Fundamental group of a point

etale-cohomology.tex

@@ -7645,91 +7645,6 @@ \section{Vanishing of finite higher direct images}
 \end{proof}
 
 
-\section{Schemes \'etale over a point}
-\label{section-schemes-etale-point}
-
-\noindent
-In this section we describe schemes \'etale over the spectrum of
-a field. Before we state the result we introduce
-the category of $G$-sets for a topological group $G$.
-
-\begin{definition}
-\label{definition-G-set-continuous}
-Let $G$ be a topological group.
-A {\it $G$-set}, sometime called a {\it discrete $G$-set},
-is a set $X$ endowed with a left action $a : G \times X \to X$
-such that $a$ is continuous when $X$ is given the discrete topology and
-$G \times X$ the product topology.
-A {\it morphism of $G$-sets} $f : X \to Y$ is simply any $G$-equivariant
-map from $X$ to $Y$.
-The category of $G$-sets is denoted {\it $G\textit{-Sets}$}.
-\end{definition}
-
-\noindent
-The condition that $a : G \times X \to X$ is continuous signifies
-simply that the stabilizer of any  $x \in X$ is open in $G$.
-If $G$ is an abstract group $G$ (i.e., a group but not a topological group)
-then this agrees with our preceding definition (see for example
-Sites, Example \ref{sites-example-site-on-group})
-provided we endow $G$ with the discrete topology.
-
-\medskip\noindent
-Recall that if $K \subset L$ is an infinite Galois extension the
-Galois group $G = \text{Gal}(L/K)$ comes endowed with a canonical
-topology. Namely the open subgroups are the subgroups of the form
-$\text{Gal}(L/K') \subset G$ where $K'/K$ is a finite subextension of $L/K$.
-The index of an open subgroup is always finite.
-We say that $G$ is a profinite (topological) group.
-
-\begin{lemma}
-\label{lemma-sheaves-point}
-Let $K$ be a field.
-Let $K^{sep}$ a separable closure of $K$.
-Consider the profinite group
-$$
-G = \text{Aut}_{\Spec(K)}(\Spec(K^{sep}))^{opp} =
-\text{Gal}(K^{sep}/K)
-$$
-The functor
-$$
-\begin{matrix}
-\text{schemes \'etale over }K &
-\longrightarrow &
-G\textit{-Sets} \\
-X/K & \longmapsto &
-\Mor_{\Spec(K)}(\Spec(K^{sep}), X)
-\end{matrix}
-$$
-is an equivalence of categories.
-\end{lemma}
-
-\begin{proof}
-A scheme $X$ over $K$ is \'etale over $K$ if and only if
-$X \cong \coprod_{i\in I} \Spec(K_i)$ with
-each $K_i$ a finite separable extension of $K$.
-The functor of the lemma associates to $X$ the $G$-set
-$$
-\coprod\nolimits_i \Hom_K(K_i, K^{sep})
-$$
-with its natural left $G$-action. Each element has an open stabilizer
-by definition of the topology on $G$. Conversely, any $G$-set $S$
-is a disjoint union of its orbits. Say $S = \coprod S_i$. Pick $s_i \in S_i$
-and denote $G_i \subset G$ its open stabilizer. By Galois theory the fields
-$(K^{sep})^{G_i}$ are finite separable field extensions of $K$, and
-hence the scheme
-$$
-\coprod\nolimits_i \Spec((K^{sep})^{G_i})
-$$
-is \'etale over $K$. This gives an inverse to the functor of the lemma.
-Some details omitted.
-\end{proof}
-
-\begin{remark}
-\label{remark-covering-surjective}
-Under the correspondence of the lemma, the coverings in the small \'etale site
-$\Spec(K)_\etale$ of $K$ correspond to surjective families of
-maps in $G\textit{-Sets}$.
-\end{remark}
 
 
 
@@ -7838,7 +7753,7 @@ \section{Galois action on stalks}
 \end{enumerate}
 Altogether we see that $\mathcal{F}_{\overline{s}}$ becomes a
 $\text{Gal}_{\kappa(s)}$-set (see
-Definition \ref{definition-G-set-continuous}).
+\'Etale Morphisms, Definition \ref{etale-definition-G-set-continuous}).
 Hence we may think of the stalk functor as a functor
 $$
 \Sh(S_\etale) \longrightarrow
@@ -7863,7 +7778,7 @@ \section{Galois action on stalks}
 
