. The paper is in the current form following the suggestions and the encouragement of a referee, whom shall be thanked for, to write an accessible account. This research was partially supported by ANR grant ANR-18-CE40-0017.

§.§ Reminders and conventions

§.§.§ Sequences
Let $X$ be a locally ringed space and

\begin{equation} \tag{$F$}
\begin{tikzcd}
\cdots \ar[r] & F_{i - 1} \ar[r, "\phi_{i-1}"] & F_i \ar[r, "\phi_i"] & F_{i + 1} \ar[r] & \cdots
\end{tikzcd}
\end{equation}

a sequence of $\cO_{X}$-modules indexed by integers $i \in \bbZ$. For morphisms of locally ringed spaces $f \colon Y \to X$ and $g \colon X \to Z$, and an $\cO_X$-module $M$, consider the sequences

\begin{gather} \tag*{$f^\ast (F)$}
\begin{tikzcd}[ampersand replacement=\&]
\cdots \ar[r] \& f^\ast F_{i - 1} \ar[r, "f^\ast \phi_{i-1}"] \& f^\ast F_i \ar[r, "f^\ast \phi_i"] \& f^\ast F_{i + 1} \ar[r] \& \cdots
\end{tikzcd}\\
\tag*{$g_\ast (F)$}
\begin{tikzcd}[ampersand replacement=\&]
\cdots \ar[r] \& g_\ast F_{i - 1} \ar[r, "g_\ast \phi_{i-1}"] \& g_\ast F_i \ar[r, "g_\ast \phi_i"] \& g_\ast F_{i + 1} \ar[r] \& \cdots
\end{tikzcd} \\
\tag*{$(F) \otimes M$}
\begin{tikzcd}[ampersand replacement=\&]
\cdots \ar[r] \& F_{i - 1} \otimes M \ar[r, "\phi_{i-1} \otimes \id"] \& F_i \otimes M \ar[r, "\phi_i \otimes \id"] \& F_{i + 1} \otimes M \ar[r] \& \cdots
\end{tikzcd}
\end{gather}

obtained from $(F)$ respectively pulling-back along $f$, pushing-forward along $g$, and taking the tensor product with $M$.

§.§.§ Analytic spaces
Let $K$ be a complete nontrivally valued non-Archimedean  field. In this paper $K$-analytic spaces are considered in the sense of Berkovich (<cit.>).
By an abuse of notation, given a $K$-analytic space $X$, an $\cO_X$-module here is what is called an $\cO_{X_{\rG}}$-module in op.cit.. As soon as the $K$-analytic space $X$ is good (that is, every point admits an affinoid neighbourhood) the  two notions coincide (op.cit. Proposition 1.3.4). For a point $x$ in a $K$-analytic space $X$, let $\cH(x)$ be the completed residue field.

For a $K$-analytic space $S$, an $S$-analytic space in groups will be called simply an $S$-analytic group. An abeloid variety over $S$ is a proper, smooth $S$-analytic group with connected fibers.[This differs, for instance, from the terminology of <cit.> where the word `abeloid' is reserved to proper smooth connected $K$-analytic groups admitting a uniformization as the one  described in section <ref>.]

§ APPLICATION TO ALGEBRAIC GROUPS

Let $k$ be a field. An algebraic group over $k$ is a finite type group $k$-scheme.

§.§ Chevalley's decomposition
Let $G$ be a connected and reduced (or, equivalently, smooth) algebraic group. By Chevalley's theorem (<cit.>), the algebraic group $G$ is an extension of an abelian variety $A$ by an affine algebraic group $L$. The algebraic groups $A$ and $L$ are respectively called the abelian and linear parts of $G$.

§.§ Commutative algebraic groups
When the algebraic group $G$ is moreover commutative, the linear part $L$ of $G$ is of the form $T \times_k U$ for a $k$-torus $T$ and a unipotent group $U$, called respectively the toric and unipotent parts of $G$.

§.§.§ Semi-abelian part
Let $\bar{k}$ denote a separable closure of $k$ and, for a $k$-scheme $X$, let $\bar{X}$ be the $\bar{k}$-scheme obtained from $X$ by extending scalars to $\bar{k}$. Since by definition tori split on finite separable extension, the $\bar{k}$-torus $\bar{T}$ is split. The semi-abelian variety

\[ 0 \too \bar{T} \too \bar{G} / \bar{U} \too \bar{A} \too 0 \]

is thus determined by a group homomorphism $c_G \colon X^\ast(\bar{T}) \to \check{A}(\bar{k})$ equivariant with respect the natural action of $\Gal(\bar{k}/k)$ on the group of characters $X^\ast(\bar{T})$ of the torus $\bar{T}$ and the set of $\bar{k}$-rational points of the dual abelian variety $\check{A}$. More precisely, for a $\bar{k}$-scheme $S$, an $S$-valued point $g$ of $\bar{G}$ is the datum of an $S$-valued point $x$ of $\bar{A}$ and, for $\chi \in \Lambda$, of an isomorphism

\[ \langle g , \chi \rangle \colon \cO_S \too x^\ast L_\chi, \]

where $L_\chi$ is the pull-back along the map $(\id, c(\chi)) \colon \bar{A} \to \bar{A} \times \check{\bar{A}}$ of the Poincaré bundle $\cL$ on $A \times \check{A}$. The trivializations $\langle g, \chi \rangle$ undergo the relation, for $\chi, \chi' \in \Lambda$,

\[ c(\chi) \otimes c(\chi') = c(\chi + \chi'),\]

meant to be understood via the isomorphism $L_\chi \otimes L_{\chi'} \cong L_{\chi + \chi'}$.

§.§.§ Vector part
Suppose that the commutative unipotent group $U$ is the vector group $\bbV(F)$ associated with a finite-dimensional $k$-vector space $F$. This is always the case when the characteristic of $k$ is $0$ (<cit.>), while it may fail in positive characteristic. Owing to <cit.>, an fppf principal bundle under a vector group admits sections Zariski-locally, whence the equalities

\[ \rH^1_{\textup{fppf}}(A, \bbV(F)) = \rH^1_{\textup{Zar}}(A, \bbV(F)) = \rH^1(A, \cO_A) \otimes_k F.\]

Via the canonical isomorphism $\Lie \check{A} \cong \rH^1(A, \cO_A)$ (see <cit.>), the isomorphism class of the vector extension

\[ 0 \too \bbV(F) \too G/T \too A \too 0,\]

seen merely as a principal $\bbV(F)$-bundle over $A$, can therefore be seen as a $k$-linear map $\phi_G \colon \omega_{\check{A}} \to F$, where $\omega_{\check{A}}$ is the dual of the Lie algebra of $\check{A}$. Consider the universal vector extension $A^\natural$ of $A$,

\[ 0 \too \bbV(\omega_{\check{A}}) \too A^\natural \too A \too 0.\]

Then, the algebraic group $G/T$ is the obtained as the push-out of $A^\natural$ along the linear map $\phi_G$ (see <ref>):

\[
\begin{tikzcd}
0 \ar[r] & \bbV(\omega_{\check{A}}) \ar[r] \ar[d, "\phi_G"] & A^\natural \ar[r] \ar[d, "\Phi_G"] & A \ar[r] \ar[d, equal] & 0\ \\
0 \ar[r] & \bbV(F) \ar[r] & G/T \ar[r] & A\ar[r]  & 0.
\end{tikzcd}
\]

§.§.§ Classification
These considerations essentially show the following:

The map $G \mapsto (A, T, F, c_G, \phi_G)$ defined above sets up a bijection between the sets of isomorphism classes of

* commutative, connected and reduced algebraic groups over $k$ whose unipotent part is a vector group;

* quintuples $(A, T, F, c, \phi)$ made of an abelian variety, a $k$-torus $T$, a finite-dimensional $k$-vector space $F$, a  $\Gal(\bar{k}/k)$-equivariant homomorphism of groups $c \colon X^\ast(\bar{T}) \to \check{A}(\bar{k})$ and a $k$-linear map $\phi \colon \omega_{\check{A}} \to F$,

the notion of isomorphism for the above quintuples being defined in the evident manner.

§.§ Anti-affine algebraic groups

§.§.§ Classification
An algebraic group $G$ over $k$ is anti-affine if $\Gamma(G, \cO_G) = k$. As recalled in the introduction, an anti-affine algebraic group is connected and reduced. Moreover, for an extension $k'$ of $k$ and an algebraic group $G$, flat base change yields the equality

\[ \Gamma(G', \cO_{G'}) =\Gamma(G, \cO_{G}) \otimes_k k', \]

where $G'$ is the algebraic group over $k'$ obtained from $G$ by extending scalars. In particular, the algebraic group $G$ is anti-affine if and only if $G'$ is.

Let $G$ be a commutative, connected, reduced algebraic group over $k$. Then, the algebraic group $G$ is anti-affine if and only if one of the following conditions is fulfilled:

* $\characteristic k = 0$, $c_G$ is injective and $\phi_G$ surjective;

* $\characteristic k >0$, $c_G$ is injective and $U = 0$.

In other words, anti-affine algebraic groups over a field of positive characteristic are anti-affine semi-abelian varieties. Moreover, if the field $k$ is an algebraic extension of a finite field, the only anti-affine semi-abelian varieties are abelian varieties (because, in this case, all $\bar{k}$-rational points of $\check{A}$ are torsion).

§.§ Affine quotient
For an algebraic group $G$ over $k$, the $k$-algebra $\Gamma(G, \cO_G)$ is of finite type and inherits a structure of Hopf algebra, so that $G_{\aff} := \Spec \Gamma(G, \cO_G)$ is an algebraic group over $k$. The canonical morphism $\pi_G \colon G \to G_{\aff}$ is faithfully flat and has the following property: given an affine algebraic group $H$ over $k$ and a morphism $f \colon G \to H$ of algebraic groups, there is an unique morphism of algebraic groups $\tilde{f} \colon G_{\aff} \to H$ such that $f = \tilde{f} \circ \pi_G$.

The kernel of the morphism $\pi_G$ is an anti-affine algebraic group over $k$.

§.§ Main theorem
Let $K$ be a complete non-trivially valued non-Archimedean  field. For a $K$-scheme finite type $X$, let $X^\an$ be the $K$-analytic space deduced from $X$. Consider:

* $A$ an abelian variety over $K$,

* $\Lambda$ a free abelian group of finite rank,

* $F$ a finite-dimensional $K$-vector space,

* $c \colon \Lambda \to \check{A}(K)$ a homomorphism of groups, and

* $\phi \colon \omega_{\check{A}} \to F$ a $K$-linear map,

where $\check{A}$ is the dual abelian variety and $\omega_{\check{A}}$ the dual of $\Lie \check{A}$. Let $G$ be the reduced, connected, commutative algebraic group over $K$ corresponding to the quintuple $(A, T, F, c, \phi)$, where $T$ is the split $K$-torus with group of characters $\Lambda$.

Suppose one of the following:

* $\characteristic K = 0$, $c$ is injective and $\phi$ surjective;

* $\characteristic K > 0$, $c$ is injective and $F = 0$.

\[\Gamma(G^\an, \cO_{G^\an}) = K.\]

§.§ Applications

§.§.§ Analytic functions on algebraic groups

A $K$-analytic function on (the analytification of) an anti-affine algebraic group over $K$ is constant.

Let $\bar{K}$ be the completion of an algebraic closure of $K$. For an anti-affine algebraic group $G$ over $K$, consider the algebraic group $\bar{G}$ over $\bar{K}$ obtained from $G$ by extending scalars to $\bar{K}$. By flat base change,

\[ \Gamma(\bar{G}, \cO_{\bar{G}}) = \Gamma(G, \cO_G) \otimes_K \bar{K},\]

so that the algebraic group $\bar{G}$ is anti-affine. Since the toric part of $\bar{G}$ is split, Theorem <ref> may be applied the algebraic group $\bar{G}$ over the non-Archimedean field $\bar{K}$ to give the identity

\[ \Gamma(\bar{G}^\an, \cO_{\bar{G}^\an}) = \bar{K}.\]

Rather generally, for a $K$-analytic space $X$, the natural homomorphism of $\bar{K}$-algebras $\Gamma(X, \cO_{X}) \otimes_K \bar{K} \to \Gamma(\bar{X}, \cO_{\bar{X}})$ is injective, where $\bar{X}$ is the $\bar{K}$-analytic space deduced from $X$ by extending scalars.

The equality $\Gamma(G^\an, \cO_{G^\an}) = K$ follows immediately.

Let $\pi \colon X \to S$ be a surjective and smooth morphism between $K$-analytic spaces. Suppose $S$ reduced and, for $s \in S$, that the natural homomorphism of $\cH(s)$-algebras

\[ \cH(s) \too \Gamma(X_s, \cO_{X_s}),\]

is an isomorphism, where $\cH(s)$ is the completed residue field at $s$.

Then, the natural homomorphism of $K$-algebras $\Gamma(S, \cO_S) \to \Gamma(X, \cO_X)$ is an isomorphism.

The morphism $\pi$ is faithfully flat, thus the map $\Gamma(S, \cO_S) \to \Gamma(X, \cO_X)$
is injective. Proving the statement amounts to exhibiting, for a $K$-analytic function $f$ on $X$, a (necessarily unique) $K$-analytic function $g$ on $S$ such that $f = g \circ \pi$. Invoking faithfully flat descent, and more precisely <cit.>, this is equivalent to proving the equality

\begin{equation}\label{Eq:FaithfullyFlatDescentFunctionConstantOnTheFibers} f \circ \pr_1 = f \circ \pr_2 \end{equation}

of $K$-analytic functions on $X \times_S X$, where $\pr_1, \pr_2 \colon X \times_S X \to X$ are the projections. (Note that faithfully flat descent can be applied because the morphism $\pi$, being smooth and surjective, is faithfully flat and without boundary, and in particular properly surjective in Daylies' terminology; see op.cit. 1.3.)

Now, it is sufficient to check condition (<ref>) only at the  only set-theoretical level because the $K$-analytic space $X \times_S X$ is reduced: indeed, the morphism $X \times_S X \to S$ is smooth and the $K$-analytic space $S$ is reduced.

Let $s$ be a point of $S$ and let us check the wanted identity on the fiber $X_s$ of $\pi$ in $s$. By hypothesis, each $\cH(s)$-analytic function on the $\cH(s)$-analytic space $X_s$ is constant. Therefore,

\[ f_{\rvert X_s} \circ \pr_1 = f_{\rvert X_s} \circ \pr_2,\]

which concludes the proof.

