Separable functors in corings
J. G\'omez-Torrecillas
Departamento de \'{A}lgebra
Universidad de Granada
E18071 Granada
SPAIN
We develop some basic functorial techniques for the study of the categories of comodules over corings. In particular, we prove that the induction functor stemming from every morphism of corings has a left adjoint, called ad-induction functor. This construction generalizes the known adjunctions for the categories of Doi-Hopf modules and entwined modules. The separability of the induction and ad-induction functors are characterized, extending earlier results for coalgebra and ring homomorphisms, as well as for entwining structures.
math0102120.tex
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\newcommand{\cat}[1]{\mathcal{#1}}
\newcommand{\coring}[1]{\mathfrak{#1}}
\newcommand{\tensor}[1]{\otimes_{#1}}
%\newcommand{\tensfun}[1]{\otimes_{{}_{#1}}}
\newcommand{\tensfun}[1]{\underset{{#1}}{\otimes}}
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\begin{document}
%Title and Authors
\title{Separable functors in corings}
\author{J. G\'omez-Torrecillas \\
Departamento de \'{A}lgebra \\
Universidad de Granada\\
E18071 Granada, SPAIN \\
e-mail: \textsf{torrecil@ugr.es}}

\date{\empty}

\maketitle
\begin{abstract}
We develop some basic functorial techniques for the study of the
categories of comodules over corings. In particular, we prove that
the induction functor stemming from every morphism of corings has
a left adjoint, called ad-induction functor. This construction
generalizes the known adjunctions for the categories of Doi-Hopf
modules and entwined modules. The separability of the induction
and ad-induction functors are characterized, extending earlier
results for coalgebra and ring homomorphisms, as well as for
entwining structures.
\end{abstract}
%\CompileMatrices
\section*{}
\textit{Keywords and prhases:} coring, comodule, cotensor product
functor, cohom functor, coring homomorphism, induction functor,
ad-induction functor, separable functor \\
\noindent \textit{2000 Mathematics Subject Classification:} 16W30

\section*{Introduction}

The notion of separable functor was introduced by C. N\u ast\u
asescu, M. Van den Bergh and F. Van Oystaeyen in
\cite{Nastasescu/VandenBergh/VanOystaeyen:1989}, where some
applications for group-graded rings were done. This notion fits
satisfactorily to the classical notion of separable algebra over a
commutative ring. Every separable functor between abelian
categories encodes a Maschke's Theorem, which explains the
interest concentrated in this notion within the module-theoretical
developments in recent years. Thus, separable functors have been
investigated in the framework of coalgebras
\cite{Castano/Gomez/Nastasescu:1997}, graded homomorphisms of
rings \cite{DelRio:1992,Castano/Gomez/Nastasescu:1998},
Doi-Koppinen modules
\cite{Caenepeel/alt:1999,Caenepeel/alt:2000unp} or, finally,
entwined modules
\cite{Brzezinski:1999unp,Brzezinski/alt:2000unp1}. These
situations are generalizations of the original study of the
separability for the induction and restriction of scalars functors
associated to a ring homomorphism done in
\cite{Nastasescu/VandenBergh/VanOystaeyen:1989}. It turns out that
that all the aforementioned categories of modules are instances of
comodule categories over suitable corings
\cite{Brzezinski:2000unp}. In fact, the separability of some
fundamental functors relating the category of comodules over a
coring and the underlying category of modules has been studied in
\cite{Brzezinski:2000unp}. Thus, we can expect that the
characterizations obtained in \cite{Brzezinski:1999unp} of the
separability of the induction functor associated to an admissible
morphism of entwining structures and its adjoint generalize to
the corresponding functors stemming from a homomorphism of
corings. This is done in this paper.

To state and prove the separability theorems, I have developed a
basic theory of functors between categories of comodules, making
the arguments independent from the Sweedler's `sigma-notation'. My
plan here is to use purely categorical methods which could be
easily adapted to more general developments of the theory. These
methods had been sketched in \cite{AlTakhman:unp2001} and
\cite{AlTakhman:1999} in the framework of coalgebras over
commutative rings and are expounded in Sections \ref{bicomodulos},
\ref{funtores} and \ref{cohom}. In Section \ref{separable}, I give
a notion of homomorphism of corings, which leads to a pair of
adjoint functors (the induction functor and its adjoint, called
here ad-induction functor). The morphisms of entwining structures
\cite{Brzezinski:1999unp} are instances of homomorphisms of
corings in our setting. Finally, the separability of these
functors is characterized.

We use essentially the categorical terminology of
\cite{Stenstrom:1975}, with the exception of the term $K$--linear
category and functor, for $K$ a commutative ring (see, e.g.
\cite[I.0.2]{SaavedraRivano:1972}). There are, however, some minor
differences: The notation $X \in \cat{A}$ for a category $\cat{A}$
means that $X$ is an object of $\cat{A}$, and the identity
morphism attached to any object $X$ will be represented by the
object itself. The notation $\rmod{K}$ stands for the category of
all unital $K$--modules. The fact that $G$ is a right adjoint to
some functor $F$ will be denoted by $F \dashv G$. For the notion
of separable functor, the reader is referred to
\cite{Nastasescu/VandenBergh/VanOystaeyen:1989}. Finally, let
$f,g : X \rightarrow Y$ be a pair of morphisms of right modules
over a ring $R$, and let $k : K \rightarrow X$ be its equalizer
(that is, the kernel of $f-g$). We will say that a left
$R$--module $Z$ preserves the equalizer of $(f,g)$ if $k
\tensor{R} Z : K \tensor{R} Z \rightarrow X \tensor{R}Z$ is the
equalizer of the pair $(f \tensor{R} Z,g \tensor{R} Z)$. Of
course, every flat module ${}_RZ$ preserves all equalizers.


\section{Bicomodules and the cotensor product
functor}\label{bicomodulos}

I first recall from \cite{Sweedler:1975} the notion of coring. The
concepts of comodule and bicomodule over a coring are
generalizations of the corresponding notions for coalgebras. We
briefly state some basic properties of the cotensor product of
bicomodules. Similar associativity properties were studied in
\cite{Guzman:1989} in the framework of coseparable corings and in
\cite{AlTakhman:unp2001} for coalgebras over commutative rings.


Throughout this paper, $A, A', A'', \dots$ denote associative and
unitary algebras over a commutative ring $K$.

\begin{punto}\label{coringdef}\textbf{Corings.}
 An $A$--\emph{coring} is a three-tuple
$(\coring{C},\Delta_{\coring{C}},\epsilon_{\coring{C}})$
consisting of an $A$--bicomodule $\coring{C}$ and two
$A$--bimodule maps
\begin{equation}
\xymatrix{ \Delta_{\coring{C}} : \coring{C} \ar[r] & \coring{C}
\tensdia{A} \coring{C} & \epsilon_{\coring{C}} : \coring{C} \ar[r]
& A}
\end{equation}
such that the diagrams
\begin{equation}
\xymatrix{ \coring{C} \ar^{\Delta_{\coring{C}}}[rr]
\ar_{\Delta_{\coring{C}}}[d]  & & \coring{C} \tensdia{A}
\coring{C}
\ar^{\coring{C} \tensfun{A} \Delta_{\coring{C}}}[d] \\
\coring{C} \tensdia{A} \coring{C} \ar^{\Delta_{\coring{C}}
\tensfun{A} \coring{C}}[rr] & & \coring{C} \tensdia{A} \coring{C}
\tensdia{A} \coring{C}}
\end{equation}
and
\begin{equation}
\xymatrix{ \coring{C} \ar^{\Delta_{\coring{C}}}[r] \ar_{\cong}[dr]
& \coring{C} \tensdia{A} \coring{C} \ar^{\coring{C} \tensfun{A}
\epsilon_{\coring{C}}}[d] \\
& \coring{C} \tensdia{A} A } \qquad \xymatrix{ \coring{C}
\ar^{\Delta_{\coring{C}}}[r] \ar_{\cong}[dr] & \coring{C}
\tensdia{A} \coring{C}
\ar^{\epsilon_{\coring{C}} \tensfun{A} \coring{C}}[d] \\
& A \tensdia{A} \coring{C} }
\end{equation}
commute.
\end{punto}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

From now on, $\coring{C}, \coring{C}', \coring{C}'', \dots$ will
denote corings over $A, A', A'', \dots$, respectively.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{punto}\label{comoddef}\textbf{Comodules.}
A \emph{right $\coring{C}$--comodule} is a pair $(M, \rho_M)$
consisting of a right $A$--module $M$ and an $A$--linear map
$\rho_M : M \rightarrow M \tensor{A} \coring{C}$ such that the
diagrams
\begin{equation}
\xymatrix{ M \ar^{\rho_M}[rr] \ar_{\rho_M}[d] & {} & M \tensdia{A}
\coring{C} \ar^{M \tensfun{A} \Delta_{\coring{C}}}[d] \\
M \tensdia{A} \coring{C} \ar^{\rho_M \tensfun{A} \coring{C}}[rr] &
{} & M \tensdia{A} \coring{C} \tensdia{A} \coring{C} } \qquad
\xymatrix{ M \ar^{\rho_M}[r] \ar_{\cong}[dr] & M \tensdia{A}
\coring{C}
\ar^{M \tensfun{A} \epsilon_{\coring{C}}}[d] \\
& M \tensdia{A} A }
\end{equation}
commute. Left $\coring{C}$--comodules are similarly defined; we
use the notation $\lambda_M$ for their structure maps. A
\emph{morphism} of right $\coring{C}$--comodules $(M,\rho_M)$ and
$(N, \rho_N)$ is an $A$--linear map $f : M \rightarrow N$ such
that the following diagram is commutative.
\begin{equation}
\xymatrix{
M \ar^f[rr] \ar^{\rho_M}[d] & & N \ar^{\rho_N}[d] \\
M \tensdia{A} \coring{C} \ar^{f \tensfun{A} \coring{C}}[rr] & & N
\tensdia{A} \coring{C}}
\end{equation}
The $K$--module of all right $\coring{C}$--comodule morphisms from
$M$ to $N$ is denoted by $\hom{\coring{C}}{M}{N}$. The $K$--linear
category of all right $\coring{C}$--comodules will be denoted by
$\rcomod{\coring{C}}$. When $\coring{C} = A$, with the trivial
coring structure, the category $\rcomod{A}$ is just the category
of all right $A$--modules, which is `traditionally' denoted by
$\rmod{A}$.

Coproducts and cokernels in $\rcomod{\coring{C}}$ do exist and
can be already computed in $\rcomod{A}$. Therefore,
$\rcomod{\coring{C}}$ has arbitrary inductive limits. If
${}_A\coring{C}$ is a flat module, then $\rcomod{\coring{C}}$ is
easily proved to be an abelian category.
\end{punto}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{punto}\textbf{Bicomodules.}
Let $\rho_M : M \rightarrow M \tensor{A} \coring{C}$ be a comodule
structure over an $A'-A$--bimodule $M$, and assume that $\rho_M$
is $A'$--linear. For any right $A'$--module $X$, the right
$A$--linear map $X \tensfun{A'} \rho_{M} : X \tensor{A'} M
\rightarrow X \tensor{A'} M \tensor{A} \coring{C}$ makes $X
\tensor{A'} M$ a right $\coring{C}$--comodule. This leads to an
additive functor $- \tensor{A'} M : \rmod{A'} \rightarrow
\rcomod{\coring{C}}$. When $A' = A$ and $M = \coring{C}$, the
functor $- \tensor{A} \coring{C}$ is left adjoint to the
forgetful functor $U_A : \rcomod{\coring{C}} \rightarrow
\rcomod{A}$ (see \cite[Proposition 3.1]{Guzman:1989}, \cite[Lemma
3.1]{Brzezinski:2000unp}). Now assume that the $A'-A$--bimodule
$M$ is also a left $\coring{C}'$--comodule with structure map
$\lambda_M : M \rightarrow \coring{C}' \tensor{A} M$. It is clear
that $\rho_M : M \rightarrow M \tensor{A} \coring{C}$ is a
morphism of left $\coring{C}'$--comodules if and only if
$\lambda_M : M \rightarrow \coring{C}' \tensor{A'} M$ is a
morphism of right $\coring{C}$--comodules. In this case, we say
that $M$ is a $\coring{C}'-\coring{C}$--bicomodule. The
$\coring{C}'-\coring{C}$--bicomodules are the objects of a
$K$--linear category $\bcomod{\coring{C}'}{\coring{C}}$ whose
morphisms are those $A'-A$--bimodule maps which are morphisms of
$\coring{C}'$--comodules and of $\coring{C}$--comodules. Some
particular cases are now of interest. For instance, when
$\coring{C}' = A'$, the objects of the category
$\bcomod{A'}{\coring{C}}$ are the $A'-A$--bimodules with a right
$\coring{C}$--comodule structure $\rho_M : M \rightarrow M
\tensor{A} \coring{C}$ which is $A'$--linear.
\end{punto}

