The Fundamental Group \\ of an Algebraic Stack
Vincent Zoonekynd
We define the notion of fundamental group of an algebraic stack, prove a comparison theorem between the fundamental group of a stack over $\CC$ and that of the associated analytic orbifold, show that this notion coincides with that defined with simplicial schemes by \cite{oda} and finally prove some short exact sequences involving these groups.
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\begin{document}
\title[The Fundamental Group of an Algebraic Stack]{The Fundamental Group \\ of an Algebraic Stack}
\author{Vincent Zoonekynd}
\address{Universit� Paris 7,
UFR de Math�matiques,
�quipe Topologie et g�om�trie alg�briques,
Case 7012,
2 place Jussieu,
75251 Paris Cedex 05}
\email{zoonek@math.jussieu.fr}
\urladdr{http://www.math.jussieu.fr/\textasciitilde zoonek/}
\keywords{Algebraic stack, fundamental group, groupoid, progroupoid}
\subjclass[2000]{Primary 14F35; % homotopy of schemes
Secondary 14A20, % stacks
18B25, % toposes
20L05, % groupoids
18D05} % 2-categories
\begin{abstract}
  We define the notion of fundamental group of an algebraic
  stack, prove a comparison theorem between the fundamental
  group of a stack over $\CC$ and that of the associated
  analytic orbifold, show that this notion coincides with
  that defined with simplicial schemes by \cite{oda} and
  finally prove some short exact sequences involving these
  groups.
\end{abstract}
\maketitle

\section{Pointless fundamental groupoid of a topos}

We shall first recall some results from \cite{leroy}.

\begin{defi}
  A \d{groupoid} is a category all of whose morphisms are
  isomorphisms. As a group is a groupoid with a single
  object, a groupoid may conversely be seen as a ``group
  with several objects''.
  
  The \d{classifying topos} of a groupoid $G$ is the
  category of presheaves on the opposite category: 
  $\cat B G = G\op\cat{Set}$; in particular, the classifying
  topos of a group $G$ is its category of (left) $G$-sets. 

  A \d{progroupoid} is a projective $1$-system 
  $(G_i)_{i\in I}$ of groupoids, \ie, a functor 
  $I \lra \cat{Groupoids}$ from a
  filtering category $I$;
  we shall often denote it as
  $\proobj G_i$.

  A \d{strict progroupoid} is a progroupoid 
  $\proobj G_i$ such that 
  \begin{theoenum}
  \item the functors $G_i \lra G_j$ induce bijections
    between the sets of objects $\ob G_i \lra \ob G_j$;
  \item for any objects $X$, $Y$, the functors 
    $G_i \lra G_j$
    induce onto maps
    $\hom_{G_i} (X,Y) \lra \hom_{G_j} (X,Y)$.
  \end{theoenum}
  All our progroupoids will be strict.

  The classifying topos of a progroupoid 
  $\proobj G_i$ is 
  $ \cat B \proobj G_i := 
  \dlimpro\nolimits^{\dcat{Topos}}
  \cat B G_i. $
  (The superscript ``$\dcat{Topos}$'' indicates the
  $2$-category in which the $2$-limit is taken.)
\end{defi}

\begin{defi}
  Let $\T$ be a topos. 
  An object $X$ is said to be \d{connected} if whenever
  $X = X_1 \amalg X_2$, 
  either $X_1$ or $X_2$ is a final object (often denoted $\nothing$).
  A \d{constant object} is a disjoint sum of copies of the
  final object $*$, \ie, it may be written
  $I_\T = \coprod_{i\in I} *$ for some set $I$. 
  The topos $\T$ is said to be connected if its final object is. 

  An object $F$ is said to be \d{trivialized by} an object
  $U$ if its \d{restriction}
  $F|_U = (F \times U \lra U)$ is a constant object of the
  topos $\T/U$. An object $F$ is said to be locally constant
  if the objects trivializing it cover the final object, 

  A \d{Galois object} in a topos $\T$ is a connected,
  locally constant object~$Y$ such that the morphism
  $Y \times (\aut Y)_\T \lra Y \times Y$ 
  is an isomorphism. 

  A category $\cat C$ is said to be \d{generated by} a set
  of objects $\cat K \subset \ob\cat C$ if 
  for any objects $x$, $y$, for any distinct morphisms $f, g
  : x \lra y$, there exists an object $z \in \cat K$ and a
  morphism $k: z \lra x$ such that $fh \neq gh$.

  A \d{Galois topos} is a topos generated by its Galois
  objects. 
\end{defi}

We shall see that Galois toposes are exactly the classifying
toposes of strict groupoids. 

\begin{defi}
  A topos is said to be \d{locally connected} if it is
  generated by its connected objects; this is equivalent to
  require that the ``constant object'' functor
  $\cat{Set} \lra \T$ have a left adjoint $\pi$, which we
  shall call the \d{connected components functor}.
\end{defi}
\begin{defi}
  A \d{projective 2-system} in a $2$-category $\dcat A$ 
  is a $2$-functor 
  $I \lra \dcat{A}$ from a small filtering $1$-category
  $I$. A projective $2$-system of toposes is said to be
  \d{strict} if the inverse image 
  functors of the topos morphisms involved are fully
  faithful.

  Let $\dcat A$ be a $2$-category and $\dcat B$ a (full)
  sub-$2$-category. A 1-morphism $f:A \lra B$ to an object of
  $\dcat B$ is said to be \d{2-universal} if
  for any morphism 
  $g : A \lra B'$ to an object of $\dcat B$, 
  there exists a morphism 
  $h : B \lra B'$ and a $2$-isomorphism 
  $\alpha : h f \lra g$ such that for any other morphism 
  $h' : B \lra B'$ and $2$-isomorphism $\alpha' : 
  h' f \lra g$, there exists a \emph{unique} $2$-isomorphism
  $\beta : h \lra h'$ such that 
  $$ \xymatrix{ hf \ar[rd]^\alpha \ar[d]_{\beta f} \\ 
    h'f \ar[r]_{\alpha'} & g.} $$
\end{defi}
\begin{theo}
  Let $\T$ be a locally connected topos.
  \begin{theoenum}
  \item The full subcategory $\SLC\T$ of disjoint sums of
    locally constant objects of $\T$ is a Galois topos and
    the inclusion functor $\SLC\T\lra\T$ is the inverse
    image functor of a topos morphism
    $\T \lra \SLC\T$ which is $2$-universal among morphisms
    to Galois toposes.
  \item The Galois toposes are the strict
    projective $2$-limits (in the $2$-category of toposes) 
    of classifying toposes of groupoids. 
  \end{theoenum}
\end{theo}
\begin{rem}
  We shall need the following points of the proof of this
  theorem. 
  \begin{theoenum}
  \item If $R$ is a covering sieve of the final object $*$
    of $\T$ (we shall write: $R \in J(*)$),
    denote $\LC(\T,R)$ the full subcategory of $\T$
    of the objects trivialized by the connected objects of
    $R$: it is a topos and the inclusion functor 
    $\LC(\T,R) \lra \T$ is the inverse image functor of a
    topos morphism.
  \item The toposes $\LC(\T,R)$ are classifying toposes of
    groupoids; more precisely, if $G$ is the groupoid of
    points of $\LC(\T,R)$, \ie, the groupoid of topos
    morphisms $\cat{Set}\lra\LC(\T,R)$ (and their $2$-isomorphisms), 
    then 
    $ \LC(\T,R) \equiv \cat B G. $
  \item The Galois topos $\SLC\T$ is the projective
    $2$-limit of these toposes: 
    $$ \SLC\T \equiv \dlimpro_{R \in J(*)}
    \!\!{}^{\dcat{Topos}}
    LC(\T,R). $$
  \end{theoenum}
\end{rem}