Let $G$ be an algebraic group. With the notation above, precomposing with $\pi^\an_G$ induces an isomorphism

\[ \Gamma(G^\an_{\aff}, \cO_{G^\an_{\aff}}) \stackrel{\sim}{\too} \Gamma(G^\an, \cO_{G^\an}). \]

Let $G_{\ant}$ be the kernel of the morphism of algebraic groups $\pi_G$.

(Reduced case.) Suppose $G$ reduced. The morphism $\pi_G \colon G \to G_{\aff}$ is smooth, thus, according to <cit.>, the morphism $\pi_G^\an$ is smooth. Given a point $s$ of $G_{\aff}^\an$, the fiber $G^\an_s$ of $\pi^\an_G$ in $s$ is isomorphic to the analytification of the algebraic group $G_{\ant} \times_K \cH(s)$ over $\cH(s)$ obtained from $G_{\ant}$ by extending scalars. The algebraic group $G_{\ant} \times_K \cH(s)$ is anti-affine by Proposition <ref>, thus Corollary <ref> implies

\[ \Gamma(G^\an_s, \cO_{G^\an_s}) = \cH(s). \]

Lemma <ref> infers the desired result.

(General case.) The case of arbitrary $G$  is reduced to the preceding one as follows. Let $G_{\red}$ be the reduced subgroup, $H$ the quotient $G / G_{\red}$ and $p \colon G \to H$ the projection onto the quotient.

The algebraic group $H$ is finite, hence affine. Therefore, the morphism $p$ factors through a surjective morphism $\bar{p} \colon G_{\aff} \to H$ sitting in the following commutative and exact diagram of algebraic groups:

\[
\begin{tikzcd}
0 \ar[r] & G_{\red} \ar[r] \ar[d, "{\pi_{G_{\red}}}"] & G \ar[r, "p"] \ar[d, "\pi_G"] & H \ar[r] \ar[d, equal] & 0 \\
0 \ar[r] & (G_{\red})_{\aff} \ar[r] & G_{\aff} \ar[r, "\bar{p}"] & H \ar[r] & 0.
\end{tikzcd}
\]

(To make sense of the diagram above, note that  the equality $(G_{\red})_{\aff} = (G_{\aff})_{\red}$ holds because the subgroup $G_{\ant}$ is anti-affine, thus reduced.)

Let $I$ be the augmentation ideal of the Hopf algebra $\Gamma(H, \cO_H)$, that is, the maximal ideal defining the neutral element of $H$. The exactness of the rows in the preceding diagram means that the sheaves of ideals of nilpotent elements of $\cO_G$ and $\cO_{G_{\aff}}$ are respectively $p^\ast I$ and $\bar{p}^\ast I$. For an integer $i \ge 0$, set

\begin{align*}
A_i:= \Gamma(H_i, \cO_{H_i}), && E_i := I^i / I^{i+1}, && F_i := p^\ast I^{i} / p^\ast I^{i+1}, && \bar{F}_i := \bar{p}^\ast I^{i} / \bar{p}^\ast I^{i+1}.
\end{align*}

Because of the equality $E_i  = I^i \otimes_{A_i} A_i/I$, the $A_i$-module $E_i$ is naturally seen as a finite-dimensional $K$-vector space. Arguing similarly, the $\cO_{G}$-module $F_i$ is supported on $G_{\red}$, the $\cO_{G_{\aff}}$-module $\bar{F}_i$ is supported on $(G_{\red})_{\aff}$, and

\begin{align*}
F_i &= E_i \otimes_K \cO_{G_{\red}}, \\
\bar{F}_i &= E_i \otimes_K \cO_{(G_{\red})_{\aff}}.
\end{align*}

For an integer $i \ge 0$, let $G_i$ and $G_{\aff, i}$ be the closed subschemes respectively of $G$ and $G_{\aff}$ defined by the sheaf of ideals $p^\ast I^{i+1}$ and $\bar{p}^\ast I^{i+1}$. In particular, $G_0 = G_{\red}$ and $G_{\aff, 0} = (G_{\red})_{\aff}$. Consider the following commutative and exact diagram of $K$-vector spaces:

\begin{equation} \label{eq:SuccessiveThickeningsAnalyticFunctionsFromAffineQuotient}
\begin{tikzcd}
0 \ar[r] & \Gamma(G_{\aff, 0}^\an, \bar{F}_i^\an) \ar[r] \ar[d, "\beta_i"] & \cO(G_{\aff, i}^\an) \ar[r] \ar[d, "\alpha_i"] & \cO(G_{\aff, i-1}^\an) \ar[r] \ar[d, "\alpha_{i-1}"] & 0 \\
0 \ar[r] & \Gamma(G_0^\an, F_i^\an) \ar[r] & \cO(G_{i}^\an) \ar[r] & \cO(G_{i-1}^\an),
\end{tikzcd}
\end{equation}

where vertical arrows are given by precomposing with $\pi^\an_G$, the rows by restricting functions and $\cO(X) := \Gamma(X, \cO_X)$  for a $K$-analytic space $X$. Remark that the first row is exact as the $k$-scheme $G_{\aff, i}$ is finite.

Since $G_i = G$ for $i$ big enough, it suffices to show that the homomorphism $\alpha_i$ is an isomorphism for each integer $i \ge 0$. Before that, note, by the reduced case and the preceding considerations, that the homomorphism of $K$-vector spaces

\[ \beta_i \colon \cO(G_{\aff, 0}^\an) \otimes_K E_i = \Gamma(G_{\aff, 0}^\an, \bar{F}_i^\an)  \too \Gamma(G_{\red}^\an, F_i^\an) =
\cO(G_0^\an) \otimes_K E_i,
\]

is an isomorphism.

Finally, arguing by induction on $i \ge 0$, the homomorphism $\alpha_i$ is seen to be an isomorphism. The reduced case settles the problem for $i = 0$, as $G_0 = G_{\red}$. For $i \ge 1$, the homomorphisms $\beta_i$ and $\alpha_{i-1}$ are isomorphisms (the second by inductive hypothesis), thus the Five Lemma applied to the commutative diagram (<ref>) leads to the result.

§.§.§ Stein spaces

A $K$-analytic space $X$ is Stein if it is separated, countable at infinity and, for each complete valued extension $K'$ of $K$, higher coherent cohomology vanishes on the $K'$-analytic space $X'$ deduced from $X$ by extension of scalars; that is, for a coherent $\cO_{X'}$-module $F$ and an integer $q \ge 1$,

\[ \rH^q(X', F) = 0.\]

A Stein exhaustion of a $K$-analytic space $X$ is a G-cover $\{ X_n \}_{n \in \bbN}$ made of Stein compact analytic domains $X_n$ of $X$ such that, for an integer $n \ge 0$, $X_n$ is contained in $X_{n + 1}$ and the restriction map $\Gamma(X_{n + 1}, \cO_X) \to \Gamma(X_n, \cO_X)$ has a dense image (<cit.>).

A compact Stein $K$-analytic space is what is called a Liu space in <cit.>, so that a Stein exhaustion is nothing but a W-exhaustion by Liu domains in terminology of op.cit., Definition 1.10.

A $K$-analytic space $X$ is

* holomorphically separated if, for distinct points $x$ and $x'$ of $X$, there is a $K$-analytic function $f$ on $X$ such that $|f(x)| \neq |f(x')|$;

* holomorphically convex if, for a compact subset $C$ of $X$, its holomorphically convex hull

\[ \hat{C}_X := \{ x \in X : |f(x)| \le \| f\|_C, f \in \Gamma(X, \cO_X)\}\]

is compact, where $\| \cdot \|_C$ is the sup norm on $C$.

A technical (but useful) characterization is the following:

The following properties, for a separated and countable at infinity  $K$-analytic space $X$, are equivalent:

* being Stein;

* admiting a Stein exhaustion;

* being holomorphically separable, holomorphically convex and having an acyclic structural sheaf $\cO_X$.

This is <cit.>. Note that, as the field $K$ is non-trivially valued, by Theorem A.5 in op.cit., the vanishing of the higher cohomology of $\cO_X$ implies that vanishing over any complete valued field extension of $K$.

For later use, let us record the following:

For $i = 1, 2$, let $X_i$ be a Stein $K$-analytic space and  $X_i \to S$ a separated morphism of $K$-analytic spaces.  Then, the fibered product $X_1 \times_S X_2$ is Stein.

The fibered product $X_1 \times_S X_2$ is a closed analytic subspace of the product $X:= X_1 \times X_2$, reducing the matter to the case when $S$ is a $K$-rational point.

When $X_1$ and $X_2$ are compact, let $K'$ be an algebraically closed complete valued extension of $K$ such that, for $i = 1, 2$, the $K'$-analytic space $X'_{i}$, deduced from $X_i$ by extension of scalars, is strict. Then, according to  Theorem 5.2 and allowing for Theorem A.5 in <cit.>, it suffices to show that the sheaf $\cO_{X}$ is acyclic and, given two distinct $K'$-points $x$, $x'$ of $X$, there is a $K'$-analytic function $f$ on $X' := X'_1 \times X_2'$ such that $|f(x)| \neq |f(x')|.$

The vanishing of higher cohomology of $\cO_X$ is granted by op.cit., Theorem A.6. For two distinct $K'$-points $x = (x_1, x_2)$, $x' = (x'_1, x'_2)$, there is $i  = 1, 2$ for which $x_i \neq x'_i$. Up to permuting the factors suppose $i = 1$. Then, the $K'$-analytic space $X_1'$ is Stein, the notion being insensible to scalar extensions, and thus holomorphically separable. Therefore, there is a $K'$-analytic function $f$ such that $|f(x_1)| \neq |f(x'_1)|$.

In the general case, according to Theorem <ref>, it suffices to exhibit a Stein exhaustion of $X_1 \times X_2$. Now, if $\{ X_{i, a}\}_{a \in \bbN}$ is a Stein exhaustion of $X_i$ for $i = 1, 2$, the preceding case implies that the collection $\{ X_{1, a} \times X_{2, b}\}_{a, b \in \bbN}$ is a Stein exhaustion of $X$.

The usual arguments of cohomological vanishing yield immediately:

A closed analytic subspace $Z$ of a Stein $K$-analytic space $X$ is Stein.

Let $X$ be a $K$-analytic space without boundary, countable at infinity and such that

\[ \sup_{x \in X} \dim_{\cH(x)} \Omega^1_{X, x} < + \infty.\]

Then, the following are equivalent:

* the $K$-analytic space $X$ is Stein;

* there is a closed immersion $X \to \bbA^{n, \an}_K$ for some integer $n \ge 0$.

(1) $\Rightarrow$ (2) This is Lütkebohmert's embedding theorem (<cit.>), granted that the $K$-analytic space $X$ admits a Stein exhaustion made of affinoids domains (as opposed to compact Stein analytic spaces). Under the absence of boundary, the latter holds by  Theorem <ref> and <cit.>.

(2) $\Rightarrow$ (1) Exhaustions of affine spaces by closed discs of increasing radii are Stein exhaustions. Coupled with Lemma <ref> this permits to conclude.

§.§.§ Stein algebraic groups

An algebraic variety $X$ over $K$ is Stein if the $K$-analytic space $X^\an$ is.

For an algebraic variety $X$, the $K$-analytic space $X^\an$ is without boundary and countable at infinity. Moreover, the bound

\[ \sup_{x \in X^\an} \dim_{\cH(x)} \Omega^1_{X^\an, x} < + \infty\]

is always satisfied. It follows from Lütkebohmert's Embedding Theorem (Theorem <ref>) that the definition of Stein algebraic variety just given agrees with the one in the introduction of this paper.

An algebraic group over $K$ is Stein if and only if it is affine.

Let $G$ be an algebraic group over $K$. According to Theorem <ref>, a morphism of $K$-analytic spaces $f \colon G^\an \to \bbA^{n, \an}_K$ factors uniquely through $G_{\aff}^\an$. It follows that, if $f$ is a closed immersion, then the morphism $\pi_G^\an$ also is. In particular, the morphism $\pi_G^\an$ is an isomorphism, being simultaneously smooth and a closed immersion.  This shows that if $G$ is Stein, then it is affine.

The converse implication is obvious.

§.§.§ Neeman's example

Let $G$ be a reduced connected algebraic group, $A$ its abelian part, $H$ be a line bundle on $A$, $\bbV(H^\vee)^\times$ the total space of $H^\vee$ deprived of its zero section, and

\[ P: = G \times_A \bbV(H^\vee)^\times.\]

The algebraic variety $P$ is the total space of the line bundle $p^\ast H^\vee$ deprived of its zero section, where $p \colon G \to A$ is the projection of $G$ onto its abelian part, and in particular a principal $\Gm$-bundle over $G$.

The following conditions are equivalent:

* $G$ is affine;

* $G$ is Stein;

* $P$ is affine;

* $P$ is Stein.

Let $X$ be a $K$-analytic space, $L$ a line bundle on $X$ and $Y$ the total space of the line bundle $L$ deprived of its zero section.

Then, the $K$-analytic space $X$ is Stein if and only $Y$ is.

The equivalence of (1) and (2) is Corollary <ref>, while that of (2) and (4) is Lemma <ref>. Clearly, if $P$ is affine, then $P$ is Stein, which settles (3) $\Rightarrow$ (4). Finally, because of the already established equivalences, if $P$ is Stein, then $G$ is affine; since the projection $P \to G$ is an affine morphism, the algebraic variety $P$ is affine.

§ MISCELLANEA

§.§ Constant and unipotent bundles

Let $f \colon X \to S$ be a morphism of schemes.

A vector bundle $V$ on $X$ is

* constant if it is isomorphic to $f^\ast E$ for some vector bundle $E$ on $S$;

* unipotent if there is an increasing filtration $0 = V_0 \subsetneq V_1 \subsetneq \cdots \subsetneq V_{d} = V$ of $V$ by $\cO_X$-submodules such that, for $i = 1, \dots, d$, the $\cO_X$-module $V_{i} / V_{i-1}$ is constant.

Note that the $\cO_X$-modules $V_i$ are automatically locally free.

The direct sum, tensor product, internal homs, symmetric and exterior powers of constant (resp. unipotent) vector bundles are constant (resp. unipotent).