\begin{punto}\label{cotensorfun}\textbf{The cotensor product.}
Let $M \in \bcomod{\coring{C}'}{\coring{C}}$ and $N \in
\bcomod{\coring{C}}{\coring{C}''}$. We consider $M \tensor{A} N$
and $M \tensor{A} \coring{C} \tensor{A} N$  as
$\coring{C}'-\coring{C}''$--bicomodules with structure maps
\begin{equation}
\xymatrix{ M \tensdia{A} N \ar^{M \tensfun{A} \rho_N}[rr] & & M
\tensdia{A} N \tensdia{A''} \coring{C}'' & &  M \tensdia{A} N
\ar^{\lambda_M \tensfun{A} M }[rr] & & \coring{C}' \tensdia{A'} M
\tensdia{A} N}
\end{equation}
\begin{equation}
\xymatrix{ M \tensdia{A} \coring{C} \tensdia{A} N \ar^{M
\tensfun{A} \coring{C} \tensfun{A} \rho_N}[rrr] & & & M
\tensdia{A} \coring{C} \tensdia{A} N \tensdia{A''} \coring{C}''}
\end{equation}
\begin{equation}
\xymatrix{
 M \tensdia{A} \coring{C} \tensdia{A} N \ar^{\lambda_M \tensfun{A}
\coring{C} \tensfun{A} N}[rrr] & & & \coring{C}' \tensdia{A'} M
\tensdia{A} \coring{C} \tensdia{A} N}
\end{equation}
The map
\begin{equation}\label{precotensor}
\xymatrix{ M \tensdia{A} N \ar^{\rho_M \tensfun{A} N - M
\tensfun{A} \lambda_N}[rrr] & & & M \tensdia{A} \coring{C}
\tensdia{A} N}
\end{equation}
is then a $\coring{C}'-\coring{C}''$-bicomodule map. Let $M
\cotensor{\coring{C}} N$ denote the kernel of \eqref{precotensor}.
 If $\coring{C}'_{A'}$ and ${}_{A''}\coring{C}''$ preserve the equalizer of
 $(\rho_M \tensor{A} N,M \tensor{A}
\lambda_N)$, then $M \cotensor{\coring{C}} N$ is both a
$\coring{C}'$ and a $\coring{C}''$--subcomodule of $M \tensor{A}
N$ and, hence, it is a $\coring{C}'-\coring{C}''$--bicomodule.
\begin{proposition*}
Assume that $\coring{C}'_{A'}$ and ${}_{A''}\coring{C}''$
preserve the equalizer of $(\rho_M \tensor{A} N,M \tensor{A}
\lambda_N)$ for every $M \in \bcomod{\coring{C}'}{\coring{C}}$
and $N \in \bcomod{\coring{C}}{\coring{C}''}$. We have an
additive bifunctor
\begin{equation}\label{cotenbifun}
\xymatrix{
 - \cotensor{\coring{C}} - :
\bcomod{\coring{C}'}{\coring{C}} \times
\bcomod{\coring{C}}{\coring{C}''} \ar[rr] & &
\bcomod{\coring{C}'}{\coring{C}''}}
\end{equation}
In particular, the cotensor product bifunctor \eqref{cotenbifun}
is defined when $\coring{C}'_{A'}$ and ${}_{A''}\coring{C}''$ are
flat modules or when $\coring{C}$ is a coseparable $A$--coring in
the sense of \cite{Guzman:1989}.
\end{proposition*}

 In the special case $\coring{C}' = A'$ and $\coring{C}'' =
A''$, we have the bifunctor
\begin{equation}
\xymatrix{
- \cotensor{\coring{C}} - : \bcomod{A'}{\coring{C}}
\times \bcomod{\coring{C}}{A''} \ar[rr] & & \bcomod{A'}{A''}}
\end{equation}
and, if we further assume $A' = A'' = K$, we have the bifunctor
\begin{equation}
\xymatrix{ - \cotensor{\coring{C}} - : \rcomod{\coring{C}} \times
\lcomod{\coring{C}} \ar[rr] & & \rmod{K}}.
\end{equation}
\end{punto}

\begin{punto}\label{coass1}\textbf{Compatibility between tensor
and cotensor.} Let $M \in \bcomod{\coring{C}'}{\coring{C}}$ and $N
\in \bcomod{\coring{C}}{\coring{C}''}$ be bicomodules. For any
right $A'$--module $W$, consider the commutative diagram
\begin{equation}\label{primeaso}
\xymatrix{ & W \tensdia{A'} (M \cotensor{\coring{C}} N) \ar[r]
\ar^{\psi}[d] & W \tensdia{A'} (M \tensdia{A} N) \ar[r]
\ar^{\cong}[d] & W
\tensdia{A'} (M \tensdia{A} \coring{C} \tensdia{A} N) \ar^{\cong}[d] \\
0 \ar[r] & (W \tensdia{A'} M) \cotensor{\coring{C}} N \ar[r] & (W
\tensdia{A'} M) \tensdia{A} N \ar[r] & (W \tensdia{A'} M)
\tensdia{A} \coring{C} \tensdia{A} N },
\end{equation}
where $\psi$ is given by the universal property of the kernel in
the second row. This leads to the following
\begin{lemma*}It follows from \eqref{primeaso} that
$W_{A'}$ preserves the equalizer of $(\rho_M \tensor{A} N, M
\tensor{A} \lambda_N)$ if and only if $\psi : W \tensor{A'} ( M
\cotensor{\coring{C}} N) \cong (W \tensor{A'} M)
\cotensor{\coring{C}} N$. In particular, $\psi$ is an isomorphism
if $W_{A'}$ is flat.
\end{lemma*}

Next, we prove a basic fact concerning with the associativity of
cotensor product.
\begin{proposition*}
Let $M \in \bcomod{\coring{C}'}{\coring{C}}, N \in
\bcomod{\coring{C}}{\coring{C}''}, L \in
\bcomod{\coring{C}'''}{\coring{C}'}$.
%\begin{enumerate}
%\item
Assume that $\coring{C}'_{A'}, L_{A'}, L \tensor{A'}
\coring{C}'_{A'}$ and ${}_{A''}\coring{C}''$ preserve the
equalizer of $(\rho_M \tensor{A} N, M \tensor{A} \lambda_N)$, and
that ${}_{A}\coring{C}, {}_AN, {}_A\coring{C} \tensor{A} N$ and
$\coring{C}'''_{A'''}$ preserve the equalizer of $(\rho_L
\tensor{A'} M, L \tensor{A'} \lambda_M)$. Then we have a canonical
isomorphism of $\coring{C}'''-\coring{C}''$--bicomodules
\[
L \cotensor{\coring{C}'} (M \cotensor{\coring{C}} N) \cong (L
\cotensor{\coring{C'}} M) \cotensor{\coring{C}} N
\]
\end{proposition*}
\begin{proof}
Since $\coring{C}'_{A'}$ and ${}_{A''}\coring{C}''$ preserve the
equalizer of $(\rho_M \tensor{A} N, M \tensor{A} \lambda_N)$, we
know that it is a $\coring{C}'-\coring{C}''$--subbicomodule.
Analogously, $L \cotensor{\coring{C}'} M$ is a
$\coring{C}'''-\coring{C}$--subbicomodule of $L \tensor{A'} M$ In
the commutative diagram
\[
\xymatrix{ 0 \ar[r] & (L \cotensor{\coring{C}'} M)
\cotensor{\coring{C}} N \ar[r] \ar[d] & (L \tensor{A'} M)
\cotensor{\coring{C}} N \ar[r] \ar[d] & (L
\tensor{A'} \coring{C}' \tensor{A'} M) \cotensor{\coring{C}} N \ar[d] \\
0 \ar[r] & (L \cotensor{\coring{C}'} M) \tensor{A} N \ar[r] & (L
\tensor{A'} M ) \tensor{A} N \ar[r] & (L \tensor{A'} \coring{C}'
\tensor{A'} M) \tensor{A} N  }
\]
the second row is exact because ${}_AN$ preserves the equalizer
of $(\rho_L \tensor{A'} M, L \tensor{A'} \lambda_M)$. The
exactness of the first row is then deduced by using that
${}_A\coring{C} \tensor{A} N$ is assumed to preserve the
equalizer of $(\rho_L \tensor{A'} M, L \tensor{A'} \lambda_M)$.
Now, consider the commutative diagram with exact rows:
\[
\xymatrix{0 \ar[r] & (L \cotensor{\coring{C}'} M)
\cotensor{\coring{C}} N \ar[r] & (L \tensor{A'} M)
\cotensor{\coring{C}} N \ar[r] & (L
\tensor{A'} \coring{C}' \tensor{A'} M) \cotensor{\coring{C}} N \\
0 \ar[r] & L \cotensor{\coring{C}'} ( M \cotensor{\coring{C}} N)
\ar[r] \ar^{\psi_1}[u] & L \tensor{A'} (M \cotensor{\coring{C}}
N) \ar[r] \ar^{\psi_2}[u] & L \tensor{A'} \coring{C}' \tensor{A'}
(M \cotensor{\coring{C}} N) \ar^{\psi_3}[u]}
\]
Lemma \ref{coass1} gives the isomorphisms $\psi_2$ and $\psi_3$,
which induce the isomorphism $\psi_1$.
\end{proof}
\end{punto}

\section{Functors between comodule categories}\label{funtores}

This section contains technical facts concerning with $K$--linear
functors between categories of comodules over corings. Part of
these tools were first developed for coalgebras over commutative
rings in \cite{AlTakhman:unp2001} and \cite{AlTakhman:1999}.
Roughly speaking, I prove an analogue to Watts theorem, which
allow to represent good enough functors as cotensor product
functors. I also include a result which states that, under mild
conditions, a natural transformation gives a bicomodule morphism
at any bicomodule. This will be used in the statement and proof
of our separability theorems in Section \ref{separable}.

Let $\coring{C}$, $\coring{D}$ be corings over $K$--algebras $A$
and $B$, respectively, and consider a $K$--linear functor
\[
\xymatrix{ F : \rcomod{\coring{C}} \ar[r] & \rcomod{\coring{D}}}
\]
\begin{punto}\label{nat1}
Let $T$ be a $K$--algebra. For every $M \in
\bcomod{T}{\coring{C}}$, consider the homomorphism of
$K$--algebras
\begin{equation}\label{homobasico}
 T \cong \rend{T}{T} \rightarrow  \hom{\coring{C}}{T \tensdia{T}
M}{T \tensdia{T} M} \cong \hom{\coring{C}}{M}{M} \rightarrow
\hom{\coring{D}}{F(M)}{F(M)}
\end{equation}
which induces a left $T$--module structure over $F(M)$ such that
$F(M)$ becomes a $T-\coring{D}$--bicomodule. We have two
$K$--linear functors
\[
\xymatrix{ - \tensdia{T} F(-), F(- \tensdia{T} -) : \rcomod{T}
\times \bcomod{T}{\coring{C}} \ar[r] & \bcomod{}{\coring{D}} }
\]
We shall construct a natural transformation
\[
\xymatrix{ \nat{-}{-} : - \tensdia{T} F(-) \ar[r] & F(-
\tensdia{T} -) }
\]
Let $\nat{T}{M}$ be the unique isomorphism of
$\coring{D}$--comodules making commutative the diagram
\[
\xymatrix{ T \tensdia{T} F(M) \ar^{\nat{T}{M}}[rr] \ar^{\cong}[dr]
& &
F(T \tensdia{T} M) \ar^{\cong}[dl] \\
& F(M)  & }
\]
To prove that $\nat{T}{M}$ is natural at $T$, consider a
homomorphism $f : T \rightarrow T$ of right $T$--modules and
define $g : F(M) \rightarrow F(M)$ by $g(x) = f(1)x$ for every $x
\in F(M)$. Since $g$ is just the image under \eqref{homobasico} of
$f(1) \in T$, it follows that $g$ is a morphism of right
$\coring{D}$--comodules. Moreover, $g$ makes the following
diagram commutative:
\[
\xymatrix{
F(M) \ar^{g}[rr] & & F(M) \\
F(T \tensdia{T} M) \ar^{\cong}[u] \ar^{F(f \tensfun{T} M)}[rr] & &
F(T \tensdia{T} M) \ar^{\cong}[u]}
\]
In the diagram
\[
\xymatrix{
& F(M) \ar^{g}[rr]  & & F(M) \\
T \tensdia{T} F(M) \ar^{\cong}[ur] \ar^{f \tensfun{T} F(M)}[rr]
\ar_{\nat{T}{M}}[dd] &
& T \tensdia{T} F(M) \ar^{\cong}[ur] \ar_{\nat{T}{M}}[dd] & \\
 &  & &  \\
 F(T \tensdia{T} M) \ar_{\cong}[uuur] \ar^{F(f \tensfun{T} M)}[rr] & &
 F(T \tensdia{T} M) \ar_{\cong}[uuur]}
\]
the commutativity of the front rectangle, which gives that
$\nat{T}{M}$ is natural, follows from the commutativity of the
rest of the diagram. From Mitchell's theorem (\cite[Theorem 3.6.5]
{Popescu:1973}) we obtain a natural transformation
\[
\xymatrix{ \nat{-}{M} : - \tensdia{T} F(M) \ar[r] & F( -
\tensdia{T} M)}.
\]
Moreover, if $F$ preserves coproducts (resp. direct limits, resp.
inductive limits) then $\nat{X}{M}$ is an isomorphism for $X_T$
projective (resp. flat, resp. any right $T$-module).
\end{punto}