\section{Pointed fundamental progroupoid of a topos}

The following lemma
 gives an explicit description of the
classifying topos of a progroupoid. In particular, the
connected objects of $\cat B \proobj G_i$ are the (images of
the) connected objects of the various $\cat B G_i$. 

\begin{lemma}
  Let $G = \proobj G_i$ be a strict progroup and 
  $\cat S_0$ the category whose objects are sets~$X$ endowed
  with an action of one of the $G_i$ (hence of any $G_k$ for
  $k \lra i$), connected as presheaves (\ie, they have a
  single orbit) and whose morphisms are 
  $$
  \raisebox{.5\depth}{\xymatrix @!0 @R=.5em @C=3pc{
      & i \\
      *!!<0pt,0.7ex>+!R!<-1.2ex,0pt>{\hom\bigl( (X, G_i), (Y, G_j) \bigl) = 
        \hom_{G_k} (X, Y) 
        \qquad \text{ for all  }\, k 
        } 
      \ar[ru] \ar[rd] \\
      & j.
      }}
  $$
  Then, the classifying topos $\cat B G$ is equivalent to
  the category $\cat S$ of disjoint sums of objects of 
  $\cat B_0$. 

  The same result holds for progroupoids.
\end{lemma}

\begin{proof}
  This results from \cite[3.3.6]{leroy}, which states that 
  the projective $2$-limit of a
  \emph{strict} projective system of classifying toposes of groupoids
  (strict means that the inverse image functors are fully faithful:
  this is the case here)
  is the topos of sheaves on $\limind^{\dcat{Cat}} \cat B G_i$ for the
  canonical topology and from 
  \cite[2.4.2, 2.4.3]{leroy} which proves that 
  the category obtained from $\limind^{\dcat{Cat}} \cat B G_i$ by
  adding disjoint sums is a topos.
\end{proof}

\begin{defi}
  A \d{base point} of a topos $\T$ is a topos morphism 
  $\cat{Set} \lra \SLC\T$. 
\end{defi}
\begin{theo}
  Let $\T$ be a locally connected topos and $P$ a set of
  points of $\SLC\T$ (\ie, base points of $\T$), 
  at least one in each connected
  component of $\SLC\T$. 
  Let $G_R$ be the groupoid whose objects are elements of $P$ seen as
  points of $\LC(\T,R)$ and whose morphisms are topos $2$-isomorphisms
  $ \smash{
    \xymatrix @C=2ex { 
      *!!<0pt,0.7ex>+!R!<-1.2ex,0pt>
      {\cat{Set}}
      \ar@/^1ex/[r]|*{\vrule height 1.5pt depth 1.5pt width 0pt}="A"
      \ar@/_1ex/[r]|*{\vrule height 1.5pt depth 1.5pt width 0pt}="B"
      \ar@{=>}"A";"B"
      & 
      *!!<0pt,0.7ex>+!L!<+1.2ex,0pt>
      {\LC(\T,R)}
      } 
    }
  $
  and let 
  $\pi_1(\T, P) := \smash{\proobj _{R \in J(*)} G_R}$.
  Then $\pi_1(\T,P)$ is a strict progroupoid and 
  $\SLC\T \equiv \cat B \pi_1(\T,P)$.
\end{theo}
\begin{proof}
  To simplify, one may assume that $\T$ is connected. 
  But as the inverse image functors of the morphisms 
  $\T \lra \SLC\T$ or $\T \lra \LC(\T,R)$ 
  preserve any decomposition 
  $* = X_1 \amalg X_2$
  and are fully faithful, $\SLC\T$ and $\LC(\T,R)$ are also
  connected. 
  We may then assume that $P$ is a single point
  and that the $G_R$ are groups. We have 
  $\LC(\T,R) \equiv \cat B G_R$, thus 
  $\SLC\T \equiv \cat B \pi_1(\T,P)$.

  The projective system $(G_R)_{R \in J(*)}$ is clearly
  filtering. It remains to prove that the morphisms 
  $f: G_R \lra G_{R'}$ are surjective: this will come from the fact
  that 
  a group morphism
  $f: G' \lra G$ is onto 
  provided $f^*: \cat B G \lra \cat B G'$ is fully faithful.
  Indeed, as $G$ is a $G$-set, one has then a bijection 
  $G = \aut _{G} G \lra \aut_{G'} G = \aut _{\im f} G$.
  But an $\im f$-automorphism of $G$ is entirely determined by a
  permutation 
  $\sigma \in \SymetricGroup_{G/\im f}$ of the orbits and by an
  automorphism of each orbit; but as the automorphism group of each orbit
  is isomorphic to $\im f$, one has 
  $$ \aut_{G'} G = \aut_{\im f} G = 
  \Biggl( \prod_{G/\im f} \im f \Biggr)
  \psd 
  \SymetricGroup_{G/\im f}^{\mathstrut}.
  $$
  But as $G \lra \aut_{G'} G$ is a bijection, one easily sees that
  $\im f = G$. 
\end{proof}
\begin{rem}\label{rem:omissionofbasepoints}%
  By abuse of notation, we shall write 
  $\pi_1\T$ instead of 
  $\pi_1(\T, P)$ when $P$ is the set of all base points of $\T$. 
\end{rem}
\begin{rem}
  All this may be generalized to profinite groupoids in the following
  way. 
  Let $\T$ be a locally connected topos. 

