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\title[Loop group schemes]{Loop group schemes and Abhyankar's lemma}
\alttitle{Sch\'emas en groupes de lacets et lemme d'Abhyankar}

\author{\firstname{Philippe} \lastname{Gille}}
\address{UMR 5208 Institut Camille Jordan - Universit\'e Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex - France }
\email{gille@math.univ-lyon1.fr}


\begin{abstract}
We define the notion of loop reductive group schemes defined over the localization of a regular henselian ring $A$ at a strict normal crossing divisor $D$. We provide a criterion for the existence of parabolic subgroups of a given type.
\end{abstract}

\begin{altabstract}
On d\'efinit la notion de sch\'emas en groupes r\'eductifs de lacets au-dessus du localis\'e d'un anneau hens\'elien $A$ en un diviseur \`a croisements normaux stricts $D$. On \'etablit un crit\`ere pour qu'un tel sch\'ema en groupes admette un sous--sch\'ema en groupes paraboliques d'un type donn\'e.
\end{altabstract}

\keywords{Reductive group schemes, normal crossing divisor, parabolic subgroups}
\altkeywords{schémas en groupes réductifs, diviseur à croisements normaux, sous-schéma en groupes paraboliques}
\subjclass{14L15, 20G15, 20G35}



\begin{document}
\maketitle
\selectlanguage{french}

\section*{Version fran\c caise abr\'eg\'ee}

Soit $A$ un anneau local hens\'elien r\'egulier muni d'un syst\`eme de param\`etres $f_1,\dots, f_r$. On note $k$ le corps r\'esiduel de $A$ et $D$ le diviseur $D=\Div(f_1)+\dots +\Div(f_r)$, il est \`a croisements normaux stricts. On pose $X=\Spec(A)$ et $U= X \setminus D=\Spec(A_D)$. La th\'eorie d'Abhyankar d\'ecrit les rev\^etements finis \'etales connexes de $U$ qui sont mod\'er\'ement ramifi\'es le long de $D$~\cite[XIII.2]{SGA1}. Un tel objet est domin\'e par un rev\^etement galoisien de la forme
\[
B_n=B[T_1^{\pm 1},\dots, T_r^{\pm 1}]/ (T_1^{n} - f_1, \dots, T_r^{n}- f_r)
\]
o\`u $n$ d\'esigne un entier $\geq 1$ premier
\`a la caract\'eristique de $k$ et $B$ est une $A$--alg\`ebre galoisienne contenant une racine primitive $n$--i\`eme de l'unit\'e. Le groupe de Galois $\Gal(B_n/A_D)$ est le produit semi-direct $\mu_n(B)^r \rtimes \Gal(B/A)$ o\`u $\mu_n(B)^r$ agit par multiplication sur les $T_1, \dots, T_r$. Si $G$ est un $A$-sch\'ema en groupes localement de pr\'esentation finie, un $1$-cocycle $z: \Gal(B_n/A_D) \to G(B_n)$ est dit de \emph{lacets} (loop en anglais) s'il est \`a valeurs dans $G(B) \subset G(B_n)$. Cette terminologie est inspir\'ee par l'analogie avec le cas des polyn\^omes de Laurent~\cite[ch. 3]{GP}.

On note $\widehat X$ l'\'eclat\'e de $X=\Spec(A)$ en son point ferm\'e, c'est un sch\'ema r\'egulier~\cite[\S 8.1, th.~1.19]{Liu} et le diviseur exceptionnel $E \subset \widehat X$ est un diviseur de Cartier isomorphe \`a $\mathbb{P}^{r-1}_k$. On note alors $R=\mathcal{O}_{\widehat X,\eta}$ l'anneau local au point g\'en\'erique $\eta$ de $E$. Cet anneau $R$ est de valuation discr\`ete, de corps des fractions $K$, et de corps r\'esiduel $F=k(E)=k(t_1, \dots, t_{r-1})$ o\`u $t_i$ d\'esigne l'image de $\frac{f_i}{f_r} \in R$ par l'application de sp\'ecialisation $R \to F$. On note alors $v: K^\times \to \ZZ$ la valuation discr\`ete associ\'ee \`a $R$ et $K_v$ le compl\'et\'e de $K$. Le r\'esultat principal de cette note est le suivant.

\begin{theo*}[extrait du th.~\ref{thm_fixed}]
On suppose que $G$ agit sur un $A$--sch\'ema propre et lisse $Z$. Soit $\phi$ un $1$--cocycle de lacets pour $G$. On note ${_\phi \! Z}/U$ le tordu par $\phi$ de $Z\times_X U$ Alors les assertions suivantes sont \'equivalentes:
\begin{enumerate}\alphenumi
\item $({_\phi \!Z)}(U) \not \not = \emptyset$;
\item $({_\phi \!Z})(K_v) \not = \emptyset$.
\end{enumerate}
\end{theo*}