\begin{proposition}\label{natpreserves} If the functor $F$
preserves coproducts, then $\nat{-}{-} : - \tensor{T} F(-)
\rightarrow F(- \tensor{T} -)$ is a natural transformation.
Moreover, if $F$ preserves direct limits, then $\nat{X}{-}$ is a
natural isomorphism for every flat right $T$--module $X$.
Finally, if $F$ preserves inductive limits, then $\nat{-}{-}$ is
a natural isomorphism.
\end{proposition}
\begin{proof}
By \ref{nat1}, $\nat{-}{M}$ is natural for every $M \in
\bcomod{T}{\coring{C}}$. Thus, we have only to show that
$\nat{X}{-}$ is natural for every $X \in \rcomod{T}$. We argue
first for $X = T$. Let $f : M \rightarrow N$ be a homomorphism in
$\bcomod{T}{\coring{C}}$. From the diagram
\[
\xymatrix{
 & F(M) \ar^{F(f)}[rr]  & & F(N) \\
T \tensdia{T} F(M) \ar^{\cong}[ur] \ar^{T \tensfun{T} F(f)}[rr]
\ar_{\nat{T}{M}}[dd] &
& T \tensdia{T} F(M) \ar^{\cong}[ur] \ar_{\nat{T}{N}}[dd] & \\
 &  & &  \\
 F(T \tensdia{T} M) \ar_{\cong}[uuur] \ar^{F(T \tensdia{T} f)}[rr] & &
 F(T \tensdia{T} N) \ar_{\cong}[uuur]}
\]
we get that $\nat{A}{-}$ is natural. Now, use a free presentation
$T^{(\Omega)} \rightarrow X$ to obtain that $\nat{X}{-}$ is
natural for a general $X_T$. The rest of the statements are easily
derived from this.
\end{proof}


\begin{lemma}\label{FunoFdos}
Let $\eta : F_1 \rightarrow F_2$ a natural transformation, where
$F_1, F_2 : \rcomod{\coring{C}} \rightarrow \rcomod{\coring{D}}$
are $K$--linear functors which preserve coproducts.
\begin{enumerate}
\item
For every $M \in \bcomod{T}{\coring{C}}$, $\eta_M : F_1(M)
\rightarrow F_2(M)$ is a $T-\coring{D}$--bicomodule homomorphism.
\item Given $X \in \rcomod{T}$ and $M \in \bcomod{T}{\coring{C}}$, the diagram
\begin{equation}\label{etas}
\xymatrix{ F_1(X \tensdia{T} M) \ar^{\eta_{X \tensdia{T} M}}[rr] &
&
F_2(X \tensdia{T} M) \\
X \tensdia{T} F_1(M) \ar^{X \tensfun{T} \eta_M}[rr]
\ar^{\nat{X}{M}}[u]& & X \tensdia{T} F_2(M) \ar^{\nat{X}{M}}[u]}
\end{equation}
is commutative.
\end{enumerate}
\end{lemma}
\begin{proof}
We need just to prove that \eqref{etas} commutes for $X = T$. In
this case, the diagram can be factored out as
\[
\xymatrix{ F_1(T \tensdia{T} M) \ar^{\eta_{T \tensdia{T} M}}[rr]
\ar^{\cong}[dr]& & F_2(T \tensdia{T} M) \ar^{\cong}[dr]&
\\
 & F_1(M) \ar^(.3){\eta_M}[rr] & \ar[u] & F_2(M)
\\
T \tensdia{T} F_1(M) \ar^{\cong}[ur] \ar^{T \tensfun{T}
\eta_M}[rr] \ar^{\nat{T}{M}}[uu] & & T \tensdia{T} F_2(M)
\ar^{\cong}[ur] \ar@{-}^(.6){\nat{T}{M}}[u]& }
\]
Since all trapezia and triangles commute, the back rectangle does,
as desired.
\end{proof}

\begin{lemma}\label{formula}
Let $T, S$ be $K$--algebras and assume that $F :
\rcomod{\coring{C}} \rightarrow \rcomod{\coring{D}}$ preserves
coproducts. Given $X \in \rcomod{S}, Y \in \bcomod{S}{T}$ and $M
\in \bcomod{T}{\coring{C}}$, the following formula holds
\[
\nat{X}{Y \tensdia{T} M} \circ (X \tensfun{S} \nat{Y}{M}) = \nat{X
\tensor{S} Y}{M}
\]
\end{lemma}
\begin{proof}
The equality will be first proved for $X = S$. Consider the
diagram
\[
\xymatrix{ S \tensdia{S} F(Y \tensdia{T} M) \ar^{\cong}[rr]
\ar^{\nat{S}{Y \tensdia{T} M}}[dd] & & F(Y
\tensdia{T} M) \ar@{=}'[d][dd] & \\
 & S \tensdia{S} Y \tensdia{T} F(M) \ar^{S \tensfun{S} \nat{Y}{M}}[ul]
 \ar^{\nat{S \tensdia{S} Y}{M}}[dl] \ar^{\cong}[rr] & & Y \tensdia{T} F(M)
 \ar^{\nat{Y}{M}}[ul] \ar^{\nat{Y}{M}}[dl] \\
 F(S \tensdia{S} Y \tensdia{T} M) \ar^{\cong}[rr] & & F(Y \tensdia{T} M) & }
\]
The back rectangle is commutative by definition of $\nat{S}{Y
\tensor{T} M}$, while the other two parallelograms are commutative
because $\nat{-}{-}$ is natural. Therefore, the right triangle is
commutative. The equality is now easily extended for $X =
S^{(\Omega)}$ and, by using a free presentation $S^{(\Omega)}
\rightarrow X \rightarrow 0$, for any $X$.
\end{proof}


\begin{punto}\textbf{Natural transformations and
bicomodule morphisms.}\label{hacebicom}  Let $M \in
\bcomod{\coring{C}'}{\coring{C}}$ a bicomodule. A functor $F:
\rcomod{\coring{C}} \rightarrow \rcomod{\coring{D}}$ is said to
be \emph{$M$--compatible} if $\nat{\coring{C}'}{M}$ and
$\nat{\coring{C}'\tensor{A'}\coring{C}'}{M}$ are isomorphisms. By
Proposition \ref{natpreserves}, the functor $F$ is
$M$--compatible for every bicomodule $M$ either if the functor
$F$ preserves inductive limits or $\coring{C}_A'$ is flat and $F$
preserves direct limits. In case that $F$ is $M$--compatible,
define $\lambda_{F(M)}$ as the unique $A'$--linear map making
commutative the diagram
\[
\xymatrix{
F(M) \ar^{\lambda_{F(M)}}[rr] \ar_{F(\lambda_M)}[dr] & &
\coring{C}' \tensdia{A'} F(M) \ar^{\nat{\coring{C}'}{M}}[dl] \\
& F(\coring{C}' \tensdia{A'} M) & }
\]
\begin{proposition*}
Let $F$ be an $M$--compatible functor which preserves coproducts.
The $A'$--linear map $\lambda_{F(M)}$ is a left
$\coring{C}'$--comodule structure on $F(M)$ such that $F(M)$
becomes a $\coring{C}'-\coring{D}$--bicomodule. Moreover, given
$F_1, F_2 : \rcomod{\coring{C}} \rightarrow \rcomod{\coring{D}}$
$M$--compatible functors and a natural transformation $\eta: F_1
\rightarrow F_2$, the map $\eta_M : F_1(M) \rightarrow F_2(M)$ is
a $\coring{C}'-\coring{D}$--bicomodule homomorphism.
\end{proposition*}
\begin{proof}
In order to prove that the coaction $\lambda_{F(M)}$ is
coassociative, let us consider the diagram:
\[
\xymatrix{ &  F(M) \ar[dl]_{\lambda_{F(M)}}
\ar[rr]^{\lambda_{F(M)}} \ar@{=}[d] & & \coring{C}' \tensor{A'}
F(M) \ar[dl]|{\Delta_{\coring{C}'} \tensfun{A'} F(M)}
\ar[ddd]^{\Upsilon_{\coring{C}',M}} \\
\coring{C}' \tensor{A'} F(M) \ar[rr]|(.6){\coring{C}' \tensfun{A'}
\lambda_{F(M)}} \ar[dr]|{\coring{C}' \tensfun{A'} F(M)}
\ar[ddd]^{\Upsilon_{\coring{C}',M}} & \ar@{=}[d] & \coring{C} '
\tensor{A'} \coring{C}' \tensor{A'} F(M) \ar[dl]|{\coring{C}'
\tensfun{A'} \Upsilon_{\coring{C}',M}}
\ar[ddd]^{\Upsilon_{\coring{C}' \tensor{A'} \coring{C}',M}} & \\
 &  \coring{C}' \tensor{A'} F(\coring{C}' \tensor{A'} M)
 \ar[ddr]|(.35){\Upsilon_{\coring{C}',\coring{C}' \tensor{A'} M}} \ar@{=}[d] & & \\
 & F(M) \ar[dl]^{F(\lambda_{M})} \ar[rr]^(.4){F(\lambda_{M})}  & & F(\coring{C}' \tensor{A'} M)
 \ar[dl]^{F(\Delta_{\coring{C}'} \tensfun{A'} M)} \\
F(\coring{C}' \tensor{A'} M) \ar[rr]^{F(\coring{C'} \tensfun{A'}
\lambda_M)} & & F(\coring{C}' \tensor{A'} \coring{C}' \tensor{A}'
M) & }
\]
We want to see that the top side is commutative. Since $F$ is
assumed to be $M$--compatible, we have just to prove that the
mentioned side is commutative after composing with the isomorphism
$\nat{\coring{C}' \tensor{A'} \coring{C}'}{M}$. This is deduced
by using Lemma \ref{formula}, in conjunction with the naturality
of $\nat{-}{M}$ and the very definition of $\lambda_{F(M)}$. The
counitary property is deduced from the commutative diagram
\[
\xymatrix{ A' \tensdia{A'} F(M) \ar^{\nat{A'}{M}}[rr] & & F(A'
\tensdia{A'} M)\\
& F(M) \ar_{\cong}[ul] \ar^{\cong}[ur] \ar_{\lambda_{F(M)}}[dl] \ar^{F(\lambda_M)}[dr]&  \\
\coring{C}' \tensdia{A'} F(M) \ar^{\epsilon_{\coring{C}'}
\tensfun{A'} F(M)}[uu] \ar^{\nat{\coring{C}'}{M}}[rr] & &
F(\coring{C}' \tensdia{A'} M) \ar_{F(\epsilon_{\coring{C}'}
\tensfun{A'} M)}[uu] }
\]
To prove the second statement, let us consider the diagram
\[
\xymatrix{ F_1(M) \ar^{\eta_M}[rr] \ar^{\lambda_{F_1(M)}}[dd]
\ar^{F_1(\lambda_M)}[dr]& & F_2(M) \ar^{F_2(\lambda_M)}[dr]
\ar@{-}^(.7){\lambda_{F_2(M)}}[d] & \\
& F_1(\coring{C}' \tensdia{A'} M) \ar^(.3){\eta_{\coring{C}'
\tensfun{A'} M}}[rr]
\ar^{\nat{\coring{C}'}{M}}[dl]& \ar[d] & F_2(\coring{C}' \tensdia{A} M) \ar^{\nat{\coring{C}'}{M}}[dl]\\
 \coring{C}' \tensdia{A'} F_1(M) \ar^{\coring{C}' \tensfun{A'}
\eta_M}[rr]& & \coring{C}' \tensdia{A'} F_2(M) &}
\]
Both triangles commute by definition of $\lambda_{F_1(M)}$ and
$\lambda_{F_2(M)}$, and the upper trapezium is commutative because
$\eta$ is natural. The bottom trapezium commutes by Proposition
\ref{FunoFdos}. Therefore, the back rectangle is commutative,
which just says that $\eta_M$ is a morphism of left
$\coring{C}'$--comodules. This finishes the proof.
\end{proof}
\end{punto}


\begin{theorem}
Assume that $\coring{C}_A$ is flat. If $F : \rcomod{\coring{C}}
\rightarrow \rcomod{\coring{D}}$ is exact and preserves direct
limits (e.g. if $F$ is an equivalence of categories), then $F$ is
naturally isomorphic to $- \cotensor{\coring{C}} F(\coring{C})$.
\end{theorem}
\begin{proof}
Let $\rho_N \rightarrow N \tensor{A} \coring{C}$ be a right
$\coring{C}$--comodule. We have the following diagram with exact
rows
\[
\xymatrix{ 0 \ar[r] & N \cotensor{\coring{C}} F(\coring{C})
\ar^{\cong}@{-->}[d]\ar[r] & N \tensdia{A} F(\coring{C})
\ar^{\rho_N \tensfun{A} F(\coring{C}) - N \tensfun{A}
\lambda_{F(\coring{C})}}[rrrr]
\ar_{\cong}^{\nat{N}{\coring{C}}}[d]& & & & N \tensdia{A}
\coring{C} \tensdia{A} F(\coring{C})
\ar_{\cong}^{\nat{N \tensdia{A} A}{\coring{C}}}[d] \\
0 \ar[r] & F(N) \ar[r] & F(N \tensdia{A} \coring{C}) \ar^{F(\rho_N
\tensfun{A} \coring{C} - N \tensfun{A} \Delta_{\coring{C}}
)}[rrrr] & & & & F(N \tensdia{A} \coring{C} \tensdia{A}
\coring{C})}
\]
where the desired isomorphism is given by the universal property
of the kernel.
\end{proof}

\section{Co-hom functors}\label{cohom}

This section contains a quick study of the left adjoint to a
cotensor product functor, if it does exist. The presentation is
inspired from the one given in \cite[1.8]{Takeuchi:1977} for
coalgebras over a field.