  A \d{finite object} is an object of the form $I_\T$ for some finite
  set $I$, \ie, it is a disjoint sum of a finite number of copies of
  the final object. 
  A \d{locally constant finite object} is a locally constant object
  $F$ such that whenever $U$ trivializes $F$, the restriction 
  $F|_U$ is finite. 
  A \d{finite Galois object} is a locally constant finite Galois
  object. 
  A \d{finite Galois topos} is a topos generated by its finite Galois
  objects. 

  The full subcategory $\SLCF \T$ of disjoint sums of 
  locally constant finite objects of $\T$ is a finite Galois topos
  and the
  inclusion functor $\SLCF \T \lra \T$ is the inverse image functor of
  a topos morphism; this topos morphism is $2$-universal among those
  to finite Galois toposes. 
  If we let $\LCF(\T,R)$ denote the category of disjoint sums of
  locally constant finite objects of $\T$ trivialized by 
  the connected elements of a sieve $R$, then 
  $\SLCF\T = \dlimpro_{R \in J(*)} \LCF(\T,R)$.

  A \d{finite groupoid} is a groupoid whose hom-sets are finite. 
  A \d{profinite groupoid} is a filtered strict projective system of
  finite groupoids. The finite Galois toposes are the strict
  projective $2$-limits of classifying toposes of finite groupoids. 

  A \d{profinite base point} of $\T$ is a topos morphism 
  $\cat{Set} \lra \SLCF\T$. If $P$ is a set of profinite base points
  of $X$, at least one in each connected component of $\SLCF\T$, and
  if $G_R$ is the groupoid whose objects are elements of $P$ seen as
  points of $\LCF(\T,R)$ and whose morphisms are topos $2$-isomorphisms
  $ \smash{
    \xymatrix @C=2ex { 
      *!!<0pt,0.7ex>+!R!<-1.2ex,0pt>
      {\cat{Set}}
      \ar@/^1ex/[r]|*{\vrule height 1.5pt depth 1.5pt width 0pt}="A"
      \ar@/_1ex/[r]|*{\vrule height 1.5pt depth 1.5pt width 0pt}="B"
      \ar@{=>}"A";"B"
      & 
      *!!<0pt,0.7ex>+!L!<+1.2ex,0pt>
      {\LCF(\T,R)}
      } 
    }
  $,
  then 
  $\pi_1(\T, P) := \proobj _{R \in J(*)}$
  is a strict finite progroupoid and 
  $\SLCF\T \equiv \cat B \pi_1(\T,P)$.

  If $\cat C$ is a Galois category and $\f F : \cat C \lra \cat{set}$
  a fiber functor (see \cite{SGA1}), then the category of disjoint
  sums of objects of $\cat C$ is a connected finite Galois topos and
  $\f F$  induces a profinite base point, still denoted $\f F$, then 
  $\wh \pi_1( \T, \{\f F\})$ is the profinite fundamental group of
  $\cat C$.
  
  A \d{quotient groupoid} of a groupoid $G$ is a groupoid morphism $f:
  G \lra H$ such that $f$ induces a bijection between the sets of
  objects of $G$ and $H$ and such that the maps $\hom_G(x,y) \lra
  \hom(fx, fy)$ be surjective for all $x$, $y \in \ob G$. The
  \d{profinite completion} of a groupoid $G$ is the progroupoid 
  $\wh G$ of its finite quotients, the profinite completion of a
  progroupoid is the progroupoid of the finite quotients of its
  components. One may check that
  $\wh\pi_1\T$ is the profinite completion of $\pi_1\T$.
\end{rem}

\section{Fundamental group of a stack}

By ``stack'', we shall mean Deligne--Mumford stack
\cite{DM}, \cite{vistoli}, unless otherwise specified.
The following proposition enables us to apply the preceeding
theory to define the fundamental group of a stack. 

\begin{prop}
  The topos of �tale sheaves on an algebraic stack is
  locally connected. 
\end{prop}
\begin{proof}
  The first lemma shows that it is generated by connected
  schemes, the second that these are connected as sheaves. 
\end{proof}
\begin{lemma}
  Let 
  $\fx F \dlra \fx G$ 
  be two distinct morphisms of �tale sheaves on an algebraic stack
  $X$. Then, there exists a connected scheme $V$, an �tale
  morphism $V \lra X$ (enabling us to see $V$ as a sheaf on
  $X$)
  and a
  morphism 
  $V \lra \fx F$ such that the two morphisms
  $V \lra \fx F \dlra \fx G$ 
  be distinct. 
\end{lemma}
\begin{proof}
  There exists an �tale morphism $U \lra X$ such that the maps
  $\fx F U \iso \hom(U,\fx F) \dlra \hom(U,\fx G) \iso \fx G U$
  be distinct. If $(U_i)_{i\in I}$ are the connected components of
  $U$, then 
  the maps 
  $\prod \hom(U_i, \fx F) \dlra \prod \hom(U_i, \fx G)$ are distinct,
  hence there exists $i \in I$ such that the maps
  $\hom(U_i, \fx F) \dlra  \hom(U_i, \fx G)$ be distinct.
\end{proof}
\begin{lemma}
  Let $X$ be a stack and $U \lra X$ an �tale morphism from a
  connected scheme. Then~$U$ is connected as an �tale sheaf
  on $X$.  
\end{lemma}
\begin{proof}
  Let $U \lra X$ be an �tale morphism from a connected scheme $U$, 
  seen as a presheaf 
  $T \lmt \hom_X(T,U)$ on $X\et$, 
  which is actually a sheaf.
  Assume $U = \fx F_1 \amalg \fx F_2$ and let us prove that either 
  $\fx F_1 = \nothing$ or $\fx F_2 = \nothing$.
  
  1. It suffices to prove that for any $Y \lra X$ in $X\et$ and for
  any geometric point 
  $x$ of $Y$, one has 
  $(\fx F)_x = \nothing$. Indeed, 
  $ 
  \{ 
  \,
%  \xymatrix{ \cat{Sh}\, Y \ar[r]^{p^*} & \cat{Set} }
  p^* : \cat{Sh}\, Y \lra \cat{Set}
  \text{ where }
  p
  \text{ is a geometric point of }
  Y
  \,
  \}
  $
  is a faithful family of fiber functors on $\cat{Sh}\,Y$, thus 
  \begin{align*}
    \{ 
    \,
    \xymatrix{ \cat{Sh}\,X \ar[r]^{f^*} & \cat{Sh}\, Y \ar[r]^{p^*} & \cat{Set} }
    \text{ where }
    &
    \xymatrix{ Y \ar[r]^f & X }
    \in X\et
    \text{ and }
    \\ &
    p
    \text{ is a geometric point of }
    Y
    \,
    \}    
  \end{align*}
  is a faithful family of fiber functors on $\cat{Sh}\,X$.