 \begin{proof}
 Let us construct the inverse to this functor. In
-Lemma \ref{lemma-sheaves-point}
+\'Etale Morphisms of Schemes, Lemma \ref{etale-lemma-sheaves-point}
 we have seen that given a $G$-set $M$ there exists an \'etale morphism
 $X \to \Spec(K)$
 such that $\Mor_K(\Spec(K^{sep}), X)$ is
@@ -7878,9 +7793,9 @@ \section{Galois action on stalks}
 
 \begin{remark}
 \label{remark-every-sheaf-representable}
-Another way to state the conclusions of
-Lemmas \ref{lemma-sheaves-point} and
-Theorem \ref{theorem-equivalence-sheaves-point}
+Another way to state the conclusion of
+Theorem \ref{theorem-equivalence-sheaves-point} and
+\'Etale Morphisms of Schemes, Lemma \ref{etale-lemma-sheaves-point}
 is to say that every sheaf on $\Spec(K)_\etale$ is representable
 by a scheme $X$ \'etale over $\Spec(K)$.
 This does not mean that every sheaf is representable in the sense of

etale.tex

@@ -2184,6 +2184,94 @@ \section{Relative morphisms}
 
 
 
+\section{Schemes \'etale over a point}
+\label{section-schemes-etale-point}
+
+\noindent
+In this section we describe schemes \'etale over the spectrum of a field.
+Before we state the result we introduce the category of $G$-sets for a
+topological group $G$.
+
+\begin{definition}
+\label{definition-G-set-continuous}
+Let $G$ be a topological group.
+A {\it $G$-set}, sometime called a {\it discrete $G$-set},
+is a set $X$ endowed with a left action $a : G \times X \to X$
+such that $a$ is continuous when $X$ is given the discrete topology and
+$G \times X$ the product topology.
+A {\it morphism of $G$-sets} $f : X \to Y$ is simply any $G$-equivariant
+map from $X$ to $Y$.
+The category of $G$-sets is denoted {\it $G\textit{-Sets}$}.
+\end{definition}
+
+\noindent
+The condition that $a : G \times X \to X$ is continuous signifies
+simply that the stabilizer of any  $x \in X$ is open in $G$.
+If $G$ is an abstract group $G$ (i.e., a group but not a topological group)
+then this agrees with our preceding definition (see for example
+Sites, Example \ref{sites-example-site-on-group})
+provided we endow $G$ with the discrete topology.
+
+\medskip\noindent
+Recall that if $K \subset L$ is an infinite Galois extension then the
+Galois group $G = \text{Gal}(L/K)$ comes endowed with a canonical
+topology, see Fields, Section \ref{fields-section-infinite-galois}.
+
+\begin{lemma}
+\label{lemma-sheaves-point}
+Let $K$ be a field. Let $K^{sep}$ a separable closure of $K$.
+Consider the profinite group $G = \text{Gal}(K^{sep}/K)$.
+The functor
+$$
+\begin{matrix}
+\text{schemes \'etale over }K &
+\longrightarrow &
+G\textit{-Sets} \\
+X/K & \longmapsto &
+\Mor_{\Spec(K)}(\Spec(K^{sep}), X)
+\end{matrix}
+$$
+is an equivalence of categories.
+\end{lemma}
+
+\begin{proof}
+A scheme $X$ over $K$ is \'etale over $K$ if and only if
+$X \cong \coprod_{i\in I} \Spec(K_i)$ with
+each $K_i$ a finite separable extension of $K$
+(Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}).
+The functor of the lemma associates to $X$ the $G$-set
+$$
+\coprod\nolimits_i \Hom_K(K_i, K^{sep})
+$$
+with its natural left $G$-action. Each element has an open stabilizer
+by definition of the topology on $G$. Conversely, any $G$-set $S$
+is a disjoint union of its orbits. Say $S = \coprod S_i$. Pick $s_i \in S_i$
+and denote $G_i \subset G$ its open stabilizer. By Galois theory
+(Fields, Theorem \ref{fields-theorem-inifinite-galois-theory})
+the fields $(K^{sep})^{G_i}$ are finite separable field extensions of $K$, and
+hence the scheme
+$$
+\coprod\nolimits_i \Spec((K^{sep})^{G_i})
+$$
+is \'etale over $K$. This gives an inverse to the functor of the lemma.
+Some details omitted.
+\end{proof}
+
+\begin{remark}
+\label{remark-covering-surjective}
+Under the correspondence of Lemma \ref{lemma-sheaves-point},
+the coverings in the small \'etale site
+$\Spec(K)_\etale$ of $K$ correspond to surjective families of
+maps in $G\textit{-Sets}$.
+\end{remark}
+
+
+
+
+
+
+
+
 
