% R. Wisbauer
% Contribution to the Nanjing conference 2004
 
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\begin{document}

\title{\bf From Galois field extensions to \\ Galois Comodules}
  
\author{Robert Wisbauer} 
 \address{ Department of Mathematics, HHU, 
 40225 D\"{u}sseldorf, Germany \\[+1mm]
 e-mail: wisbauer@math.uni-duesseldorf.de \\
 web site: http://math.uni-duesseldorf.de/{\small$\sim$}wisbauer 
 }  
\date{}
\maketitle

\abstracts{ 
Given a finite automorphism group $G$ of a field extension $E\supset K$, $E$
can be considered as module over the group algebra $K[G]$. Moreover,
$E$ can also be viewed as a comodule over the bialgebra $K[G]^*$
and here a canonical isomorphism involving the subfield fixed 
under the action of $G$ arises.  
This isomorphism and its consequences were extended and studied for
group actions on commutative rings, for actions of Hopf algebras 
on noncommutative algebras, then for corings with grouplike elements and 
eventually to comodules over corings. The purpose of this note is to report  
about this development and to give the reader some idea about the notions 
and results involved in this theory (without claiming  
to be comprehensive).} 
 
\section{\large Preliminaries}

To begin with we recall the algebraic structures for which 
Galois type conditions are applied. 
 We follow the notation in [\refcite{BW}].
Throughout $R$ will denote a commutative associative ring with unit.
 
\begin{thm} \em {\bf Algebras and modules.}  
$A$, or more precisely $(A,\mu,1_A)$, stands for an associative 
$R$-algebra with multiplication $\mu:A\ot_R A\to A$ and unit $1_A$.
Right $A$-modules are defined as $R$-modules $M$ with an action
$\varrho_M:M\ot_R A\to M$.

For the category of right $A$-modules we write $\cM_A$ and 
denote the morphisms between $M,N\in \cM_A$ by $\Hom_A(M,N)$.
It is well known that $A$ is a projective generator in $\cM_A$.
\end{thm}

\begin{thm} \em {\bf Coalgebras and comodules.}  
An {\em $R$-coalgebra} is a triple $(C,\Delta,\vareps)$
where $C$ is an $R$-module, $\Delta:C\to C\ot_R C$ 
is the coproduct and $\vareps:C\to R$ is the counit. 
Right $C$-comodules are $R$-modules $M$ with a coaction 
$\varrho^M:M\to M\ot_R C$. 

The category of right $C$-comodules is denoted by $\cM^C$
and the morphisms between $M,N\in \cM^C$ are written as $\Hom^C(M,N)$. 
 As a right comodule,  $C$ is a subgenerator in 
$\cM^C$, that is, every right $C$-comodule is a subcomodule of a 
$C$-generated comodule. Note that $\cM^C$ need not have projectives 
even if $R$ is a field.  
\end{thm}

Left (co)modules and their categories are defined
and denoted  in an obvious way. 

\begin{thm} \em {\bf Bialgebras and Hopf modules.}  
An {\em $R$-bialgebra} is a quintuple $(B,\Delta,\vareps,\mu,1_B)$
where $(B,\Delta,\vareps)$ is an $R$-coalgebra and $(B,\mu,1_B)$
is an $R$-algebra such that $\Delta$ is an algebra morphism 
(equivalently $\mu$ is a coalgebra morphism). 

An $R$-module $M$ that is a right $B$-module by $\varrho_M:M\ot_R B\to M$
and a right $B$-comodule by $\varrho^M:M\to M\ot_R B$ is called 
a {\em right $B$-Hopf module} provided for any $m\in M$ and $b\in B$,
 $\varrho^M(mb)=\varrho^M(m)\Delta(b)$.
 The category of all right $B$-Hopf modules is denoted by $\cM^B_B$.  
The module $B\ot_R B$ allows for a right $B$-Hopf module structure
and with this it is a subgenerator in $\cM^B_B$.  
For $M\in \cM_B^B$ the {\em coinvariants} are defined as
 $$M^{coB} =\{ m\in M\,|\, \varrho^M(m)= m\ot_R 1_B\} \simeq \Hom^B_B(A,M).$$

An $R$-bialgebra $B$ is called a {\em Hopf algebra} if there is an antipode, 
that is, an $R$-linear map $S:B\to B$ which is the inverse of the
identity of $B$ with respect to the convolution product in $\End_R(B)$
(see also \ref{Hopf}). 
\end{thm}

For any $R$-algebra $A$ which is finitely generated and projective as $R$-module,
the dual $A^*=\Hom_R(A,R)$ can be considered as an $R$-coalgebra 
with natural comultiplication and counit. Here we are interested 
in the following special case. 

\begin{thm}\label{group} \em {\bf Group algebras and their dual.}  
Let $G$ be a finite group of order $n\in \N$ and $R[G]$ the group algebra, that is,
$R[G]$ is a free $R$-module with basis the group elements 
$\{g_1,\ldots,g_n\}$ and the product given by the group multiplication.  
Furthermore, $R[G]$ is an $R$-coalgebra with coproduct induced by 
$\Delta(g)=g\ot g$ and counit $\vareps(g)=1_R$, for $g\in G$.
With these structures $R[G]$ is an $R$-bialgebra, and even a Hopf algebra
with antipode $S$ induced by $S(g)=g^{-1}$ for $g\in G$.
\smallskip

The $R$-dual $R[G]^*=\Hom_R(R[G],R)$ is also a Hopf algebra. 
The multiplication of $f,g\in R[G]^*$ is given by $f*g(x)= f(x)g(x)$ for $x\in G$.
To describe the coalgebra structure
let $\{p_g\}_{g\in G}\subset R[G]^*$ be the dual basis to 
$\{g\}_{g\in G}$. Then coproduct and counit are defined by
  $$\Delta (p_g) = \sum_{kh=g} p_k\ot p_h, \quad \vareps (P_g)=\delta_{1,g}.$$   
The antipode $S$ of $R[G]^*$ is induced by $S(p_g)=p_{g^{-1}}$ for $g\in G$. 
\end{thm}

\begin{thm}\label{comod.AB}\em {\bf Comodule algebras and relative Hopf modules.}  
Let $B$ be an $R$-bialgebra. An $R$-algebra $A$ is called 
{\em right $B$-comodule algebra} if $A$ is a right $B$-comodule 
by $\varrho^A:A\to A\ot_R B$ such that  $\varrho^A$ is an  
algebra morphisms.