  2. If $x$ is a geometric point of $U$, then $U_x = *$ and 
  $U_x = (\fx F_1)_x \amalg (\fx F_2)_x$, thus 
  $(\fx F_1)_x = *$ and $(\fx F_2)_x = \nothing$ (or the converse).

  3. If $x : \spec k \lra U$ is a geometric point of $U$ such that 
  $(\fx F_1)_x = *$  and $(\fx F_2)_x = \nothing$, and if 
  $x' : \spec k' \lra U$ is a geometric point of $U$ above $x$, \ie, 
  $$ \xymatrix{\spec k' \ar[rd]^{x'} \ar[d] \\
    \spec k \ar[r]_x & U, } $$
  then there are maps 
  $U_{x'} \lra U_x$, 
  $(\fx F_1 )_{x'} \lra (\fx F_1)_x$ and 
  $(\fx F_2 )_{x'} \lra (\fx F_2)_x$, thus 
  $(\fx F_2 )_{x'} = \nothing$ and $(\fx F_1)_x=*$.
  As a result, the condition 
  ``$(\fx F_1)_x = *$ and $(\fx F_2)_x = \nothing$'' does not depend
  on the choice of a geometric point $x$ above a given point $x_0 \in
  X$. 

  4. The condition ``$(\fx F_1)_x = *$ and $(\fx F_2)_x = \nothing$''
  is open. Indeed, if 
  $(\fx F_1)_x \neq \nothing$, the, as
  $(\fx F_1)_x = \limind \fx F_1 (V)$, there exists 
  $V \lra U$ �tale such that $\fx F_1(V)\neq \nothing$, and thus 
  $(\fx F_1)_y \neq \nothing$ for any $y$ in the image of 
  $V \lra U$ (which is open). 
  Similarily, the condition 
  ``$(\fx F_2)_x = *$ and $(\fx F_1)_x = \nothing$''
  is open, thus the condition 
  ``$(\fx F_1)_x = *$ and $(\fx F_2)_x = \nothing$''
  is closed. 
  As $U$ is connected, one has 
  $
  \forall x \in U 
  \quad 
  (\fx F_1)_x = * \text{ and } (\fx F_2)_x = \nothing
  $
  (or the converse).

  5. Let $V \lra X$ be an object of 
  $X\et$ and $x : \spec k \lra V$ be a geometric point of $V$. In the
  following diagram, the bold arrow are �tale
  $$ \xymatrix{
    & U_x \ar[rr] \ar[ld] && 
    \smash[b]{U \Times_X V}
    \vrule width 0pt height 0pt depth 4pt
    \ar[ld]\ar[rd]^p \\
    \spec k \ar[rr]_x && V \ar[rd] && U \ar[ld] \\
    &&& X. }
  $$
  One may identify $U_x$ with the set of points $y$ of $U
  \times_X V$ over $x$.
  $$ \xymatrix{ 
    && \smash[b]{U \Times_X V} \ar[ld]\ar[rd] \\
    \spec k \ar@{.>}[rru]^y \ar[r]_x & V \ar[rd] && U \ar[ld] \\ 
    && X } $$
  As the decomposition 
  $U = \fx F_1 \amalg \fx F_2$ is preserved by pull-backs, 
  $U_x = \coprod_{y \in U_x} U_y$
  gives 
  $(\fx F_2)_x = \coprod_{y \in U_x} (\fx F_2) _y$. 
  But if 
  $y \in U_x$, one has a map 
  $(\fx F_2)_y \lra (\fx F_2)_{p(y)} = \nothing$, 
  thus 
  $(\fx F_2)_y = \nothing$.
\end{proof}

\begin{rem}
  This result also holds for the topos of �tale sheaves on
  Artin stacks.
\end{rem}

\begin{rem}
  One may also define the profinite fundamental group of a
  stack $X$ using �tale coverings, \ie, finite �tale stack
  morphisms $Y \lra X$ instead of locally constant finite
  sheaves: see \cite{Z:ThVK}.
\end{rem}
\section{Comparison theorem}

We shall now compare the fundamental group of a stack over
$\CC$ and that of the associated (generalized) orbifold. 

\begin{defi}
  A \d{groupoid in a category} $\cat C$ (with finite projective
  limits) is the datum of 
  two objects $G_1$ and $G_0$, and of morphisms 
  $s, t: G_1 \lra G_0$, 
  $m : G_1 
  \times_{s, G_0, t} 
  G_1
  \lra G_1$, 
  $e: G_0 \lra G_1$ 
  and $i: G_1 \lra G_1$
  such that, for any object $T$,
  $(\hom(T,G_1), \hom(T,G_0), s, t, m, e, i)$ be a groupoid,
  where the maps $s$, $t$, $m$, $e$, $i$ are respectively 
  source, target, composition, identity and inverse. 
  A \d{topological groupoid} is a groupoid in the category of
  topological spaces.

  An \d{�tale groupoid} is a topological groupoid whose
  source morphism (and hence, all of whose structure
  morphisms) is �tale. 
  A \d{proper groupoid} is a topological groupoid such that
  the morphism
  $(s,t): G_1 \lra G_0 \times G_0$ be proper.
  One may also consider �tale or proper groupoids in the category of
  schemes. 

  A \d{generalized orbifold} is an �tale proper groupoid
  (see \cite{MP} for the notion of orbifold).
\end{defi}

\begin{prop}
  Let $X$ be a stack of finite type over $\CC$ 
  and $R \dlra U$ a presentation of $X$.
  The groupoid $R(\CC) \dlra U(\CC)$ is �tale and proper. 
  We call it a (generalized) \d{orbifold associated to}~$X$.
\end{prop}
\begin{proof}
  The groupoid is clearly �tale and properness results from 
  \cite[3.22]{LMB}.
\end{proof}

\begin{defi}
  An \d{equivariant sheaf} on an �tale groupoid 
  $G_1 \dlra G_0$
  is a sheaf $\fx F$ on $G_0$ and a morphism 
  $\alpha : G_1 
  \mathbin{  {}_s{\mathop{\times}\limits_{G_0}} }
  \fx F \lra \fx F$
  such that 
  $$ \raisebox{.5\depth}{\xymatrix{
      G_1 \Times_{G_0} G_1 \Times_{G_0} \fx F \ar[r]^-{1\times\alpha}
      \ar[d]_-{m\times 1} & G_1 \Times_{G_0} \fx F \ar[d]^-\alpha \\
      G_1 \Times_{G_0}\fx F \ar[r]_\alpha & \fx F}}
  $$
  $$
  \raisebox{.5\depth}{\xymatrix{\relax
      \fx F = \smash{G_0 \Times_{G_0} \fx F}\vrule width 0pt depth 1ex
      \ar@{=}[rd]\ar[r]^-{e\times1} &
      G_1 \Times_{G_0} \fx F \ar[d]^\alpha \\ & \fx F}}
  \qquad
  \raisebox{.5\depth}{\xymatrix{ 
      G_1 \Times_{G_0} \fx F \ar[r]^\alpha \ar[d] & \fx F
      \ar[d] \\ G_1 \ar[r]_b & G_0. }} $$ 
  Equivalently, one may require that 
  $(\pr_2, \alpha) : G_1 \times G_0 \fx F \dlra \fx F$ 
  be the source and target maps of a groupoid such that 
  $$ \xymatrix{
    G_1 \times \fx F \ar[d] \dar[r] & \fx F \ar[d] \\ 
    G_1 \dar[r] & G_0 } $$
  be a groupoid morphism. 