C'est assez proche d'un r\'esultat sur les polyn\^omes de Laurent~\cite[\S, th.~7.1]{GP}. L'application principale concerne le cas d'un sch\'ema en groupes r\'eductifs de lacets. Par d\'efinition, un $U$-sch\'ema en groupes r\'eductifs $G$ est \emph{de lacets} si il est isomorphe \`a un tordu de sa forme d\'eploy\'ee $G_0$ par un $1$--cocycle de lacets \`a valeurs dans le sch\'ema en groupes des automorphismes $\Aut(G_0)$. On applique alors le r\'esultat ci-dessus
\`a des $A$--sch\'emas de sous-groupes paraboliques de $G_0$ d'un type donn\'e (th.~\ref{thm_parabolic}). On en d\'eduit par exemple que si $G$ est un $U$-sch\'ema en groupes r\'eductifs \emph{de lacets}, alors $G$ admet un $U$--sch\'ema en groupes de Borel si et seulement si le $K_v$--sch\'ema en groupes $G_{K_v}$ est quasi-d\'eploy\'e. Plus g\'en\'eralement l'isotropie de $G$ est contr\^ol\'ee par l'indice de Tits de $G_{K_v}$.

\selectlanguage{english}

\section{Introduction}\label{section_intro}

In the reference~\cite{GP}, we investigated a theory of loop reductive group schemes over the ring of Laurent polynomials $k[t_1^{\pm 1}, \dots, t_n^{\pm 1}]$. Using Bruhat--Tits' theory, this permitted to relate the study of those group schemes to that of reductive algebraic groups over the field of iterated Laurent series $k((t_1)) \dots ((t_n))$. The main issue of this note is to start a similar approach for reductive group schemes defined over the localization $A_D$ of a regular henselian ring $A$ at a strict normal crossing divisor $D$ and to relate with algebraic groups defined over a natural field associated to $A$ and $D$, namely the completion $K_v$ of the fraction field $K$ with respect to the valuation arising from the blow-up of $\Spec(A)$ at its maximal ideal. The example which connects the two viewpoints is $k\llbracket t_1,\dots, t_n\rrbracket[\frac{1}{t_1}, \dots \frac{1}{t_n}]$ where $K_v \cong k\bigl(\frac{t_1}{t_n}, \dots, \frac{t_1}{t_{n-1}})((t_n))$.

After defining the notion of loop reductive group schemes in this setting, we show that for this class of group schemes, the existence of parabolic subgroups over the localization $A_D$ is controlled by the parabolic subgroups over $K_v$ (Theorem~\ref{thm_parabolic}).

\section{Tame fundamental group}

\subsection{Abhyankar's lemma}

Let $X=\Spec(A)$ be a regular local scheme (not assumed henselian at this stage). Let $k$ be the residue field of $A$ and $p \geq 0$ be its characteristic. We put $\widehat \ZZ'= \prod_{l \not =p} \ZZ_l$. Let $K$ be the fraction field of $A$, and let $K_s$ be a separable closure of $K$. It determines a base point $\xi: \Spec(K) \to X$ so that we can deal with the Grothendieck fundamental group $\Pi_1(X, \xi)$~\cite{SGA1}.

Let $(f_1, \dots, f_r)$ be a regular sequence of $A$ and consider the divisor $D= \sum D_i= \sum \Div(f_i)$, it has strict normal crossings. We put $U = X \setminus D= \Spec(A_D)$.

We recall that a finite \'etale cover $V \to U$ is \emph{tamely ramified} with respect to $D$ if the associated \'etale $K$--algebra $L=L_1 \times \dots \times L_a$ is tamely ramified at the $D_i's$, that is, for each $i$, there exists $j_i$ such that for the Galois closure $\widetilde L_{j_i}/K$ of $L_{j_i}/K$, the inertia group associated to $v_{D_i}$ has order prime to $p$~\cite[XIII.2.0]{SGA1}.

Grothendieck and Murre defined the tame (\emph{mod\'er\'e} in French) fundamental group $\Pi_1^D(U, \xi)$ with respect to $U \subset X$ as defined in~\cite[XIII.2.1.3]{SGA1} and~\cite[\S 2]{GM}. This is a profinite quotient of $\Pi_1(U, \xi)$ whose quotients by open subgroups provides finite Galois tame covers of $U$.