A {\em right $(A,B)$-Hopf module} is an $R$-module $M$ which is 
a right $A$-module and a right $B$-comodule by $\varrho^M:M\to M\ot_R B$
such that for all $m\in M$ and $a\in A$, 
$\varrho^M(ma)= \varrho^M(m)\varrho^A(a)$.
The category of these modules is denoted by $\cM_A^H$ and it has
$A\ot_R H$ as a subgenerator. 
For $M\in \cM_A^H$ the coinvariants are defined as
 $$M^{coB}=\{ m\in M\,|\,\varrho^M(m)= m\ot_R 1_B\}\simeq \Hom^B_A(A,M).$$
 
 Note that in the above construction the right $(A,B)$-Hopf modules
 may be replaced by the category $\cM(B)^D_A$ of {\em right $(A,D)$-Hopf modules}  
 where $D$ is a right $B$-module coalgebra  and the objects 
 are right $D$-comodules which are also right $A$-modules satisfying 
 some compatibility condition. Then $A\ot_R D$ is a subgenerator $\cM(B)^D_A$
 (see [\refcite{Doi}], [\refcite{MenZuc}]).
\end{thm}

Under weak (projectivity) conditions,
for all the structures considered above the related (co)module categories 
can be understood as module categories over some algebra subgenerated
by a suitable module. We refer to [\refcite{Wis:sur}] for more details. 
All this settings are subsumed as special cases of 
  
\begin{thm} \em {\bf Corings and comodules.}  
 An $A$-coring is a triple $(\cC,\uDelta,\ueps)$ where
 $\cC$ is an $(A,A)$-bimodule with coproduct
$\uDelta:\cC\to \cC\ot_A\cC$ and counit $\ueps: \cC\to A$.
Associated to this there are the right and  left  dual rings
$\cC^*=\Hom_A(\cC,A)$ and ${^*\cC}={_A\Hom(\cC,A)}$ with 
the convolution products.

A  {\em right $\cC$-comodule} is a right $A$-module $M$ together with an 
$A$-linear $\cC$-coaction 
 $\varrho^M: M\to M\ot_A\cC$. 
These comodules form a category which we denote by $\cM^\cC$. 
It is an additive category with coproducts and cokernels, 
 and $\cC$ is a subgenerator in it.   The functor 
$-\ot_A\cC: \cM_A\to \cM^\cC$ is right adjoint to the forgetful functor
by the isomorphisms, for $M\in \cM^\cC$ and $X\in \cM_A$,
        $$ \Hom^\cC  (M, X\ot_A \cC ) \to \Hom_A( M, X), \; 
         f \mapsto  (\id_{X}\ut  \ueps)\circ f, $$        
         with inverse map \;  
	$h\mapsto (h\ut  \id_{\cC})\circ \varrho^M$. 

Notice that for any monomorphism (injective map) $f:X\to Y$ in $\cM_A$, the
colinear map
$f\ot\id_\cC:X\ot_A\cC\to Y\ot_A\cC$ is a monomorphism in $\cM^\cC$
but need not be injective. In case $_A\cC$ is flat, monomorphisms in 
$\cM^\cC$ are injective maps and in this case $\cM^\cC$ is a Grothendieck
category (see 18.14 in [\refcite{BW}]).

Any right $\cC$-comodule $(M,\varrho^M)$ allows for a left $^*\cC$-module structure
by putting  $f\rhu m = (\id_M\ot f)\circ\varrho^M(m)$, for any $f\in {^*\cC}$,
$m\in M$. This 
yields a faithful functor  $\Phi: \cM^\cC \to {_{^*\cC}\cM}$ 
 which is a full embedding if and only if  the map
 $$\alpha_{K}: K\ot_A \cC  \to \Hom_A(^*\cC,K), \quad n\ot
c\mapsto [f\mapsto nf(c)],$$ 
is injective for any $K\in \cM_A$. 
This is called the
{\em left $\alpha$-condition} on $\cC$ and it
holds if and only if 
$_A\cC$ is locally projective. 
In this case $\MC$ can be identified with
$\sigma[{_{^*\cC} \cC}]$, the full subcategory of ${_{^*\cC} \cM}$
whose objects are subgenerated by $\cC$. 
\end{thm}


  \begin{thm}\label{Acomod} \em  {\bf $A$ as a $\cC$-comodule.} 
   An element $g$ of an $A$-coring $\cC$
  is called a {\em grouplike element}
   if $\uDelta(g) = g\ut g$ and  $\ueps(g) = 1_{A}$.
   Such a grouplike element $g$ exists if and only if $A$ is
      a right or left $\cC$-comodule, by the coactions
   $$ \varrho^A: A\to  \cC,\; a\mapsto ga, \quad
        {^A\!\varrho}: A\to \cC,\;  a\mapsto ag.$$
  Write  $A_g$ or ${_gA}$ to 
  consider $A$ with the right or left comodule structure induced by $g$.
  Given an $A$-coring $\cC$ with a grouplike element $g$ and $M\in\bM^\cC$,
  the {\em $g$-coinvariants} of $ M$  are defined as the $R$-module
  $$ M^{co\cC}_g = \{m\in M \;|\; \varrho^ M(m) = m\ut g\}
     = \Ke (\varrho^M - (-\ot g)), $$
  and there is an isomorphism
  $$\Hom^\cC(A_g,M) \to M^{co\cC}_g, \quad f\mapsto f(1_A).$$
 The bijectivity of this map is clear by the fact that any  $A$-linear map with
source $A$ is uniquely determined by the image of $1_A$.
As special cases we have the coinvariants 
\begin{zlist}
\item $\End^\cC(A_g) \simeq A^{co\cC}_g= \{a\in A_g\,|\,ga=ag\}$,
      the centraliser of $g$ in $A$.
\item For any $X\in \bM_A$,
      $(X\ot_A\cC)^{co\cC}\simeq\Hom^\cC(A_g,X\ot_A\cC)\simeq X,$ \\
      and for $X=A$,
      $$\cC^{co\cC}\simeq\Hom^\cC(A_g,\cC)\simeq \Hom_A(A_g,A)\simeq A,$$
      which is a left $A$- and right $\End^\cC(A_g)$-morphism.
\end{zlist}