Suppose that the morphism $f \colon X \to S$ is faithfully flat and that the natural homomorphism $\cO_S \to f_\ast \cO_X$ is an isomorphism. Then, for a vector bundle $V$ on $X$, the following are equivalent:

* the vector bundle $V$ is constant;

* the natural homomorphism of $\cO_X$-modules $f^\ast f_\ast V \to V$
is an isomorphism.

In particular, for a constant vector bundle $V$, the $\cO_S$-module $f_\ast V$ is a vector bundle.

(1) $\Rightarrow$ (2) Needless to say, it suffices to show the statement when $V$ is equal (as opposed to merely isomorphic) to $f^\ast E$ for some vector bundle $E$ on $S$. By the projection formula, the natural homomorphism

\[ E \otimes f_\ast \cO_X \too f_\ast f^\ast E = f_\ast V\]

of $f_\ast \cO_X$-modules is an isomorphism. Composing it with the natural homomorphism $\cO_S \to f_\ast \cO_X$ of $\cO_S$-modules, which is an isomorphism by hypothesis, yields an isomorphism of $\cO_S$-modules $E \to f_\ast V$. Identifying $E$ with $f_\ast V$ by means of the latter, the natural homomorphism $f^\ast E \to V$ is nothing but the identity.

(2) $\Rightarrow$ (1) It suffices to set $E := f_\ast V$ as soon as it is established that $f_\ast V$ is a vector bundle on $S$. The $\cO_X$-module $f^\ast f_\ast V$ is finitely presented and flat, as it is isomorphic to the vector bundle $V$. Since the morphism $f$ is faithfully flat (hence universally injective at the level of local rings <cit.>), the $\cO_S$-module $f_\ast V$ is finitely presented and flat (<cit.>), that is, a vector bundle over $S$. This also proves the last statement.

Suppose that the morphism $f \colon X \to S$ is faithfully flat and that the natural homomorphism $\cO_S \to f_\ast \cO_X$ is an isomorphism. Let $p \colon S' \to S$ be a finite and locally free morphism of schemes. Consider the following cartesian square

\[
\begin{tikzcd}
X' \ar[r, "q"] \ar[d, "f'", swap] & X \ \ar[d, "f"]\\
S' \ar[r, "p"] & S.
\end{tikzcd}
\]

Then, for a constant vector bundle $V$ on $X'$, the vector bundle $q_\ast V$ on $X$ is constant.

According to Proposition <ref>, it suffices to show that the homomorphism of $\cO_X$-modules
\[\alpha \colon f^\ast f_\ast q_\ast V \too q_\ast V\]
is an isomorphism. The morphism $f'$ is faithfully flat because being faithful flatness is a property stable under base change. Moreover, by flatness of $p$, the homomorphism of $\cO_{S'}$-modules $p^\ast f_\ast \cO_X \to f'_\ast \cO_{X'}$ obtained by base change is an isomorphism. It follows, that the natural homomorphism $\cO_{S'} \to f'_\ast \cO_{X'}$ is an isomorphism.

Proposition <ref>, applied to the constant vector bundle $V$ on $X'$, states that the natural homomorphism of $\cO_{X'}$-modules
\[\beta \colon f'^\ast f'_\ast V \too V\]
is an isomorphism. Now, by flat base change along $f$, the natural homomorphism of $\cO_{X}$-modules $f^\ast p_\ast f'_\ast V \to q_\ast f'^\ast f'_\ast V$ is an isomorphism. The equality $f^\ast p_\ast f'_\ast V  = f^\ast f_\ast q_\ast V $ implies that the map $\alpha$ is the push-forward of $\beta$ along the morphism $q$, thus an isomorphism.

Suppose that the natural homomorphism $\cO_S \to f_\ast \cO_X$ is an isomorphism. Let $V$ be a unipotent vector bundle on $X$ and $F$ an $\cO_X$-module such that $f_\ast F$ vanishes. Then,

\[ f_\ast (F \otimes V) = 0. \]

Consider a short exact sequence of $\cO_X$-modules

\[ 0 \too V' \too V \too V'' \too 0, \]

where $V'$ and $V''$ are vector bundles such that $f_\ast(F \otimes V')$ and $f_\ast (F \otimes V'')$ vanish. Then, the sequence of $\cO_X$-modules

\[ 0 \too F \otimes V' \too F \otimes V \too F \otimes V'' \too 0\]

is exact and, by left exactness of the push-forward along $f$, the $\cO_S$-module $f_\ast V$ vanish. By induction on the rank of $V$ and by definition of a unipotent vector bundle, one reduces to the case $V = f^\ast E$ for some vector bundle $E$ on $S$. The projection formula then yields

\[ f_\ast (F \otimes f^\ast E) = f_\ast F \otimes E = 0,\]

which concludes the proof.

§.§ Connections

§.§.§ Vector bundles on first order thickenings

Let $X_0$, $X_1$ be schemes endowed with a closed immersion $s \colon X_0 \to X_1$ and a morphism $f \colon X_1 \to X_0$ such that $f \circ s = \id_{X_0}$. Suppose that the sheaf of ideals $I := \Ker (\cO_{X_1} \to \cO_{X_0})$ is of square zero. For a quasi-coherent $\cO_{X_1}$-module $F$, consider the short exact sequence

\begin{equation} \tag{$F$} 0 \too IF \too F \too  F /I F \too 0. \end{equation}

The sequence of $\cO_{X_0}$-modules

\begin{equation} \tag*{$f_\ast (F)$} 0 \too f_\ast (IF) \too f_\ast F \too s^\ast F \too 0  \end{equation}

obtained by pushing-forward $(F)$ along $f$ is short exact because the morphism $f$ is affine (affineness only depends on the underlying reduced structure).

Pushing forward a homomorphism of $\cO_{X_1}$-modules $\phi \colon F \to F'$ yields a homomorphism of short exact sequences of $\cO_{X_0}$-modules $f_\ast \phi \colon f_\ast (F) \to f_\ast (F')$. The so-defined functor

\[
\begin{array}{rcl}
\vspace{5pt}\left\{ \cO_{X_1}\textup{-modules}\right\} &\too&
\begin{array}{c}
\textup{short exact sequences} \\
\textup{of $\cO_{X_0}$-modules}
\end{array} \right\}} \\
F &\longmapsto& f_\ast (F)
\end{array}
\]

is faithful. Moreover, an isomorphism of $\cO_{X_1}$-modules $f^\ast s^\ast F \to F$ induces a splitting of the short exact $f_\ast (F)$.

For a vector bundle $F$ on $X_1$, the bijection

\[ \Hom_{\cO_{X_1}\textup{-mod}}(f^\ast s^\ast F, F) \too \Hom_{\cO_{X_0}\textup{-mod}}(s^\ast F, f_\ast F)\]

given by adjunction induces a bijection

\[
\left\{
\begin{array}{c}
\textup{isomorphisms $\rho \colon f^\ast s^\ast F \to F$} \\
\textup{such that $s^\ast \rho = \id_{s^\ast F}$}
\end{array} \right\}
\stackrel{\sim}{\too}
\left\{
\begin{array}{c}
\textup{splittings of the short} \\
\textup{exact sequence $f_\ast (F)$}
\end{array} \right\}.
\]

The only thing to show is that, for a splitting $\phi \colon s^\ast F \to f_\ast F$ of the short exact sequence $f_\ast (F)$, the homomorphism $\Phi \colon f^\ast s^\ast F \to F$ obtained by extending  $\cO_{X_1}$-linearly $\phi$,  is an isomorphism. To check this, one may reason locally on $X_0$ and split the short exact sequence $f_\ast (F) $:

\[ f_\ast F \cong s^\ast F \oplus s^\ast F \otimes f_\ast I, \]

allowing for the identities $I F = I \otimes F$ and $f_\ast (IF) = f_\ast I \otimes s^\ast F$, which hold respectively by the flatness of $F$ and because the ideal $I$ is of square zero.

Via these identifications, the splitting $\phi$ is of the form $v \mapsto (v, \epsilon(v))$ for a homomorphism of $\cO_X$-modules $\epsilon \colon s^\ast F \to s^\ast F \otimes f_\ast I$. By writing a section of $f^\ast s^\ast F$ as $(v , v')$, for sections $v$ of $s^\ast F$ and $v'$ of $s^\ast F \otimes f_\ast I$, the map $\Phi$ is defined as

\[ (v, v') \longmapsto (v, \epsilon(v) + v'),\]

where the term $\epsilon(v')$ vanished because the ideal $I$ is of square $0$. Such an expression defines clearly an isomorphism, which concludes the proof.

§.§.§ Tensor product and Baer sums
For simplicity, assume $X_1$ to be the first order thickening of $\bbV(E^\vee)$ along its zero section $s$, where $E$ is a vector bundle on $X_0$ and $\bbV(E^\vee)$ the total space of its dual. Concretely, the scheme $X_1$ is the spectrum of the $\cO_{X_0}$-module $\cO_{X_0} \oplus E$ endowed with $\cO_{X_0}$-algebra structure

\[(a, v).(b,w) = (ab, aw+bv).\]

In what follows, it will be important to distinguish whether tensor products are taken with respect to $\cO_{X_0}$ or $\cO_{X_1}$. To mark the difference, but at the same time make notation lighter, for $i = 0, 1$ and $\cO_{X_i}$-modules $V$ and $V'$, write $V \otimes_i V'$ instead of $V \otimes_{\cO_{X_i}} V'$. For a vector bundle $V$ on $X_1$, set $V_0 := s^\ast V$ so that the vector bundle  $f_\ast V$ on $X_0$ is an extension of $V_0$ by $V_0 \otimes_0 E$.

In order to make sense of the following statement, remark that, for vector bundles $V$, $V'$ on $X_1$, the $\cO_{X_0}$-modules

\begin{align*}
f_\ast V \otimes_0 V_0', && f_\ast V' \otimes_0 V_0,&& f_\ast (V \otimes_{1} V')
\end{align*}

are all extensions of $V_0 \otimes_0 V'_0$ by $E \otimes_0 V_0 \otimes_0 V'_0$.

The extension $f_\ast (V \otimes_{1} V')$ is the Baer sum of the extensions $f_\ast V \otimes_0 V_0'$ and $f_\ast V' \otimes_0 V_0$.

Unfortunately, the argument is quite clumsy and goes through the explicit construction of the Baer sum in question. To recall it,  let $p \colon f_\ast V \to V_0$ and $p' \colon f_\ast V' \to V_0'$ denote the homomorphisms given by restriction to $X_0$, and consider the $\cO_{X_0}$-submodule
\[ W\subseteq   (V_0  \otimes_0 f_\ast V' ) \oplus (f_\ast V \otimes_0 V_0')\]
made of couples whose components have the same image in $V_0 \otimes_0 V'_0$ respectively via $p \otimes \id_{V'_0}$ and $\id_{V_0} {\otimes} p'$. The vector bundle $W$ on $X_0$ fits into the following short exact sequence of $\cO_{X_0}$-modules:

\begin{equation} \tag{$W$}
0 \too (E \otimes_0 V_0 \otimes_0 V'_0)^{\oplus 2} \too W \too V_0 \otimes_0 V'_0 \too 0.
\end{equation}

The Baer sum mentioned in the statement is the push-out of the short exact sequence $(W)$ along the sum map
\[\sigma \colon (E \otimes_0 V_0 \otimes_0 V'_0)^{\oplus 2} \too E \otimes_0 V_0 \otimes_0 V'_0.\]

This being said, consider the natural epimorphism of $\cO_{X_0}$-modules

\[ \phi \colon f_\ast V \otimes_{0} f_\ast V' \too f_\ast (V \otimes_{1} V')\]

given by universal property of tensor product. The vector bundle $E$, seen as a sheaf of ideals of $f_\ast \cO_{X_1}$, is of square zero, hence the kernel of the map $\phi$ is

\[ \Ker \phi = (E \otimes_0 V_0) \otimes_0 (E \otimes_0 V'_0).\]

The right-hand side of the above equality is seen to be also the intersection of the kernels of the homomorphisms $p \otimes \id_{f_\ast V'}$ and $\id_{f_\ast V'}  {\otimes} p'$. That is, upon setting

\[ \psi = (p \otimes \id_{f_\ast V'}, \id_{f_\ast V'}  {\otimes} p') \colon f_\ast V \otimes_{0} f_\ast V'  \too (V_0 \otimes_0 f_\ast V') \oplus ( f_\ast V \otimes_0 V_0'), \]

the homomorphisms $\phi$ and $\psi$ share the same kernel. Besides, the identity

\[ (p \otimes \id_{V_0'}) \circ (\id_{f_\ast V} {\otimes} p') = p \otimes p' =  (\id_{V_0}  \otimes p') \circ ( p {\otimes}  \id_{f_\ast V'})  \]

of maps $f_\ast V \otimes_0 f_\ast V' \to V_0 \otimes_0 V_0'$ implies that the image of $\psi$ is $W$. Consequently, the homomorphism $\phi$ factors uniquely through an epimorphism of $\cO_{X_0}$-modules

\[  \tilde{\phi} \colon W \too f_\ast (V \otimes_1 V'). \]

The kernel of $\phi$ is contained in that of $p \otimes p'$ and the homomorphism $\tilde{\phi}$ acts on the quotient

\[
\frac{\ker (p \otimes p')}{\ker \phi} = \frac{K + K'}{K \cap K'} \cong \frac{K}{K \cap K'} \oplus \frac{K'}{K \cap K'}= (E \otimes_0 V_0 \otimes_0 V_0')^{\oplus 2}
\]

as the sum map $\sigma$, where $K$ and $K'$ are respectively the kernels of the homomorphisms $p \otimes \id_{f_\ast V'}$ and $\id_{f_\ast V} \otimes p'$. In other words, the homomorphism $\tilde{\phi}$ fits into the following commutative and exact diagram of $\cO_{X_0}$-modules:

\[
\begin{tikzcd}
0 \ar[r] & (E \otimes_0 V_0 \otimes_0 V'_0)^{\oplus 2} \ar[d, "\sigma"] \ar[r]  & W \ar[d, "\tilde{\phi}"]  \ar[r] & V_0 \otimes_0 V'_0 \ar[d, equal] \ar[r] & 0\ \\
0 \ar[r] & E \otimes_0 V_0 \otimes_0 V'_0 \ar[r] & f_\ast (V \otimes_{1} V') \ar[r] & V_0 \otimes_0 V'_0 \ar[r] & 0,
\end{tikzcd}
\]

where the upper row is the short exact sequence $(W)$.