 \section{Galois categories}
 \label{section-galois}
@@ -2215,20 +2303,16 @@ \section{Galois categories}
 
 \begin{example}
 \label{example-galois-category-G-sets}
-Let $G$ be a topological group. A $G$-set, sometimes called a discrete
-$G$-set is a set $X$ endowed with a left action $a : G \times X \to X$
-such that $a$ is continuous when $X$ is given the discrete topology
-and $G \times X$ the product topology. A morphism of $G$-sets $f : X \to Y$
-is a $G$-equivariant map from $X$ to $Y$. The category of $G$-sets
-is denoted $G\textit{-Sets}$.
-An important example will be the forgetful functor
+Let $G$ be a topological group. An important example will be the
+forgetful functor
 \begin{equation}
 \label{equation-forgetful}
 \textit{Finite-}G\textit{-Sets} \longrightarrow \textit{Sets}
 \end{equation}
-where $\textit{Finite-}G\textit{-Sets}$
-is the full subcategory of $G\textit{-Sets}$
-whose objects are the finite $G$-sets.
+where $\textit{Finite-}G\textit{-Sets}$ is the full subcategory of
+$G\textit{-Sets}$ whose objects are the finite $G$-sets.
+The category $G\textit{-Sets}$ of $G$-sets is defined in
+Definition \ref{definition-G-set-continuous}.
 \end{example}
 
 \noindent
@@ -3116,6 +3200,64 @@ \section{Finite \'etale morphisms}
 transformation of functors to get (3).
 \end{proof}
 
+\begin{lemma}
+\label{lemma-fundamental-group-Galois-group}
+Let $k$ be a field and let $\overline{k}$ be an algebraic closure.
+Set $X = \Spec(k)$ and denote $\overline{x} : \Spec(\overline{k}) \to X$
+be the geometric point corresponding to our chose algebraic closure.
+Let $k \subset k^{sep} \subset \overline{k}$ be the separable
+algebraic closure. There is a canonical isomorphism
+$$
+\text{Gal}(k^{sep}/k) \longrightarrow \pi_1(X, \overline{x})
+$$
+of profinite topological groups.
+\end{lemma}
+
+\begin{proof}
+We first carefully construct the map. Observe that
+$\text{Gal}(k^{sep}/k) = \text{Aut}(\overline{k}/k)$
+as $\overline{k}$ is the perfection of $k^{sep}$.
+Then recall that $\pi_1(X, \overline{x}) = \text{Aut}(F_{\overline{x}})$
+where $F_{\overline{x}}$ is the functor
+$$
+Y \longmapsto F_{\overline{x}}(Y) = \Mor_X(\Spec(\overline{k}), Y)
+$$
+Consider the map
+$$
+\text{Aut}(\overline{k}/k) \times F_{\overline{x}}(Y)
+\to F_{\overline{x}}(Y),\quad
+(\sigma, \overline{y}) \mapsto
+\sigma \cdot \overline{y} = \overline{y} \circ \Spec(\sigma)
+$$
+This is an action because
+$$
+\sigma\tau \cdot \overline{y} =
+\overline{y} \circ \Spec(\sigma\tau) =
+\overline{y} \circ \Spec(\tau) \circ \Spec(\sigma) =
+\sigma \cdot (\tau \cdot \overline{y})
+$$
+The action is functorial in $Y \in \textit{F\'Et}_X$ and we
+obtain the canonical map.
+
+\medskip\noindent
+Using our map above for every object $Y$ in $\textit{F\'Et}_X$
+the finite set $F_{\overline{x}}(Y)$ gets a canonical
+$\text{Gal}(k^{sep}/k)$-action. To finish the proof it suffices
+to show that each $F_{\overline{x}}(Y)$ is an object of
+$\textit{Finite-}\text{Gal}(k^{sep}/k)\textit{-Sets}$
+and that in this way we obtain an equivalence of categories
+$\textit{F\'Et}_X \to \textit{Finite-}\text{Gal}(k^{sep}/k)\textit{-Sets}$.
+This is sufficient by the recognition results in
+Proposition \ref{proposition-galois} and
+Lemma \ref{lemma-single-out-profinite}.
+To see this one shows that the construction given here
+is the same as the construction in the equivalence
+Lemma \ref{lemma-sheaves-point}
+and that the equivalence of that lemma induces an equivalence between
+the category of finite \'etale schemes over $\Spec(K)$
+and finite $G$-sets. We omit the details.
+\end{proof}
+