\smallskip
  
  Given any right $B$-module $M$,
   $M\otimes_BA$ is a right $\cC$-comodule via the coaction
  $$\varrho^{M\otimes_BA}: M\otimes_BA\to M\otimes_BA\otimes_A\cC \cong
       M\otimes_B\cC, \quad m\ut a\mapsto m\ut ga.$$
  This yields a functor $-\ot_B A:\cM_B\to \cM^\cC$. 
  Right adjoint to this is the {\em $g$-coinvariants functor}
   $\Hom^\cC(A_g,-): \bM^{\cC}\to \bM_B.$ 
   
    For $N\in \bM_B$ the unit of the adjunction is given by
  $$ N\to (N\otimes_BA)^{co\cC},\quad n\mapsto n\ut 1_A,$$
  and for  $ M\in \bM^{\cC}$, the counit reads
 $$ M^{co\cC}\otimes_BA\to  M, \quad m\ut a\mapsto ma.$$
\end{thm}
 


\begin{thm}\label{pmod.coring} \em 
{\bf Coring of a projective module.}  
 For $R$-algebras $A,B$, let $P$ be a $(B,A)$-bimodule that is  finitely
generated and projective as a right $A$-module. Let $p_1, \ldots,p_n\in P$ 
and $\pi_1,\ldots, \pi_n\in P^*=\Hom_A(P,A)$ be a dual basis for $P_A$.
 Then the $(B,B)$-bimodule $P\ot_A P^*$ is an algebra by the isomorphism 
  $$P\ot_A P^* \to \End_A(P),\quad p\ot f\mapsto [q\mapsto pf(q)],$$ 
and the $(A,A)$-bimodule $P^*\ot_BP$ is an $A$-coring with  
coproduct and counit  
 $$ \begin{array}{c}
 \uDelta:P^*\ot_B P \to (P^*\ot_B P)\ot_A (P^*\ot_BP), \quad
      f\ot p \mapsto \sum_i f\ot p_i \ot \pi_i \ot p,\\[+2mm]
 \ueps:  P^*\ot_BP \to A,\quad f\ot p \mapsto f(p).
\end{array}$$ 
As a special case, for the $(A,A)$-bimodule $P=A^n$,  $n\in \N$,
 $P^*\ot_AP$ can be
identified  with the $n\times n$-matrices $M_n(A)$ over $A$, 
endowed with an $A$-coring structure (matrix coring).  
\end{thm}

\begin{thm}\label{Sweedler} \em {\bf The Sweedler coring.}  
Given an $R$-algebra morphism $\phi:B\to A$, the tensor product 
$\cC = A\otimes_B A$ is an $A$-coring with coproduct 
$$\DC : \cC\to  \cC\otimes_A\cC\simeq 
A\otimes_BA\otimes_BA, \quad a\ut a'\mapsto a\ut 1_A\ut  a',$$ 
and 
counit $\eC(a\ut a') = aa'$. $\cC$ is called the {\em Sweedler $A$-coring} associated to the algebra (or ring) morphism $\phi:B\to A$.  
Clearly $1_A\ot 1_A$ is a grouplike element in $\cC$. 

Since $A$ is finitely generated and projective as right $A$-module,
in view of \ref{Acomod}
this is a special case of \ref{pmod.coring}.
\end{thm} 


\begin{thm}\label{entw} \em  {\bf Entwining structures.}  
Given an $R$-algebra $A$ and an $R$-coalgebra $C$ one may think about 
compatibility conditions between these two structures. This led to the
notion of  
a (right-right) {\em  entwining structure} which is given by  
an {\em entwining map}, that is, an $R$-module map 
$ \psi: C\ot _{R} A\to A\ot _{R} C$  satisfying the conditions
\begin{zlist}
\item  $
\psi\circ(I_{C}\ot \mu) = (\mu\ot I_{C})\circ
       (I_A\ot \psi)\circ(\psi\ot 
 I_{A}), $
\item $(I_{A}\ot \Delta)\circ\psi = (\psi\ot
      I_C)\circ(I_{C}\ot\psi)\circ(\Delta\ot I_{A}),$
\item  $
\psi\circ (I_{C}\ot \iota) = \iota\ot I_{C},$
\item  $(I_{A}\ot \eps)\circ\psi =
\eps\ot I_A.$
\end{zlist}

Associated to  any entwining structure $(A,C,\psi)$ is the  category 
of (right-right) {\em $(A,C,\psi)$-entwined modules}
denoted by $\cM_{A}^{C}(\psi)$. An 
object $M\in \cM_{A}^{C}(\psi)$ is  a right $A$-module with 
multiplication $\varrho_{M}$ and a right $C$-comodule with coaction 
$\varrho^{M}$ satisfying 
$$\varrho^M\circ \varrho_M =
(\varrho_{M}\ut I_C)\circ (I_M\ut\psi )\circ (\varrho^{M}\ut I_A),$$
and morphisms in $\cM^{C}_{A}(\psi)$ are maps which are right $A$-module 
as well as right $C$-comodule morphisms. 
\smallskip

Entwining structures were introduced in [\refcite{BrzMaj}]
in the context of gauge theory on noncommutatice spaces. 
It then turned out that they are instances of corings since -
with the data given above - $A\otimes_{R} C$ is an $A$-coring
with $(A,A)$-bimodule struture 
\begin{center}
 $b(a'\ut c) a = ba'\psi(c\ut a)$, for $a,a',b\in A$, $c\in C$,
\end{center} 
coproduct $\DC = \id_A\ut \Delta$ and counit $\ueps=\id_A\ut \eps$
(see  32.6 in [\refcite {BW}]).
With this correspondence the category $\cM^C_A(\psi)$ can be identified
with the comodule category $\cM^{A\ot_RC}$.
\end{thm}


\begin{thm}\label{bialg} \em {\bf Bialgebras and corings} 
  (see  33.1 in [\refcite{BW}]).
Let $(B,\Delta_B,\vareps_B)$ be an $R$-bialgebra.
 Then  $B\ot_R B$ is a $B$-coring by the coproduct $\uDelta:\id_B\ot \Delta_B$,
 the counit $\ueps= \id_B\ot \vareps_B$, and the 
 $(B,B)$-bimodule structure
 \begin{center}
 $a(c\ot d)b = (ac \ot d)\Delta_B(b)$ where $a,b,c,d\in B$.
 \end{center}      
     With this structure the right $B$-Hopf modules can be identified 
     with the right $B\ot_R B$-comodules, that is, $\cM^B_B=\cM^{B\ot_RB}$.      
   Clearly $1_B\ot 1_B$ is a grouplike element in $B\ot_RB$ and
   the ring of $B\ot_RB$-covariants of $B$ is isomorphic to $R$.       
\end{thm}