To put it differently, the lower row is the push-out of the upper one along the sum map, that is, the extension $f_\ast (V \otimes_1 V')$ is the desired Baer sum.

§.§.§ Connections
Let $S$ be a scheme, $f \colon X \to S$ a separated morphism of schemes, $\Delta_X \colon X \to X \times_S X$ the diagonal morphism and $I := \Ker (\cO_{X \times_S X} \to \Delta_{X \ast} \cO_X)$ its augmentation ideal. With this notation, the sheaf of differentials (denoted $\Omega^1_{f}$ or $\Omega^1_{X/S}$) relative to $f$ is the $\cO_X$-module $\Delta_X^\ast I$.

Let $\Delta_{X, 1}$ be the first infinitesimal neighbourhood of the diagonal, that is, the closed subscheme of $X \times_S X$ defined by the sheaf of ideals $I^2$.

For $i = 1, 2$, let $p_i \colon \Delta_{X, 1} \to X$ be the morphism induced by the $i$-th projection.  The $\cO_X$-module
\[\cJ_f^1 := p_{1 \ast} \cO_{\Delta_{X, 1}}\]
is called the sheaf of $1$-order jets. Each $p_i$ induces a homomorphism of $f^{-1} \cO_S$-algebras $j_i \colon \cO_X \to \cJ^1_f$. The homomorphism of $f^{-1}\cO_S$-modules $j_2 - j_1$ is called the canonical derivation and written

\[ \rd_f \colon \cO_{X} \to \Omega^1_f. \]

Let $F$ be a vector bundle on $X$.

A connection on $F$ is an isomorphism $\nabla \colon p_1^\ast F \to p_2^\ast F$ of vector bundles over $\Delta_{X, 1}$ whose restriction to the diagonal $\Delta_X^\ast \nabla$ is the identity of $F$.

§.§.§ Atiyah extension
The kernel of the restriction map $\cO_{\Delta_{X, 1}}  \to i_\ast \cO_X$, where $i \colon X \to \Delta_{X, 1}$ is the closed immersion induced by the diagonal, is by definition of square zero. This allows to adopt the notation introduced in <ref> with

\begin{align*}
X_0 = X, &&X_1 = \Delta_{X, 1} , && s = i,  && f = p_1.
\end{align*}

Pushing-forward the $\cO_{\Delta_{X, 1}}$-module $p_2^\ast F$ along $p_1$ yields the short exact sequence

\begin{equation}
\tag{$\cJ^1_{f}(F)$}
0 \too \Omega^1_{f} \otimes F\too \cJ^1_{f}(F) \too F \too 0
\end{equation}

of $\cO_S$-modules, where  $\cJ^1_{f}(F) := p_{1 \ast} p_2^\ast F$ is the $\cO_{X}$-module of first order jets of $F$. Applied to the $\cO_{\Delta_{X,1}}$-module $p_2^\ast F$, Proposition <ref> implies:

The bijection

\[ \Hom_{\cO_{\Delta_{X, 1}}\textup{-mod}}(p_1^\ast F, p_2^\ast F) \too \Hom_{\cO_{X}\textup{-mod}}(F, \cJ^1_{X/S}(F))\]

given by adjunction induces a bijection

\[
\left\{
\begin{array}{c}
\textup{connections on $F$}
\end{array} \right\}
\too
\left\{
\begin{array}{c}
\textup{splittings of the short} \\
\textup{exact sequence $(\cJ^1_{X/S}(F))$}
\end{array} \right\}.
\]

Consider the short exact sequence of $\cO_X$-modules

\begin{equation} \tag{$\At_{f}(F)$} 0 \too \Omega^1_{f} \otimes \cEnd F \too \At_{f} (F) \too \cEnd F \too 0,
\end{equation}

where $\At_{f}(F) := \cHom(E, \cJ^1_{f}(F))$, obtained by taking the tensor product of the short exact sequence $(\cJ^1_{f}(F))$ with $F^\vee$.

The short exact sequence $\At_{f}(F)$ (alternatively denoted $\At_{X/S}(F)$) is called the relative Atiyah extension of $F$.

When $F$ is a line bundle, the vector bundle $\cEnd F$ is canonically identified with $\cO_X$ and splittings of the Atiyah extension $(\At_f(F))$ are in bijection with those of $(\cJ^1_f(F))$, thus with connections on $F$.

§.§.§ Infinitesimal rigidifications
Let $\pr_1, \pr_2 \colon X \times_{S} X \to X$ be respectively the first and the second projection and $x \colon S \to X$ a section of the structural morphism $f \colon X \to S$. Let $X_1$ be the first order thickening of $X$ along $x$, $\iota \colon X_1 \to X$ the closed immersion and $\pi = f \circ \iota \colon X_1 \to S$ the structural morphism.

An infinitesimal rigidification of $F$ at $x$  is an isomorphism of $\cO_{X_1}$-modules $\rho \colon \pi^\ast x^\ast F \to \iota^\ast F$ such that $x^\ast \rho$ is the identity.

The augmentation ideal of the closed immersion $x_1 \colon S \to X_1$ induced by $x$ is of square zero. This permits to employ the conventions introduced in section <ref> with $X_0 = S$, $s = x_1$ and $f = \pi$. For instance, if $F$ is a vector bundle, then pushing-forward the $\cO_{X_1}$-module $\iota^\ast F$ along $\pi$ yields the short exact sequence

\begin{equation}
\tag*{$\pi_\ast (\iota^\ast F)$}
0 \too x^\ast \Omega^1_{X/S} \otimes x^\ast F\too \pi_\ast \iota^\ast F \too x^\ast F \too 0
\end{equation}

of $\cO_S$-modules. Applied to $\cO_{X_1}$-module $\iota^\ast F$, Proposition <ref> reads as:

The bijection

\[ \Hom_{\cO_{X_1}\textup{-mod}}(\pi^\ast x^\ast  F, \iota^\ast F) \too \Hom_{\cO_{S}\textup{-mod}}(x^\ast F, \pi_\ast \iota^\ast F)\]

given by adjunction induces a bijection

\[
\left\{
\begin{array}{c}
\textup{infinitesimal} \\
\textup{rigidifications of $F$ at $x$}
\end{array} \right\}
\too
\left\{
\begin{array}{c}
\textup{splittings of the short} \\
\textup{exact sequence $\pi_\ast (\iota^\ast F)$}
\end{array} \right\}.
\]

Let $\tau \colon X_1 \to \Delta_{X, 1}$ be the morphism determined by $p_{1} \circ \tau = x \circ \pi$ and $p_{2} \circ \tau = \iota$.

If $\nabla \colon p_1^\ast F \to p_2^\ast F$ is a connection on $F$, then the homomorphism of $\cO_{X_1}$-modules $\tau^\ast \nabla$ is an infinitesimal rigidification of $F$ at $x$.

Consider the homomorphism $\psi \colon p_2^\ast F \to \tau_\ast \iota^\ast F$ of $\cO_{\Delta_{X, 1}}$-modules given by adjunction (note the equality $\iota^\ast F = \tau^\ast p_{2}^\ast F$). Pushing it forward along the morphism $p_1$ gives a homomorphism

\[ \pi_\ast \psi \colon (\cJ^1_{X/S}(F)) \too p_{1 \ast} \tau_\ast (\iota^\ast F)\]
of short exact sequences of $\cO_X$-modules. Now, by definition of $\tau$, the following square is commutative:

X_1 [r, "π"] [d, "τ"]     S [d, "x"]

Δ_X, 1 [r, "p_1"]     X.

Therefore, the short exact sequence $p_{1 \ast} \tau_\ast (\iota^\ast F)$ of $\cO_X$-modules is nothing but the push-forward along the closed immersion $x$ of the short exact sequence $\pi_\ast(\iota^\ast F)$ of $\cO_S$-modules. Let

\[ \phi \colon x^\ast (\cJ^1_{X/S}(F)) \too \pi_\ast(\iota^\ast F)\]

the homomorphism of short exact sequences of $\cO_S$-modules adjoint to $\pi_\ast \psi$.

The homomorphism $\phi$ of short exact sequence of $\cO_S$-modules defined above is an isomorphism.

The homomorphism $\psi \colon p_2^\ast F \to \tau_\ast \iota^\ast F$, when restricted to the diagonal, is the evaluation at $x$. Therefore, the homomorphism $\pi_\ast \psi$ of short exact sequence of $\cO_X$-modules is the following commutative diagram

\[
\begin{tikzcd}
0 \ar[r] & \Omega^1_{X / S} \otimes F \ar[r] \ar[d, "\ev_x"] & \cJ^1_{X/ S}(F) \ar[r] \ar[d, "\pi_\ast \psi"]& F \ar[r] \ar[d, "\ev_x"] & 0 \\
0 \ar[r] & x_\ast x^\ast \Omega^1_{X / S} \otimes x_\ast x^\ast F \ar[r] & x_\ast \pi_\ast \iota^\ast F \ar[r] & x_\ast x^\ast F \ar[r] & 0
\end{tikzcd}
\]

where $\ev_x$ stays for the evaluation at $x$. The homomorphism $\phi$, which is the adjoint to $\pi_\ast \psi$, reads as the following commutative diagram

\[
\begin{tikzcd}
0 \ar[r] &  x^\ast \Omega^1_{X / S} \otimes  x^\ast F \ar[r] \ar[d, equal] & x^\ast \cJ^1_{X/ S}(F) \ar[r] \ar[d, "\tilde{\phi}"]&  x^\ast F \ar[r] \ar[d, equal] & 0 \\
0 \ar[r] &  x^\ast \Omega^1_{X / S} \otimes  x^\ast F \ar[r] &  \pi_\ast \iota^\ast F \ar[r] &  x^\ast F \ar[r] & 0
\end{tikzcd}
\]

of $\cO_S$-modules. The Five Lemma implies that $\phi$ is an isomorphism.

§.§ Weil restriction of scalars
Let $\pi \colon S' \to S$ be a finite and locally free morphism of schemes.

§.§.§ Definition

For an $S'$-scheme $X'$, let $h_{X'} \colon (\Sch/S') \to (\Sets)$ denote the contravariant functor of points of $X'$:

\[ h_{X'}(T') = \Hom_{S'}(T', X').\]

Consider the functor $\pi_\ast h_{X'} \colon (\Sch/S') \to (\Sets)$ defined, for an $S$-scheme $T$,

\[ \pi_\ast h_{X'}(T) := h_{X'}(T_{S'}) = \Hom_{S'}(T_{S'}, X').\]

An $S$-scheme representing the functor (unique up to a unique isomorphism, if it exists) is called the Weil restriction of $X'$ and written $\Res_{\pi} X'$ (or $\Res_{S'/S} X'$).

Let $X' \to S'$ be an $S'$-scheme such that, for each $s' \in S'$ and each finite subset $P$ of the fiber $X'_{s'}$ of $X'$ at $s'$, there is an affine open subset of $X'$ containing $P$. Then, the Weil restriction of $X'$ exists.

Let $E'$ be a vector bundle on $S'$. The $S'$-scheme $\GL(E')$ is affine, so that its Weil restriction $\Res_{S'/S} \GL(E')$ exists. For an $S$-scheme $\alpha \colon T \to S$, consider the cartesian square

\[
\begin{tikzcd}
T' := T \times_{S} S' \ar[d, "\alpha'"] \ar[r, "\pi_T"] & T \ar[d, "\alpha"] \\
S' \ar[r, "\pi"] & S
\end{tikzcd}
\]

If the morphism $\alpha$ is flat, then by flat base change the natural homomorphism of $\cO_T$-modules

\[ \alpha^\ast \pi_\ast E' \too \pi_{T \ast} \alpha'^\ast E'\]

is an isomorphism. In particular, for an automorphism $\phi$ of the $\cO_{T'}$-module $\alpha'^\ast E'$, its push-forward $\pi_{T \ast}$ is an automorphism of the $\cO_T$-module $\alpha^\ast \pi_\ast E'$. This gives rise to a homomorphism of groups

\[ j_{E', T} \colon (\Res_{S'/S}  \GL(E'))(T) = \GL(E')(T') \too \GL(\pi_\ast E')(T).\]

Suppose the $S$-scheme $\Res_{S'/S}  \GL(E')$ flat. Then, the preceding construction yields to a morphism of group $S$-schemes

\[ j_{E'} \colon \Res_{S'/S}  \GL(E') \too \GL(\pi_\ast E').\]

The morphism $j_{E'}$ is a closed immersion and identifies $\Res_{S'/S}  \GL(E')$ with the closed subgroup of $\GL(\pi_\ast E')$ made of $\pi_\ast \cO_{S'}$-linear automorphisms of $\pi_\ast E'$.

In what follows, suppose that the Weil restriction $\Res_{S'/S} X'$ exists.

§.§.§ Base change
Let $T \to S$ be a morphism of schemes and $T' := T_{S'}$. Then, the natural projection $\pi_{T} \colon T' \to T$ is finite and locally free, the Weil restriction of $X'_{T'}$ along $\pi_{T}$ exists and

\[ \Res_{T'/T} X'_{T'} = (\Res_{S'/S} X') \times_{S} T. \]

§.§.§ Canonical projection
Consider the morphism of $S'$-schemes

\[ p_{X'} \colon (\Res_{S'/S} X') \times_S S' = \Res_{S' \times S'/S'} X'_{S' \times S'} \too X', \]

where $S' \times S'$ stays for $S' \times_S S'$, defined as follows. For an $S'$-scheme $T' \to S'$ and a morphism of $(S'\times S')$-schemes $t \colon T'_{S' \times S'} \to X'_{S' \times S'}$, the $T'$-valued point $p_{X'}(t)$ is the morphism of $S'$-schemes $T' \to X'$ obtained from $t$ by base change along the diagonal $S' \to S' \times_S S'$.

§.§.§ Canonical injection
Suppose $X' = X_{S'}$ for some $S$-scheme $X$. `Adjunction' induces a morphism of $S$-schemes

\[ \epsilon_X \colon X \too \Res_{S'/S} X'.\]

Namely, for an $S$-scheme $T$ and a morphism of $S$-schemes $x \colon T \to X$, the morphism $\epsilon_X$ sends $x$ to the morphism $x \times \id_{S'} \colon T_{S'} \to X_{S'}$.