Let $V \to U$ be a finite \'etale tame cover. In this case Abhyankar's lemma states that there exists a flat Kummer cover $X'=\Spec(A') \to X$ where
\[
A'= A[T_1,\dots, T_r]/ (T_1^{n_1} - f_1, \dots, T_r^{n_r}- f_r)
\]
and the $n_i$'s are coprime to $p$ such that $V'= V \times_X X' \to X'$ extends uniquely to a finite \'etale cover $Y' \to X'$~\cite[XIII.5.2]{SGA1}.

\begin{lemm}\label{lem_picard}
Let $V \to U$ be a finite \'etale cover which is tame. Then $\Pic(V)=0$.
\end{lemm}

\begin{proof}
We use the same notation as above. We know that $X'$ is regular~\cite[XIII.5.1]{SGA1} so a fortiori locally factorial. It follows that the restriction maps $\Pic(X') \to \Pic(V') \to \Pic(V)$ are surjective~\cite[21.6.11]{EGA4}. Since $A'$ is finite over the local ring $A$, it is semilocal so that $\Pic(A')=\Pic(X')=0$. Thus $\Pic(V)=0$ as desired.
\end{proof}

From now on we assume that $A$ is henselian. According to~\cite[18.5.10]{EGA4}, the finite $A$--ring $A'$ is a finite product of henselian local rings. We observe that $A' \otimes_A k= k[T_1,\dots, T_r]/ (T_1^{n_1}, \dots, T_r^{n_r})$ is a local Artinian algebra so that $A'$ is connected. It follows that $A'$ is a henselian local ring. Its maximal ideal is $\gm'= \gm \otimes_A A' + \langle T_1, \dots, T_r \rangle$ so that $A' / \gm'= k$. Since there is an equivalence of categories between finite \'etale covers of $A$ (resp.\, $A'$) and \'etale $k$--algebras~\cite[18.5.15]{EGA4}, the base change from $A$ to $A'$ provides an equivalence of categories between the category of finite \'etale covers of $A$ and that of $A'$.

It follows that $Y' \to X'$ descends uniquely to a finite \'etale cover $\widetilde f: \widetilde Y \to X$. From now on, we assume that $V$ is furthermore connected, it implies that
\[
H^0(V, \cO_V)=B[T_1,\dots, T_r]/ (T_1^{n} - f_1, \dots, T_r^{n}- f_r)
\]
where $B$ is a finite connected \'etale cover of $A$. It follows that $V \to U$ is a quotient of a Galois cover of the shape
\[
%B_n=B[T_1^{\pm 1},\dots, T_r^{\pm 1}]/ (T_1^{n} - f_1, \dots, T_r^{n}- f_r)
B_n= \bigl( B[T_1,\dots, T_r]/ ( T_1^{n} - f_1, \dots, T_r^{n}- f_r) \bigr) \otimes_A A_D \cong B[T_1^{\pm 1},\dots, T_r^{\pm 1}]/ ( T_1^{n} - f_1, \dots, T_r^{n}- f_r)
\]
where $B$ is Galois cover of $A$ containing a primitive $n$--th root of unity. We notice that $B_n$ is the localization at $T_1\dots T_r$ of $B_n'=B[T_1,\dots, T_r]/ (T_1^{n} - f_1, \dots, T_r^{n}- f_r)$. We have
\[
\Gal(B_n/A_D)= \Bigg(\prod_{i=1}^r \mu_n(B) \Biggr) \rtimes \Gal(B/A).
\]
Passing to the limit we obtain an isomorphism
\[
\pi_1^t(U, \xi) \cong \Bigg(\prod_{i=1}^r \widehat \ZZ'(1) \Biggr)
\rtimes \pi_1(X, \xi).
\]
We denote by $f: U^{sc, t} \to U$ the profinite \'etale cover associated to the quotient $\pi_1^t(U, \xi)$ of $\pi_1(U,\xi)$. According to~\cite[Thm.~2.4.2]{GM}, it is the universal tamely ramified cover of $U$. It is a localization of the inductive limit $\widetilde B'$ of the $B_n'$. On the other hand we consider the inductive limit $\widetilde B$ of the $B$'s and observe that $\widetilde B'$ is a $\widetilde B$-ring.

\subsection{Blow-up}\label{subsec_blow_up}

We follow a blowing-up construction arising from~\cite[Lem.~15.1.1.6]{EGA4}. We denote by $\widehat X$ the blow-up of $X=\Spec(A)$ at its closed point, this is a regular scheme~\cite[\S 8.1, Thm.~1.19]{Liu} and the exceptional divisor $E \subset \widehat X$ is a Cartier divisor isomorphic to $\mathbb{P}^{r-1}_k$. We denote by $R=\mathcal{O}_{\widehat X,\eta}$ the local ring at the generic point $\eta$ of $E$. The ring $R$ is a DVR of fraction field $K$ and of residue field $F=k(E)=k(t_1, \dots, t_{r-1})$ where $t_i$ is the image of $\frac{f_i}{f_r} \in R$ by the specialization map. We denote by $v: K^\times \to \ZZ$ the discrete valuation associated to $R$.

We deal now with a Galois extension $B_n$ of $A_D$ as above. Since $B$ is a connected finite \'etale cover of $A$, $B$ is regular and local; it is furthermore henselian~\cite[18.5.10]{EGA4}. We denote by $L$ the fraction field of $B$ and by $L_n$ that of $B_n$. We have $[L_n:L]= n^r$. We want to extend the valuation $v$ to $L$ and to $L_n$.