 \begin{thm}\label{com.cor}  \em {\bf Comodule algebras and corings} 
  (see 33.2 in [\refcite{BW}]).    
 Let $(B,\Delta_B,\vareps_B)$ be an $R$-bialgebra.
 Then for a right $B$-comodule algebra $A$, $A\ot_R B$ is an
 $A$-coring with coproduct $\uDelta=\id_A\ot \Delta_B$, counit 
 $\ueps=\id_A\ot \vareps_B$, and $(A,A)$-bimodule structure
 \begin{center}
 $a(c\ot b)d = (ac\ot b)\varrho^A(d)$, for $a,c,d\in A$ and $b\in B$.
 \end{center}     
Here the right relative $(A,B)$-Hopf modules are just the right 
$A\ot_R B$-comodules, that is,  $\cM^B_A=\cM^{A\ot_RB}$.  
 \end{thm}
  

\begin{thm}\label{coint} \em {\bf Cointegrals.}  
An $(A,A)$-bilinear map $\delta:\cC\ot_A\cC\to \cC$ is called a {\em cointegral
in $\cC$} if   
$$(\id_\cC \ot\delta)\circ (\Delta \ut \id_\cC  )= 
      (\delta\ot \id_\cC)\circ (\id_\cC  \ot\Delta).$$
Cointegrals are characterised by the fact that 
 for any $M\in \cM^\cC$, the map 
 $$(\id_M\ot\delta)\circ (\varrho^M\ot\id_\cC): M\ot_A \cC \to M$$ 
     is a comodule morphism (or by the 
corresponding property for left $\cC$-comodules). 

In [\refcite{CaGrVe}], Section 5, these maps are related to the counit for the adjoint
pair of functors $-\ot_A\cC$ and the forgetful functor.
For $R$-coalgebras $C$ over a commutative ring $R$ with $C_R$ locally
projective, a cointegral is precisely a $C^*$-balanced $R$-linear map $C\ot_R
C\to R$  (e.g., 6.4 in[\refcite{BW}]).
\end{thm}

Recall some properties of relative injectivity 
from [\refcite{Wis:cogal}], Section 2:

\begin{thm}\label{rel.inj} \em {\bf Relative injectivity.}  
 Let $M\in \bM^\cC$ and $S= \End^\cC(M)$.  

$M$ is {\em $(\cC,A)$-injective} provided 
the structure map $\varrho^M:M\to M\ot_A\cC$ is split by a $\cC$-morphism 
$\lambda: M\ot_A\cC\to M$. 

 $M$ is called {\em strongly $(\cC,A)$-injective} if this $\lambda$ is
$\cC$-co\-linear and $S$-linear.
Given a subring $B\subseteq S$,  $M$ is said to be
{\em $B$-strongly $(\cC,A)$-injective} if $\lambda$ is 
$\cC$-co\-linear and $B$-linear.
 
 $M$ is called {\em fully $(\cC,A)$-injective} if there is 
 a cointegral $\delta_M:\cC\ot_A\cC\to \cC$ such that 
$\roM$ is split by $(\id_M\ot \delta_M)\circ (\varrho^M\ot \id_\cC)$.

The notions for left $\cC$-comodules are defined symmetrically. 
\end{thm}
 
For $R$-coalgebras $C$, $B$-strongly $(C,R)$-injective comodules are named 
{\em $B$-equivariantly $C$-injective}
  (see Definition 5.1 in [\refcite{SchSch}]).

\pagebreak[3]
 
\begin{thm}\label{full.prop}
{\em {\bf Fully $(\cC,A)$-injective comodules.}}  Let 
$M\in \cM^\cC$ with $S=\End^\cC(M)$. 
\begin{zlist}
\item $M$ is fully $(\cC,A)$-injective if and only if 
   $$ (\id_M\ot \widetilde\delta_M) \circ\varrho^M =\id_M\;\mbox{  
  where }\;\widetilde\delta_M=\delta_M \circ \Delta: \cC\to A.$$ 
\item $\cC$ is a fully $(\cC,A)$-injective right (left) comodule 
      if and only if $\cC$ is a coseparable coring.
\item Let $M$ be fully $(\cC,A)$-injective. Then:
\begin{rlist}
 \item Every comodule in $\sM$ is fully $(\cC,A)$-injective.   
 \item If $M$ is a subgenerator in $\MC$ then $\cC$ is a coseparable coring. 
 \item For any subring $B\subset S$ and $X\in \cM_B$, $X\ot_B M$ is  
      fully $(\cC,A)$-injective.
 \item If $M_A$ is finitely generated and projective, then 
      $M^*$ is a fully $(\cC,A)$-injective left $\cC$-comodule. 
\end{rlist}
\end{zlist} 
\end{thm}
 


\section{\large Galois extensions and comodules}


Classical Galois theory studies the action of a finite automorphism group $G$  
on a field $E$ and then considers $E$ as extension of the subfield of the 
elements which are left unchanged by the action of $G$.
This can be understood as a comodule situation (compare [\refcite{Mont}], Chapter 8).  

\begin{thm}\label{sep.field} \em 
{\bf Galois field extension.}  
Let $G$ be a finite automorphism group of a field extension $E\supset K$ 
and let $F=E^G$ be the fixed field of $G$. Thus the 
group algebra $K[G]$ acts on $E$ and so its dual, the 
Hopf algebra $H=\Hom_K(K[G],K)=K[G]^*$ coacts on $E$. 

To describe this let $G=\{g_1,\ldots,g_n\}$ and choose   
$\{b_1,\ldots,b_n\}\subset E$ as a basis of the $F$-vectorspace $E$. 
Denote by $\{p_1,\ldots,p_n\}\subset K[G]^* $ the dual basis to 
$\{g_1,\ldots,g_n\}\subset K[G]$.
Then $E$ is a right $K[G]^*$-comodule by the coaction
$$\varrho^E: E\to E\ot_K K[G]^*, \quad 
a\mapsto \sum_{i=1}^n (g_i\cdot a) \ot p_i,$$
and we can define the {\em Galois map} 
$$\gamma: E\ot_F E\to E\ot_K K[G]^*, \quad a\ot b \mapsto 
\sum_{i=1}^n a(g_i\cdot b) \ot p_i.$$

For any $w=\sum_j a_j\ot b_j\in \Ke \gamma$, we have 
$\sum_{j,i} a_j(g_i \cdot b_j)\ot p_i=0$ and by the independence 
of the $p_1,\ldots ,p_n$, $\sum_{j} a_j(g_i \cdot b_j)=0$ for all $i$.
Now Dedekind's lemma on the independence of automorphisms 
implies that all $a_j=0$ and thus $w=0$.  
This shows that $\gamma$ is injective and for dimension reasons it is 
in fact bijective. 