§.§.§ Group schemes
If $X'$ is a group $S'$-scheme, then $\Res_{S'/S} X'$ inherits the structure of a group $S$-scheme from $X$.

§.§.§ Restriction along first order thickenings
Let $E$ be a vector bundle over $S$ and $\pi \colon S' \to S$ the first order thickening of the $S$-scheme $\bbV(E^\vee)$ along the zero section $S \to \bbV(E^\vee)$. Concretely, the scheme $S'$ is the spectrum of the $\cO_S$-module $\cO_S \oplus E$ endowed with the $\cO_S$-algebra structure

\begin{equation} \label{Eq:RingStructureOnFirstInfinitesimalNeigh}(f, v).(g,w) = (fg, fw+gv).
\end{equation}

Let $f \colon X \to S$ be an $S$-scheme and $\cT_{X/S}$ be the dual of the $\cO_X$-module $\Omega^1_{X/S}$. When the $\cO_X$-module $\Omega^{1}_{X/S}$ is a vector bundle, the $S$-scheme $\bbV(\cT_{X/S} \otimes f^\ast E)$ represents the functor associating to an $S$-scheme $\alpha \colon T \to S$ the set of couples $(x, \phi)$ made of an $T$-valued point $x$ of $X$ and a homomorphism of $\cO_{T}$-modules $\delta \colon x^\ast \Omega^1_{X/S} \to \alpha^\ast E$.

Let $X$ be an $S$-scheme such that $\Omega^{1}_{X/S}$ is a vector bundle. Then, the $S$-scheme $\bbV(\cT_{X/S} \otimes f^\ast E)$ is the Weil restriction of the $S'$-scheme $X_{S'}$. Moreover, suppose the $S$-scheme $X$ has one of the following properties:

* quasi-compact;

* quasi-separated;

* flat.

Then, the $S$-scheme $\Res_{S'/S} X_{S'} = \bbV(\cT_{X/S} \otimes f^\ast E)$ has it too.

This is done by computing the set $\pi_\ast h_{X_{S'}}(T)$ for a $T$-scheme $S$. However, since the situation is compatible with base change, one is led back to the case $T = S$.

Now, an $S'$-valued point of $X_{S'}$ corresponds to the datum of an $S$-valued point $x$ of $X$ and of an $\cO_S$-linear derivation $\cO_X \to x_\ast E$. By universal property of the module of Kähler differentials, the latter factors through unique a homomorphism of $\cO_X$-modules $\Omega^1_{X/S} \to x_\ast E$ or, equivalenty by adjunction, a homomorphism of $\cO_S$-modules $x^\ast \Omega^1_{X/S}  \to E$, which can be in turn be seen as a global section of the vector bundle $x^\ast \cT_{X/S} \otimes E$ over $S$.

The permanence of the properties is clear, as $\bbV(\cT_{X/S} \otimes f^\ast E)$ is the total space of the vector bundle $\cT_{X/S} \otimes f^\ast E$ over $X$.

The canonical morphism $\epsilon_X \colon X \to \Res_{S'/S} X_{S'} = \bbV(\cT_{X/S} \otimes f^\ast E)$ is the zero section.

Let $g \colon Y \to S$ be an $S$-scheme, $F \colon Y_{S'} \to X_{S'}$ a morphism of $S'$-schemes and $F_0 \colon Y \to X$ the morphism of $S$-schemes obtained by base change along the zero section $S \to S'$. Suppose that the $\cO_X$-module $\Omega^1_{X/S}$ and the $\cO_Y$-module $\Omega^1_{Y/S}$ are vector bundles. Then, Proposition <ref> implies that the diagram of $S$-schemes

\[
\begin{tikzcd}[column sep=35pt]
\bbV(\cT_{Y/S} \otimes f^\ast E) \ar[d, "q"] \ar[r, "\Res_{\pi} F"] & \bbV(\cT_{X/S} \otimes g^\ast E) \ar[d, "p"] \\
Y \ar[r, "F_0"] & X
\end{tikzcd}
\]

is commutative. Let $\cT F_0 \colon  \cT_{Y/S}  \to F_0^\ast \cT_{X/S}$ be the tangent map to $F_0$.

Consider the global section $\delta$ of the vector bundle $F_0^\ast \cT_{X/S} \otimes g^\ast E$ on $Y$ corresponding to the morphism of $S$-schemes

\[ \Res_{\pi} F \circ \epsilon_Y \colon Y \too \bbV(\cT_{X/S} \otimes g^\ast E).\]

With the notation above,

\[ \Res_{\pi} F = \cT F_0 \otimes \id_E + \delta.  \]

The map $\cT F_0 \otimes \id_E$ can also be seen as $\Res_\pi \pi^\ast F_0$, where $\pi^\ast F_0 \colon X_{S'} \to Y_{S'}$ is the morphism obtained from $F_0$ by base change. The statement is local, therefore the schemes $S$, $X$ and $Y$ may be assumed to be respectively the spectrum of rings $R_0$, $A_0$ and $B_0$. The vector bundle $E$ therefore corresponds to a locally free $R_0$-module $M$, and the schemes $S'$, $X_{S'}$ and $Y_{S'}$ are respectively the spectrum of the $R$-algebras

\begin{align*}
R := R_0 \oplus M, && A := A_0 \oplus M \otimes_R A_0, &&B := B_0 \oplus M \otimes_R B_0,
\end{align*}

whose ring structure are defined by the formula (<ref>). The morphisms $F$ and $F_0$ are respectively defined by an $R$-algebra homomorphism $\phi \colon A \to B$ and an $R_0$-algebra homomorphism $\phi_0 \colon A_0 \to B_0$, fitting in the following commutative and exact diagram of $R_0$-modules

\[
\begin{tikzcd}
0 \ar[r] & M \otimes_R A_0 \ar[d, "\id_M \otimes \phi_0"] \ar[r] & A \ar[r] \ar[d, "\phi"]& A_0 \ar[r] \ar[d, "\phi_0"]&  0\ \\
0 \ar[r] & M \otimes_R B_0 \ar[r] & B \ar[r] & B_0 \ar[r] &  0.
\end{tikzcd}
\]

Note that the leftmost vertical arrow is $\id_M \otimes \phi_0$ because $\phi$ is an $R$-algebras homomorphism (while it would be false if $\phi$ were merely an $R_0$-algebra map). It follows that $\phi$ is of the form, for $a \in A_0$ and $v \in M \otimes_R A_0$,

\[ \phi(a, v) = (\phi_0(a), \delta(a) + v)\]

for an $R_0$-linear derivation $\delta \colon A_0 \to M \otimes_R B_0$.

Let $\tilde{\phi}_0 \colon A \to B$ the $R$-algebra homomorphism obtained from $\phi_0$ by extending scalars from $R_0$ to $R$. Then, for $a \in A_0$ and $v \in M \otimes_R A_0$,

\[ \tilde{\phi}_0(a, v) =  (\phi_0(a), v),\]

whence $\phi = \tilde{\phi}_0 + \delta$ as desired.

§.§ Principal bundles
Let $S$ be a scheme, $f \colon X \to S$ an $S$-scheme and $G$ a faithfully flat and finitely presented group $S$-scheme.

§.§.§ Definition
A principal $G$-bundle on $X$ is the datum of a faithfully flat $X$-scheme $P$ endowed with an action of $G$ such that the morphism of $S$-schemes

\[ (\sigma, \pr_P) \colon G \times_S P \too P \times_X P\]

is an isomorphism, where $\sigma, \pr_P \colon G \times_S P \to P$ are respectively the morphism defining the action and the projection onto $X$. Let $\rH^1_{\textup{fppf}}(X, G)$ denote the set of isomorphism classes of principal $G$-bundles on $X$.

§.§.§ Pull-back
For a morphism of $S$-schemes $\pi \colon X' \to X$ and a principal $G$-bundle $P$ on $X$, the fibered product $\pi^\ast P := P \times_X X'$ inherits the structure of a principal $G$-bundle on $X'$.

§.§.§ Push-out along group maps
Let $G$ and $G'$ be faithfully flat, finitely presented and affine group $S$-schemes. For a morphism of group $S$-schemes $\rho \colon G \to G'$ and a principal $G$-bundle $P$ on $X$, the quotient $\rho_\ast P$ of $S$-scheme $G' \times_S X$ via the action
\[g(g',x) = (\rho(g)g', gx)\]
of $G$ exists and is a principal $G'$-bundle on $X$ (see Proposition 4.5.6 and 12.1.2 in <cit.>).

§.§.§ Sum of principal bundles
Suppose that $G$ is moreover affine and commutative, so that the group law $\mu_G \colon G \times_S G \to G$ is a group $S$-schemes morphism. For principal $G$-bundles $P$ and $P'$ on $X$, the $X$-scheme $P \times_X P'$ is a principal $(G \times_S G)$-bundle, thus it makes sense to consider the principal $G$-bundle

\[ P \wedge P' := \mu_{G\ast} (P \times_X P').\]

This operation endows the set $\rH^1_{\textup{fppf}}(X, G)$ with the structure of an abelian group with neutral element (the isomorphism class of) $G \times_S X$ and inverse $P \mapsto [-1]_\ast P$, where $[-1]$ is the inverse map on $G$.

Furthermore, the push-out an endomorphism of $G$ as a group $S$-scheme induces an endomorphism of the abelian group $\rH^1_{\textup{fppf}}(X, G)$. This equips $\rH^1_{\textup{fppf}}(X, G)$ with a structure of module on the ring $\End(G)$ of endomorphisms of $G$. For instance, if $G = \bbV(F)$ for some vector bundle $F$ on $S$, the set $\rH^1_{\textup{fppf}}(X, G)$ is naturally a $\Gamma(S, \cO_S)$-module.

Suppose $G = \Gm$. For a line bundle $L$ on $X$, let $\bbV(L)^\times$ denote the principal $\Gm$-bundle associated with $L$, that is the total space of $L$ deprived of its zero section. Owing to Grothendieck's generalization of Hilbert's Satz 90 (<cit.>), the so defined map

\[
\begin{array}{rcl}
\Pic(X) &\too& \rH^1_{\textup{fppf}}(X, \Gm) \\
L &\longmapsto & \bbV(L)^\times
\end{array}
\]

is an isomorphism of groups. In particular, for line bundles $L$ and $L'$ on $X$, the morphism $(s, s') \mapsto s \otimes s'$ induces an isomorphism

\[ \bbV(L)^\times \wedge \bbV(L')^\times \cong \bbV(L \otimes L')^\times.\]

Suppose $G = \bbV(F)$ for some vector bundle $F$ on $S$ and let

\begin{equation} \tag{$\cF$} 0 \too f^\ast F \too \cF \stackrel{p}{\too} \cO_X \too 0, \end{equation}

be a short exact sequence of $\cO_X$-modules.

The $X$-scheme $\bbA(\cF) := \bbP(\cF) \smallsetminus \bbP(f^\ast F)$ represents the functor associating to an $X$-scheme $\pi \colon X' \to X$ the set of splittings of the short exact sequence of $\cO_{X'}$-modules $\pi^\ast (\cF)$.

By definition, an $X'$-valued point of $\bbP(\cF)$ is the datum of a surjective homomorphism of $\cO_{X'}$-modules $\phi \colon \pi^\ast V^\vee \to L$ where $L$ is a line bundle on $X'$. It defines an $X'$-valued point of $\bbP(\cF) \smallsetminus \bbP(f^\ast F)$ if and only if the composite homomorphism of $\cO_{X'}$-modules

\[ \cO_{X'} \stackrel{s}{\too} f^\ast \cF^\vee \stackrel{\phi}{\too} L\]

is an isomorphism, where $s$ is the dual of the projection $p$. The latter happens if and only if the homomorphism $p \circ \phi^\vee \colon L^\vee \to \cO_{X'}$ is an isomorphism. The datum of of the latter is equivalent to that of a splitting of $\pi^\ast(V)$.

According to <cit.>, the so-defined map

\[
\begin{array}{rcl}
\rH^1(X, f^\ast F) &\too& \rH^1_{\textup{fppf}}(X, \bbV(F)) \\
(\cF) &\longmapsto & \bbA(\cF)
\end{array}
\]

is an isomorphism of $\Gamma(S, \cO_S)$-modules. More explicitly, given extensions $(\cF)$ and $(\cF')$ of $\cO_X$ by $f^\ast F$, the additivity of the above map follows from the isomorphism

\[ \bbA(\cF) \wedge \bbA(\cF') \cong \bbA((\cF) +_{\rB} (\cF')),\]

where $(\cF) +_\rB (\cF')$ is Baer sum of $(\cF)$ and $(\cF')$.

§.§ Extensions of commutative group schemes
Keep the notation introduced in <ref> and suppose now that $X$ is also a faithfully flat, finitely presented, commutative group $S$-scheme, with group law $\mu_X$ and zero section $e_X$.

§.§.§ Definition
An extension of $X$ by $G$ is the datum of a short exact sequence of commutative group $S$-schemes

\begin{equation} \tag{$E$} 0 \too G \stackrel{i_E}{\too} E \stackrel{p_E}{\too} X \too 0,
\end{equation}

where the morphism $p_E$ is faithfully flat.  Note that an extension $E$ of $X$ by $G$ is naturally a principal $G$-bundle over $X$.

When the natural homomorphism $\cO_S \to f_\ast \cO_X$ is an isomorphism (for instance, when $X$ is an abelian scheme), the commutativity of $E$ is automatic.

Morphisms of extensions are defined in the evident way. An isomorphism of extensions is a morphism inducing the identity on $X$ and $G$. Let $\Ext(X, G)$ be set of isomorphism classes of extensions of $X$ by $G$.

§.§.§ Push-out
Given a morphism of group $S$-schemes $\rho \colon G \to G'$, where the group $S$-scheme $G'$ is assumed to be faithfully flat, finitely presented, affine and commutative, and an extension $(E)$ of $X$ by $G$, the cokernel of the morphism

\[ (i_E, \rho) \colon G \too E \times_S G'\]

is isomorphic to the principal $G'$-bundle $\rho_\ast E$ and is an extension of $X$ by $G'$.

§.§.§ Baer sum
The Baer sum of extensions $(E)$ and $(E')$ of $X$ by $G$ is the push-out of $E \times_X E$ along the sum morphism $G \times_S G \to G$. The Baer sum endows $\Ext(X, G)$ with the structure of an abelian group.

