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\title{Galois correspondences}
\author{David Pierce}
\date{2004.12.16}

\begin{document}
  \maketitle

This note gives a unified treatment to several mathematical
observations. 

For notation, let $2$ be considered as the universe $\{0,1\}$ of an
abelian group.  For each $e$ in $2$, let $A_e$ be
a set, and let $*$ be a relation from $A_0$ to $A_1$.  This means that~$*$
is a subset of $A_0\times A_1$; if $*$ contains $(x_0,x_1)$, then we write
\begin{equation*}
  x_0* x_1.
\end{equation*}
If $x_e\in A_e$, let
\begin{equation*}
  [x_e]=\{x_{e+1}\in A_{e+1}:x_0* x_1\}\in\pow{A_{e+1}}.
\end{equation*}
If $X\in\pow{A_e}$, let
\begin{equation*}
  X'=\bigcap\{[x]:x\in X\}\in\pow{A_{e+1}}.
\end{equation*}
Let $A_{e+1}^*$ be the image of $\pow{A_e}$ under the map $X\mapsto
X'$. 

\begin{theorem}
Let $e\in 2$.
  \begin{mylist}
    \item
If $X,Y\in\pow{A_e}$ and $X\included Y$, then $Y'\included X'$.
\item
If $X\in\pow{A_e}$, then $X\included X''$.
\item
The map $X\mapsto X'$ is a bijection from $A_e^*$ to $A_{e+1}^*$ with
inverse $X\mapsto X'$.
  \end{mylist}
\end{theorem}

\begin{proof}
  Exercise.  (For the last point, see \cite[ch.~V,
  Lemma~2.6]{MR82a:00006}.) 
\end{proof}

Regardless of how the maps $X\mapsto X'$ are originally defined, if
they meet the conditions established in the theorem, they constitute a
\defn{Galois correspondence} between $A_0^*$ and $A_1^*$.  (This
definition is in \cite[p.~35]{MR1800596}.)  There are several
examples, as you should verify: 

\subsection*{Field-theory}

The usual Galois correspondence in field-theory is the case when $A_0$
is a field $L$ that is a finite Galois extension of a field $K$, and
$A_1$ is $\Aut{L/K}$, and
\begin{equation*}
  x*\sigma\Iff x^{\sigma}=x.
\end{equation*}
Then $A_0^*$ comprises the subfields $F$ of $L$ that include $K$, and
$A_1^*$ comprises the subgroups $H$ of $\Aut{L/K}$, and
$F'=\Aut{L/F}$, and $H'=\Fix H$.

\subsection*{The Zariski topology}

Suppose $A_0$ is a ring $R$ (commutative with $1$), and $A_1$ is
$\Spec R$, that is, the set of prime ideals of $R$.  Let $*$ be $\in$.
Then 
\begin{equation*}
  [x]\cup[y]=[xy]
\end{equation*}
if $x,y\in R$.  Hence the sets $[x]$ are the basic closed sets for a
topology, the \defn{Zariski topology} on $\Spec R$.  (See for example
\cite[pp.~54--55]{MR97a:13001} or \cite[\S~II.2]{MR57:3116}.)  The
topology is compact, although possibly not Hausdorff.  In this
topology, if
$X\included\Spec R$, then $X''$ is the closure of $X$.  If $X\included
R$, then $X''$ is the radical (in the sense of \cite[ch.~VIII,
  Definition 2.5]{MR82a:00006}) of the (possibly improper) ideal
$(X)$.  In general, $A_0^*$
comprises the radical ideals of $R$, and $A_1^*$ comprises the closed
subsets of $\Spec R$.

\subsection*{The Stone space}

Now suppose in particular that $A_0$ is a Boolean ring or algebra $B$,
and $A_1$ is its \defn{Stone space} $S(B)$, the set of ultrafilters of
$B$.  The ultrafilters are dual to the prime ideals, all of which are
maximal.  Let $*$ be $\in$ again.  Then
\begin{equation*}
  [x]\cap[y]=[xy]=[x\land y]
\end{equation*}
when $x,y\in B$, and also
\begin{equation*}
  [x]\comp=[x+1]=[\lnot x],
\end{equation*}
so that
\begin{equation*}
  [x]\cup[y]=[x+y+xy]=[x\lor y].
\end{equation*}
Hence the sets $[x]$ are basic open and closed sets for a topology on
$S(B)$.  This topology is compact as before, but also Hausdorff.  The
elements of $A_1^*$ are still just the closed subsets of
$S(B)$; the elements of $A_0^*$ are just the filters of $B$.  If
$X\included B$, then $X''$ is the filter generated by $X$; if
$X\included S(B)$, then $X''$ is its closure.

\subsection*{Model-theory}

Suppose $\lang$ is a signature for first-order logic. Let $A_0$ be the
\emph{class} $\Mod{\lang}$ of $\lang$-structures, let $A_1$ be $\Sn$,
and let $*$ be $\models$.  Then $A_1^*$ is the set of theories of
$\lang$, and $A_0^*$ is the set of \defn{elementary classes} of
$\lang$-structures.  (See \cite[\S~3.4]{MR1800596}.)

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