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\TopicFR{Géométrie algébrique}
\TopicEN{Algebraic geometry}

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\title{On the restriction of finite torsors}
\alttitle{Sur la restriction de torseurs finis}

\author{\firstname{Andrea} \lastname{Marrama}\CDRorcid{0000-0002-8601-0810}}
\address{Dipartimento di Matematica ``Tullio Levi-Civita'', Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy}
\email{andrea.marrama@unipd.it}

\begin{abstract}
We give a detailed account of the proof of a purity theorem for finite flat torsors due to Laurent Moret-Bailly. In addition, we show that a part of this result, namely the full faithfulness, holds also under weaker assumptions.
\end{abstract}

\begin{altabstract}
Nous exposons de façon détaillée la démonstration d'un théorème de pureté pour les torseurs finis et plats dû à Laurent Moret-Bailly. Nous montrons aussi qu'une partie de ce résultat, à savoir la pleine fidélité, reste également valable sous des hypothèses moins restrictives.
\end{altabstract}

\thanks{The author was funded by the European Union, Next Generation EU, through the Ministero dell'Università e della Ricerca, project PRIN-20222B24AY}
\CDRGrant[Ministero dell'Università e della Ricerca]{PRIN-20222B24AY}

\subjclass{14L30, 14M17, 14F20}

\keywords{\kwd{Torsors} \kwd{finite group schemes} \kwd{purity}}
\altkeywords{\kwd{Torseurs} \kwd{schémas en groupes finis} \kwd{pureté}}

\COI{The author does not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and has declared no affiliations other than their research organizations.}

\dateposted{2026-07-27}
\begin{document}
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\maketitle

\section*{Introduction}

The purpose of this text is to give a more concise and convenient form to some contents of the author's Master thesis~\cite{Ma}, together with an additional result which is surely known to the experts, but was left as an open question in loc.~cit. More specifically, we give an account of the proof of a purity theorem for finite flat torsors due to Laurent Moret-Bailly (Theorem~\ref{Pur4tors}), which was originally published without proof in~\cite{MB}. The statement is analogous to the Zariski--Nagata purity theorem for finite étale coverings (see~\cite[X, \S 3]{SGA2}). The proof is based on Auslander's method from~\cite{Au} and relies on a crucial result from loc.~cit.\ (Theorem~\ref{Au}). As hinted above, this proof follows an argument suggested by Moret-Bailly. We remark that a slightly more general theorem was recently obtained in~\cite[Theorem~7.1.3]{CS} using a similar strategy. The goal of this note is to provide a fully detailed proof of Moret-Bailly's result and to show that a part of it, namely the full faithfulness, holds also under weaker assumptions (Proposition~\ref{Ff4tors}).

\begin{nota*}
For any scheme~$S$, we denote by $\O_S$ its structure sheaf. Let $S$ be a scheme, $G$ an affine $S$-group-scheme. We denote by $\Sch_{/S}$ the category of $S$-schemes and by $\Tors(S,G)$ the category of fppf $G$-torsors over $S$. If $T \to S$ is a morphism of schemes and $X$ is any $S$-scheme, we write $X_T$ for the pullback~$X\times_S T$; we will consider $G_T$ as a $T$-group-scheme and, if $X$ is an fppf $G$-torsor over $S$, we will consider $X_T$ as a $G_T$-torsor over $T$. If $X$ and $Y$ are two $S$-schemes, we denote by $\Hom_S(X,Y)$ the set of homomorphisms of $S$-schemes from $X$ to $Y$; if $X$ and $Y$ are fppf $G$-torsors over $S$, we write $\Hom_{(S,G)}(X,Y)$ for the set of homomorphisms from $X$ to $Y$ in $\Tors(S,G)$. Regular schemes are assumed to be locally Noetherian.
\end{nota*}

We first address the full faithfulness statement, which is an easy consequence of the following two lemmas; the first one is simply recalled from~\cite{EGAII}.