Similarly to the case of principal $G$-bundles, the abelian group $\Ext(X, G)$ is endowed with the structure of an $\End(G)$-module.

§.§.§ Comparison with principal bundles
Seeing an extension as a principal bundle gives rise to a homomorphism of $\End(G)$-modules

\[ \lambda_G \colon \Ext(X, G) \too \rH^1_{\textup{fppf}}(X, G). \]

A rigidification of a principal $G$-bundle $P$ over $X$ is an isomorphism

\[ \phi \colon \pr_1^\ast P \wedge \pr_2^\ast P \too  \mu_X^\ast P \]

of principal $G$-bundles over $X \times_S X$ such that

* the diagram of $X \times_S X \times_S X$-schemes

\[
\begin{tikzcd}[column sep=50pt]
\pr_1^\ast P \wedge \pr_2^\ast P \wedge \pr_3^\ast P \ar[r, "\pr_{12}^\ast \phi \wedge \id"] \ar[d, "\id \wedge \pr_{23}^\ast \phi"']&  \mu_{X, 12}^\ast P \wedge \pr_3^\ast P \ar[d, "(\mu_{X, 12} \times \pr_3)^\ast \phi"]  \\
\pr_1^\ast P \wedge \mu_{X, 23}^\ast P \ar[r, "({\pr_1} \times \mu_{X, 23})^\ast \phi"] & \mu_{X, 123}^\ast P
\end{tikzcd}
\]

is commutative, where $\mu_{X, ij} = \mu_X \circ \pr_{ij}$ for $(i, j) = (1, 2), (2, 3)$ and $\mu_{X, 123} $ is the morphism $(x_1, x_2, x_3) \mapsto x_1+x_2+x_3$;

* $\sigma^\ast \phi = \phi$ where $\sigma$ is the symmetry automorphism of the principal $G$-bundle $\pr_1^\ast P \wedge \pr_2^\ast P$.

According to <cit.>, the datum of a rigidification on $P$ amounts to that of the structure of an extension of $X$ by $G$ compatible with the action of $G$ on $P$. Moreover, when $X$ is an abelian scheme over $S$, the conditions (1) and (2) above are automatically fulfilled and the map $\lambda_G$ is injective (cf. the proof of <cit.>).

Suppose $G = \Gm$ and identify principal $\Gm$-bundles with line bundles as in Example <ref>. A rigidification of a line bundle $L$ on $X$ is an isomorphism of $\cO_{X \times_S X}$-modules

\[ \phi \colon \pr_1^\ast L \otimes \pr_2^\ast L    \stackrel{\sim}{\too}  \mu_X^\ast L . \]

Suppose $G = \bbV(F)$ for some vector bundle $F$ on $S$. Identify principal $\bbV(F)$-bundles with extensions of $\cO_X$ by $f^\ast F$ as in Example <ref>. For a short exact sequence of $\cO_X$-modules

\begin{equation} \tag{$\cF$} 0 \too f^\ast F \too \cF \too \cO_X \too 0, \end{equation}

a rigidification of $(\cF)$ is an isomorphism of short exact sequences of $\cO_{X \times_S X}$-modules

\[ \phi \colon \pr_1^\ast (\cF) +_\rB \pr_2^\ast (\cF) \stackrel{\sim}{\too}  \mu_X^\ast (\cF),\]

where $\pr_1^\ast (\cF) +_\rB \pr_2^\ast (\cF)$ is the Baer sum of the extensions $\pr_1^\ast (\cF)$ and $\pr_2^\ast (\cF)$.

For a vector bundle $F$  on $S$, let $S'$ be the first order thickening of $\bbV(F^\vee)$ along its zero section. Let $L$ be a line bundle on the group $S'$-scheme $X':= X \times_S S'$ endowed with a rigidification $\phi \colon \pr_1^\ast L \otimes \pr_2^\ast L \to \mu_{X'}^\ast L$,  with the obvious notation.

Let $s \colon X \to X'$ be the closed immersion induced by the zero section of $F^\vee$ and $\pi \colon X' \to X$ the projection onto $X$. The $\cO_X$-module $\cF := \pi_\ast L$ sits into the following short exact sequence of $\cO_X$-modules:

\begin{equation} \tag{$\cF$}
0 \too f^\ast F \otimes s^\ast L \too \cF \too s^\ast L \too 0.
\end{equation}

According to Proposition <ref>, the push-forward of the $\cO_{X' \times X'}$-module $\pr_1^\ast L \otimes \pr_2^\ast L$ along the morphism  $ \pi \times \pi \colon X' \times_{S'} X' \to X \times_S X $
is the Baer sum of the extensions $\pr_1^\ast \cF \otimes \pr_2^\ast s^\ast L$ and $\pr_1^\ast s^\ast L \otimes \pr_2^\ast \cF $.

Assume further that the line bundle $s^\ast L$ is isomorphic to $\cO_X$. Then, the vector bundle $\cF$ is an extension of $\cO_X$ by $f^\ast F$ and the isomorphism

\[ (\pi \times \pi)_\ast \phi \colon \pr_1^\ast \cF +_\rB \pr_2^\ast \cF  \too \mu_X^\ast \cF\]

is a rigidification as the one considered in the previous example.

Suppose that $X$ is an abelian scheme over $S$. Then, the map $\lambda_G$ is injective and its image is the set of isomorphism classes of principal $G$-bundles admitting a rigidification.

§.§ Linearizations
Let $S$ be a scheme, $G$ a group $S$-scheme, $X$ an $S$-scheme endowed with an action of $G$. Let $\gamma \colon G \to S$ denote the structural morphism, $\mu \colon G \times_S G \to G$ the morphism defining the group law of $G$ and $\sigma \colon G \times_S X \to X$ the morphism defining the action of $G$ on $X$.

§.§.§ Definition
A $G$-linearization of a quasi-coherent $\cO_X$-module $F$ is an isomorphism of $\cO_{G \times X}$-modules $\lambda \colon \pr_X^\ast F \to \sigma^\ast F$ such that the  diagram

\begin{equation} \label{eq:CommutativeDiagramLinearization}
\begin{tikzcd}[column sep = 35pt]
\pr_{23}^\ast \pr_X^\ast F \ar[r, "{\pr_{23}^\ast \lambda}"] \ar[d, equal] & \pr_{23}^\ast \sigma^\ast F \ar[r, equal] & (\id_G \times \sigma)^\ast \pr_X^\ast F \ar[d, "{(\id_G \times \sigma)^\ast \lambda}"] \\
(\mu \times \id_X)^\ast \pr_X^\ast F \ar[r, "{(\mu \times \id_X)^\ast \lambda}"] & (\mu \times \id_X)^\ast \sigma^\ast F \ar[r, equal]& (\id_G \times \sigma)^\ast \sigma^\ast F
\end{tikzcd}
\end{equation}

of $\cO_{G \times G \times X}$-modules is commutative, where $\pr_X \colon G \times_S X \to X$ is the projection and $\pr_{23} \colon G \times_S G \times_S X \to G \times_S X$ is the projection onto the product of the second and third factor (<cit.>).

A $G$-linearized quasi-coherent $\cO_X$-module is the datum of a quasi-coherent $\cO_X$-module together with a $G$-linearization. Homomorphisms between $G$-linearized quasi-coherent $\cO_X$-modules are $G$-equivariant homomorphisms of $\cO_X$-modules.

§.§.§ Representations
Let $F$ be a vector bundle on $S$. A representation of $G$ is a morphism of group $S$-schemes $\rho \colon G \to \GL(F)$.

A $G$-linearization $\lambda$ of $F$ is an automorphism of the vector bundle $\gamma^\ast F$ on $G$. The isomorphism $\lambda$ defines a morphism of $S$-schemes $\rho_\lambda \colon G \to \GL(F)$. The commutativity of the diagram (<ref>) is equivalent to $\rho_\lambda$ being a morphism of group $S$-schemes. The so-defined map

\[
\begin{array}{rcl}
\{
\begin{array}{c} \textup{$G$-linearization of $F$} \end{array}\}
&\too& \{ \begin{array}{c} \textup{representations $G \to \GL(F)$} \end{array}\} \\
\lambda &\longmapsto & \rho_\lambda
\end{array}
\]

is bijective (<cit.>).

§.§.§ Invariants
Suppose that the action of $G$ on $X$ is trivial, that is, the morphisms $\sigma$ and $\pr_X$ coincide. A $G$-linearization $\lambda$ of a quasi-coherent $\cO_X$-module $F$ is thus an automorphism of the $\cO_{G \times X}$-module $\pr_X^\ast F$. Its push-forward $\pr_{X \ast } \lambda$ is thus an automorphism of the $\cO_X$-module $\pr_{X \ast} \pr_X^\ast F$. The $\cO_X$-module $F^G$ of invariant sections of $F$ is the equalizer of the homomorphism of $\cO_X$-modules

\[
\begin{tikzcd}
F \ar[r] & \pr_{X \ast} \pr_X^\ast F \ar[r, "\id", swap, yshift = -3pt] \ar[r, "\pr_{X \ast} \lambda", yshift=3pt]& \pr_{X \ast} \pr_X^\ast F
\end{tikzcd}
\]

If the $S$-scheme $G$ is quasi-compact and quasi-separated, the morphism $\pr_X$ shares the same properties and the $\cO_X$-module $\pr_{X \ast} \pr_X^\ast F$ is quasi-coherent. It follows that the $\cO_X$-module $F^G$ is quasi-coherent.

§.§.§ Pull-back
Let $Y$ be an $S$-scheme endowed with an action of $G$ and $f \colon X \to Y$  a $G$-equivariant morphism of $S$-schemes. Let $\pr_Y, \tau \colon G \times_S Y \to Y$ denote respectively the second projection and the morphism defining the action of $G$ on $Y$. The equalities $\pr_Y  \circ (\id_G, f) = f \circ \pr_X$ and $\tau  \circ (\id_G, f) = f \circ \sigma$ induce, for an $\cO_Y$-module $F$, canonical isomorphisms

\begin{align*}
(\id_G,  f)^\ast \pr_X^\ast F &= \pr_Y^\ast f^\ast F \\
(\id_G,  f)^\ast \sigma^\ast F &= \tau^\ast f^\ast F.
\end{align*}

Taking heed of the previous identifications, the isomorphism

\[ (\id_G, f)^\ast \lambda \colon \pr_Y^\ast f^\ast F \too \tau^\ast f^\ast F, \]

is a $G$-linearization of the $\cO_Y$-module $f^\ast F$.

§.§.§ Push-forward
Keep the notation introduced in the preceding paragraph and let $E$ be a quasi-coherent $\cO_X$-module. Suppose the morphism $f$ quasi-compact and quasi-separated, so that the $\cO_Y$-module $f_\ast E$ is quasi-coherent. Assume that the group $S$-scheme $G$ is flat and that the $S$-scheme $Y$ is quasi-compact and quasi-separated. By flat base change, the natural homomorphisms of $\cO_{G \times Y}$-modules

\begin{align*}
\pr_Y^\ast f_\ast E &\too (\id_G \times f)_\ast \pr_X^\ast E,\\
\tau^\ast f_\ast E &\too (\id_G \times f)_\ast \sigma^\ast E,
\end{align*}

are isomorphisms. Given a $G$-linearization $\lambda$ of $E$, the isomorphism of $\cO_{G \times Y}$-modules

\[ (\id_G \times f)_\ast \lambda \colon \pr_Y^\ast E \too \tau^\ast E, \]

where the preceding isomorphisms have been taken into consideration, is a $G$-linearization of the quasi-coherent $\cO_Y$-module $f_\ast E$ (<cit.>).

For a $G$-linearized quasi-coherent $\cO_X$-module $E$ and a $G$-linearized quasi-coherent $\cO_Y$-module $F$, the homomorphisms $f^\ast f_\ast E \to E$ and $F \to f_\ast f^\ast F$ given by adjunction are $G$-equivariant with respect to the $G$-linearizations on $f^\ast f_\ast E$ and $f_\ast f^\ast F$ introduced above.

Suppose $G$ flat, and $f \colon X \to S$ quasi-compact, quasi-separated, faithfully flat and such that the natural homomorphism $\cO_S \to f_\ast \cO_X$ is an isomorphism. According to Proposition <ref>, for a constant vector bundle $V$ on $X$, the $\cO_S$-module $f_\ast V$ is a vector bundle and the natural homomorphism $\phi_V \colon f^\ast f_\ast V \to V$ is an isomorphism.

If $V$ is moreover $G$-linearized, then the isomorphism $\phi_V$ is $G$-equivariant. In particular, the functors

\[
\begin{tikzcd}[ row sep=0pt]
{ \left\{
\begin{array}{c}
\textup{representations} \\
\textup{of $G$}
\end{array}
\right\}
\ar[r, yshift=2.5pt, "f^\ast"]  &
{ \left\{
\begin{array}{c}
\textup{$G$-linearized constant} \\
\textup{vector bundles on $X$}
\end{array}
\right\}
} \ar[l, yshift=-2.5pt, "f_\ast"]
\end{tikzcd}
\]

are mutually quasi-inverse. The resulting equivalence of categories is preserves direct sums, tensor products, internal homs, symmetric and exterior powers, and is compatible with arbitrary base change.

Suppose $f$ admits a section $x \colon S \to X$. For a $G$-linearized constant vector bundle $V$ on $X$, endow the vector bundle $x^\ast V$ on $S$ with linear action of $G$ deduced from that of $f_\ast V$ via the isomorphism of $\cO_S$-modules $f_\ast V \to x^\ast V$ given by evaluation at $x$. The functor associating to a $G$-linearized constant vector bundle $V$ the so-defined representation $G \to \GL(x^\ast V)$ is a quasi-inverse to the functor $f^\ast$.

§.§.§ Linearization as a descent datum
Suppose that the group $S$-scheme $G$ is faithully flat and finitely presented. Let $Y$ be a quasi-separated and quasi-compact $S$-scheme (endowed with the trivial action of $G$) and $f \colon X \to Y$ a quasi-compact, quasi-separated and $G$-invariant morphism.

A quasi-coherent $\cO_Y$-module $F$ admits the identity of $\pr_Y^\ast F$ as a $G$-linearization, thus its pull-back $f^\ast F$ is naturally $G$-linearized. Conversely, given a $G$-linearized quasi-coherent $\cO_X$-module $E$, its push-forward $f_\ast E$ has a natural $G$-linearization. Since the action of $G$ on $Y$ is trivial, one can consider the $\cO_Y$-module of invariant sections $(f_\ast E)^G$. If the $S$-scheme $G$ is quasi-compact and quasi-separated, then the $\cO_Y$-module $(f_\ast E)^G$ is quasi-coherent.