Let $\coring{C}$, $\coring{D}$ be corings over $K$--algebras $A$
and $B$, respectively.

\begin{definition}
A bicomodule $N \in \bcomod{\coring{C}}{\coring{D}}$ is said to be
\emph{quasi-finite} as a right $\coring{D}$--comodule if the
functor $- \tensor{A} N : \rcomod{A} \rightarrow
\rcomod{\coring{D}}$ has a left adjoint $\cohom{\coring{D}}{N}{-}
: \rcomod{\coring{D}} \rightarrow \rcomod{A}$. This functor is
called the \emph{co-hom} functor associated to $N$.
\end{definition}

The natural isomorphism which gives the adjunction will be denoted
by
\begin{equation}\label{adjuncion}
\xymatrix{ \adiso{X}{Y} : \hom{A}{\cohom{\coring{D}}{N}{X}}{Y}
\ar[rr] & & \hom{\coring{D}}{X}{Y \tensdia{A} N},}
\end{equation}
for $Y \in \rcomod{A}$, $X \in \rcomod{\coring{D}}$.


\begin{punto}\label{casichico}
Let $\theta : Id \rightarrow \cohom{\coring{D}}{N}{-} \tensor{A}
N$ be the unit of the adjunction \eqref{adjuncion}. Therefore,
the isomorphism $\adiso{X}{Y}$ is given by the assignment $f
\mapsto (f \tensor{A} N) \theta_X$. In particular, the map
\[
\xymatrix{ X \ar^(.3){\theta_X}[r]  & \cohom{\coring{D}}{N}{X}
\tensdia{A} N \ar^{id \tensfun{A} \lambda_N }[rr]  &
&\cohom{\coring{D}}{N}{X} \tensdia{A} \coring{C} \tensdia{A} N}
\]
determines an $A$--linear map
\[
\xymatrix{ \rho_{\cohom{\coring{D}}{N}{X}} :
\cohom{\coring{D}}{N}{X} \ar[r] & \cohom{\coring{D}}{N}{X}
\tensdia{A} \coring{C}}
\]
such that $(id \tensor{A} \lambda_N) \theta_X =
(\rho_{\cohom{\coring{D}}{N}{X}} \tensor{A} id) \theta_X$. The
coaction $\rho_{\cohom{\coring{D}}{N}{X}}$ makes
$\cohom{\coring{D}}{N}{X}$ a right $\coring{C}$--comodule.
Therefore we have a functor $\cohom{\coring{D}}{N}{-} :
\rcomod{\coring{D}} \rightarrow \rcomod{\coring{C}}$.
\end{punto}

\begin{punto}\label{adjcohom}
The following is a basic tool in our investigation.

\begin{proposition*}
Let $N$ be a $\coring{C}-\coring{D}$--bicomodule. Assume that
${}_B\coring{D}$ preserves the equalizer of $(\rho_Y \tensor{A} N,
Y \tensor{A} \lambda_N)$ for every right $\coring{C}$--comodule
$Y$ (e.g., ${}_B\coring{D}$ is flat or $\coring{C}$ is a
coseparable $A$--coring in the sense of \cite{Guzman:1989}).
\begin{enumerate}
\item
If $N$ is quasi-finite as a right $\coring{D}$--comodule, then the
natural isomorphism \eqref{adjuncion} restricts to an isomorphism
$\hom{\coring{C}}{\cohom{\coring{D}}{N}{X}}{Y} \cong
\hom{\coring{D}}{X}{Y \cotensor{\coring{C}} N}$. Therefore,
$\cohom{\coring{D}}{N}{-}$ is left adjoint to $-
\cotensor{\coring{C}} N$.
\item Conversely, if
$- \cotensor{\coring{C}} N : \rcomod{\coring{C}} \rightarrow
\rcomod{\coring{D}}$ has a left adjoint, then $N$ is quasi-finite
as a right $\coring{C}$--comodule.
\end{enumerate}
\end{proposition*}
\begin{proof}
1. We need to prove that if $f \in
\hom{A}{\cohom{\coring{D}}{N}{X}}{Y}$, then $f$ is
$\coring{C}$--colinear if and only if the image of $(f \tensor{A}
N)\theta_X$ is included in $Y \cotensor{\coring{C}} N$. But these
are straightforward computations in view of the definition of the
contensor product. \\
2. Since the inclusion $\coring{C} \cotensor{\coring{C}} N
\subseteq \coring{C} \tensor{A} N$ splits-off, we get from
\ref{coass1} the natural isomorphism $(- \tensor{A} \coring{C})
\cotensor{\coring{C}} N \cong - \tensor{A} ( \coring{C}
\cotensor{\coring{C}} N) \cong - \tensor{A} N$. Now, the functor $
- \tensor{A} \coring{C}$ is right adjoint \cite[Lemma 3.1]
{Brzezinski:2000unp} to the forgetful functor $U_A :
\rcomod{\coring{C}} \rightarrow \rcomod{A}$ and, by hypothesis, $
- \cotensor{\coring{C}} N$ is right adjoint to the functor
$\cohom{\coring{D}}{N}{-} : \rcomod{\coring{D}} \rightarrow
\rcomod{\coring{C}}$. This implies that $- \tensor{A} N$ is right
adjoint to $U_A \cohom{\coring{D}}{N}{-}$, as desired.
\end{proof}
\end{punto}

\begin{example}
Let $N \in \bcomod{\coring{C}}{B}$. Then $N_B$ is quasi-finite if
and only if $- \tensor{A} N: \rcomod{A} \rightarrow \rcomod{B}$
has a left adjoint, that is, if and only if ${}_AN$ is finitely
generated and projective. In such a case, the left adjoint is $-
\tensor{B} \hom{A}{N}{A} : \rcomod{B} \rightarrow
\rcomod{\coring{C}}$. Notice that taking $B = K$, we obtain a
canonical structure of right $\coring{C}$--comodule on $N^* =
\hom{A}{N}{A}$ for every left $\coring{C}$--comodule $N$ such
that $N$ finitely generated and projective as a left $A$--module.
\end{example}

\begin{example}\label{restr}
Given an $A$--coring $\coring{C}$ and an $K$--algebra homomorphism
$\rho : A \rightarrow B$, we can consider the functor $-
\tensor{A} \coring{C}: \rcomod{B} \rightarrow \rcomod{\coring{C}}$
which is already the composite of the restriction of scalars
functor $(-)_{\rho} : \rcomod{B} \rightarrow \rcomod{A}$ and the
functor $- \tensor{A} \coring{C} : \rcomod{A} \rightarrow
\rcomod{\coring{C}}$. Since these functors have both left
adjoints, given, respectively, by the induction functor $ -
\tensor{A} B : \rcomod{A} \rightarrow \rcomod{B}$ and the
underlying functor $U_A : \rcomod{\coring{C}} \rightarrow
\rcomod{A}$, we get that the composite functor $- \tensor{A} B :
\rcomod{\coring{C}} \rightarrow \rcomod{B}$ is a left adjoint to
$- \tensor{A} \coring{C} : \rcomod{B} \rightarrow
\rcomod{\coring{C}}$. Clearly, $- \tensor{A} \coring{C} \cong -
\tensor{B} (B \tensor{A} \coring{C})$ and, thus, $B \tensor{A}
\coring{C}$ becomes a quasi-finite comodule.
\end{example}

\section{Separable homomorphisms of corings}\label{separable}

I propose a notion of homomorphism of corings which generalizes
both the concept of morphism of entwining structures
\cite{Brzezinski:1999unp} and the coring maps originally
considered in \cite{Sweedler:1975}. An induction functor is
constructed, which is shown to have a right adjoint, called
ad-induction functor. The separability of these two functors is
characterized in terms which generalize both the previous results
on rings \cite{Nastasescu/VandenBergh/VanOystaeyen:1989} and on
coalgebras \cite{Castano/Gomez/Nastasescu:1997}. Our approach
rests on the fundamental characterization of the separability of
adjoint functors given in \cite{Rafael:1990} and
\cite{DelRio:1992}.

Consider an $A$--coring $\coring{C}$ and a $B$--coring
$\coring{D}$, where $A$ and $B$ are $K$--algebras.

\begin{definition}
 A \emph{coring homomorphism} is a pair
$(\varphi,\rho)$, where $\rho: A \rightarrow B$ is a homomorphism
of $K$--algebras and $\varphi: \coring{C} \rightarrow \coring{D}$
is a homomorphism of $A$--bimodules and such that the following
diagrams are commutative:
\[
\xymatrix{
\coring{C} \ar^{\Delta_{\coring{C}}}[rr]
\ar^{\varphi}[dd] & & \coring{C}
\tensdia{A} \coring{C} \ar^{\varphi \tensfun{A} \varphi}[dr] & \\
 & & & \coring{D} \tensdia{A} \coring{D} \ar^{\omega_{\coring{D},\coring{D}}}[dl]\\
\coring{D} \ar^{\Delta_{\coring{D}}}[rr] & & \coring{D}
\tensdia{B} \coring{D} & } \qquad \xymatrix{ \coring{C}
\ar^{\epsilon_{\coring{C}}}[rr]
\ar^{\varphi}[dd]& & A \ar^{\rho}[dd] \\
& & & \\
\coring{D} \ar^{\epsilon_{\coring{D}}}[rr] & & B },
\]
where $\omega_{\coring{D},\coring{D}} : \coring{D} \tensdia{A}
\coring{D} \rightarrow \coring{D} \tensdia{B} \coring{D}$ is the
canonical map induced by $\rho : A \rightarrow B$.
\end{definition}

Throughout this section, we consider a coring homomorphism
$(\varphi,\rho) : \coring{C} \rightarrow \coring{D}$. We will
define the induction and ad-induction functors connecting the
categories of comodules $\rcomod{\coring{C}}$ and
$\rcomod{\coring{D}}$.