We denote by $l=B/\gm_B$ the residue field of $B$, this is a finite Galois field extension of $k$. Also $(t_1,\dots, t_r)$ is a system of parameters for $B$. We denote by $w:L^\times \to \ZZ$ the discrete valuation associated to the exceptional divisor of the blow-up of $\Spec(B)$ at its closed point. Then $w$ extends $v$ and $L_w/K_v$ is an unramified extension of degree $[L:K]$ and of residual extension $F_l= l(t_1, \dots, t_{r-1})/k(t_1, \dots, t_{r-1})$.

On the other hand we denote by $w_n:L_n^\times \to \ZZ$ the discrete valuation associated to the exceptional divisor of the blow-up of $\Spec(B_n)$ at its closed point. We put $l_n= B_n/ \gm_{B_n}$, we have $l=l_n$. The valuation $\frac{w_n}{n}$ on $L_n$ extends $w$ and its residual extension is $F_{l,n}= l\Bigl(t_1^{1/n},\dots, t_{r-1}^{1/n} \Bigr) /k\bigl(t_1, \dots, t_{r-1} \bigr)$ so that $[F_{l,n}:F_l]=n^{r-1}$. Furthermore the ramification index $e_n$ of $L_n/L$ is $\geq n$. Since $n^r \leq e_n \, {[F_{l,n}:F_l]} \leq [L_n:K]=n^r$(where the last inequality is~\cite[\S VI.3, Prop.~2]{BAC}) it follows that $e_n=n$. The same statement shows that the map $L_w \otimes_L L_n \to L_{w_n}$ is an isomorphism. To summarize $L_{w_n}/L_w$ is tamely ramified of ramification index $n$ and of degree $n^r$. Altogether we have $L_{w_n}=L_w \otimes_K L_n$ so that $L_{w_n}$ is Galois over $K_v$ of group ${\prod_i \mu_n(B) \rtimes \Gal(B/A)} = {\prod_i \mu_n(l) \rtimes \Gal(l/k)}$.

We denote by $\Delta$ the diagonal embedding $\mu_n(l) \subset \prod_i \mu_n(l)$. We put $L_{w_n}^{\Delta}= L_n^{\Delta(\mu_n(B))}$. Since $t_r$ is an uniformizing parameter of $K_v$ and since $\Delta(\zeta) \,. \, t_r= \zeta. t_r$ for each $\zeta \in \mu_n(B)$, it follows that $(L_{w_n})^{\Delta}$ is the maximal unramified extension of $L_{w_n}/K_v$.

\subsection{Loop cocycles and loop torsors}

Let $G$ be an $X$--group scheme locally of finite presentation. A loop cocycle is an element of $Z^1\bigl(\pi_1^t(U), G(\widetilde B) \bigr)$ and it defines a Galois cocycle in $Z^1(\pi_1^t(U), G(U^{sc, t}))$. We denote by $Z^1_{loop}(\pi_1^t(U), G(U^{sc, t}))$ the image of the map $Z^1\bigl(\pi_1^t(U), G(\widetilde B) \bigr) \to Z^1(\pi_1^t(U), G(U^{sc, t}))$ and by $H^1_{loop}(U, G)$ the image of the map
\[
Z^1\bigl(\pi_1^t(U), G(\widetilde B) \bigr) \to H^1(\pi_1^t(U), G(U^{sc, t})) \to H^1(U, G).
\]
We say that a $G$-torsor $E$ over $U$ (resp.\, an fppf sheaf $G$-torsor) is a loop torsor if its class belongs to $H^1_{loop}(U, G) \subset H^1(U, G)$.

A given class $\gamma \in H^1_{loop}(U, G)$ is represented by a $1$--cocycle $\phi: \Gal(B_n/A_D) \to G(B)$ for some cover $B_n/A$ as above. Its restriction $\phi^{ar}: \Gal(B/A) \to G(B)$ to the subgroup $\Gal(B/A)$ of $\Gal(B_n/A_D)$ is called the ``arithmetic part'' and the other restriction $\phi^{geo}: \prod_i \mu_n(B) \to \gG(B)$ is called the geometric part. We observe that $\phi^{geo}$ is a $B$-group homomorphism.