Notice that the coinvariants of the $K[G]^*$-comodule $E$ are
   $$ \{a\in E\,|\,    \sum_{i=1}^n (g_i\cdot a) \ot p_i = a\ot \vareps\}=E^G,$$
since for each such $a\in E$ and $g_i\in G$, 
$ g_i\cdot a=  (g_i\cdot a)  p_i(g_i)= a\vareps(g_i)=a$.    
\end{thm}


The definition of Hopf Galois extensions goes back to 
Chase-Harrison-Rosenberg [\refcite{CHR}] where the classical Galois
theory of fields was extended to groups acting on commutative rings.  
This was generalised in Chase-Sweedler [\refcite{CS}] to 
coactions of Hopf algebras on commutative $R$-algebras and then,
in Kreimer-Takeuchi [\refcite{KT}], to coactions on noncommutative $R$-algebras.

\begin{thm}\label{comod.A} \em {\bf Comodule algebras.} 
Let $H$ be a Hopf $R$-algebra and  
 $A$ a right $H$-comodule algebra with structure map 
 $\varrho^A: A\to A\ot_R H$ and $B = A^{coH}$.  Then 
$B\subset A$ is called {\em right $H$-Galois} if the following map is bijective:
 $$\gamma: A\ot_B A\to A\ot H, \quad a\ot b\mapsto (a\ot 1)\varrho^A(b).$$
 

For examples and more information about such extensions we refer to [\refcite{Mont}], Section 8. Further investigation on such structures were done in particular 
by Doi, Takeuchi and Schneider [\refcite{DoiTak:cle}], [\refcite{DoiTak:miy}], 
[\refcite{Sch:pri}], [\refcite{Sch:rep}], [\refcite{Sch:nor}].


Generalising results about the action of an affine algebraic group
scheme on an affine scheme 
the following theorem was proved in [\refcite{Sch:pri}]. This shows (again) that 
$H$-Galois extensions are closely related to modules inducing  equivalences.
\smallskip

{\bf Schneider's Theorem.} {\em
Let $H$ be a Hopf algebra over a field $R$ with bijective antipode.  
 Then for a right $H$-comodule algebra  $A$ and $B=A^{coH}$ the following are equivalent:
 \begin{blist}
 \item $B\subset A$ is a $H$-Galois extension and $A$ 
       is faithfully flat as a left \linebreak[3] $B$-module;
 \item $B\subset A$ is a $H$-Galois extension and $A$ 
       is faithfully flat as a right $B$-module;
 \item  $-\ot_B A: \cM_B \to \cM^H_A$ is an equivalence;
 \item  $A\ot_B -: {_B\cM} \to {_A\cM^H}$ is an equivalence.           
 \end{blist}
}
 
Notice that the above theorem shows a left right symmetry which will not be maintained in (most of) the subsequent generalisations.
\smallskip

As mentioned in \ref{comod.AB}, the $(A,H)$-Hopf modules
can be generalised to $(A,D)$-Hopf modules where $D$ is a right 
$H$-module coalgebra yielding the category $\cM(H)^D_A$. 
If there is a {\em grouplike element} $x\in D$, then 
$A$ is in $\cM(H)^D_A$ and for any $M \in  \cM(H)^D_A$ coinvariants 
can be defined as $\Hom^D_A(A,M)$. Then $B= \Hom^D_A(A,A)$ is 
a subring of $A$ and the inclusion $B\hra A$ is called a {\em right Hopf-Galois extension}
provided the canonical map  
$$A\ot_B A\to A\ot_R D, \quad a\ot b\mapsto (a\ot x)\varrho^A(b)$$ 
is bijective.
For this setting an extension of Schneider's Theorem 
is proved by Menini and Zuccoli (see Theorem 3.29 in [\refcite{MenZuc}]).
\end{thm}



\begin{thm} \em {\bf Coalgebra-Galois extensions.}  
\label{definition.Galois}
Let $C$ be an $R$-coalgebra and  $A$ an $R$-algebra and a right 
$C$-comodule with 
coaction $\varrho^A:A\to A\otimes_R C$.  
 Define the {\em coinvariants} of $A$ as
 $$B= \{b\!\in\! A\; | \; \mbox{for all } a\!\in\!
A, \; \varrho^A(ba) = b\varrho^A(a)\}.$$ 
 The
extension $B\hookrightarrow A$ is called a
{\em coalgebra-Galois extension} (or a {\em $C$-Galois extension})
if the following left $A$-module, right
$C$-comodule map is bijective:
$$
\gamma: A\otimes _BA\to A\otimes_R C,\qquad a\ut a'\mapsto a\varrho^A(a').
$$
 
Notice that here the definition of covariants does not require the 
existence of a grouplike element in $C$ and thus coalgebra-Galois  
extensions are defined for arbitrary coalgebras. 
This notion was introduced in [\refcite{BrzHaj:coa}], following their appearance as
generalised principal bundles in [\refcite{BrzMaj}].
 The main geometric motivation for this was
 the need for principal bundles with coalgebras 
playing the role of a structure group. 
The main result Theorem 2.7 in [\refcite{BrzHaj:coa}] shows how coalgebra Galois extensions are related to entwining structures.
\smallskip

 {\bf Theorem.} {\em Let $R$ be a field and
 $A$ a $C$-Galois extension of $B$ (as defined above).
Then there exists a unique entwining map $\psi: C\ot_R A\to A\ot_R C$
such that $A\in \cM^C_A(\psi)$ with structure map $\varrho^A$. }
\end{thm}


 

\begin{thm}\label{Gal.char}  \em {\bf Galois corings.}  
 Let $\cC$ be an $A$-coring  with a grouplike element  $g$ and
  $B=A^{co\cC}_g$.
Following Definition 5.3 in [\refcite {Brz:str}], $(\cC,g)$ is called
a {\em Galois coring}   if the canonical map   
$$\chi:A\ot_SA\to \cC, \quad a\ot a'\mapsto aga',$$
is an isomorphism (of corings). 
It was pointed out in [\refcite{Wis:gal}]
that this can be seen as the evaluation map 
 $$\Hom^\cC(A_g,\cC)\ot_S A \to \cC,\quad f\ut a\mapsto f(a).$$

 
{\em The following assertions are equivalent 
  {\em (4.6 in [\refcite{Wis:gal}])}:
   
  \begin{blist}
  \item  $(\cC,g)$ is a Galois coring;   
  \item for every $(\cC,A)$-injective comodule $N\in \bM^\cC$, the 
  evaluation
     $$\Hom^\cC(A_g,N)\ot_B A \to N, \quad f\ut a \mapsto 
  f(a),$$
    is an isomorphism.
  \end{blist}
 }
   
 
 
  Notice that here the canonical isomorphism can be extended to 
  related isomorphisms for the class of all relative injective comodules.
   