\begin{punto}\label{sigma}
We start with some unavoidable technical work. For every
$B$--bimodule $X$, let us denote by $\sigma_X : B \tensor{A} X
\rightarrow X \tensor{B} B$ the $B$--bimodule morphism given by $b
\tensor{A} x \mapsto bx \tensor{B} 1$. Given $B$--bimodules $X$,
$Y$, a straightforward computation shows that the diagram
\begin{equation}\label{xy1}
\xymatrix{ B \tensdia{A} X \tensdia{A} Y \ar^{B \tensfun{A}
\omega_{X,Y}}[d]
\ar^{\sigma_X \tensfun{A} Y}[rr]&  & X \tensdia{B} B \tensdia{A} Y \ar^{X \tensfun{B} \sigma_Y}[d]\\
B \tensdia{A} X \tensdia{B} Y \ar^{\sigma_{X \tensdia{B} Y}}[rr] &
& X \tensdia{B} Y \tensdia{B} B}
\end{equation}
commutes, where $\omega_{X,Y} : X \tensor{A} Y \rightarrow X
\tensor{B} Y$ is the obvious map. We have as well that for every
homomorphism of $B$--bimodules $f : X \rightarrow Y$ the following
diagram is commutative:
\begin{equation}\label{xy2}
\xymatrix{ B \tensdia{A} X \ar^{B \tensfun{A} f}[rr]
\ar^{\sigma_X}[d]& &
B \tensdia{A} Y \ar^{\sigma_Y}[d]\\
X \tensdia{B} B \ar^{f \tensfun{B} B }[rr] & & Y \tensdia{B} B}
\end{equation}
\end{punto}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{punto}\label{labm}\textbf{The induction functor.}
Let $\lambda_M : M \rightarrow \coring{C} \tensor{A} M$ be a left
$\coring{C}$--comodule. Define $\widetilde{\lambda}_M : M
\rightarrow \coring{D} \tensor{A} M$ as the composite map
\[
\xymatrix{ M \ar_{\widetilde{\lambda}_M}[rr] \ar_{\lambda_M}[dr]&
& \coring{D}
\tensdia{A} M \\
& \coring{C} \tensdia{A} M \ar_{\varphi \tensfun{A} M}[ur]&}
\]
and $\lambda_{B \tensor{A} M} : B \tensor{A} M \rightarrow
\coring{D} \tensor{B} B \tensor{A} M$ as
\[
\xymatrix{ B \tensdia{A} M \ar^{\lambda_{B \tensdia{A} M}}[rr]
\ar_{B \tensfun{A} \widetilde{\lambda}_M}[dr]& & \coring{D} \tensdia{B} B \tensdia{A} M\\
 & B \tensdia{A} \coring{D} \tensdia{A} M \ar_{\sigma_{\coring{D}} \tensfun{A} M}[ur]& }
\]
\begin{proposition*}
The homomorphism of left $B$--modules $\lambda_{B \tensor{A} M}$
endows $B \tensor{A} M$ with a structure of left
$\coring{D}$--comodule. This gives a functor $B \tensor{A} - :
\lcomod{\coring{C}} \rightarrow \lcomod{\coring{D}}$.
\end{proposition*}
\begin{proof}
Consider the diagram
\[
\xymatrix{ M \ar^{\lambda_M}[ddr]
\ar^{\widetilde{\lambda}_{M}}[rrrr] \ar@<0.4pt>[rrrr]
\ar_{\widetilde{\lambda}_M}[dddd] \ar@<0.4pt>[dddd]
\ar^{\lambda_M}[drr]& \ar@{}[dddr]|{(2)} & \ar@{}[dddr]|{(3)}&
&\coring{D} \tensdia{A} M \ar_{\coring{D} \tensfun{A}
\widetilde{\lambda}_M}[ddl] \ar@<0.4pt>[ddl]
\ar^{\coring{D} \tensfun{A} \lambda_M}[d]\\
 & & \coring{C} \tensdia{A} M \ar^{\varphi \tensfun{A} M}[urr]
 \ar^{\coring{C} \tensfun{A} \lambda_M}[dd] & &
 \coring{D} \tensdia{A} \coring{C} \tensdia{A} M \ar^{\coring{D} \tensfun{A} \varphi \tensfun{A} M}[dl]\\
 &\coring{C} \tensdia{A} M \ar^{\varphi \tensfun{A} M}[ddl] \ar^{\Delta_{\coring{C}} \tensfun{A} M }[dr] &
 &
 \coring{D} \tensdia{A} \coring{D} \tensdia{A} M
 \ar_{\omega_{\coring{D},\coring{D}} \tensfun{A} M}[ddr]
 \ar@<0.4pt>[ddr]& \\
 & & \coring{C} \tensdia{A} \coring{C} \tensdia{A} M
 \ar^{\varphi \tensfun{A} \varphi \tensfun{A} M}[ur] \ar@{}[d]|{(1)} & & \\
\coring{D} \tensdia{A} M \ar_{\Delta_{\coring{D}} \tensfun{A} M
}[rrrr] \ar@<0.4pt>[rrrr]& & & & \coring{D} \tensdia{B} \coring{D}
\tensdia{A} M}
\]
The pentagon labeled as (1) commutes since $\varphi$ is a
homomorphism of corings. The four-edged diagram (2) is commutative
since $M$ is a left comodule. The commutation of the quadrilateral
(3) follows easily from the displayed decomposition of $\coring{D}
\tensor{A} \widetilde{\lambda}_M$. Therefore, the pentagon with
bold arrows commutes. Now consider the diagram
\[
\xymatrix{
B \tensor{A} M \ar@/^2pc/[rrr]|{\lambda_{B \tensfun{A} M}} %%
\ar[dr]|{B \tensfun{A} \widetilde{\lambda}_M} %%
\ar[dd]|{B \tensfun{A} \widetilde{\lambda}_M} %%
\ar@/_4pc/[dddd]|(.4){\lambda_{B \tensor{A} M}} %%
\ar@{}[dddr]|{(4)} &
 & &   \coring{D} \tensor{B} B \tensor{A} M
\ar[dd]|{\coring{D} \tensfun{B} B \tensfun{A} \widetilde{\lambda}_M} %%
\ar@/^4pc/[dddd]|(.6){\coring{D} \tensfun{B} \lambda_{B \tensor{A} M}} \\ %%%%%%%%%%%%%%%%%%%%
& B \tensor{A} \coring{D} \tensor{A} M
\ar^{\sigma_{\coring{D}} \tensfun{A} M}[urr] %%
\ar^{B \tensfun{A} \coring{D} \tensfun{A}
\widetilde{\lambda}_M}[d] %%
\ar@{}[rr]|{(5)}&
& &  \\ %%%%%%%%%%%%%%%%%%%
B \tensor{A} \coring{D} \tensor{A} M
\ar[dd]|{\sigma_{\coring{D} \tensfun{A} M}} %%
\ar_{B \tensfun{A} \Delta_{\coring{D}} \tensfun{A} M}[dr] & B
\tensor{A} \coring{D} \tensor{A} \coring{D} \tensor{A} M
\ar^{B \tensfun{A} \omega_{\coring{D},\coring{D}} \tensfun{A} M}[d] %%
\ar^{\sigma_{\coring{D}} \tensfun{A} \coring{D} \tensfun{A} M}[rr]
& & \coring{D} \tensor{B} B \tensor{A} \coring{D} \tensor{A} M
\ar[dd]|{\coring{D} \tensfun{B} \sigma_{\coring{D}} \tensfun{A} M}
  \\ %%%%%%%%%%%%%%%%%%%
& B \tensor{A} \coring{D} \tensor{B} \coring{D} \tensor{A} M
\ar[drr]|{\sigma_{\coring{D} \tensor{B} \coring{D}} \tensfun{A} M} %%
\ar@{}[rr]|{(6)} & &  \\ %%%%%%%%%%%%%%%%%%%
\coring{D} \tensor{B} B \tensor{A} M
\ar@/_2pc/[rrr]|{\Delta_{\coring{D}}
\tensfun{B} B \tensfun{A} M} %%
\ar@{}[ur]|{(7)} & & &   \coring{D} \tensor{B} \coring{D}
\tensor{B} B \tensor{A} M }
\]

We have proved before that (4) is commutative. Moreover, (5) is
obviously commutative and (6) and (7) commute by \eqref{xy1} and
\eqref{xy2}. It follows that the outer curved diagram commutes,
which gives the pseudo-coassocitative property for the coaction
$\lambda_{B \tensor{A} M}$. To check the counitary property, let
$\iota_M : M \rightarrow A \tensor{A} M$ be the canonical
isomorphism. We get from the commutative diagram
\[
\xymatrix{ M \ar^{\widetilde{\lambda}_M}[rr] \ar_{\iota_M}[dd]
\ar_{\lambda_M}[dr] & &
\coring{D} \tensor{A} M \ar^{\epsilon_{\coring{D}} \tensfun{A} M}[dd] \\
& \coring{C} \tensor{A} M \ar^{\epsilon_{\coring{C}} \tensfun{A}
M}[dl]
\ar_{\varphi \tensfun{A} M}[ur] & \\
A \tensor{A} M \ar_{\rho \tensfun{A} M}[rr] & & B \tensor{A} M }
\]
that the diagram
\[
\xymatrix{ B \tensor{A} M \ar^{\lambda_{B \tensor{A} M}}[rrrr] %%
\ar|{B \tensfun{A} \widetilde{\lambda}_M}[drr] %%
\ar@/_1pc/|(.7){\iota_{B \tensor{A} M}}[dddrrrr] %%
\ar^{B \tensfun{A} \iota_M}[ddd] %%
& & & & \coring{D} \tensor{B} B \tensor{A} M
\ar^{\epsilon_{\coring{D}} \tensfun{B} B \tensfun{A} M}[ddd] \\
& & B \tensor{A} \coring{D} \tensor{A} M \ar_{\sigma_{\coring{D}} \tensfun{A} M}[urr] %%
\ar|(.6){B \tensfun{A} \epsilon_{\coring{D}} \tensfun{A} M}[dd] & & \\
& & & & \\
B \tensor{A} A \tensor{A} M \ar_{B \tensfun{A} \rho \tensfun{A}
M}[rr] & & B \tensor{A} B \tensor{A} M \ar_{\sigma_{B} \tensfun{A}
M}[rr] & & B \tensor{B} B \tensor{A} M}
\]
commutes, where $\iota_{B \tensor{A} M} : B \tensor{A} M
\rightarrow B \tensor{B} B \tensor{A} M$ denotes the canonical
isomorphism. Therefore, $\lambda_{B \tensor{A} M} : B \tensor{A} M
\rightarrow \coring{D} \tensor{B} B \tensor{A} M$ is a left
$\coring{D}$--comodule structure map. In order to show that the
assignment $M \mapsto B \tensor{A} M$ is functorial, we will prove
that $B \tensor{A} f$ is a homomorphism of left
$\coring{D}$--comodules for every morphism $f : M \rightarrow N$
in $\lcomod{\coring{C}}$. So, we have to show that the outer
rectangle in the following diagram is commutative
\[
\xymatrix{ B \tensor{A} M \ar^{B \tensfun{A} f}[rrrr] %%
\ar|{B \tensfun{A} \widetilde{\lambda}_M}[dr] %%
\ar|{\lambda_{B \tensor{A} M}}[dd]& & & &
B \tensor{A} N \ar|{B \tensor{A} \widetilde{\lambda}_N}[dl] %%
\ar|{\lambda_{B \tensor{A} N}}[dd] \\
&  B \tensor{A} \coring{D} \tensor{A} M %%
\ar|{\sigma_{\coring{D}} \tensfun{A} M}[dl] %%
\ar^{B \tensfun{A} \coring{D} \tensfun{A} f}[rr]
& & B  \tensor{A} \coring{D} \tensor{A} N \ar|{\sigma_{\coring{D}} \tensfun{A} N}[dr]  & \\
\coring{D} \tensor{B} B \tensor{A} M %%
\ar_{\coring{D} \tensfun{B} B \tensfun{A} f}[rrrr] %%
 & & & & \coring{D} \tensor{B} B \tensor{M} }
\]
From the definition of $\widetilde{\lambda}_M,
\widetilde{\lambda}_N$ and the fact that $f$ is a morphism of
$\coring{C}$--comodules, it follows that the upper trapezium
commutes. The lower trapezium is commutative by \eqref{xy2}. Since
the two triangles commute by definition, we get that the outer
rectangle is commutative and $B \tensor{A} f$ is a morphism in
$\lcomod{\coring{D}}$.
\end{proof}
\end{punto}

\begin{punto}\label{numb}
Proposition \ref{labm} also implies by symmetry that for every
right $\coring{C}$--comodule $\rho_{M} : M \rightarrow M
\tensor{A} \coring{C}$, the right $B$--module $M \tensor{A} B$ is
endowed with a right $\coring{D}$--comodule structure $\rho_{ M
\tensor{A} B } : M \tensor{A} B \rightarrow  M \tensor{A} B
\tensor{B} \coring{D}$ given by $\rho_{M \tensor{A} B} = (M
\tensor{A} \delta_{\coring{D}})(\widetilde{\rho}_M \tensor{A} B)$,
where $\widetilde{\rho}_M = (M \tensor{A} \varphi) \rho_M$ and
$\delta_{\coring{D}} : \coring{D} \tensor{A} B \rightarrow B
\tensor{B} \coring{D}$ is the obvious map. We can already state
that
\begin{proposition*}
The assignment $M \mapsto M \tensor{A} B$ establishes a functor $-
\tensor{A} B : \rcomod{\coring{C}} \rightarrow
\rcomod{\coring{D}}$.
\end{proposition*}
\end{punto}

\begin{punto}\textbf{The ad-induction functor.}
Let us consider the left $\coring{D}$--comodule structure
\[
\xymatrix{\lambda_{B \tensor{A} \coring{C}} : B \tensor{A}
\coring{C} \ar[r] & \coring{D} \tensor{B} B \tensor{A} \coring{C}}
\]
defined on $B \tensor{A} \coring{C}$ in \ref{labm}.

We have as well a canonical structure of right
$\coring{C}$--comodule
\[
\xymatrix{ B \tensor{A} \Delta_{\coring{C}} : B \tensor{A}
\coring{C} \ar[r] & B \tensor{A} \coring{C} \tensor{A} \coring{C}
}
\]
\begin{proposition*}
The $B-A$--bimodule $B \tensor{A} \coring{C}$ is a
$\coring{D}-\coring{C}$--bicomodule, which is quasi-finite as a
right $\coring{C}$--comodule. Therefore, if ${}_A\coring{C}$
preserves the equalizer of $(\rho_Y \tensor{B} B \tensor{A}
\coring{C},Y \tensor{B} \lambda_{B \tensor{A} \coring{C}})$ for
every  right $\coring{D}$--comodule $Y$, then the functor
\[
\xymatrix{- \cotensor{\coring{D}} (B \tensor{A} \coring{C})
:\rcomod{\coring{D}} \ar[r] & \rcomod{\coring{C}}}
\]
is right adjoint to
\[
\xymatrix{- \tensor{A} B : \rcomod{\coring{C}} \ar[r] &
\rcomod{\coring{D}}}.
\]
\end{proposition*}
\begin{proof}
Since the comultiplication $\Delta_{\coring{C}}$ is coassociative,
we get that the diagram
\[
\xymatrix{\coring{C} \ar^{\widetilde{\lambda}_{\coring{C}}}[rr] %%
\ar_{\Delta_{\coring{C}}}[dr] %%
\ar^{\Delta_{\coring{C}}}[ddd] & & \coring{D} \tensor{A}
\coring{C} \ar^{\coring{D} \tensfun{A} \Delta_{\coring{C}}}[ddd]\\
& \coring{C} \tensor{A} \coring{C} \ar_{\varphi \tensfun{A} \coring{C}}[ur] %
\ar^{\coring{C} \tensfun{A} \Delta_{\coring{C}}}[d]& \\
& \coring{C} \tensor{A} \coring{C} \tensor{A} \coring{C} %%
\ar^{\varphi  \tensfun{A} \coring{C} \tensfun{A} \coring{C}}[dr]& \\
\coring{C} \tensor{A} \coring{C} \ar^{\Delta_{\coring{C}} \tensfun{A} \coring{C}}[ur] %%
\ar^{\widetilde{\lambda}_{\coring{C}} \tensfun{A} \coring{C}}[rr]&
& \coring{D} \tensor{A} \coring{C} \tensor{A} \coring{C}}
\]
is commutative. This implies that the left trapezium of the
following diagram is commutative.
\[
\xymatrix{ B \tensor{A} \coring{C} \ar^{\lambda_{B \tensor{A} \coring{C}}}[rr] %%
\ar_{B \tensfun{A} \widetilde{\lambda}_{\coring{C}}}[dr] %%
\ar^{B \tensfun{A} \Delta_{\coring{C}}}[ddd]
& & \coring{D} \tensor{B} B \tensor{A} \coring{C} %
\ar^{\coring{D} \tensfun{B} B \tensfun{A} \Delta_{\coring{C}}}[ddd] \\
& B \tensor{A} \coring{D} \tensor{A} \coring{C} %
\ar^{B \tensfun{A} \coring{D} \tensfun{A} \Delta_{\coring{C}}}[d] %%
\ar_{\sigma_{\coring{D}} \tensfun{A} \coring{C}}[ur]&  \\
& B \tensor{A} \coring{D} \tensor{A} \coring{C} \tensor{A} \coring{C} %
\ar^{\sigma_{\coring{D}} \tensfun{A} \coring{C} \tensfun{A} \coring{C}}[dr]&  \\
B \tensor{A} \coring{C} \tensor{A} \coring{C} %
\ar^{B \tensfun{A} \widetilde{\lambda}_{\coring{C}} \tensfun{A} \coring{C}}[ur] %%
\ar^{\lambda_{B \tensor{A} \coring{C}} \tensfun{A}
\coring{C}}[rr]& & \coring{D} \tensor{B} B \tensor{A} \coring{C}
\tensor{A} \coring{C}}
\]
Since we know that the rest of inner diagrams are also
commutative, we obtain that the outer diagram commutes, too. This
proves that $B \tensor{A} \coring{C}$ is a
$\coring{D}-\coring{C}$--bicomodule.