  The equivariant sheaves and their morphisms form a topos,
  denoted $\cat{Sh}(G_1 \dlra G_0)$.  

  The \d{fundamental groupoid} of an �tale groupoid 
  $G_1 \dlra G_0$
  is defined to be 
  $$ 
  \pi_1 ( G_1 \dlra G_0 ) := 
  \pi_1 \cat{Sh}(G_1 \dlra G_0).
  $$
\end{defi}

\begin{rem}
  \cite{vistoli} shows that if $R\dlra U$ is a presentation
  of a stack $X$, then $\cat{Sh} \, X \equiv \cat{Sh} (R
  \dlra U)$.
\end{rem}
\begin{rem}
  As there is an equivalence of categories between the
  category of coverings of an orbifold and that of �tale
  coverings of one of its presentations, this definition
  generalizes that of the fundamental group of an orbifold
  \cite[13.2.5]{thurston}.
\end{rem}

\begin{theo}
  Let $X$ be a separated algebraic stack, locally of finite
  type over $\CC$, such that there exists a surjective �tale
  and \emph{finite} morphism $U \lra X$ from a scheme.  We
  let $R \dlra U$ be the groupoid defined by this morphism.
  Then one has
  $$\wh\pi_1 X
  \equiv 
  \wh\pi_1 \bigl( R(\CC)\dlra U(\CC) \bigr).$$
\end{theo}

\begin{proof}
  \cbstart
  We shall identify locally constant finite
  �tale sheaves on the groupoid $R \dlra U$ with its equivariant �tale
  coverings. 
  It suffices to show that the functor 
  \begin{equation}
    \label{eq:functcompgr}
    \definefunction
    {\SLCF\cat{Sh}(R\dlra U)}%
    {\SLCF\cat{Sh}\bigl(R(\CC)\dlra U(\CC)\bigr)}%
    {\XYMATRIX{
        R \Times_U F \ar[d]
        \ar@<+2pt>[r]^-{\pr_2}
        \ar@<-2pt>[r]_-{\alpha}
        & F \ar[d]^f \\
        R
        \ar@<+2pt>[r]^s
        \ar@<-2pt>[r]_t
        & U
        }}%
    {\XYMATRIX{
        R(\CC) \Times_{U(\CC)} F(\CC) \ar[d]
        \ar@<+2pt>[r]
        \ar@<-2pt>[r]
        & F(\CC) \ar[d]^f \\
        R(\CC)
        \ar@<+2pt>[r]
        \ar@<-2pt>[r]
        & U(\CC)
        }}%
  \end{equation}
  is an equivalence of categories. 

  Let $(F, f, \alpha)$ and $(F', f', \alpha')$ be two �tale coverings
  of $R \dlra U$. Let us first show that the map 
  \begin{equation}
    \label{eq:mapproofcompth}
    \raisebox{.5\depth}{\xymatrix{
        \hom\bigl(
        (F,f,\alpha), 
        (F', f', \alpha')
        \bigr)
        \ar[d]
        \\
        \hom\bigl(
        (F(\CC),f(\CC),\alpha(\CC)), 
        (F'(\CC), f'(\CC), \alpha'(\CC))
        \bigr)
        }}
  \end{equation}
  is onto. Indeed, any $\phi_0: F(\CC) \lra F'(\CC)$ with 
  $ f(\CC) = f'(\CC) \circ \phi_0$
  is of the form 
  $\phi_0 = \phi(\CC)$ for some 
  $\phi : F \lra F'$ with 
  $f = f ' \circ \phi$
  (see \cite[XII 5.1]{SGA1}); 
  further, as 
  the diagram 
  $$ \xymatrix{ 
    R(\CC) \Times_{U(\CC)} F(\CC) 
    \ar[r]^-\alpha \ar[d]_{1\times\phi} & 
    F(\CC) \ar[d]^\phi \\
    R(\CC)\Times_{U(\CC)} F'(\CC) 
    \ar[r]_-{\alpha'} & F'(\CC) } $$
  commutes and as that the functor 
  \begin{equation}
    \label{eq:functproofcompesptopassoc}
    \definefunction
    {\cat{Sch}/\CC}{\cat{Top}}{X}{X(\CC)}
  \end{equation}
  is faithful, we see that the diagram 
  $$ \xymatrix{ 
    R \Times_U F 
    \ar[r]^\alpha \ar[d]_{1\times\phi} 
    & F \ar[d]^\phi \\
    R\Times_U F' \ar[r]_{\alpha'} & F' } $$
  commutes.
  The faithfulness of \eqref{eq:functproofcompesptopassoc}
  also implies that the map \eqref{eq:mapproofcompth} is
  one-to-one. Hence, the functor \eqref{eq:functcompgr} is fully
  faithful. 