Let $\QCoh_{X}^G$ the category of $G$-linearized quasi-coherent $\cO_X$-modules (whose arrows are $G$-equivariant homomorphisms of $\cO_X$-modules) and $\QCoh_Y$ the category of quasi-coherent $\cO_Y$-modules (whose arrows are homomorphism of $\cO_Y$-modules). The preceding constructions define functors

\[
\begin{tikzcd}[ row sep=0pt]
f^\ast \colon \QCoh_Y \ar[r, yshift=2.5pt]  & \QCoh_{X}^G \ar[l, yshift=-2.5pt] \colon (f_\ast-)^G \\
\hspace{43pt} F  \ar[r, mapsto]& f^\ast F \hspace{55pt} \\
\hspace{18pt} (f_\ast E)^G & E \hspace{66pt} \ar[l, mapsto]
\end{tikzcd}
\]

Assume that the $S$-schemes $G$, $X$ and $Y$ are quasi-separated and quasi-compact, and that the following hold:

* the $S$-scheme $G$ is flat;

* the $G$-invariant morphism $f \colon X \to Y$ is faithfully flat and finitely presented, and the morphism of $S$-schemes $(\sigma, \pr_X) \colon G \times_S X \to X \times_Y X$ is an isomorphism.

Then, the functors $f^\ast$ and $(f_\ast -)^G$ are quasi-inverse to each other. Moreover, they preserves with direct sums, tensor products, internal homs, symmetric and exterior powers and are compatible with arbitrary base change $S' \to S$.

Hypothesis (2) in the preceding statement can be reformulated by saying that $X$ is an fppf principal $G$-bundle over $Y$.

This is a straightforward consequence of faithfully flat descent.

Given a quasi-coherent $\cO_X$-module $F$, a descent datum for $F$ is an isomorphism of $\cO_{X \times_Y X}$-modules

\[ \phi \colon \pr_2^\ast F \too \pr_1^\ast F,\]

where $\pr_1, \pr_2 \colon X \times_Y X \to X$ are respectively the first and second projection, such that following diagram of $\cO_{X \times_Y X \times_Y X}$-modules

\begin{equation} \label{eq:CommutativeDiagramDescentDatum}
\begin{tikzcd}[column sep = 35pt]
\pr_{23}^\ast \pr_2^\ast F \ar[r, "{\pr_{23}^\ast \phi}"] \ar[d, equal] & \pr_{23}^\ast \pr_1^\ast F \ar[r, equal] & \pr_{12}^\ast \pr_2^\ast F \ar[d, "{\pr_{12}^\ast \phi}"] \\
\pr_{13}^\ast \pr_2^\ast F \ar[r, "{\pr_{13}^\ast \phi}"] &\pr_{13}^\ast \pr_1^\ast F \ar[r, equal]& \pr_{12}^\ast \pr_1^\ast F
\end{tikzcd}
\end{equation}

is commutative, where $\pr_{12}, \pr_{13}, \pr_{23} \colon X \times_Y \times X \times_Y X \to X \times_Y X$ are the projections onto the product respectively of the first and second, first and third, and second and third factors.

The striking resemblence of (<ref>) with (<ref>) suggests that descent data and linearizations are closely related. That is indeed the case, for a descent datum $\phi$ the isomorphism of $\cO_{G \times_S X}$-modules

\[ \lambda := (\sigma, \pr_X)^\ast \phi \colon \pr_X^\ast F \too \sigma^\ast F \]

being a $G$-linearization of $F$. The commutativity of (<ref>) for $\lambda$ is obtained by pulling back (<ref>) along the isomorphism of $S$-schemes

\[
\begin{array}{rcl}
G \times_S X \times_S X &\too& X \times_Y X \times_Y X \\
(g, h, x) &\longmapsto & (ghx, hx, x).
\end{array}
\]

Vice versa, for a $G$-linearization $\lambda$ of $F$, the pull-back of $\lambda$ along the inverse of the isomorphism $(\sigma, \pr_X)$ is a descent datum.

This sets up a bijection between descent data and $G$-linearizations on $F$, and eventually leads to an equivalence between the category of $\cO_X$-modules equipped with a descent datum and that of $G$-linearized $\cO_X$-modules. Faithfully flat descent permits to conclude.

Let $\alpha$ and $\beta$ be structural morphisms respectively of the $S$-schemes $X$ and $Y$. Suppose the $S$-schemes $X$ and $Y$ quasi-compact, quasi-separated and faithfully flat, the group $S$-scheme $G$ is faithfully flat and finitely presented, and that the natural homomorphism $\cO_S \to \alpha_\ast \cO_X$ is an isomorphism. Assume that $f \colon X \to Y$ is an fppf principal $G$-bundle and that there is a section $x \colon S \to X$ of $\alpha$.

Let $V$ be a vector bundle on $Y$ such that the vector bundle $f^\ast V$ on the $S$-scheme $X$ is constant (that is, the vector bundle $f^\ast V$ is isomorphic to $\alpha^\ast E$ for some vector bundle $E$ on $S$). Then, the natural $G$-linearization of $f^\ast V$ induces a $G$-linearization on the vector bundle $\alpha_\ast f^\ast V$. Via the isomorphism of $\cO_S$-modules

\[ \alpha_\ast f^\ast V \too x^\ast f^\ast V = f(x)^\ast V\]

given by evaluation at $x$, the vector bundle $f(x)^\ast V$ acquires a $G$-linearization, which in turn corresponds to a representation

\[ \rho_V \colon G \too \GL (f(x)^\ast V).\]

Theorem <ref> yields immediately:
The functor

\[
\begin{tikzcd}[ row sep=0pt]
\rho \colon \left\{ \begin{array}{c} \textup{vector bundles on $Y$} \\ \textup{such that $f^\ast V$ is constant} \end{array} \right\} \ar[r]  & \left\{ \begin{array}{c} \textup{representations of $G$} \end{array} \right\}\\
\hspace{143pt} V  \ar[r, mapsto]& \rho_V \hspace{99pt}
\end{tikzcd}
\]

is an equivalence of categories. Furthermore,

* the functor $\rho$ preserves direct sums, tensor products, internal homs, symmetric and exterior powers, and is compatible with arbitrary base change;

* for a vector bundle $V$ on $Y$ such that $f^\ast V$ is constant, the homomorphism of $\cO_S$-modules $\beta_\ast V \to f(x)^\ast V$ given by evaluation at $f(x)$ induces an isomorphism of $\cO_S$-modules

\[ \beta_\ast V \stackrel{\sim}{\too} (f(x)^\ast V)^{G}. \]

§.§.§ Base change
Let $\pi \colon S' \to S$ be a morphism of $S$-schemes. Let $G'$, $X'$ and $\sigma'$ denote respectively the $S'$-schemes $G_{S'}$, $X_{S'}$ and the morphism of $S'$-schemes $\sigma_{S'}$ deduced by base change from $S$ to $S'$. The morphism $\sigma'$ defines an action of $G'$ on $X'$. Let $q_G \colon G' \to G$ and $q_X \colon X' \to X$ be the base change morphisms.

Let $\lambda$ be a $G$-linearization of a quasi-coherent $\cO_X$-module $F$. Then, the isomorphism of $\cO_{G' \times X'}$-modules $(q_G \times q_X)^\ast \lambda$ is a $G'$-linearization of the quasi-coherent $\cO_{X'}$-module $q_X^\ast F$.

Conversely, let $\lambda'$ be a $G$-linearization of a quasi-coherent $\cO_{X'}$-module $F'$. Suppose the morphism $\pi$ quasi-separated and quasi-compact, so that the morphisms $q_X$ has the same properties, hence the $\cO_X$-module $q_{X \ast} F'$ is quasi-coherent. Assume further that the $S$-scheme $G$ is flat, quasi-separated and quasi-compact. By flat base change along the morphisms of $\pr_X$ and $\sigma$, the natural homomorphism of $\cO_{G \times X}$-modules

\begin{align*}
\pr_{X}^\ast q_{X \ast} F &\too (q_G \times q_X)_\ast \pr_{X'}^\ast F, \\
\sigma^\ast q_{X \ast} F &\too (q_G \times q_X)_\ast \sigma'^\ast F,
\end{align*}

are isomorphisms. Then, the isomorphism of $\cO_{G \times X}$-modules

\[ (q_G \times q_X)_\ast \lambda' \colon \pr_{X}^\ast q_{X \ast} F \too \sigma^\ast q_{X \ast} F,\]

the preceding isomorphism being borne in mind, is a $G$-linearization of the quasi-coherent $\cO_{X}$-module $q_{X \ast} F'$.

§.§.§ Restriction of scalars
Let $\pi \colon S \to S'$ be a finite and locally free morphism of schemes. Let $F$ be a quasi-coherent $\cO_X$-modules endowed with a $G$-linearization $\lambda$. Suppose that the Weil restriction $G'$ and $X'$ respectively of $G$ and $X$ along $\pi$ exist. Via the canonical isomorphism of $S'$-schemes

\[ \Res_{S/S'}(G \times_S X) = G' \times_{S'} X',\]

the morphism $\sigma' \colon G' \times_{S'} X' \to X'$ induced by $\sigma$ defines an action of the group $S'$-scheme $G'$ on $X'$. Let $p_{X} \colon X'_{S} \to X$ be the natural morphism (defined in <ref>) and $q_{X} \colon X'_{S} \to X'$ the base change morphism, and let $p_{G}$, $q_G$ be the analogous morphisms for $G$.

The morphism $p_X \colon X'_S \to X$ is $G'_S$-equivariant with respect to the action of $G'_S$ on $X$ deduced by the one of $G$ on $X$ and the morphism of group $S'$-schemes $p_{G}$. Therefore, the isomorphism of $\cO_{G'_S \times X'_S}$-modules $(p_G \times p_X)^\ast \lambda$ is a $G'_S$-linearization of the quasi-coherent $\cO_{X'_S}$-module $p_X^\ast F$.

Suppose the $S'$-scheme $G'$ flat, quasi-compact and quasi-separated. Then, the isomorphism of $\cO_{G' \times X'}$-modules

\[ \lambda' := (q_G \times q_X)_\ast (p_G \times p_X)^\ast \lambda \]

is a $G'$-linearization of the quasi-coherent $\cO_{X'}$-module $F' := q_{X \ast} p_X^\ast F$.

§ THE UNIVERSAL VECTOR EXTENSION OF AN ABELIAN SCHEME

Let $S$ be a scheme, $\alpha \colon A \to S$ an abelian scheme, $e \colon S \to A$ the zero section and $\omega_{A} = e^\ast \Omega_{A/S}^1$. Let $\mu, \pr_1, \pr_2 \colon A \times_S A \to A$ be respectively the group law, the first and the second projection.

§.§ Connections on homogeneous line bundles

A homogeneous line bundle on $A$ is the datum of a line bundle on $A$ and of an isomorphism of $\cO_A$-modules $\phi \colon  \pr_1^\ast L \otimes \pr_2^\ast L \to \mu^\ast L$. The isomorphism $\phi$ is called rigidification.

The isomorphism $(e, e)^\ast \phi \colon e^\ast L \otimes e^\ast L \to e^\ast L$ induces a trivialization $u \colon \cO_S \to e^\ast L$ of the line bundle $e^\ast L$ on $S$.

Before going into connections on homogeneous line bundles, let us record the following fact, which will become indispensable later on.

Let $L$ be a homogeneous line bundle on $A$ and $s \colon \cO_A \to L$ a global section of $L$. Then, the following conditions are equivalent:

* the homomorphism of $\cO_A$-modules $s$ is an isomorphism;

* there is a morphism of schemes $x \colon S' \to A$ such that the composite morphism $\alpha \circ x \colon S' \to S$ is faithfully flat and the homomorphism of $\cO_{S'}$-modules $x^\ast s \colon \cO_{S'} \to x^\ast L$ is an isomorphism.

The implication (1) $\Rightarrow$ (2) is clear, for it suffices to take $S' = A$ and $x = \id$. Thus, let us prove the converse one.

The homogeneity of $L$ implies that the line bundle $[-1]^\ast L$ is isomorphic to the dual one $L^\vee$. By means of this, the pull-back of the section $s$ along he morphism $[-1]$ defines a global section $t = [-1]^\ast s$ of the line bundle $L^\vee$. The endomorphism $f$ of $\cO_A$ obtained as the composite homomorphism

\[ f \colon \cO_A \stackrel{s}{\too} L \stackrel{t}{\too} \cO_A\]

is of the form $\alpha^\ast g$ for a global section $g$ of $\cO_S$, since the natural homomorphism $\cO_S \to \alpha_\ast \cO_A$ is an isomorphism.

The hypothesis implies that the homomorphism of $\cO_{S'}$-modules $x^\ast t$, as well as $x^\ast s$, is an isomorphism. It follows that the function $x^\ast f$ on $S'$ is invertible and, because of the equality $x^\ast f = x^\ast \alpha^\ast g$ and  the faithful flatness of the morphism $\alpha \circ x$, that so is the function $g$ on $S$. The latter permits to conclude that $s$ is necessarily an isomorphism.

As long as the scheme $S$ is integral, the previous statement can be reformulated by saying that non-trivial homogeneous line bundles have no non-zero sections. As soon as $S$ fails to be irreducible or reduced, this is no longer true.

Now, let us turn our attention to connections. In order to do so, let $\Delta_1$ be the first order thickening of $A \times_S A$ along the diagonal, $p_i \colon \Delta_1 \to A$ the $i$-th projection  for $i = 1, 2$, $A_1$ the first order thickening of $A$ along $e$, $\iota \colon A_1 \to A$ the closed immersion, $\pi \colon A_1 \to S$ the structural morphism and $\tau \colon A_1 \to \Delta_1$ the morphism defined by $p_{1} \circ \tau = x \circ \pi$ and $p_{2} \circ \tau = \iota$.