We will see that $B \tensor{A} \coring{C}$ is a quasi-finite
 right $\coring{C}$--comodule, i.e., that $\xymatrix{- \tensor{A}
B : \rcomod{\coring{C}} \ar[r] & \rcomod{B}}$ is left adjoint to
$\xymatrix{- \tensor{B} B \tensor{A} \coring{C} : \rcomod{B}
\ar[r] & \rcomod{\coring{C}}}$.
 Now, these functors fit in the
commutative diagrams
\[
\xymatrix{ \rcomod{A} \ar[rr]^{- \tensfun{A} \coring{C}}& &
\rcomod{\coring{C}}
\\ \rcomod{B} \ar[u]^{- \tensfun{B} B}
\ar[urr]_{- \tensfun{B} B \tensfun{A} \coring{C} }& & } \qquad
\xymatrix{\rcomod{\coring{C}}  \ar[drr]_{- \tensfun{A} B}
\ar[rr]^{U_A} & & \rcomod{A} \ar[d]^{- \tensfun{A} B}\\ & &
\rcomod{B}},
\]
where $U_A$ denotes the forgetful functor. Since $- \tensor{A} B$
is left adjoint to $ - \tensor{B} B$ and $U_A$ is left adjoint to
$ - \tensor{A} \coring{C}$ we get the desired adjunction. The
rest of the proposition follows from Proposition \ref{adjcohom}.
\end{proof}

\begin{remark*}
This proposition applies in the case that ${}_A\coring{C}$ is flat
or when $\coring{D}$ is a coseparable $B$--coring in the sense of
\cite{Guzman:1989}.
\end{remark*}
\end{punto}


\begin{punto}\textbf{The unit.}
Let $\overline{\Delta}_{\coring{C}} : \coring{C} \rightarrow
\coring{C} \tensor{A} B \tensor{B} B \tensor{A} \coring{C}$ be
the composite map
\[
\xymatrix{ \coring{C} \ar[r]^(.3){\Delta_{\coring{C}}} &
\coring{C} \tensor{A} \coring{C} \ar[r]^(.3){\iota} & \coring{C}
\tensor{A} B \tensor{B} B \tensor{A} \coring{C} }
\]
where $\iota$ maps $c \tensor{} c' \in \coring{C} \tensor{A}
\coring{C}$ to $c \tensor{} 1 \tensor{} 1 \tensor{} c'$. This map
is a homomorphism of $\coring{C}$--bicomodules. The unit $\theta$
of the adjunction $- \tensor{A} B \dashv - \tensor{B} B \tensor{A}
\coring{C}$ at a right $\coring{C}$--comodule $M$ is given by the
composite map
\[
\xymatrix{M \ar[r]^(.3){\rho_M} & M \tensor{A} \coring{C}
\ar[rr]^(.4){\varsigma_M \tensfun{A} \coring{C}} & & M \tensor{A}
B \tensor{B} B \tensor{A} \coring{C}}
\]
where $\varsigma_M : M \rightarrow M \tensor{A} B \tensor{B} B$
maps $m \in M$ to $m \tensor{A} 1 \tensor{B} 1$. We see, in
particular, that $\theta_{\coring{C}} =
\overline{\Delta}_{\coring{C}}$. By Proposition \ref{adjcohom},
$\theta_M$ factorizes throughout $(M \tensor{A} B )
\cotensor{\coring{D}} (B \tensor{A} \coring{C})$ and, therefore,
it gives the unit $\theta_M$ for the adjunction $ - \tensor{A} B
\dashv - \cotensor{\coring{D}} ( B \tensor{A} \coring{C})$ at $M$.
So, the multiplication $\Delta_{\coring{C}}$ finally induces a map
\[
\xymatrix{\overline{\Delta}_{\coring{C}} : \coring{C} \ar[r] &
(\coring{C} \tensor{A} B) \cotensor{\coring{D}} (B \tensor{A}
\coring{C})}
\]
which is a homomorphism of $\coring{C}$--bicomodules.
\end{punto}

We are now ready to state the characterization of the separability
of the induction functor.

\begin{theorem}\label{sepind}
Assume that $\coring{A}_A$ preserves the equalizer of $(\rho_Y
\tensor{B} B \tensor{B} \tensor{A} \coring{C}, Y \tensor{B}
\lambda_{B \tensor{A} \coring{C}})$ for every $Y \in
\rcomod{\coring{D}}$, and that $X_A$ preserves the equalizer of
$(\rho_{\coring{C} \tensor{A} B} \tensor{B} B \tensor{A}
\coring{C}, \coring{C} \tensor{A} B \tensor{B} \lambda_{B
\tensor{A} \coring{C}})$ for every $X \in \rcomod{\coring{C}}$.
The functor $ \xymatrix{- \tensor{A} B : \rcomod{\coring{C}}
\ar[r] & \rcomod{\coring{D}}}$ is separable if and only if there
is a homomorphism if $\coring{C}$--bicomodules
\[
\xymatrix{\omega_{\coring{C}} : (\coring{C} \tensor{A} B)
\cotensor{\coring{D}} (B \tensor{A} \coring{C}) \ar[r] &
\coring{C}}
\]
such that $\omega_{\coring{C}} \overline{\Delta}_{\coring{C}} =
\coring{C}$.
\end{theorem}
\begin{proof}
Assume that $ - \tensor{A} B$ is separable. By \cite[Theorem
4.1]{DelRio:1992} or \cite[Theorem 1.2]{Rafael:1990}, the unit of
the adjunction
\[
\xymatrix{\theta : 1_{\rcomod{\coring{C}}} \ar[r] & (- \tensor{A}
B) \cotensor{\coring{D}} (B \tensor{A} \coring{D})}
\]
is split-mono, that is, there is a natural transformation
$\xymatrix{\omega : (- \tensor{A} B) \cotensor{\coring{D}} (B
\tensor{A} \coring{C}) \ar[r]
 & 1_{\rcomod{\coring{C}}}}$ such that $\omega \theta =
1_{\rcomod{\coring{C}}}$. By Lemma \ref{coass1} the functor
$(-\tensor{A} B) \cotensor{\coring{D}} (B \tensor{A} \coring{C})$
is $\coring{C}$--compatible. By Proposition \ref{hacebicom},
$\omega_{\coring{C}}$ is a $\coring{C}$--bicomodule map.
Obviously, $\omega_{\coring{C}} \overline{\Delta}_{\coring{C}} =
\coring{C}$.\\
To prove the converse, we need to construct a natural
transformation $\omega$ from the bicomodule map
$\omega_{\coring{C}}$. Given a right $\coring{C}$--comodule $M$,
consider the diagram
\[
\xymatrix{ M \tensor{A} B \tensor{B} \coring{D} \tensor{B} B \tensor{A} \coring{C}%%
\ar^{\rho_M \tensfun{A} B \tensfun{B} \coring{D} \tensfun{B} B
\tensfun{A} \coring{C}}[rrrr]&
 & &
  & M \tensor{A} \coring{C} \tensor{A} B \tensor{B} \coring{D} \tensor{B} B \tensor{A} \coring{C}\\
   & & & & \\
M \tensor{A} B \tensor{B} B \tensor{A} \coring{C}%%
\ar^{\rho_M \tensfun{A} B \tensfun{B} B \tensfun{A}
\coring{C}}[rrrr]%%
\ar^{\rho_{M \tensor{A} B} \tensfun{B} B \tensfun{A}
\coring{C}}@<1ex>[uu]%%
\ar_{M \tensfun{A} B \tensfun{B} \lambda_{B \tensor{A}
\coring{C}}}@<-1ex>[uu]&
& & &  M \tensor{A} \coring{C} \tensor{A} B \tensor{B} B \tensor{A} \coring{C}%%
\ar^{M \tensfun{A} \rho_{\coring{C} \tensor{A} B} \tensfun{B} B \tensfun{A} \coring{C}}@<1ex>[uu]%%
\ar_{M \tensor{A} \coring{C} \tensor{A} B \tensor{B} \lambda_{B \tensor{A} \coring{C}}}@<-1ex>[uu]\\
 & & & & \\
(M \tensor{A} B) \cotensor{\coring{D}} (B \tensor{A} \coring{C})
\ar@{..>}^{\kappa_M}[rrrr] \ar[uu]& & & & M \tensor{A}
((\coring{C} \tensor{A} B) \cotensor{\coring{D}} (B \tensor{A}
\coring{C})) \ar[uu]},
\]
where the vertical are equalizer diagrams (here, we are using the
definition of the cotensor product and the fact that $M_A$
preserves the equalizer of $(\rho_{\coring{C} \tensor{A} B}
\tensor{B} B \tensor{A} \coring{C}, \coring{C} \tensor{A} B
\tensor{B} \lambda_{B \tensor{A} \coring{C}})$). If we prove that
the top rectangle commutes, then there is a unique dotted arrow
$\kappa_M$ making the bottom rectangle commute. The identity
\[
(M \tensfun{A} \coring{C} \tensfun{A} B \tensfun{B} \lambda_{B
\tensor{A} \coring{C}})(\rho_M \tensfun{A} B \tensfun{B} B
\tensfun{A} \coring{C}) = (\rho_M \tensfun{A} B \tensfun{B}
\coring{D} \tensfun{B} B \tensfun{A} \coring{C})(M \tensfun{A} B
\tensfun{B} \lambda_{B \tensor{A }\coring{C}})
\]
is obvious; so we need just to prove that
\[
(M \tensor{A} \rho_{\coring{C} \tensor{A} B} \tensfun{B} B
\tensfun{A} \coring{C})(\rho_M \tensfun{A} B \tensfun{B} B
\tensfun{A} \coring{C}) = (\rho_M \tensfun{A} B \tensfun{B} B
\tensfun{B} \coring{D} \tensfun{B} B \tensfun{A}
\coring{C})(\rho_{M \tensor{A} B} \tensfun{B} B \tensfun{A}
\coring{C}),
\]
which is equivalent to
\[
(M \tensfun{A} \rho_{\coring{C} \tensor{A} B})(\rho_M \tensfun{A}
B) = ( \rho_M \tensfun{A} B \tensfun{B} \coring{D})\rho_{M
\tensor{A} B},
\]
and this last identity is easy to check. Now, consider the diagram
\[
\xymatrix{ M \ar[ddd]^{\rho_M} \ar@{=}[dr] \ar[rrr]^{\theta_M} & &
&
 M \tensor{A} B \tensor{B} B \tensor{A} \coring{C} %%
\ar[ddd]_{\rho_M \tensfun{A} B \tensfun{B} B \tensfun{A}
\coring{C}}\\
& M \ar[d]^{\rho_M} \ar[r]^(.3){\theta_M} & (M \tensor{A} B)
\cotensor{\coring{D}} (B \tensor{A}
\coring{C}) \ar@{^{(}->}[ur] \ar[d]^{\kappa_M} &  \\
& M \tensor{A} \coring{C} \ar[r]^(.3){M \tensor{A}
\overline{\Delta}_{\coring{C}}} \ar@{=}[dl] & M \tensor{A}
((\coring{C} \tensor{A} B) \cotensor{\coring{D}} (B \tensor{A}
\coring{C}))
\ar@{^{(}->}[dr] & \\
M \tensor{A} \coring{C} \ar[rrr]^{M \tensfun{A}
\overline{\Delta}_{\coring{C}}} & & & M \tensor{A} \coring{C}
\tensor{A} B \tensor{B} B \tensor{A} \coring{C}}
\]