  The following is a one-sided generalization of Schneider's theorem 
    (see 4.8 in [\refcite{Wis:gal}]). 
 \smallskip
 \pagebreak[3]
     
 {\bf The Galois Coring Structure Theorem.}  {\em 
 
  \begin{zlist}
  \item The following are equivalent:
   \begin{blist}
   \item $(\cC,g)$ is a Galois coring and $_BA$ is flat;
   \item $_A\cC$ is flat and $A_g$ is a generator in $\bM^\cC$.
   \end{blist}
  \item The following are equivalent:
    \begin{blist}
   \item $(\cC,g)$ is a Galois coring and $_BA$ is faithfully flat;
   \item $_A\cC$ is flat and $A_g$ is a projective generator in $\bM^\cC$;
   \item  $_A\cC$ is flat and $\Hom^\cC(A_g,-): \bM^\cC \to \bM_B$ is an
          equivalence   \\
          with inverse $-\ot_BA:\bM_B\to \bM^{\cC}$ {\em (cf. \ref{Acomod})}.
  \end{blist}
  \end{zlist} }
  
  If the base ring $A$ is injective as right $A$-module, then
  $\cC$ is injective as right $\cC$-comodule and thus (see 4.9 in [\refcite{Wis:gal}])
  we obtain the  
\smallskip  
  
  {\bf Corollary.} {\em  
   Assume $A$ to be a right self-injective ring and let $\cC$ be an 
  $A$-coring with grouplike element $g$.
  \begin{zlist}
  \item The following are equivalent:
    \begin{blist}
    \item $(\cC,g)$ is a Galois coring;
    \item for every injective comodule $N\in \cM^\cC$, the evaluation
         $$\Hom^\cC(A_g,N)\ot_B A \to N, \quad f\ut a \mapsto 
         f(a),$$
    is an isomorphism.
    \end{blist}
  \item The following are equivalent:
    \begin{blist}
    \item $(\cC,g)$ is a Galois coring and $_BA$ is (faithfully) flat;
    \item  $_BA$ is (faithfully) flat and
  for every injective comodule $N\in \cM^\cC$, the following evaluation
  map is an isomorphism:
 $$\Hom^\cC(A_g,N)\ot_B A \to N, \quad f\ut a \mapsto f(a).$$
    \end{blist}
  \end{zlist} }
\end{thm}
 
  
  \begin{thm}\label{Hopf} \em 
  {\bf Hopf algebras.}  
  Given an $R$-bialgebra $B$, by definition the $B$-coring $B\ot_R B$ is Galois
  provided the canonical map 
 $$\gamma :B\ot_{R} B\to B\ot_R B, \quad a\ut b\mapsto (a\ut 
  1)\Delta(b)$$
  is an isomorphism. Since bijectivity of this map is equivalent to 
  the existence of an antipode (see 15.2 in [\refcite{BW}]) we 
  have:
 \smallskip
 
{\em  For a bialgebra $B$ the following are equivalent:
 \begin{blist}
   \item  $B\ot_RB$ is a Galois $B$-coring;
   \item $B$ is a Hopf algebra (has an antipode);
   \item $\Hom^B_B(B,-): \cM^B_B \to \cM_R$ is an equivalence (with 
         inverse $-\ot_R B$).
  \end{blist}
   If (any of) these conditions hold, $B$ is a projective generator in 
  $\bM^B_B$. }
  \end{thm}
   
 \bigskip 
  
 
The notion of Galois corings was extended to comodules   
by El Kaoutit and G\'omez-Torrecillas in [\refcite{ElGo.com}], where
to any bimodule $_SP_A$ with $P_A$ finitely generated and projective, 
a coring $P^*\ot_S P$ was associated (see \ref{pmod.coring}) and it 
was shown that the map  
$$\varphi:\Hom_A(P,A)\ot_SP \simeq  \Hom^\cC(P,\cC)\ot_SP \to  \cC$$
is a coring morphism provided $P$ is also a right $\cC$-comodule and
$S=\End^\cC(P)$.

In [\refcite{BW}], 18.25, such comodules $P$ are termed {\em Galois comodules}
provided $\varphi$ is bijective, and it is proved 
in [\refcite{BW}], 18.26, that this condition implies that 
the functors $\Hom_A(P,-)\ot_SP$ and $-\ot_A\cC$ from $\cM_A$ to 
 $\cM^\cC$ are isomorphic.  


\begin{thm} \label{Gal.com} \em {\bf Galois comodules.}   
 Let $P$ be a right $\cC$-comodule such that $P_A$ is finitely generated and
projective  and let $S=\End^\cC(P)$. 
Then $P$ is called a {\em Galois comodule} 
if the evaluation map  
 $$ \Hom^\cC(P,\cC)\ot_S P\to \cC, \; f\ot m\mapsto f(m),$$
is an isomorphism of right $\cC$-comodules. 
\smallskip
 
{\em
Considering $P^*\ot_SP$ as an $A$-coring (via \ref{pmod.coring}),  
the following are equivalent:
\begin{blist}
\item  $P$ is a Galois comodule;
\item there is a (coring) isomorphism
 $$ P^*\ot_S P \to \cC, \quad \xi\ot m \mapsto \sum
      (\xi\ot \id_\cC)\, \varrho^P(m);$$  
\item for every $(\cC,A)$-injective comodule $N\in \cM^\cC$, the evaluation
   $$\Hom^\cC(P,N)\ot_S P \to N, \quad f\ut m \mapsto f(m),$$
  is a (comodule) isomorphism;
\item for every right $A$-module $X$, the map
   $$\Hom_A(P,X)\ot_S P \to X\ot_A\cC, \quad 
    g\ut m \mapsto  (g\ot \id_\cC)\, \varrho^P(m),$$
  is a (comodule) isomorphism.
\end{blist}
}
\smallskip