  The functor \eqref{eq:functcompgr} is essentially surjective. 
  Indeed, if 
  $$ \xymatrix{ 
    R(\CC) \Times_{U(\CC)} F(\CC) \ar[d]_{1\times f_0}
    \ar@<+2pt>[r]^-{\pr_2}
    \ar@<-2pt>[r]_-{\alpha_0}
    & F(\CC) \ar[d]^{f_0} \\
    R(\CC)
    \ar@<+2pt>[r]^s
    \ar@<-2pt>[r]_t
    & U(\CC)
    }
  $$
  is an equivariant �tale cover of $R(\CC)\dlra U(\CC)$, then 
  by \cite[XII 5.1]{SGA1}
  there exists an �tale cover 
  $f : F \lra U$ such that 
  $F(C) \iso F_0$ (we shall thus identify $F(\CC)$ and $F_0$) and 
  $f(\CC) = f_0$. 
  As 
  $U \lra X$ is finite, so is 
  $(R \times_U F)(\CC) \iso R(\CC) \times_{U(\CC)} F_0 \lra U(\CC)$
  and thus 
  by \cite[XII 5.1]{SGA1}
  there exists 
  $\alpha : R \times_U F \lra F$ such that $\alpha(\CC) = \alpha_0$. 
  One may then show as above that 
  $(F, f, \alpha)$ is an equivariant covering, 
  again from the fact that $(F_0, f_0, \alpha_0)$ is one and the fact that 
  the functor \eqref{eq:functproofcompesptopassoc} is faithful. 
  \cbend
\end{proof}

\section{Simplicial schemes}

We shall now show that our notion of fundamental groupoid of
a stack coincides with that of \cite{oda}.

\begin{defi}
  Let $\Delta$ be the \d{category of simplices}, \ie, the
  category whose objects are the sets $[n] = \{0,\dots,n\}$
  and whose morphisms are nondecreasing maps, and 
  $\Delta_n$ the full subcategory of $\Delta$ containing 
  $[0],\dots,[n]$.

  Let $\T$ be a locally connected topos and denote by $\pi$
  its ``connected components'' functor. 

  A \d{simplicial object} in $\T$ is a functor 
  $\Delta\op \lra \T$; we shall denote 
  $\Delta\op\T$ the category of simplicial objects of $\T$. 
  The restriction functor 
  $\Delta\op\T \lra \Delta_n\op\T$ has left and right
  adjoints, called $n$th \d{skeleton} and \d{coskeleton}
  denoted $\sk_n$ and $\cosk_n$.

  A \d{hypercovering} is a simplicial object $X_\bullet$
  such that the morphisms 
  \begin{align*}
    X_0       &\lra e \\
    X_{n+1}   &\lra \bigl(\cosk_{n}(X_\bullet)\bigr)_{n+1}
  \end{align*}
  be epic. 

  If $I$ is a set and $X$ an object of $\T$, set 
  $I \otimes X := \coprod_{i\in I} X$.
  If $I_\bullet$ is a simplicial set ans $X_\bullet$ a
  simplicial object, set
  $I_\bullet \otimes X_\bullet := (n \lmt I_n \otimes X_n).$

  A \d{strict homotopy} between two morphisms 
  $f,\,g : X_\bullet \lra Y_\bullet$ is a morphism 
  $[1] \otimes X_\bullet \lra Y_\bullet$ whose restrictions
  to 
  $[0]\otimes X_\bullet = X_\bullet$ are $f$ and $g$. 
  We call \d{homotopy} the equivalence relation 
  generated by strict homotopy. 
  We denote  $\cat{HR}\T$ the category of hypercoverings of
  $\T$ and their morphisms up to homotopy. One may show that
  $(\cat{HR}\T)\op$ is filtering \cite{AM}.

  Finally, let $\H$ denote the category of CW-complexes and
  their morphisms modulo homotopy (which may be seen as a
  category whose objects are simplicial sets) and 
  $\pro\H$ the category of its proobjects. 
\end{defi}

\begin{defi}
  The \d{homotopy type} of a locally connected topos $\T$ is
  $$ \{\T\} := \proobj_{Y_\bullet \in \cat{HR}\T}
  \pi Y_\bullet 
  \in \pro\H.
  $$
  Its \d{fundamental groupoid} is 
  $$ \pi_1 \{\T\} := \proobj_{Y_\bullet \in \cat{HR}\T}
  \pi_1 \pi Y_\bullet
  $$
  (with the same convention as in remark
  \ref{rem:omissionofbasepoints}
  for the omission of base
  points). 
\end{defi}

\begin{prop}
  If $\T$ is a locally connected topos, then 
  $$ \cat B \pi_1 \{ \T \} \equiv \SLC \T. $$
\end{prop}
\begin{proof}
  One has
  \begin{align*}
    \cat B \pi_1 \{ \T \} 
    &= \cat B \pi_1 \proobj_{U_\bullet \in \cat{HR}\,\T} 
    \pi U_\bullet
    &&\text{by the definition of $\{\T\}$}
    \\
    &= \cat B \proobj_{U_\bullet \in \cat{HR}\,\T} 
    \pi_1 \pi U_\bullet 
    &&\vtop{\hbox{by the definition of the fundamental}%
      \vskip -4pt
      \hbox{group of a pro-simplicial set}}
    \\
    &= \dlimpro_{U_\bullet \in \cat{HR}\,\T} 
    \cat B \pi_1 \pi U_\bullet 
    &&\vtop{\hbox{by the definition of the 
        classifying}%
      \vskip -4pt
      \hbox{space of a progroup}}
    \\
    &\equiv \dlimpro_{U_\bullet \in \cat{HR}\,\T} \LC(\T, U_0) 
    &&\text{from \cite[10.6]{AM}} \\
    &\equiv \dlimpro_{R \in J(*)} \LC(\T, R). 
  \end{align*}
  To prove the last equivalence, it suffices to show that
  any sieve $R$ is finer than a sieve generated by a single
  object $U_0$. One may be tempted to set $U_0 =
  \coprod_{(U\to*)\in R} U$, but this coproduct need not be
  small. Instead, as there exists a \emph{set} $\cat K
  \subset \ob\T$, generating $\T$, one may set
  $$ U_0 = \coprod_{\substack{
      C \in \cat K \text{ such that} \\
      C\to* \text{ factors} \\
      \text{through a } (U\to*)\in R
      }}
  C.
  $$
\end{proof}

\section{Short exact sequences}

\begin{theo}
  Let $\T$ be a connected locally connected topos, $x$ a
  point of $\SLC\T$ and $\f F : \SLC\T \lra \cat{Set}$ the
  corresponding fiber functor.  Let~$Y$ be a Galois object
  of $\SLC\T$.
  \begin{theoenum}
  \item The category $\SLC\T /Y$ is a Galois topos and a
    point $a \in \f F(Y)$ defines a fiber functor $\SLC\T
    \lra \cat{Set}$.
  \item The functor 
    $$\definefunction{\SLC\T }{(\SLC\T )/Y}%
    X{(X\times Y\lra Y)} 
    $$
    induces a topos morphism
    $(\SLC\T )/Y \lra \SLC\T $.
  \item There is a short exact sequence
    $$ 1 \lra \pi_1(Y, a) \lra \pi_1(\T, x) 
    \lra \aut Y \lra 1. $$
  \end{theoenum}  
\end{theo}
\begin{proof}
  One may assume that $\T$ is the classifying topos of a
  progroup and that $\f F$ is the forgetful functor
  $G\text{-}\cat{Set} \lra \cat{Set}$.  The following lemma
  identifies $G\text{-}\cat{Set}/Y$ as the classifying topos
  of a subprogroupoid $H$ of~$G$, thereby proving $(i)$ and
  $(ii)$.
  \begin{lemma}
    Let $G$ be a progroup, $Y$ a connected $G$-set, 
    $a \in Y$ and $H$ the stibilizer of $a$. 
    Then, there is an equivalence of categories 
    $$ G\text{-}\cat{Set} / Y \equiv H\text{-}\cat{Set}. $$
  \end{lemma}
  \begin{proof}
    Consider the functor 
    $$\definefunction
    {G\text{-}\cat{Set}/Y}
    {H\text{-}\cat{Set}}
    {(\xymatrix{Z \ar[r]^\pi & Y})}
    {\pi^{-1}(\{a\}).}
    $$