\begin{lemm}[{\cite[Corollaire~6.1.14]{EGAII}}]\label{lemma}
Let $S$ be a locally Noetherian normal scheme, $T \to S$ an integral morphism of schemes, $U\subseteq S$ a dense open subscheme. The restriction map
\[
\Hom_S(S,T) \longrightarrow \Hom_S(U,T)
\]
is bijective.
\end{lemm}

\begin{lemm}\label{repr}
Let $S$ be a scheme, $G$ an affine $S$-group-scheme, $X$ and $Y$ two fppf $G$-torsors over $S$. The following fppf sheaf of sets (with the obvious restriction maps):
\[
\begin{aligned}
	\HHom_G(X,Y)\colon\Sch_{/S} & \longrightarrow \Sets
\\	(T \to S) & \longmapsto \Hom_{(T,G_T)}(X_T,Y_T)
\end{aligned}
\]
is representable by an affine $S$-scheme, which is fppf locally isomorphic to $G$. In particular, if $G$ is finite over $S$, then $\HHom_G(X,Y)$ is representable by a finite $S$-scheme.
\end{lemm}

\begin{proof}
We may argue in a similar way as for the representability of fppf torsors with respect to an affine group scheme. Namely, the question is fppf local on $S$ by effective descent of affine schemes (see e.g.~\cite[\href{https://stacks.math.columbia.edu/tag/0245}{Tag~0245}]{SP}). Thus, we are reduced to the situation where $X$ and $Y$ are trivial $G$-torsors, in which case $\HHom_G(X,Y)$ is isomorphic to $G$ as a presheaf on $\Sch_{/S}$ and the claim is clear. The final statement of the lemma follows because the property of being finite is fppf local on the base.
\end{proof}

\begin{prop}\label{Ff4tors}
Let $S$ be a locally Noetherian normal scheme, $U\subseteq S$ a dense open subscheme, $G$ a finite $S$-group-scheme. The pullback functor
\[
\Tors(S,G) \longrightarrow \Tors(U,G_U)
\]
is fully faithful.
\end{prop}

\begin{proof}
Let $X$, $Y$ be fppf $G$-torsors over $S$. We have to show that the restriction map
\begin{equation}\label{res-hom-tors}
\tag{$*$}
\Hom_{(S,G)}(X,Y) \longrightarrow \Hom_{(U,G_U)}(X_U,Y_U)
\end{equation}
is bijective. Let $Z$ be the finite $S$-scheme representing $\HHom_G(X,Y)$ by Lemma~\ref{repr}. The map~\eqref{res-hom-tors} may then be identified with the restriction map:
\[
\Hom_S(S,Z) \longrightarrow \Hom_S(U,Z),
\]
which is bijective by Lemma~\ref{lemma} because $Z$ is integral over $S$.
\end{proof}

\begin{rema}
The proposition could also be proved bypassing Lemma~\ref{repr}. Indeed, the bijectivity of~\eqref{res-hom-tors} may be checked fppf locally, thus reducing the matter to the bijectivity of the restriction map~$\Hom_S(S,G) \to \Hom_S(U,G)$, which holds by Lemma~\ref{lemma}.
\end{rema}

Regarding the essential surjectivity of the pullback functor above, we now report a result due to Laurent Moret-Bailly giving a sufficient condition. The statement is analogous to the Zariski--Nagata purity theorem for finite étale coverings (see~\cite[X, \S 3]{SGA2}). We provide here a proof, omitted in the original publication~\cite{MB}, based on a sketch which the author of loc.~cit.\ communicated to us. The argument relies on a crucial result by Maurice Auslander, which we recall for the reader's convenience.

\begin{theo}[{\cite[Theorem~1.3]{Au}}]\label{Au}
Let $R$ be a regular local ring, $M$ a finitely generated reflexive $R$-module. Suppose that the $R$-module of endomorphisms~$\End_R(M)$ is isomorphic to a direct sum of copies of $M$. Then, $M$ is a free $R$-module.
\end{theo}

We now come to Moret-Bailly's purity theorem for finite flat torsors.

\begin{theo}[{\cite[Lemme~2]{MB}}]\label{Pur4tors}
Let $S$ be a regular scheme, $U\subseteq S$ an open subscheme, $G$ a finite flat $S$-group-scheme. Suppose that the complement of $U$ in $S$ has codimension at least $2$. Then, the pullback functor
\[
\Tors(S,G) \longrightarrow \Tors(U,G_U)
\]
is an equivalence of categories.
\end{theo}

\begin{proof}
Full faithfulness follows from Proposition~\ref{Ff4tors}. To show that we have an equivalence of categories, we will exhibit, for any fppf $G_U$-torsor over $U$, an object of $\Tors(S,G)$ which restricts to the given one over $U$.