Furthermore for $\sigma \in \Gal(B/A)$ and $\tau \in
\prod_i \mu_n(B)$ the computation of~\cite[p.~16]{GP} shows that $\phi^{geo}(\sigma \tau \sigma^{-1})=
\phi^{ar}(\sigma) \, {^\sigma \! \phi}(\tau) \,
\phi^{ar}(\sigma)^{-1}$ so that $\phi^{geo}$ descends to a homomorphism of $A$-group schemes $\phi^{geo}: \mu_n^r \to {_{\phi^{ar}}\!G}$. This provides a parameterization of loop cocycles.

\begin{lemm} \label{lem_dico}
\leavevmode
\begin{enumerate}
\item For $B_n/A_D$ as above, the map $\phi \mapsto (\phi^{ar}, \phi^{geo})$ provides a bijection between $Z^1_{loop}\bigl(\Gal(B_n/A_D), G(B) \bigr) $ and the couples $(z, \eta)$ where $z \in Z^1\bigl(\Gal(B/A), G(B))$ and $\eta: \prod_i \mu_n \to {_zG}$ is an $A$--group homomorphism.
\item The map $\phi \mapsto (\phi^{ar}, \phi^{geo})$ provides a bijection between $Z^1_{loop}\bigl(\pi^1(U, \xi)^t, G(\widetilde B)\bigr)$ and the couples $(z, \eta)$ where $z \in Z^1\bigl(\pi^1(X, \xi), G(\widetilde B))$ and $\eta: \prod_{i=1}^r \widehat \ZZ' \to {_zG}$ is an $A$--group homomorphism.
\end{enumerate}
\end{lemm}

\begin{proof}
This is similar with~\cite[Lem.~3.7]{GP}.
\end{proof}

We examine more closely the case of a finite \'etale $X$--group scheme $\gF$ of constant degree $d$.

\begin{lemm}\leavevmode
\begin{enumerate}
\item \label{3.1} $\gF(\widetilde B)= \gF(X^{sc})= \gF(U^{sc,t})$.
\item \label{3.2} We assume that $d$ is prime to $p$. We have $H^1_{loop}(U, \gF)= H^1(U, \gF)$.
\item \label{3.3} We assume that $d$ is prime to $p$. Let $f: \gF \to \gH$ be a homomorphism of $A$--group schemes (locally of finite type). Then $f_*\Bigl(H^1(U, \gF) \Bigr) \subset H^1_{loop}(U, \gH)$.
\end{enumerate}
\end{lemm}

\begin{proof}
\begin{proof}[\mref{3.1}]
We are given a cover $B_n/A_D$ as above such that $\gF_{B_n} \cong \Gamma_{B_n}$ is finite constant. as above. Since $B$ and $B_n$ are connected, the map $\gF(B) \to \gF(B_n)$ reads as the identity $\Gamma \cong \gF(B) \to \gF(B_n) \cong \Gamma$ so is bijective. By passing to the limit we get $\gF(\widetilde B)= \gF(U^{sc,t})$.
\let\qed\relax
\end{proof}

\begin{proof}[\mref{3.2}]
Let $\gE$ be a $\gF$--torsor over $U$. This is a finite \'etale $U$--scheme. Since $U$ is noetherian and connected, we have a decomposition $\gE= V_1 \times_U \cdots \times_U V_l$ where each $V_i$ is a connected finite \'etale $U$--scheme of constant degree $d_i$. We have $d_1+ \dots +d_l=d$ so that we can assume that $d_1$ is prime to $p$. We have then $\gE(V_1) \not = \emptyset$.

It follows that $f_1: V_1 \to U$ is a finite \'etale cover so that there exists a factorization $ U^{sc, t} \to V_1 \xrightarrow{h} U$ of $f$ so that $\gE(U^{sc, t}) \not = \emptyset$. Therefore $[\gE]$ arises from $H^1(\pi_1^t(U, \xi), \gF(U^{sc, t})) \subset H^1(U, \gF)$. It follows that $H^1(\pi_1^t(U, \xi), \gF(U^{sc, t})) \simlgr H^1(U, \gF)$. We use now (1) and obtain the desired bijection $H^1(\pi_1^t(U, \xi), \gF(B)) \simlgr H^1(U, \gF)$.
\let\qed\relax
\end{proof}

\begin{proof}[\mref{3.3}]
This follows readily from (2).
\end{proof}
\let\qed\relax
\end{proof}

\subsection{Twisting by loop torsors}

We assume that the $A$--group scheme $G$ acts on an $A$--scheme $Z$. Let $\phi: (\prod_i^r \mu_n)(B) \rtimes \Gal(B/A) \to G(B)$ be a loop cocycle. It gives rise to an $A$--action of $\mu_n^r$ on $_{\phi^{ar}}\!Z$. We denote by $(_{\phi^{ar}}\!Z)^{\phi^{geo}}$ the fixed point locus for this action, it is representable by a closed $A$--subscheme of $_{\phi^{ar}}\!Z$~\cite[A.8.10.(1)]{CGP}. We have a closed embedding $(_{\phi^{ar}}\!Z)^{\phi^{geo}} \times_X U \subset {_\phi \! Z}$ of $U$-schemes.



\section{Fixed points method }


\begin{theo}\label{thm_fixed}
Let $X=\Spec(A)$ be a henselian regular local scheme and $U=X \setminus D$ as above. We denote by $v: K^\times \to \ZZ$ the discrete valuation associated to the exceptional divisor $E$ of the blow-up of $X$ at its closed point.