 The next theorem - partially proved in [\refcite{ElGo.com}] -
 shows which additional conditions on a Galois comodule 
are sufficient to make it a (projective) generator in $\cM^\cC$
(see 18.27 in [\refcite{BW}]).
\smallskip

\pagebreak[3]

{\bf The Galois comodule structure theorem.}
{\em 
\begin{zlist}
\item The following are equivalent:
 \begin{blist}
 \item $P$ is a Galois comodule and $_SP$ is flat;
 \item $_A\cC$ is flat and $P$ is a generator in $\cM^\cC$.
 \end{blist}
\item The following are equivalent:
  \begin{blist}
 \item $M$ is a Galois comodule and $_SP$ is faithfully flat; 
 \item $_A\cC$ is flat and $P$ is a projective generator in $\cM^\cC$;
 \item  $_A\cC$ is flat and $\Hom^\cC(P,-): \cM^\cC \to \cM_S$ is an
        equivalence     
        with the inverse $-\ot_SP:\cM_S\to \cM^{\cC}$.
\end{blist}
\end{zlist} }
\end{thm}

These Galois comodules are further investigated in Brzezi\'nski [\refcite{Brz:gal}]
and their relevance for  descent theory,  vector bundles,  
and non-commutative geometry is pointed out there.
In particular {\em principal comodules} are considered, that is, 
Galois comodules in the above sense which are projective as modules over their
endomorphism rings. 
Related questions are, for example, also 
considered by Caenepeel, De Groot and Vercruysse in [\refcite{CaGrVe}].



\section{\large General Galois comodules}

 
Recall that for a Galois $\cC$-comodule $P$ in the sense of 
\ref{Gal.com} (where $P_A$ is finitely generated and projective)
 the functors $-\ot_A\cC$ and $\Hom_A(P,-)\ot_S P$ are isomorphic.
 In [\refcite{Wis:cogal}] it is suggested to take this property as definition
 without further condition on the $A$-module structure of $P$.
 
 Throughout this section let $\cC$ be an $A$-coring,
$P\in \cM^\cC$ and $S=\End^\cC(P)$, $T=\End_A(P)$. 
  
\begin{thm}\label{gal.comod} \em {\bf Galois comodules.}  
 We call $P$ a {\em Galois comodule} if 
  $$-\ot_A\cC \simeq \Hom_A(P,-)\ot_S P \mbox{ as functors}:\cM_A\to \cM^\cC.$$    
 
  {\em The following are equivalent {\em ([\refcite{Wis:cogal}], 2.1)}:
  
  \begin{blist}
\item   $P$ is a Galois comodule;
\item  $\Hom_A(P,-)\ot_SP$ is right adjoint to the forgetful 
    functor $\cM^\cC\to \cM_A$, that is, for $K\in \cM_A$ and $M\in \cM^\cC$,
    there is a (bifunctorial) isomorphism
    $$\Hom^\cC(M, \Hom_A(P,K)\ot_S P) \to \Hom_A(M,K);$$
\item for any $K\in \cM_A$ there is a functorial isomorphism of comodules
  $$\Hom_A(P,K)\ot_S P\to K\ot_A\cC,\;g\ot p\mapsto(g\ot\id_\cC)\varrho^P(p);$$
\item for every $(\cC,A)$-injective $N\in \cM^\cC$,
    $$\Hom^\cC(P,N)\ot_S P\to N, \; f\ot p\mapsto f(p),$$
   is an isomorphism (in $\cM^\cC$).
\end{blist} }
\end{thm}
 

  These comodules have good properties 
(see 2.2 in [\refcite{Wis:cogal}]):

\begin{thm}\label{prop.gal} {\em {\bf Isomorphisms for Galois comodules.}}
 Let $P\in \cM^\cC$ be a Galois comodule. %Then:
\begin{zlist}
\item For any $(\cC,A)$-injective $N\in \cM^\cC$, there is a canonical isomorphism
   $$\Hom^\cC(P,N)\to\Hom^\cC(P,\Hom^\cC(P,N)\ot_SP).$$  
\item For any $K\in \cM_A$, there is a canonical isomorphism
   $$\Hom_A(P,K)\to \Hom^\cC(P,\Hom_A(P,K)\ot_SP).$$  

\item There are right $\cC$-comodule isomorphisms 
   $$\Hom^\cC(P,\cC)\ot_SP\simeq \cC \simeq  \Hom_A(P,A)\ot_SP.$$
   
\item There is a $T$-linear isomorphism
 $$
T\ot_S P \to P\ot_A\cC,\quad t\ot p \mapsto (t\ot \id_\cC)\varrho^P(p).$$

\item For any $K\in \cM_A$ and index set $\Lambda$,
   $$\Hom^\cC(P, (K\ot_A\cC)^{\Lambda})\ot_SP\simeq
     \Hom_A(P,K)^{\Lambda}\ot_SP\simeq 
        K^{\Lambda}\ot_A\cC.$$
\end{zlist}
\end{thm}
 
It is clear from the definition that $(\cC,A)$-injective modules 
are of particular interest in this setting (see 2.3 in [\refcite{Wis:cogal}]):

\begin{thm}\label{CA-inj} {\em {\bf $(\cC,A)$-injective modules.}} %{full.gal}
 Let $P$ be a Galois comodule.
\begin{zlist}
\item For $N\in \cM^\cC$ the following are equivalent:
\begin{blist}
\item $N$ is $(\cC,A)$-injective;
\item $\Hom^\cC(P,\varrho^N):\Hom^\cC(P,N)\to \Hom^\cC(P,N\ot_A\cC)$ 
      is a coretraction in $\cM_S$. 
\end{blist}
\item For $P$ the following are equivalent:
\begin{blist}
\item $P$ is $(\cC,A)$-injective;
\item the inclusion $S\hra T$ is split by a right $S$-linear map.
\end{blist}
\item For $P$ the following are equivalent:
  \begin{blist}
  \item $P$ is  strongly $(\cC,A)$-injective;
  \item the inclusion $S\hra T$ is split by an $(S,S)$-bilinear map.
  \end{blist}
  