Let $j\colon U \to S$ denote the inclusion morphism, let $X$ be an fppf $G_U$-torsor over $U$ and denote by $\rho\colon G_U\times_U X \to X$ the $G_U$-action on $X$. Because $X \to U$ is finite flat, it corresponds to a coherent $\O_U$-algebra~$\P$, which is locally free as an $\O_U$-module. Consider the quasi-coherent $\O_S$-algebra~$\Q\coloneqq j_*\P$, which satisfies $\restr{\Q}{U}=\P$. Then, $\Q$ corresponds to an affine $S$-scheme~$\tilde{X}$ such that $\tilde{X}_U=X$. Let now $\G$ denote the coherent $\O_S$-algebra corresponding to $G \to S$. Then, $\restr{\G}{U}$ is the coherent $\O_U$-algebra corresponding to $G_U \to U$ and $\rho$ corresponds to a homomorphism of $\O_U$-algebras $\rho^\#\colon\P \to \P \otimes_{\O_U} \restr{\G}{U}$. We set $\tilde{\rho}^\#\coloneqq j_*\rho^\#\colon\Q \to \Q \otimes_{\O_S}\G$, where $j_*(\P \otimes_{\O_U} \restr{\G}{U})\cong\Q \otimes_{\O_S}\G$ because $\G$ is finite locally free as an $\O_S$-module. We have that $\tilde{\rho}^\#$ is a homomorphism of $\O_S$-algebras satisfying $\restr{\tilde{\rho}^\#}{U}=\rho^\#$, hence it corresponds to a homomorphism of $S$-schemes $\tilde{\rho}\colon G\times_S\tilde{X} \to \tilde{X}$ whose pullback along $j$ is $\rho$. By functoriality of $j_*$ (and of the identification used in the definition of $\tilde{\rho}^\#$), $\tilde{\rho}$ verifies the rules of a $G$-action on $\tilde{X}$. It remains to see that $\tilde{\rho}$ makes $\tilde{X}$ an fppf $G$-torsor over $S$.

We first observe that $\Q$ is a coherent $\O_S$-module by~\cite[Corollaire~5.11.4]{EGAIV2}, because $S$ is regular, $\P$ is finite locally free and the codimension of $S\setminus U$ in $S$ is at least $2$. In other words, $\tilde{X}$ is a finite $S$-scheme. Next, we claim that $\Q$ is a locally free $\O_S$-module. In order to prove this, consider the $\O_S$-modules of homomorphisms (respectively endomorphisms) $\F\coloneqq\Homm_{\O_S}(\G,\Q)$, $\H\coloneqq\Endd_{\O_S}(\Q)$. We define a homomorphism of $\O_S$-modules $\theta\colon\F \to \H$ by setting, for $V\subseteq S$ open and $u\colon \restr{\G}{V} \to \restr{\Q}{V}$ a homomorphism of $\O_V$-modules:
\[
\theta_V(u) \colon \restr{\Q}{V}
\xlongrightarrow[\hphantom{(\id,u)}]{\tilde{\rho}^\#\rvert_V}\restr{\Q}{V} \otimes_{\O_V} \restr{\G}{V}
\xlongrightarrow{(\id,u)}\restr{\Q}{V},
\]
where the last map uses the $\O_V$-algebra structure of $\restr{\Q}{V}$. We argue that $\restr{\theta}{U}$ is an isomorphism. Indeed, let $\Omega^\#\colon\P \otimes_{\O_U}\P \to \P \otimes_{\O_U} \restr{\G}{U}$ be the homomorphism of $\O_U$-algebras given by the canonical map~$\P \to \P \otimes_{\O_U} \restr{\G}{U}$ on the first component and $\rho^\#$ on the second component. Because $X$ is a $G_U$-torsor over $U$ (with respect to $\rho$), we have that $\Omega^\#$ is an isomorphism. This allows to find an inverse~$(\restr{\theta}{U})^{-1}\colon \restr{\H}{U} \to \to \restr{\F}{U}$ of $\restr{\theta}{U}$, namely, for $V\subseteq U$ open and $v$ an endomorphism of the $\O_V$-module~$\restr{\Q}{V} = \restr{\P}{V}$:
\[
(\restr{\theta}{U})^{-1}_V(v) \colon \restr{\G}{V}
\xlongrightarrow{\hphantom{(\Omega^\#)\rvert_V^{-1}}}	\restr{\P}{V}
\xlongrightarrow{\hphantom{(\Omega^\#)\rvert_V^{-1}}} \otimes_{\O_V} \restr{\G}{V} \xlongrightarrow{(\Omega^\#)\rvert_V^{-1}} \restr{\P}{V}
\xlongrightarrow{\hphantom{(\Omega^\#)\rvert_V^{-1}}} \otimes_{\O_V} \restr{\P}{V} \xlongrightarrow{(\id,v)} \restr{\P}{V},
\]
where the first arrow is the canonical map of $\O_V$-algebras. Note now that, because $\Q$ is the pushforward of an $\O_U$-module along $j$ and it is coherent, then, by~\cite[Théorème~5.10.5]{EGAIV2}, $\Q$ has depth at least $2$ at all points of $S\setminus U$. Therefore, the same holds for $\F$ and $\H$. But then, by loc.~cit.\ (now used in the opposite direction than above), the fact that $\theta$ is an isomorphism over $U$ is enough to deduce that $\theta$ is an isomorphism of $\O_S$-modules. In particular, for every point $s\in S$, we have an isomorphism of the stalks $\End_{\O_{S,s}}(\Q_s)=\H_s\cong\F_s=\Hom_{\O_{S,s}}(\G_s,\Q_s)$. Because $\G$ is locally free, it follows that $\End_{\O_{S,s}}(\Q_s)$ is isomorphic to a direct sum of copies of $\Q_s$. Moreover, $\Q_s$ is a finitely generated reflexive $\O_{S,s}$-module, because $\Q$ is coherent, locally free over $U$ and, as observed above, it has depth at least $2$ on $S\setminus U$ (then one uses the criterion~\cite[\href{https://stacks.math.columbia.edu/tag/0AVA}{Tag~0AVA}]{SP}). Thus, for every $s\in S$, we have that $\Q_s$ is a free $\O_{S,s}$-module by Auslander's result (Theorem~\ref{Au}), since $S$ is regular. This proves that $\Q$ is a locally free $\O_S$-module, hence that $\tilde{X}$ is a flat $S$-scheme. In fact, $\tilde{X} \to S$ is an fppf covering. Indeed, it only remains to see that it is surjective, but this follows from the fact that $\tilde{X} \to S$ is a finite, hence closed morphism and its image contains the dense subset $U\subseteq S$, as $X \to U$ is an fppf $G_U$-torsor, hence faithfully flat.