Let $G$ be an affine $A$-group scheme of finite presentation acting on a proper smooth $A$--scheme $Z$. Let $\phi$ be a loop cocycle for $G$. Then $Y= \bigl(_{\phi^{ar}}Z\bigr)^{\phi^{geo}}$ is a smooth proper $A$--scheme and the following are equivalent:
\begin{enumerate}\romanenumi
\item \label{4i} $(_\phi\!Z)(K_v) \not = \emptyset$;
\item \label{4ii} $Y(k) \not \not = \emptyset$;
\item \label{4iii} $Y(U) \not \not = \emptyset$;
\item \label{4iv} $(_\phi\!Z)(U) \not = \emptyset$.
\end{enumerate}
\end{theo}

This is quite similar with the fixed point theorem~\cite[\S, Thm.~7.1]{GP}. The following example makes the connection.

\begin{exam}
We assume that $A=k\llbracket t_1,\dots, t_r\rrbracket$ for a field $k$ and $k[U]=k\llbracket t_1,\dots, t_n\rrbracket\bigl[\frac{1}{t_1}, \dots, \frac{1}{t_r} \bigr]$. We consider an affine algebraic $k$--group $G$ acting on a smooth proper $k$--scheme $Z$. In this case $K=k((t_1,\dots, t_r))$ and $A$ embeds in $k\bigl(\frac{t_1}{t_r}, \dots, \frac{t_{r-1}}{t_r}\bigr)\llbracket t_r\rrbracket$ so that $K$ embeds in $k\bigl(\frac{t_1}{t_n}, \dots, \frac{t_{r-1}}{t_r}\bigr)((t_r))$ which is nothing but the complete field $K_v$. If $Q$ is a loop $G$-torsor over $U$, the statement is then that ${^QZ}(U) \not = \emptyset$ if and only if ${^QZ}(K_v) \not = \emptyset$. Taking a cocycle $\phi \in Z^1(\pi_1(U)^t, G(k_s))$ for $E$, this rephrases by the equivalence between $(_{\phi}Z)(U) \not = \emptyset$ and $(_{\phi}Z)(K_v) \not =\emptyset$.

What we have from~\cite[Thm.~7.1]{GP} (in characteristic zero but this extends to this tame setting) is the equivalence between $(_{\phi}Z)(k[t_1^{\pm 1}, \dots, t_r^{\pm 1}]) \not = \emptyset$ and $(_{\phi}Z)\bigl(k((t_1)) \dots ((t_r)) \bigr)\not = \emptyset$. Since $(_{\phi}Z)(k[t_1^{\pm 1}, \dots, t_r^{\pm 1}]) \subset (_{\phi}Z)(U)$ and $(_{\phi}Z)\bigl(K_v \bigr) \subset (_{\phi}Z)\bigl(k((t_1)) \dots ((t_r)) \bigr) $, it follows that this special case of Theorem~\ref{thm_fixed} is a consequence of the fixed point result of~\cite{GP}.
\end{exam}

We proceed to the proof of Theorem~\ref{thm_fixed}.

\begin{proof}
According to~\cite[A.8.10.(1)]{CGP}, $Y= \bigl(_{\phi^{ar}}(Z^{\phi^{geo}}) \bigr)$ is a closed $A$--scheme of $_{\phi^{ar}}Z$ so it is proper. It is smooth over $X$ according to point (2) of the same reference. Let $\phi: \Gal(B_n/A_D) \to G(B)$ be the loop $1$-cocycle for some Galois cover $B_n/A_D$ as above for some $n$ prime to $p$. Up to replacing $G$ by $_{{\phi^{ar}}\!G}$ and $G$ by $_{\phi^{ar}Z}$, we can assume that $\phi^{ar}=1$ without lost of generality.

\begin{proof}[\mref{4ii}$\Rightarrow$\mref{4iii}]
Since $Y_k$ is the special fiber of the smooth $X$--scheme $Y$, Hensel's lemma shows that $Y(A) \to Y(k)$ is onto. Since $Y(k)$ is not empty, it follows that $Y(A)$ is not empty and so is $Y(U)$.
\let\qed\relax
\end{proof}

\begin{proof}[\mref{4iii}$\Rightarrow$\mref{4iv}]
Since $Y(U) \subset {_\phi\!Z}(U)$, $Y(U) \not = \emptyset$ implies that ${_\phi\!Z}(U)\not = \emptyset$.
\let\qed\relax
\end{proof}

\begin{proof}[\mref{4iv}$\Rightarrow$\mref{4i}]
This is obvious.
\let\qed\relax
\end{proof}