    Let $Z$ and $Z'$ be objects of 
    $G\text{-}\cat{Set}/Y$. The map
    $$ \hom_{Y,G}(Z,Z') \lra \hom_H\Bigl(
    \pi^{-1}\bigl(\{a\}\bigr), 
    \pi'{}^{-1}\bigl(\{a\}\bigr)
    \Bigl) $$
    is injective, because a morphism
    $f : Z \lra Z'$ is entirely determined by its fiber
    $f_a$. Indeed, as $Y$ is connected, for any $z \in Z$,
    there exists $g \in G$ such that $\pi(z) = g \cdot a$, 
    thus $f(z) = f(gg^{-1}\cdot z)  = 
    g \cdot f(g^{-1}\cdot z))$, for 
    $g^{-1} \cdot z \in \pi^{-1}(\{a\})$.
    
    To show that the map is onto, let 
    $f_a : \pi^{-1}(\{a\}) \lra \pi'{}^{-1}(\{a\})$
    be an $H$-morphisms and set 
    $f(z) = g \cdot f_a( \gamma^{-1} \cdot z )$ 
    if 
    $\pi(z) = g \cdot a$. 
    This is well-defined, for if 
    $\pi(z) = g \cdot a = g' \cdot a$, 
    then 
    $g^{-1} g' a =  a$
    thus  $h = g^{-1} g' \in H$, 
    but as $f_a$ is $H$-equivariant, 
    $h^{-1} f_a (g^{-1} \cdot z ) =
    f_a( h^{-1} g^{-1} z )$, 
    thus 
    $g f_a( g^{-1} z ) = g' f_a( g'{}^{-1} z )$.
    Further, the map $f$ is equivariant: 
    $g' \cdot f(z) =  
    g' g \cdot f_a( (g' g)^{-1} \cdot g' z ) =
    f(g' \cdot z)$.

    Our functor is thus fully faithful. As for essential
    surjectivity, let $U$ be an $H$-set and define a
    $G$-action on 
    $Y \times U$ as follows: 
    for all $y \in Y$, choose 
    $\gamma_y \in G$ such that 
    $y = \gamma _y \cdot a$ (one may set $\gamma_a=1$) and
    set 
    $g \cdot(y,u) = (g \cdot y, 
    \gamma_{g y} ^{-1} g \gamma_y u)$. 
    This is clearly a $G$-action and the action
    of $H$ on the fiber is the desired one. 
  \end{proof}

  The following lemma shows that 
  $\pi_1(Y, a)$ is a normal subgroup of 
  $\pi_1(\T, x)$ and identifies the quotient, thereby
  establishing 
  $(iii)$.
\end{proof}

\begin{lemma}
  Let $G$ be a progroup, 
  $Y$ a Galois $G$-set, 
  $a \in Y$ and $H$ the stabilizer of $a$. 
  Then, the subprogroup $H$ is normal and there is an
  isomorphism
  $$ G / H \iso \aut Y.$$
\end{lemma}
\begin{proof}
  Let us first assume that $G$ is a group. 
  As $Y$ is Galois, the map 
  $$ \Psi : \definefunction 
  {Y \times \aut Y}
  {Y \times Y}
  {(y,\phi)}
  {(y,\phi(y))}
  $$
  is bijective so we may define 
  $$ \Phi : \definefunction
  {G}
  {\aut Y}
  {g}
  {\phi \text{ such that } \Psi(a,\phi) = (a, g\cdot a).}
  $$
  This map is onto: as $Y$ is connected, an automorphism 
  $\phi \in \aut Y$ is entirely determined by 
  $\phi(a)$, but there exists a 
  $g \in G$ such that 
  $\phi(a) = g \cdot a$, thus 
  $\phi = \Phi(g)$. This map is clearly a group morphism
  and its kernel is readily seen to be $H$. 
  
  In the case $G = \proobj G_i$ is a progroup, one has
  short exact sequences 
  $$ 1 \lra H_i \lra G_i \lra \aut Y \lra 1$$
  thus, by \cite[I.8.9.2]{SGA4}, 
  $$ 1 \lra H \lra G \lra \aut Y \lra 1.$$
\end{proof}

\begin{coro}
  Let $X$ be a connected scheme, $G$ a finite group actiong on $X$ and
  $x$ a geomertic point of $X$. Then, there is a short exact sequence 
  $$ 1 \lra \wh\pi_1 (X, x) \lra \wh\pi_1( [X / G], x) \lra G \lra 1. $$
\end{coro}
\begin{proof}
  Let us apply the theorem to
  $\T = \SLCF \cat{Sh}(G \times X \dlra X)$, 
  the topos
  of disjoint sums of $G$-equivariant �tale coverings of $X$ and the
  object $Y = G \times X$, where the action of $G$ and the morphism 
  $Y \lra X$ are 
  $$ \definefunction
  {G \times Y}
  {Y}
  {(h,g,x)}
  {(gh^{-1}, hx)}
  \qquad\qquad
  \pi : \definefunction
  {Y}{X}
  {(g,x)}
  {g \cdot x}
  $$

  To prove that 
  $G = \aut Y$, remark that the one-to-one morphism 
  $$ \Phi : \definefunction
  {G}
  {\aut Y}
  {h}
  { \Phi(h) : \definefunction{Y}{Y}
    {(g,x)}{(h^{-1}g, h \cdot x)}
    }
  $$
  is onto: indeed, if 
  $\phi \in \aut Y$, $x \in X$, 
  as 
  $\phi$ is an automorphism over~$X$, one may write 
  $\phi(1,x) = (h^{-1}, h \cdot x)$ with 
  $h \in G$: thus 
  $\phi$ and $\Phi(h)$ coincide at a point $(1,x)$ of $Y$, hence on
  the connected component 
  $\{1\}\times X$, hence on its orbit $Y$.