By Lemma \ref{FunoFdos}, we have that $M \tensor{A} \theta_{M} =
\theta_{M \tensor{A} \coring{C}}$. Since $\theta$ is natural, and
$\theta_M = \overline{\Delta}_{\coring{C}}$, this implies that the
external rectangle commutes. We know that the four trapezia
commute, whence the internal rectangle commutes as well. Define
$\omega_M = (M \tensor{A} \epsilon_{\coring{C}})(M \tensor{B}
\omega_{\coring{C}})\kappa_M$, which gives a natural
transformation $\omega : (- \tensor{A} B) \cotensor{\coring{D}} (B
\tensor{A} \coring{C})  \rightarrow 1_{\rcomod{\coring{C}}}$.
Moreover,
\begin{multline*}
\omega_M \theta_M = (M \tensor{A} \epsilon_{\coring{C}})(M
\tensor{A} \omega_{\coring{C}})\kappa_M\theta_M = \\ (M \tensor{A}
\epsilon_{\coring{C}})(M \tensor{A}\omega_{\coring{C}})(M
\tensor{A} \theta_M)\rho_M = (M \tensor{A} \epsilon_M)\rho_M = M
\end{multline*}
Therefore, $\omega \theta = 1_{\rcomod{\coring{C}}}$ and, by
\cite[Theorem 1.2]{Rafael:1990}, the functor $- \tensor{A} B$ is
separable.
\end{proof}

The counit map $\epsilon_{\coring{C}} : \coring{C} \rightarrow A$
is a homomorphism of $A$--corings, where we consider on $A$ the
canonical $A$--coring structure. When applied to
$\epsilon_{\coring{C}}$, Theorem \ref{sepind} boils down to

\begin{corollary}\cite[Theorem 3.5]{Brzezinski:2000unp}\label{sepolvido}
For an $A$--coring $\coring{C}$, the forgetful functor $U_A :
\rcomod{\coring{C}} \rightarrow \rcomod{A}$ is separable if and
only if there is an $A$--bimodule map $\gamma : \coring{C}
\tensor{A} \coring{C} \rightarrow A$ such that $\gamma
\Delta_{\coring{C}} = \epsilon_{\coring{C}}$ and, in Sweedler's
sigma notation,
\[
c_{(1)}\gamma(c_{(2)} \tensor{A} c') = \gamma  (c \tensor{A}
c'_{(1)})c'_{(2)} \qquad \text{for all } \; c, c' \in \coring{C}
\]
\end{corollary}
\begin{proof}
Obviously, the forgetful functor coincides with $ - \tensor{A} A$,
so that we get from Theorem \ref{sepind} a characterization of the
separability of this functor in terms of the existence of a
$\coring{C}$--bicomodule map $\omega_{\coring{C}} : \coring{C}
\tensor{A} \coring{C} \rightarrow \coring{C}$ such that
$\omega_{\coring{C}} \Delta_{\coring{C}} = \coring{C}$. Now,
notice that the adjointness isomorphism
\[
\hom{\coring{C}}{\coring{C} \tensor{A} \coring{C}}{\coring{C}}
\cong \hom{A}{\coring{C} \tensor{A} \coring{C}}{A}
\]
transfers faithfully the mentioned properties of
$\omega_{\coring{C}}$ to the desired properties of $\gamma =
\epsilon_{\coring{C}}\omega_{\coring{C}}$.
\end{proof}

\begin{punto}\textbf{The counit of the adjunction.}
Let $\hat{\epsilon}_{\coring{C}}$ be the homomorphism of
$B$--bimodules that makes commute the following diagram
\[
\xymatrix{B \tensor{A} \coring{C} \tensor{A} B
\ar[rr]^{\hat{\epsilon}_{\coring{C}}} %%
\ar[d]^{B \tensor{A} \epsilon_{\coring{C}} \tensor{A} B} & & B \\
B \tensor{A} A \tensor{A} B \ar[rr]^{B \tensfun{A} \rho
\tensfun{A} B} & & B \tensor{A} B \tensor{A} B \ar[u]^{m}},
\]
where $m : B \tensor{A} B \tensor{A} B \rightarrow B$ is the
obvious multiplication map. Define, for every right $B$--module
$Y$, $\chi_Y = \mu_Y (Y \tensor{A} \hat{\epsilon}_{\coring{C}})$,
where $\mu_Y : Y \tensor{B} B \rightarrow Y$ is the canonical
isomorphism. This natural transformation $\chi$ gives the counit
of the adjunction $- \tensor{A} B \dashv - \tensor{B} ( B
\tensor{A} \coring{C})$. By \ref{adjcohom}, the counit of the
adjunction $- \tensor{A} B \dashv - \cotensor{\coring{D}} ( B
\tensor{A} \coring{C})$ is given by the restriction of $\chi$ to
$(- \cotensor{\coring{D}} (B \tensor{A} \coring{C}) \tensor{A} B$.
We shall use the same notation for this counit. Now, define
$\hat{\varphi}$ as the $B$--bimodule map completing the diagram
\[
\xymatrix{B \tensor{A} \coring{C} \tensor{A} B
\ar[r]^(.7){\hat{\varphi}} \ar[d]_{B \tensfun{A} \varphi \tensfun{A} B} & \coring{D} \\
B \tensor{A} \coring{D} \tensor{A} B \ar[ur]_{m_{\coring{D}}} & }
\]
where $m_{\coring{D}} : B \tensor{A} \coring{D} \tensor{A} B
\rightarrow \coring{D}$ is the obvious multiplication map given
by the $B$--bimodule structure of $\coring{D}$. We claim that
$\hat{\varphi}$ is a $\coring{D}$--comodule map. To prove this, we
first show that the diagram
\begin{equation}\label{phihat}
\xymatrix{(\coring{D} \cotensor{\coring{D}} (B \tensor{A}
\coring{C})) \tensor{A} B \ar[r] & \coring{D} \tensor{B} B
\tensor{A} \coring{C} \tensor{A} B \ar[d]^{\chi_{\coring{D}}} \\
B \tensor{A} \coring{C} \tensor{A} B \ar[u]^{\lambda_{B \tensor{A}
\coring{C}} \tensfun{A} B} \ar[r]^(.6){\hat{\varphi}} &
\coring{D}}
\end{equation}
is commutative. This is done by the following computation, where
$\widetilde{m} : B \tensor{B} B \tensor{A} B \rightarrow B$
denotes the obvious multiplication map:

\begin{multline*}
\chi_{\coring{D}} (\lambda_{B \tensor{A} \coring{C}} \tensfun{A} B
)  = \mu_{\coring{D}}(\coring{D} \tensfun{A}
\hat{\epsilon}_{\coring{C}})(\sigma_{\coring{D}} \tensfun{A}
\coring{C} \tensfun{A} B)(B \tensfun{A}
\widetilde{\lambda}_{\coring{C}} \tensfun{A} B) = \\
   \mu_{\coring{D}}(\coring{D} \tensfun{B} m)(\coring{D}
 \tensfun{B} B \tensfun{A} \rho \tensfun{A} B)(\coring{D} \tensfun{B} B
 \tensfun{A} \epsilon_{\coring{C}} \tensfun{A}
 \coring{D})(\sigma_{\coring{D}} \tensfun{A} \coring{C}
 \tensfun{A} B)(B \tensfun{A} \varphi \tensfun{A} \coring{C}
 \tensfun{A} B)(B \tensfun{A} \Delta_{\coring{C}} \tensfun{A} B)
 =\\
 \mu_{\coring{D}}(\coring{D} \tensfun{B} m)(\coring{D} \tensfun{B}
 B \tensfun{A} \epsilon_{\coring{D}} \tensfun{A} B)(\coring{D}
 \tensfun{B} B \tensfun{A} \varphi \tensfun{A}
 B)(\sigma_{\coring{D}} \tensfun{A} \coring{C} \tensfun{A} B)(B
 \tensfun{A} \varphi \tensfun{A} \coring{C} \tensfun{A} B)(B
 \tensfun{A} \Delta_{\coring{C}} \tensfun{A} B) = \\
\mu_{\coring{D}}(\coring{D} \tensfun{B} m)(\coring{D} \tensfun{B}
 B \tensfun{A} \epsilon_{\coring{D}} \tensfun{A}
 B)(\sigma_{\coring{D}} \tensfun{A} \varphi \tensfun{A} B)(B
 \tensfun{A} \varphi \tensfun{A} \coring{C} \tensfun{A} B)(B
 \tensfun{A} \Delta_{\coring{C}} \tensfun{A} B) = \\
 \mu_{\coring{D}}(\coring{D} \tensfun{B} m)(\coring{D} \tensfun{B}
 B \tensfun{A} \epsilon_{\coring{D}} \tensfun{A}
 B)(\sigma_{\coring{D}} \tensfun{A} \coring{D} \tensfun{A} B)(B
 \tensfun{A} \varphi \tensfun{A} \varphi \tensfun{A} B)(B
 \tensfun{A} \Delta_{\coring{C}} \tensfun{A} B)) = \\
 \mu_{\coring{D}}(\coring{D} \tensfun{B} \widetilde{m})(\coring{D}
 \tensfun{B} B \tensfun{B} \epsilon_{\coring{D}} \tensfun{A}
 B)(\sigma_{\coring{D}} \tensfun{B} \coring{D} \tensfun{A} B)(B
 \tensfun{A} \omega_{\coring{D},\coring{D}} \tensfun{A} B)(B
 \tensfun{A} \varphi \tensfun{A} \varphi \tensfun{A} B)(B
 \tensfun{A} \Delta_{\coring{D}} \tensfun{A} B) = \\
 \mu_{\coring{D}}(\coring{D} \tensfun{B} \widetilde{m})(\coring{D}
 \tensfun{B} B \tensfun{B} \epsilon_{\coring{D}} \tensfun{A}
 B)(\sigma_{\coring{D}} \tensfun{B} \coring{D} \tensfun{A} B)(B
 \tensfun{A} \Delta_{\coring{D}} \tensfun{A} B)(B \tensfun{A}
 \varphi \tensfun{A} B) = \\
 \mu_{\coring{D}}(\coring{D} \tensfun{B}
 \widetilde{m})(\sigma_{\coring{D}} \tensfun{B}
 \epsilon_{\coring{D}} \tensfun{A} B)(B \tensfun{A}
 \Delta_{\coring{D}} \tensfun{A} B)(B \tensfun{A} \varphi
 \tensfun{A} B) = \\
 \mu_{\coring{D}}(\coring{D} \tensfun{B} \widetilde{m})(\sigma_{\coring{D}} \tensfun{B} B \tensfun{A}
 B)(B \tensfun{A} \coring{D} \tensfun{B} \epsilon_{\coring{D}}
 \tensfun{A} B) ( B \tensfun{A} \Delta_{\coring{D}} \tensfun{A}
 B)(B \tensfun{A} \varphi \tensfun{A} B) = \\
 \mu_{\coring{D}}(\coring{D} \tensfun{B}
 \widetilde{m})(\sigma_{\coring{D}} \tensfun{B} B \tensfun{A} B)(B
 \tensfun{A} \varphi \tensfun{A} B) = \\
 \mu_{\coring{D}}(\coring{D} \tensfun{B} m)(\sigma_{\coring{D}} \tensfun{A} B \tensfun{A} B)(B
 \tensfun{A} \varphi \tensfun{A} B) = \hat{\varphi}
\end{multline*}
%
We know that $\lambda_{B \tensor{A} \coring{C}} \tensor{A} B$ is a
homomorphism of $\coring{D}$--bicomodules and, by \ref{hacebicom},
$\chi_{\coring{D}}$ is $\coring{D}$--bicolinear, too. This proves
that $\hat{\varphi} : B \tensor{A} \coring{C} \tensor{A} B
\rightarrow \coring{D}$ is a homomorphism of
$\coring{D}$--bicomodules.
\end{punto}

We are now in a position to prove our separability theorem for the
ad-induction functor.

\begin{theorem}\label{sepcores}
Assume that ${}_AB$ and ${}_A\coring{C}$ preserve the equalizer of
$(\rho_M \tensor{B} B \tensor{A} \coring{C}, M \tensor{B}
\lambda_{B \tensor{A} \coring{C}})$ for every right
$\coring{D}$--comodule $M$ (e.g. ${}_AB$ and ${}_A\coring{C}$ are
flat or $\coring{D}$ is a coseparable $B$--coring in the sense of
\cite{Guzman:1989}). The functor $\xymatrix{-
\cotensor{\coring{D}} (B \tensor{A} \coring{C}) :
\rcomod{\coring{D}} \ar[r] & \rcomod{\coring{C}}}$ is separable if
and only if there exists a $\coring{D}$--bicomodule homomorphism
$\xymatrix{\hat{\nu}_{\coring{D}} : \coring{D} \ar[r]& B
\tensor{A} \coring{C} \tensor{A} B }$ such that $\hat{\varphi}
\hat{\nu}_{\coring{D}} = \coring{D}$.
\end{theorem}
\begin{proof}
If $- \cotensor{\coring{D}} (B \tensor{A} \coring{C})$ is
separable then, by \cite[Theorem 1.2]{Rafael:1990}, there exists a
natural transformation
\[
\xymatrix{\nu : 1_{\rcomod{\coring{D}}} \ar[r] & ( -
\cotensor{\coring{D}} (B \tensor{A} \coring{C})) \tensor{A} B}
\]
such that $\chi \nu = 1_{\rcomod{\coring{D}}}$. In particular,
$\chi_{\coring{D}}\nu_{\coring{D}} = \coring{D}$ and, by
Proposition \ref{hacebicom}, $\nu_{\coring{D}}$ is a bicomodule
map (in fact, we easily get that the functor $(-
\cotensor{\coring{D}} (B \tensor{A} \coring{C})) \tensor{A} B$ is
$\coring{D}$--compatible from the fact that $\coring{D}
\cotensor{\coring{D}} (B \tensor{A} \coring{C})$ is a direct
summand of $\coring{D} \tensor{B} B \tensor{A} \coring{C}$ as a
left $B$--module). The map $\lambda_{B \tensor{A} \coring{C}}
\tensor{A} B$ gives an isomorphism of $\coring{D}$--bicomodules
$B \tensor{A} \coring{C} \tensor{A} B \cong (\coring{D}
\cotensor{D} (B \tensor{A} \coring{C})) \tensor{A} B$ which
implies, after \eqref{phihat}, that
\[
\coring{D} = \chi_{\coring{D}}\nu_{\coring{D}}  =
\hat{\varphi}(\lambda_{B \tensor{A} \coring{C}} \tensor{A}
B)^{-1}\nu_{\coring{D}}
\]
Thus, $\hat{\nu}_\coring{D} = (\lambda_{B \tensor{A} \coring{C}}
\tensor{A} B)^{-1}\nu_{\coring{D}}$ is the desired
$\coring{D}$--bicomodule map.