Following Grothendieck, a connection on $L$ is an isomorphism of $\cO_{\Delta_1}$-modules $\nabla \colon p_1^\ast L \to p_2^\ast L$ whose restriction to $A$ is the identity. Similarly, an infinitesimal rigidification[Mazur-Messing call an `infinitesimal rigidification' simply `rigidification'. Here, the adjective `infinitesimal' is added in order to distinguish the concept from that of rigidification of a homogeneous line bundle.] at $e$ is an isomorphism of $\cO_{A_1}$-modules $\rho \colon \pi^\ast e^\ast \to \iota^\ast L$ whose restriction to the zero section is the identity. (Refer to <ref> and <ref> for basics on connections and infinitesimal rigidifications.) In particular, for a connection $\nabla$ on $L$, $\tau^\ast \nabla$ is an infinitesimal rigidifcation of $L$ at $e$ (see Remark <ref>).

Let $(L, \phi)$ be a homogeneous line bundle on $A$. The map

\[
\begin{array}{rcl}
\left\{ \textup{connections on $L$} \right\} &\too&
{ \left\{
\begin{array}{c}
\textup{infinitesimal} \\
\textup{rigidifications of $L$ at $e$}
\end{array} \right\}
} \\
\nabla &\longmapsto & \tau^\ast \nabla
\end{array}
\]

is a bijection.

The argument is borrowed from (2) and (3) in the proof of <cit.>.

(Injective.) Let $\nabla, \nabla' \colon p_1^\ast L \to p_2^\ast L$ be connections on $L$. Since both isomorphisms $\nabla$ and $\nabla'$ are the identity when restricted to the diagonal, they differ by an homomorphism of $\cO_X$-modules $L \to \Omega^1_{\alpha} \otimes L$. The latter is equivalent to the datum of a global section $\omega$ of $\Omega^1_{\alpha}$, as $L$ is a line bundle. On the other hand, the homomorphism $\epsilon  \colon \alpha_\ast \Omega^1_\alpha \to \omega_A$ evaluation at $e$ is also an isomorphism. Therefore, the isomorphisms $\tau^\ast \nabla$ and $\tau^\ast \nabla'$ differ by $\epsilon(\omega)$, which is $0$ if and only if $\omega$ is.

(Surjective.) Let $\rho \colon \pi^\ast e^\ast L \to \iota^\ast L$ be an infinitesimal rigidification of $L$ at $e$. The morphism $p_2 - p_1 \colon \Delta_1 \to A$ factors  through the closed immersion $\iota \colon A_1 \to A$. Indeed, when restricted to the diagonal, the map $p_2 - p_1$ has constant value $e$. As $A_1$ (resp. $\Delta_1$) is the first order thickening of $A$ along $e$ (resp. of $X \times_S X$ along the diagonal $\Delta$), the morphism $p_2 - p_1$ induces a morphism $\eta \colon \Delta_1 \to A_1$ between first order thickenings such that $\iota \circ \eta = p_2 - p_1$.

The pull-back of the rigidification $\phi$ along the morphisms of $S$-schemes

\[
(e \circ \pi \circ \eta, p_1), (p_2 - p_1, p_1) \colon \Delta_1 \too A \times A,
\]

furnishes the following isomorphisms of line bundles over $\Delta_1$,

\begin{align*}
(e \circ \pi \circ \eta, p_1)^\ast \phi \colon \eta^\ast \pi^\ast e^\ast L \otimes p_1^\ast L &\too p_1^\ast  L, \\
(p_2 - p_1, p_1)^\ast \phi \colon  (p_2 - p_1)^\ast L \otimes p_1^\ast L &\too p_2^\ast L.
\end{align*}

On the other hand, taking the tensor product of $\eta^\ast \rho$ with $p_1^\ast L$  gives rise the isomorphism of $\cO_{\Delta_1}$-modules

\[ \eta^\ast \rho \otimes \id \colon \eta^\ast \pi^\ast e^\ast L \otimes p_1^\ast L \too (p_2 - p_1)^\ast L \otimes p_1^\ast L.\]

Consider the unique homomorphism of $\cO_{\Delta_1}$-modules $\nabla \colon p_1^\ast L \to p_2^\ast L$ making the following diagram commutative:

\begin{equation} \label{eq:DiagramDefiningConnectionFromRigidifcation}
\begin{tikzcd}[column sep=35pt]
\eta^\ast \pi^\ast e^\ast L \otimes p_1^\ast L \ar[r, "{\eta^\ast \rho \otimes \id}"]  \ar[d, "{(e \circ \pi \circ \eta, p_1)^\ast \phi}", swap] & (p_2 - p_1)^\ast L \otimes p_1^\ast L \ar[d, "{(p_2 - p_1, p_1)^\ast \phi}"] \\
p_1^\ast  L \ar[r, "\nabla"] & p_2^\ast L
\end{tikzcd}
\end{equation}

To conclude, one has to show that the infinitesimal rigidification $\tau^\ast \nabla$ is $\rho$.

The endomorphism $\eta \circ \tau$ of the $S$-scheme $A_1$ is the identity.

Since $\iota$ is a closed immersion (thus a monomorphism of schemes), it suffices to show the equality $\iota \circ \eta \circ \tau = \iota$. By definition of $\eta$,

\[ \iota \circ \eta \circ \tau = (p_2 - p_1) \circ \tau = \iota - e \circ \pi = \iota,\]

as $A_1$-valued points of $A$, because $e \circ \pi$ is the neutral element of the group $A(A_1)$.

Pulling-back the diagram (<ref>) along $\tau$ gives the following commutative diagram of $\cO_{A_1}$-modules:

\[
\begin{tikzcd}[column sep=35pt]
\pi^\ast e^\ast L \ar[r, "\tau^\ast \nabla"]\ar[d, "{\id \otimes \pi^\ast u}", swap] & \iota^\ast L \ar[d, "\id \otimes \pi^\ast u"] \\
\pi^\ast e^\ast L \otimes \pi^\ast e^\ast L \ar[r, "{\rho \otimes \id}"]  & \iota^\ast L \otimes \pi^\ast e^\ast L
\end{tikzcd}
\]

where $u \colon \cO_S \to e^\ast L$ is the trivialization introduced in Remark <ref>. (Here we used the equalities  $p_{1} \circ \tau = x \circ \pi$ and $p_{2} \circ \tau = \iota$ holding by definition of $\tau$.) The equality $\tau^\ast \nabla = \rho$ follows and concludes the proof.

For a homogeneous line bundle $L$ on $A$, let

\begin{equation}
\tag{$\At_{A/S}(L)$} 0 \too \Omega^1_{A/S} \too \At_{A/S}(L) \too \cO_A \too 0
\end{equation}

be its Atiyah extension (see <ref>). Recall that connections on $L$ correspond to splitting of the Atiyah extension of $L$.

Let $(L, \phi)$ be a homogeneous line bundle on $A$.

* If the cohomology group $\rH^1(S, \omega_A)$ vanishes, then the line bundle $L$ admits a connection.

* The sequence of $\cO_S$-modules

\begin{equation}
\tag*{$\alpha_\ast (\At_{A/S}(L))$}
0 \too \alpha_\ast \Omega^1_{A/S} \too \alpha_\ast \At_{A/S}(L) \too \alpha_\ast \cO_A \too 0
\end{equation}

is short exact.

* The homomorphism $\alpha_\ast \alpha^\ast (\At_{A/S}(L)) \to (\At_{A/S}(L))$ of short exact sequences of $\cO_A$-modules obtained by adjunction is an isomorphism.

(1) According to Proposition <ref> it suffices to show that the line bundle $L$ admits an infinitesimal rigidification at $e$. With the notation of Proposition <ref>, an infinitesimal rigidification corresponds to a splitting of the short exact sequence of $\cO_S$-modules

\begin{equation}\tag*{$\pi_\ast (\iota^\ast L)$}
0 \too \omega_A \otimes e^\ast L \too \pi_\ast \iota^\ast L \too e^\ast L \too 0.
\end{equation}

The latter is an extension of the line bundle $e^\ast L$ on $S$ by the vector bundle $\omega_A \otimes e^\ast L$, therefore its isomorphism class lies in the cohomology group

\[ \rH^1(S, \cHom(e^\ast L, \omega_A \otimes e^\ast L)) = \rH^1(S, \omega_A).\]

By hypothesis, the cohomology group $\rH^1(S, \omega_A)$ vanishes, hence the short exact sequence $\pi_\ast (\iota^\ast L)$ splits and the line bundle $L$ admits a connection.

(2) Pushing-forward along $\alpha$ is a left-exact functor, therefore it suffices to show that the natural map $p \colon \alpha_\ast \At_{A/S}(L) \to \alpha_\ast \cO_A$ is surjective. The statement is local on $S$, thus the scheme $S$ may be supposed to be affine. Under this assumption, the cohomology group $\rH^1(S, \omega_A)$ vanishes and, because of (1), the line bundle $L$ admits a connection. In other words, by Proposition <ref>,  the Atiyah extension $(\At_{A/S}(L))$ of $L$  admits a splitting $s \colon \cO_A \to \At_{A/S}(L)$.

The homomorphism of $\cO_S$-modules $\alpha_\ast s \colon \alpha_\ast \cO_A \to \alpha_\ast \At_{A/S}(L)$ is a section of $p$, that is, it satisfies $p \circ \alpha_\ast s = \id$. In particular, the homomorphism $p$ is surjective.

(3) The homomorphism of short exact sequences of $\cO_A$-modules in question is the commutative diagram

\[
\begin{tikzcd}
0 \ar[r] & \alpha^\ast \alpha_\ast \Omega^{1}_{A/S} \ar[r] \ar[d] & \alpha^\ast \alpha_\ast \At_{A/S}(L) \ar[r] \ar[d] & \alpha^\ast \alpha_\ast \cO_A \ar[r] \ar[d] & 0 \\
0 \ar[r] & \Omega^{1}_{A/S} \ar[r] & \At_{A/S}(L) \ar[r] & \cO_A \ar[r] & 0
\end{tikzcd}
\]

where the three vertical arrows are given by adjunction. The vector bundles $\cO_A$ and $\Omega^1_{A/S}$ are constant, thus the leftmost and the rightmost vertical arrows are isomorphisms (see Proposition <ref>). The Five Lemma implies that the central vertical arrow is an isomorphism too.

The Atiyah extension $(\At_{A/S}(L))$ is obtained as the tensor product with $L^\vee$ of the short exact sequence

\begin{equation}
\tag{$\cJ^1_{A/S}(L)$} 0 \too \Omega^{1}_{A/S} \otimes L \too \cJ^1_{A/S}(L) \too L \too 0,
\end{equation}

where $\cJ^1_{A/S}(L)$ is the $\cO_X$-module of first order jets of $L$. Proposition <ref> furnishes an isomorphism
$\phi \colon e^\ast (\cJ^1_{A/S}(L)) \to \pi_\ast (\iota^\ast L)$ of short exact sequences of $\cO_S$-modules. Taking the tensor product with $e^\ast L^\vee$ induces an isomorphism

\[ e^\ast (\At_{A/S}(L)) \too \pi_\ast (\iota^\ast L) \otimes e^\ast L^\vee\]

of short exact sequences of $\cO_S$-modules. Note also that, by Remark <ref>, the line bundle $e^\ast L$ on $S$ is trivial, whence an isomorphism

\[ e^\ast (\At_{A/S}(L)) \cong \pi_\ast (\iota^\ast L).\]

§.§ The canonical extension of the trivial line bundle

The functor associating to an $S$-scheme $S'$ the set of isomorphism classes of homogeneous line bundles on $A \times_S S'$ is representable by an abelian scheme $\check{\alpha} \colon \check{A} \to S$, called the dual abelian scheme.

See <cit.> bearing in mind <cit.>.

Let $\cL$ denote the Poincaré bundle, that is, the universal homogeneous line bundle on $A \times_S\check{A}$.

Let $q \colon A \times_S \check{A} \to \check{A}$ be the projection onto the second factor. Let $(\At_q(\cL))$ be the Atiyah extension relative to $q$ of the Poincaré bundle $\cL$ and set $\cU_{\check{A}} := q_\ast \At_q(\cL)$. The sequence of $\cO_{\check{A}}$-modules

\begin{equation} \tag{$\cU_{\check{A}}$}
0 \too \check{\alpha}^\ast \omega_{A} \too \cU_{\check{A}} \too \cO_{\check{A}} \too 0,\end{equation}

obtained by pushing forward the short exact sequence $(\At_q(\cL))$ along $q$, is short exact by Proposition <ref> (2) being applied to the to the abelian scheme $A \times_S \check{A}$ over $\check{A}$.

The extension $(\cU_{\check{A}})$ is called the canonical extension of $\cO_{\check{A}}$.

The pull-back of the Atiyah extension of $\cL$ along the unramified morphism $(\id_{A}, \check{e})$ is the Atiyah extension of the trivial line bundle $\cO_{A}$. Therefore, the canonical derivation $\rd_{A/S} \colon \cO_{A} \to \Omega^1_{A/S}$, being a connection on the trivial bundle, defines a splitting of $\check{e}^\ast (\cU_{\check{A}})$, called the canonical splitting.

Let $A_1$ be the first order thickening of $A$ along the zero section $e$, $\iota \colon A_1 \times_S \check{A} \to A \times_S \check{A}$ the morphism obtained from the closed immersion $A_1 \to A$ by base change along $\check{\alpha}$ and $\pi \colon A_1 \times_S \check{A} \to \check{A}$ the second projection. Remark <ref> (applied to the abelian scheme $A \times_S \check{A}$ over $\check{A}$) furnishes an isomorphism of sequences of $\cO_{\check{A}}$-modules

\[ (e, \id_{\check{A}})^\ast (\At_q(\cL)) \stackrel{\sim}{\too} \pi_\ast (\iota^\ast \cL).  \]

On the other hand, by Proposition <ref> (3), the evaluation at the zero section,

\[ (\cU_{\check{A}}) = q_\ast (\At_q(\cL)) \too (e, \id_{\check{A}})^\ast (\At_q(\cL)), \]

is an isomorphism of sequences of $\cO_{\check{A}}$-modules. Composing these isomorphisms
yields to an isomorphism of sequences of $\cO_{\check{A}}$-modules

\[ (\cU_{\check{A}}) \cong \pi_\ast (\iota^\ast \cL). \]

§.§ Moduli space of connections

§.§.§ Definition

The $S$-scheme $\check{A}^\natural = \bbP(\cU_{\check{A}}) \smallsetminus \bbP(\check{\alpha}^\ast \omega_{A})$ is called the moduli space of rank $1$ connections on $A$.