To conclude that $\tilde{X}$ is an fppf $G$-torsor over $S$ with respect to $\tilde{\rho}$, it is now sufficient to show that the homomorphism of $S$-schemes $\tilde{\Omega}\colon G\times_S\tilde{X} \to \tilde{X}\times_S\tilde{X}$, given by $\tilde{\rho}$ on the first component and the second projection on the second component, is an isomorphism. Equivalently, we shall prove that the homomorphism of $\O_S$-algebras $\tilde{\Omega}^\#\colon\Q \otimes_{\O_S}\Q \to \Q \otimes_{\O_S}\G$, given by the canonical map~$\Q \to \Q \otimes_{\O_S}\G$ on the first component and $\tilde{\rho}^\#$ on the second component, is an isomorphism. Note that $\restr{\tilde{\Omega}^\#}{U}=\Omega^\#$ is an isomorphism of $\O_U$-algebras. Moreover, the coherent $\O_S$-modules~$\Q \otimes_{\O_S}\Q$ and $\Q \otimes_{\O_S}\G$ have depth at least $2$ at all points of $S\setminus U$, because $\Q$ and $\G$ are finite locally free, $S$ is regular and $S\setminus U$ has codimension at least $2$ in $S$. Therefore, $\tilde{\Omega}^\#$ is an isomorphism by~\cite[Théorème~5.10.5]{EGAIV2} and this concludes the proof of the theorem.
\end{proof}

A consequence of the previous theorem is that, given a regular base scheme, the obstruction to extend a finite flat torsor from a dense open subscheme lies exclusively in codimension~$1$. A~slightly more general version of this result, based on a similar proof, is given in~\cite[Theorem~7.1.3]{CS}. For a non-Noetherian analogue (in the context of algebraic spaces over Prüfer bases), see~\cite[Theorem~6.8]{GL}.

\section*{Acknowledgments}

The author wishes to thank Laurent Moret-Bailly for kindly sharing a sketch of the proof of Theorem~\ref{Pur4tors} and for several comments that helped improving the exposition of the subject. The author is also grateful to Gabriel Zalamansky for suggesting to work on this topic in the first place, Matteo Longo for helpful comments on an earlier version of the manuscript and the anonymous referee for useful remarks on the structure of the paper.

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