For the converse, assume there is a $\coring{D}$--bicomodule map
$\hat{\nu}_{\coring{D}} : \coring{D} \rightarrow B \tensor{A}
\coring{C} \tensor{A} B$ such that $\hat{\varphi}
\hat{\nu}_{\coring{D}} = \coring{D}$. For each right
$\coring{D}$--comodule $M$, let us prove that the homomorphism of
right $\coring{D}$--comodules
\begin{equation}\label{prenu}
\xymatrix{M \ar[r]^(.4){\rho_M} & M \tensor{B} \coring{D}
\ar[rr]^(.4){M \tensfun{B} \hat{\nu}_{\coring{D}}} & & M
\tensor{B} B \tensor{A} \coring{C} \tensor{A} B}
\end{equation}
factorizes throughout $(M \cotensor{\coring{D}} (B \tensor{A}
\coring{C})) \tensor{A} B$. Since ${}_AB$ preserves the equalizer
of $(\rho_M \tensor{B} B \tensor{A} \coring{C}, M \tensor{B}
\lambda_{B \tensor{A} \coring{C}})$, we know that
\[
(M \cotensor{\coring{D}} (B \tensor{A} \coring{C})) \tensor{A} B
\cong M \cotensor{\coring{D}} (B \tensor{A} \coring{C} \tensor{A}
B).
\]
Therefore, we need just to check the equality
\begin{equation}\label{Rolanda}
(\rho_M \tensor{B} B \tensor{A} \coring{C} \tensor{A} B)(M
\tensor{B} \hat{\nu}_{\coring{D}})\rho_M = (M \tensor{B}
\lambda_{B \tensor{A} \coring{C} \tensor{A} B})(M \tensor{B}
\hat{\nu}_{\coring{D}})\rho_M
\end{equation}
This is done by the following computation:
\begin{multline*}
(M \tensor{B} \lambda_{B \tensor{A} \coring{C} \tensor{A} B})(M
\tensor{B} \hat{\nu}_{\coring{D}})\rho_M = \\
(M \tensor{B} \coring{D} \tensor{B} \hat{\nu}_{\coring{D}})(M
\tensor{B} \Delta_{\coring{D}})\rho_M = (M \tensor{B} \coring{D}
\tensor{B}
\hat{\nu}_{\coring{D}})\rho_M = \\
(\rho_M \tensor{B} \hat{\nu}_{\coring{D}})\rho_M = (\rho_M
\tensor{B} B \tensor{A} \coring{C} \tensor{A} B)(M \tensor{B}
\hat{\nu}_{\coring{D}})\rho_M
\end{multline*}
Thus, we have proved that the natural transformation given in
\eqref{prenu} factorizes throughout a natural transformation
\begin{equation*}\label{nu}
\xymatrix{\nu_M : M \ar[r] & (M \cotensor{\coring{D}} (B
\tensor{A} \coring{C})) \tensor{A} B}
\end{equation*}
This means that we have a commutative diagram
\begin{equation}\label{rombo}
\xymatrix{ M \ar^{\rho_M}[r] \ar^{\nu_M}[drr]&%%
 M \tensor{B} \coring{D} \ar^(.4){M \tensfun{B} \hat{\nu}_{\coring{D}}}[r]&%%
  M \tensor{B} B \tensor{A} \coring{C} \tensor{A} B \ar^{\chi_M}[drr]& & \\
& & (M \cotensor{\coring{D}} (B \tensor{A} \coring{C}))
\tensor{A} B \ar_(.7){\chi_M}[rr] \ar[u]& &M }
\end{equation}
Finally, we will show that $\nu_M$ splits off $\chi_M$ by means
of the following computation:
\begin{alignat*}{2}
\rho_M \chi_M \nu_M & = \rho_M\chi_M(M \tensfun{B}
\hat{\nu}_{\coring{D}})\rho_M & \\
& = \chi_{M \tensor{B} \coring{D}}(\rho_M \tensfun{B} B
\tensfun{A} \coring{C} \tensfun{A} B)(M \tensfun{B}
\hat{\nu}_{\coring{D}})\rho_M & \qquad (\chi \; \text{is
natural})\\
 & = (M \tensfun{B} \chi_{\coring{D}})(\rho_M \tensfun{B} B
\tensfun{A} \coring{C} \tensfun{A} B)(M \tensfun{B}
\hat{\nu}_{\coring{D}})\rho_M & \qquad (\text{by Lemma
\ref{FunoFdos})} \\
 & = (M \tensfun{B} \chi_{\coring{D}})(M \tensfun{B} \lambda_{B
 \tensor{A} \coring{C} \tensor{A} B})(M \tensfun{B}
\hat{\nu}_{\coring{D}})\rho_M & \qquad (\hat{\nu}_{\coring{D}} \;
\text{is bicolinear}) \\
& = (M \tensfun{B} \hat{\varphi})(M \tensfun{B}
\hat{\nu}_{\coring{D}})\rho_M  = \rho_M &
\end{alignat*}

Since $\rho_M$ is a monomorphism, we get $\chi_M \nu_M = M$. By
\cite[Theorem 1.2]{Rafael:1990}, $- \cotensor{D} ( B \tensor{A}
\coring{C})$ is a separable functor.
\end{proof}

By applying the stated theorem to $\epsilon_{\coring{C}} :
\coring{C} \rightarrow A$ we obtain:

\begin{corollary}\cite[Theorem 3.3]{Brzezinski:2000unp}
Let $\coring{C}$ be an $A$--coring. Then the functor $- \tensor{A}
\coring{C}$ is separable if and only if there exists an invariant
$e \in \coring{C}$ (that is, $e \in \coring{C}$ satisfying $ae =
ea$ for every $a \in A$) such that $\epsilon_{\coring{C}}(e) = 1$.
\end{corollary}
\begin{proof}
This follows from Theorem \ref{sepcores} taking that the
$A$--bimodule homomorphisms from $A$ to $\coring{C}$ correspond
bijectively with the invariants of $\coring{C}$ into account.
\end{proof}
\begin{punto}\textbf{Final Remarks.}
T. Brzezi\'nski showed \cite[Proposition 2.2]{Brzezinski:2000unp}
that if $(A,C)_{\psi}$ is an entwining structure over $K$ then $A
\tensor{K} C$ can be endowed with an $A$--coring structure in
such a way that the category $\rcomod{A \tensor{K} C}$ is
isomorphic with the category $\cat{M}_A^C(\psi)$ of entwined
$(A,C)_{\psi}$--modules. It turns out that every morphism of
entwining structures $(f,g) : (A,C)_{\psi} \rightarrow
(B,D)_{\gamma}$ in the sense of \cite{Brzezinski:1999unp} gives a
coring morphism $(f \tensor{K} g, f) : A \tensor{K} C \rightarrow
B \tensor{K} D$. Some straightforward computations show that the
statements of the separability theorem \cite[Theorem
3.4]{Brzezinski:1999unp} correspond to our theorems \ref{sepind}
and \ref{sepcores}. In fact, one can give the notions of totally
integrable and totally cointegrable morphism of entwining
structures to the framework of morphisms of corings $(\phi,\rho):
\coring{C} \rightarrow \coring{D}$ by requiring the existence of
the splitting bicomodule maps $\omega_{\coring{C}}$ and
$\hat{\nu}_{\coring{D}}$, respectively. In the first case, the
existence of $\omega_{\coring{C}}$ can be transferred, if desired,
to the existence of certain bimodule map with extra properties by
means of the adjointness isomorphism
\[
\hom{\coring{C}}{(\coring{C} \tensor{A} B) \cotensor{\coring{D}}
(B \tensor{A} \coring{C})}{\coring{C}} \cong \hom{A}{(\coring{C}
\tensor{A} B) \cotensor{\coring{D}} (B \tensor{A} \coring{C})}{A}
\]
\end{punto}

\noindent \textbf{Acknowledgement:} I would like to thank Tomasz
Brzezi{\'n}ski for some helpful comments.

\providecommand{\bysame}{\leavevmode\hbox
to3em{\hrulefill}\thinspace}
\begin{thebibliography}{10}

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\end{thebibliography}


%\bibliographystyle{amsplain}
%\bibliography{C:/texdoc/bib/nueva}


\end{document}

math0102120.extracted_bib
\begin{thebibliography}{10}

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T.~Brzezi{\'n}ski, S.~Caenepeel, G.~Militaru, and S.~Zhu,
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\bibitem{Caenepeel/alt:2000unp}
S.~Caenepeel, B.~Ion, G.~Militaru, and Z.~Shenglin,
\emph{Separable functors
  for the category of {Doi-Hopf} modules {II}}, in ``Hopf algebras
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  Lecture Notes Pure Appl. Math. \textbf{209}, Marcel Dekker, New
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\bibitem{Caenepeel/alt:1999}
S.~Caenepeel, B.~Ion, G.~Militaru, and S.~Zhu, \emph{Separable
functors for the
  category of {Doi-Hopf} modules, applications}, Adv. Math. \textbf{145} (1999),
  239--290.

\bibitem{Castano/Gomez/Nastasescu:1997}
F.~{Casta{\~n}o Iglesias}, J.~{G{\'o}mez-Torrecillas}, and C.~{N\u
ast\u
  asescu}, \emph{Separable functors in coalgebras. {Applications}.}, Tsukuba J.
  Math. \textbf{21} (1997), 329--344.

\bibitem{Castano/Gomez/Nastasescu:1998}
F.~{Casta{\~n}o Iglesias}, J.~{G{\'o}mez-Torrecillas}, and {N\u
ast\u asescu,
  C.}, \emph{Separable functors in graded rings.}, J. Pure Appl. Algebra
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\bibitem{DelRio:1992}
A.~Del~R{\'\i}o, \emph{Categorical methods in graded ring theory},
Publicacions
  Mat. \textbf{36} (1992), 489--531.

\bibitem{Guzman:1989}
F.~Guzman, \emph{Cointegrations, relative cohomology for
comodules, and
  coseparable corings}, J. Algebra \textbf{126} (1989), 211--224.

\bibitem{Nastasescu/VandenBergh/VanOystaeyen:1989}
C.~N{\u a}st{\u a}sescu, M.~Van~den Bergh, and F.~Van~Oystaeyen,
  \emph{Separable functors applied to graded rings}, J. Algebra \textbf{123}
  (1989), 397--413.

\bibitem{Popescu:1973}
N.~Popescu, \emph{Abelian categories with applications to rings
and modules},
  Academic Press, London, 1973.

\bibitem{Rafael:1990}
M.~D. Rafael, \emph{Separable functors revisited}, Comm. Algebra
\textbf{18}
  (1990), 1445--1459.

\bibitem{SaavedraRivano:1972}
N.~Saavedra~Rivano, \emph{Categories tannakiennes}, Lecture Notes
in
  Mathematics, vol. 265, Springer, Berlin, 1972.

\bibitem{Stenstrom:1975}
B.~{Stenstr\"om}, \emph{Rings of {Quotients}}, Springer, Berlin,
1975.

\bibitem{Sweedler:1975}
M.~Sweedler, \emph{The predual theorem to the {Jacobson-Bourbaki
Theorem}},
  Trans. Amer. Math. Soc. \textbf{213} (1975), 391--406.

\bibitem{Takeuchi:1977}
M.~Takeuchi, \emph{Morita theorems for categories of comodules},
{J. Fac. Sci.
  Univ. Tokyo} \textbf{24} (1977), 629--644.

\end{thebibliography}