\begin{proof}[\mref{4i}$\Rightarrow$\mref{4ii}]
We assume that $(_\phi\!Z)(K_v) \not = \emptyset$. By definition we have
\[
(_\phi Z)(K_v) = \bigl\{ z \in Z(L_{w_n})
\, \big| \, \phi(\sigma). \sigma(z)=z \enskip
\forall \sigma \in \Gal(L_n/K) \bigr\}
\]
and our assumption is that this set is non-empty. Let $O_{w_n}$ be the valuation ring of $L_{w_n}$. Since $Z$ is proper over $X$, we have the specialization map $Z(L_{w_n}) \to Z_k(F_{l,n})$. We get that the set
\[
\bigl\{ z \in Z_k(F_{l,n}) \,\big| \, \phi(\sigma). \sigma(z)=z, \enskip
\forall \sigma \in \Gal(L_{w_n}/K_v) \bigr\}
\]
is not empty. Since we have an embedding
\[
F_{l,n}=l\bigl(t_1^{1/n},\dots, t_{r-1}) \enskip \hookrightarrow
\enskip l\bigl(\bigr(t_1^{1/n} \bigr)\bigr) \dots
\bigl(\bigl(t_{r-1}^{1/n} \bigr)\bigr)
\]
in a higher field of Laurent series, successive specializations along the coordinates $t_1^{1/n}$,\dots, $t_{r-1}^{1/n}$ show similarly that the set
\begin{equation}\label{eq_omega}
\Bigl\{ z \in (Z_k)\bigl(l \bigr)
\,\Big| \, \phi(\sigma). \sigma(z)=z, \enskip
\forall \sigma \in \Gal(L_{w_n}/K_v) \Bigr\}
\end{equation}
is not empty. Since $\eta^{ar}=1$, this set is $(Z_k)^{\eta^{geo}}(k)$. Thus $Y(k)=(Z_k)^{\eta^{geo}}(k)$ is non empty.
\end{proof}
\let\qed\relax
\end{proof}


\section{Parabolic subgroups of loop reductive group schemes}

\subsection{Chevalley groups}

Let $G_0$ be Chevalley group defined over $\ZZ$. Let $T_0$ be a maximal split $\ZZ$-subtorus of $G_0$ together with a Borel subgroup $B_0$ containing it. We denote by $\Delta_0$ the Dynkin diagram of $(G_0,B_0,T_0)$. We denote by $G_{0,ad}$ the adjoint quotient of $G_0$ and by $G_0^{sc}$ the simply connected covering of $DG_0$. We have a map $\Aut(G_0) \to \Aut(G_0^{sc}) \simlgr \Aut(G_{0,ad})$ and a fundamental exact sequence
\[
1 \to G_{0,ad} \to \Aut(G_{0,ad}) \to \Out(G_{0,ad}) \to 1
\]
where $\Out(G_{0,ad}) \simlgr \Aut(\Delta_0)$. We recall that there is a bijection~$I \mapsto P_{0,I}$ between the finite subsets of $\Delta_0$ and the parabolic subgroups of $G_0$ containing $B_0$~\cite[XXVI.3.8]{SGA3}; it is increasing for the inclusion order, in particular $B_0= P_{0,\emptyset}$ and $G_0= P_{0,\Delta_0}$. We consider the total scheme $\Par_{G_0}$ of parabolic subgroups of $G_0$, it is a projective smooth $\ZZ$--scheme equipped with a type map $\mathbf{t}: \Par_{G_0} \to \Ouf(\Delta_0)$ where $\Ouf(\Delta_0)$ stands for the finite constant scheme attached to the set of subsets of $\Delta_0$~\cite[XXVI.3]{SGA3}. The fiber at $I$ is denoted by $\Par_{G_0,I}$, it has connected fibers and is the scheme of parabolic subgroups of $G_0$ of type $I$. We have a natural action of $\Aut(G_0)$ on $\Par_{G_0}$. As in~\cite[\S 5.1]{Gi}, we denote by $\Aut_I(G_0)$ the stabilizer of $I$ for this action. By construction $\Aut_I(G_0)$ acts on $\Par_{G_0,I}$.

\subsection{Definition}

Let $G$ be a reductive $U$-group scheme in the sense of Demazure--Grothendieck~\cite[XIX]{SGA3}. Since $U$ is connected and $G$ is locally splittable~\cite[XXII.2.2]{SGA3} for the \'etale topology, $G$ is an \'etale form of a Chevalley group $G_0$ as above defined over $\ZZ$.

We say that $G$ is a \emph{loop group scheme} if the $\Aut(G_0)$-torsor $Q=\Isom(G_0,G)$ (defined in~\cite[XXIV.1.9]{SGA3}) is a loop $\Aut(G_0)$-torsor. We denote by $G_{0,ad}$ the adjoint quotient of $G_0$ and by $G_0^{sc}$ the simply connected covering of $DG_0$. We have a map $\Aut(G_0) \to \Aut(G_0^{sc}) \simlgr \Aut(G_{0,ad})$ which permits to see $G_{ad}$ (resp.\ $G^{sc}$) as twisted forms of $G_{0,ad}$ (resp.\ $G^{sc}_0$) so that $G_{ad}$ and $G^{sc}$ are also loop reductive group schemes. We consider the map $\Aut(G_0) \to \Aut(G_{0,ad}) \to
\Out(G_{0,ad}) \simlgr \Aut(\Delta_0)$.