  To show that $Y$ is Galois, it suffices to remark that 
  \begin{align*}
    Y \Times_X (\aut X) &=
    \{ (g,h,x,y) \in G\times X\times G\times X \ :\ 
    g\cdot x = y \}
    \\
    Y \Times_X Y &=
    \{ (g,h,x,y) \in G\times X\times G\times X \ :\ 
    g\cdot x = h\cdot y \}.
  \end{align*}

  Finally, as 
  $\T/Y \equiv \SLCF\cat{Sh}\,X$, the short exact sequence of the
  theorem 
  $$ 1 \lra \wh\pi_1 \T/Y \lra 
  \wh\pi_1\T \lra
  G \lra 1 $$
  reads 
  $$ 1 \lra \wh\pi_1 X \lra 
  \wh\pi_1 [X/G] \lra
  G \lra 1. 
%  \hfill \qedhere
  $$
\end{proof}
\begin{rem}
  The same result holds for stacks quotiented by a finite group.
\end{rem}
\begin{coro}
  Let $X$ be a connected quasi-compact algebraic stack on a field $k$,
  let $\bar k$ be a separable closure of $k$ and let 
  $x$ be a point of $\bar X = X \tens_k \bar k$. 
  Then, one has a short exact sequence of profinite
  fundamental groups 
  $$ 1 \lra \wh\pi_1(\bar X, x) \lra 
  \wh\pi_1 (X,x) \lra 
  \gal(\bar k : k ) \lra 1. $$
\end{coro}
\begin{proof}
  By lemma \ref{lemma:dsfsdkfgkjjd1},
  for each Galois extension 
  $k':k$, 
  $X \tens_k k'$ is a Galois object in 
  $\SLCF\cat{Sh}\,X$
  and the short exact sequence of the theorem reads 
  $$ 
  1 \lra \wh \pi_1(X \tens_k k') \lra \wh\pi_1(X) 
  \lra \gal(k':k) \lra 1,
  $$
  thus, from \cite[I.8.9.2]{SGA4}, 
  $$ 
  1 \lra \limpro_{k':k} \wh \pi_1(X \tens_k k') 
  \lra \limpro_{k':k} \wh\pi_1(X) 
  \lra \limpro_{k':k} \gal(k':k) \lra 1,
  $$
  but lemma \ref{lemma:dsfsdkfgkjjd2} identifies the first term,
  yielding 
  $$ 1 \lra \wh\pi_1(X \tens_k k') 
  \lra \wh \pi_1(X) \lra \gal(\bar k : k) \lra 1$$
  as desired.
\end{proof}
\begin{lemma}\label{lemma:dsfsdkfgkjjd1}%
  Let $X$ be a connected algebraic stack over a field $k$. Then 
  $\smash{X \Otimes_k k' \lra X}$ is Galois. 
  and 
  $ \gal (k' : k ) \iso \aut_X (X \otimes_k k') $
\end{lemma}
\begin{proof}
  Let $s,t : R \dlra U$ be a presentation of $X$, 
  $\spec A_i$ affine open sets covering $U$, 
  $\spec B_{ijk}$ affine open sets covering 
  $s^{-1} \spec A_i \inter t^{-1} \spec A_j$.
  An automorphism of 
  $X \tens _k k'$ over $X$ induces 
  automorphisms of 
  $\spec k' \tens_k B_{ijk}$ over 
  $\spec B_{ijk}$
  and of 
  $\spec k' \tens_k A_i$ over $\spec A_i$, 
  compatible with $s$ and $t$; 
  each of these uniquely determines an automorphism of $k'$ over~$k$,
  but as the groupoid 
  $R \dlra U$ is connected, it is always the same automorphism of
  $k':k$. 

  To show that $X'$ is Galois, just compute:
  \begin{align*}
    X' \Times_X X' &=
    \left(
      X \Times_{\spec k} \spec k'
    \right)
    \Times_X
    \left(
      X \Times_{\spec k} \spec k'
    \right)
    \\
    &=
    X
    \Times_{\spec k}
    \left(
      \spec k' \Times_{\spec k} \spec k'
    \right)
    \\
    &=
    X
    \Times_{\spec k}
    \left(
      \spec k' \times \gal(k':k)
    \right)
    &&\text{for $k':k$ is Galois}
    \\
    &=
    X
    \Times_{\spec k}
    \aut_X (X \tens_k k').    
  \end{align*}
\end{proof}
\begin{lemma}\label{lemma:dsfsdkfgkjjd2}%
  Let $X$ be a connected
  algebraic stack over a field $k$, with a quasi-compact presentation,
  and let $\bar k$ be a separable closure of $k$. Then, there is an
  isomorphism of profinite groups 
  $$ \wh\pi_1( X \otimes_k \bar k) \iso \proobj_{\substack{
      k':k \text{ finite}\\\text{Galois extension}}}
  \wh\pi_1( X \otimes_k k' ). $$
\end{lemma}
\begin{proof}
  It suffices to prove that any �tale covering of 
  $\bar X = X \tens_k \bar k$ comes from a covering of 
  $X' = X \tens_k k'$ for some \emph{finite} Galois extension 
  $k':k$. Let $s,t : R \dlra U$ be a presentation of~$X$, with $R$ and
  $U$ quasi-compact, 
  let 
  $\alpha: R\times_{\bar U} F \lra F$
  be an equivariant �tale cover of 
  $\bar R \dlra \bar U$,
  let 
  $\spec A_i$ be a finite number of open affine sets covering~$U$, 
  let $\spec B_{ij}$ be a finite number of open affine sets covering 
  $\alpha^{-1} \spec A_i$. 
  The morphisms 
  $F \lra \bar U$ and $\alpha$ may be written, locally, 
  \begin{align*}
    \spec \dfrac{A_i \tens \bar k [ x_1, \dots, x_{n_i} ]}%
    {(f_{i,1}, \dots, f_{i,n_i})} 
    &\lra 
    \spec A_i\tens \bar k,
    \\
    \spec B_{ij} \Otimes_{A_i} 
    \dfrac{A_i \tens \bar k [ x_1, \dots, x_{n_i} ]}%
    {(f_{i,1},\dots,f_{i,n_i})} 
    &\lra
    \spec     
    \dfrac{A_i \tens \bar k [ x_1, \dots, x_{n_i} ]}%
    {(f_{i,1},\dots,f_{i,n_i})}
  \end{align*}
  and only involve a finite number of elements of 
  $\bar k$: thus, $F$ is defined over the finite Galois extension
  generated by these elements. 
\end{proof}
\bibliographystyle{amsplain}
\nocite{*}
\bibliography{1}
\end{document}

1.bbl
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\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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