\item For $P$ the following are equivalent:
\begin{blist}
\item $P$ is fully $(\cC,A)$-injective; 
\item $\cC$ is a coseparable $A$-coring.
\end{blist}
\end{zlist}
\end{thm}
 
 
Notice that so far we did not make any assumptions neither on the $A$-module
nor on the $S$-module structure of $P$. Of course special properties of this type 
influence the behaviour of Galois comodules. For the $S$-module
structure we get (see 4.8 in [\refcite{Wis:cogal}]): 

\begin{thm}\label{S.prop} {\em {\bf Module properties of $_SP$.}} 
 Let $P\in \cM^\cC$ be a Galois comodule.  
\begin{zlist}
\item If $_SP$ is finitely generated, then $_A\cC$ is finitely generated.
\item If $_SP$ is finitely presented, then $_A\cC$ is finitely presented.
\item If $_SP$ is projective, then $_A\cC$ is projective. 
\item If $_TP$ is finitely generated and
      $_SP$ is locally projective, then $_A\cC$ is locally projective. 
\item  If $_SP$ is flat, then $_A\cC$ is flat and $P$ is a generator in
      $\cM^\cC$.
\item If $_SP$ is faithfully flat, then $_A\cC$ is flat and $P$ is a
projective       generator in $\cM^\cC$.
\end{zlist}
\end{thm}
 
If $_A\cC$ is flat as an $A$-module then $\cM^\cC$ is a Grothendieck category 
(see 18.14 in [\refcite{BW}]) and the endomorphism ring of any semisimple
 right $\cC$-comodule is a (von Neumann) regular ring. This implies part 
 of the next proposition (see 4.11 in [\refcite{Wis:cogal}]).
 
 
\begin{thm}\label{sem.gal}{\em {\bf Semisimple Galois comodules.}}
Assume $_A\cC$ to be flat. For a semisimple right $\cC$-comodule $P$,
the following are equivalent:
\begin{blist}
 \item $P$ is a Galois comodule;
 \item $P$ is a generator in $\cM^\cC$;
 \item $\mu_\cC:\Hom^\cC(P,\cC)\ot_SP\to \cC$ is surjective. 
\end{blist}
In this case $\cC$ is a right semisimple coring (and $_A\cC$ is projective).
\end{thm}
 
 
 Recall that $P^*\ot_S P$ has a coring structure provided 
 $P_A$ is finitely generated and projective (see \ref{pmod.coring}).
Moreover, $P^*=\Hom_A(P,A)$ is a left $\cC$-comodule canonically
and we have a left-right symmetry for Galois comodules
(see 5.3 in [\refcite{Wis:cogal}]): 


\begin{thm}\label{gal.Afg} {\em {\bf Galois comodules with $P_A$ f.g. projective.}} 
Assume $P_A$ to be finitely generated and projective.
Then the following are equivalent:
\begin{blist}
\item $P$ is a Galois right $\cC$-comodule;
\item $\Hom^\cC(P,\cC)\ot_R P\simeq \cC$ as right $\cC$-comodule;
\item $P^*$ is a Galois left $\cC$-comodule;
\item ${^{\cC}\Hom}(P^*,\cC)\ot P^*\simeq \cC $ as left $\cC$-comodule;
\item $P^*\ot_S P \simeq \cC$ as $A$-corings.
\end{blist}
\end{thm}
 
 
In case $A$ is a $\cC$-comodule, that is, there is a grouplike element 
$g\in\cC$,  and $S=\End^\cC(A)$, it is a Galois (right) comodule 
($(\cC,g)$ is a Galois coring) if and only if the map 
     $$A\ot_S A\to \cC,\quad a\ot a'\mapsto aga',$$
is an isomorphism. Under the given conditions, $A\ot_S A$ has a 
canonical  co\-ring structure (Sweedler coring, \ref{Sweedler}) 
and the map is a co\-ring isomorphisms (see 28.18 in [\refcite{BW}]).   

 At various places we have observed a nice behaviour of strongly $(\cC,A)$-injective 
comodules. For Galois comodules this property is symmetric in the 
following sense - an observation also proved in [\refcite{Brz:gal}], Theorem 7.2. 

\begin{thm}\label{cov.symm} {\em {\bf Strongly $(\cC,A)$-injective Galois
comodules.} }
Let $P$ be a Galois comodule with $P_A$ finitely generated and
projective. Then the following are equivalent:
\begin{blist}
\item $P$ is strongly $(\cC,A)$-injective;
\item $P^*$ is strongly $(\cC,A)$-injective;
\item the inclusion $S\hra T$ is split by an $(S,S)$-bilinear map.
\end{blist}
\end{thm}

{\bf Proof.} This follows from \ref{CA-inj} and symmetry. 
\smallskip

Finally we consider various conditions which imply that
a Galois comodule induces an equivalence (see 5.7 in [\refcite{Wis:cogal}]).  

\begin{thm}\label{equival} {\em {\bf Equivalences.}} Let $P\in \cM^\cC$ be a 
Galois comodule with $P_A$ finitely generated and projective.
Then
$$\Hom^\cC(P,-): \cM^\cC\to \cM_S$$ 
is an equivalence with inverse functor $-\ot_SP$ provided that
\begin{rlist}
 \item $P$ is strongly $(\cC,A)$-injective,  or  
 \item $P^*$ is $(\cC,A)$-injective and $_SP$ is flat, or 
 \item $P^*$ is coflat and $_SP$ is flat, or 
 \item $\cC$ is a coseparable coring. 
\end{rlist}
\end{thm}
 
\begin{thm}\label{rem.Br}\em  {\bf Remarks.}  
(1)  Entwining structures can be 
   considered as corings and hence the assertions in  
   \ref{CA-inj} may be compared with Lemma 4.1 and Remarks 4.2 and 5.3 in
   Schauenburg and Schneider [\refcite{SchSch}].  

(2) {\em Weak Galois corings} are considered in
  [\refcite{Wis:wea}], 2.4. For such corings the action of $A$ on $\cC$ is not
  required to be unital.

(3) For a deeper study of weak entwining and weak coalgebra-Galois extensions
    the reader may consult Brzezi\'nski, Turner and Wrightson [\refcite{BTW}].

(4) For recent investigation of the Galois theory for Hopf algebroids 
  we refer to B\"ohm [\refcite{bohm}].
\end{thm} 
\smallskip

{\bf Acknowledgement.} The author appreciates useful remarks 
on the manuscript by Tomasz Brzezi\'nski.

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\end{document}