If $\phi: \Gal(B_n/A_D) \to \Aut(G_0)(B)$ is a loop cocycle, we get an action of $\Gal(B_n/A_D)$ on $\Delta_0$ called the star action. If $I$ is stable under the star action, we can twist $\Par_{G_0,I}$ by $\phi$ and deal with the scheme ${_\phi\!\Par_{G_0,I}}$ which is the scheme of parabolic subgroup schemes of $G$ of type $I$.

\subsection{Parabolics}

\begin{theo}\label{thm_parabolic}
Assume that $G$ is a loop reductive $U$-group scheme and let $\phi: \Gal(B_n/A_D) \to \Aut(G_0)(B)$ be a loop cocycle such that $G \cong {_\phi \! G_0}$. Let $I \subset \Delta_0$ be a subset stable under the star action defined by $\phi$. Then the following are equivalent:
\begin{enumerate}\romanenumi
\item \label{6i} $G$ admits a $U$--parabolic subgroup of type $I$;
\item \label{6ii} the $k$--morphism $\eta^{geo}_k: \mu_n^r \to
\Aut(_{\eta^{ar}}\! G_0)_k = \bigl({_{\eta^{ar}} \!\Aut(G_0)}\bigr)_k$ normalizes a parabolic $k$--subgroup of $_{\eta^{ar}}G_{0,k}$ of type $I$;
\item \label{6iii} $G_{K_v}$ admits a parabolic subgroup of type $I$.
\end{enumerate}
\end{theo}

\begin{proof}
Without loss of generality we can assume that $G$ is adjoint. Our assumption on the star action is rephrased by saying that $\phi$ takes values in $\Aut_I(G_0)$. We apply Theorem~\ref{thm_fixed} to the action of $\Aut_I(G_0)$ on the proper $A$-scheme $\Par_{G_0,I}$. We consider the $A$-scheme $Y= ({_{\phi^{ar}}}\!\Par_{G_0,I})^{\phi^{geo}}$. Theorem~\ref{thm_fixed} shows that the following statements are equivalent.
\begin{enumerate}
\renewcommand*\theenumi{\roman{enumi}$'$}\relabel
\item \label{6i'} $(_\phi\!\Par_{G_0,I})(U) \not = \emptyset$;
\item \label{6ii'} $Y(k) \not = \emptyset$.
\item \label{6iii'} $(_\phi\!\Par_{G_0,I})(K_v) \not = \emptyset$.
\end{enumerate}

Clearly \eqref{6i'} is equivalent to condition \eqref{6i} of the Theorem and similarly we have $\eqref{6iii'} \Longleftrightarrow \eqref{6iii}$. It remains to establish the equivalence between \eqref{6ii} and \eqref{6ii'}.

Assume that $({_{\phi^{ar}}}\!\Par_{G_0,I})^{\phi^{geo}}(k)$ is not empty and pick a $k$--point $z$. Then the stabilizer $({_{\phi^{ar}}}\!G_0)_z$ is a $k$--parabolic subgroup of ${_{\phi^{ar}}}\!G_0$ of type $I$ which is stabilized by the action $\phi^{geo}_k$. In other words, $\phi^{geo}_k$ normalizes $({_{\phi^{ar}}}\!G)_z$. Conversely we assume that ${_{\phi^{ar}}}\!G$ admits a $k$--parabolic subgroup of type $I$ normalized by $\phi^{geo}$. It defines then a point $z \in ({_{\phi^{ar}}}\!\Par_{G_0,I})(k)$ which is fixed by $\phi^{geo}$.
\end{proof}

\subsection{An example}

Assume that the residue field $k$ is not of characteristic $2$ and consider the diagonal quadratic form of dimension~$2^r$
\[
q = \sum_{I \subset \{1, \dots,r\}} u_I \, t^I (x_I)^2
\]
where $t_I= \prod_{i \in I} t_i$ and $u_I \in A^\times$. Then $\SO(q)$ is a loop reductive group scheme over $U$. Since the projective quadric $\{q =0 \}$ is a scheme of parabolic subgroups of $\SO(q)$, Theorem~\ref{thm_parabolic} shows that $q$ is isotropic over $A_D$ if and only if $q$ is isotropic over $K_v$. The $2$-dimensional case is related with~\cite[proof of Theorem~3.1]{CTPS}.


\section*{Acknowledgements}

We thank R. Parimala for sharing her insight about the presented results. Finally we thank the referee for a careful reading of the manuscript.

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