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\begin{document}

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\section*{coring}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{base_ring_extension}{Base ring extension}\dotfill \pageref*{base_ring_extension} \linebreak
\noindent\hyperlink{morphisms_of_corings_over_different_bases}{Morphisms of corings over different bases}\dotfill \pageref*{morphisms_of_corings_over_different_bases} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{sweedler_corings}{Sweedler corings}\dotfill \pageref*{sweedler_corings} \linebreak
\noindent\hyperlink{matrix_corings}{Matrix corings}\dotfill \pageref*{matrix_corings} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of coring is a generalization of a $k$-[[coalgebra]]. While for a coalgebra $k$ must be a \textbf{commutative} ring (often a [[field]]), a coring is defined over a general noncommutative ring $k$ or even an associative algebra $A$.

Whereas a coalgebra structure is defined on a $k$-module (if $k$ is a field, it is a [[vector space]]) -- which may be regarded as a central $k$-[[bimodule]] -- a coring structure is defined on a general bimodule over a general [[ring]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

An \textbf{$A$-coring} is a [[comonoid]] in the [[monoidal category]] of central [[bimodule]]s over a fixed (typically noncommutative) unital [[ring]] $A$.

This generalizes the notion of $A$-[[coalgebra]]s which are defined only if $A$ is commutative and where the bimodules in question are [[central bimodule|central]].

\hypertarget{base_ring_extension}{}\subsection*{{Base ring extension}}\label{base_ring_extension}

More generally, fix a ground commutative ring $R$. Corings will be now over $R$-algebras. So a coring will mean a pair $(A,C)$ where $A$ is an $R$-algebra and $C$ an $A$-coring.

Let $\alpha:A\to B$ be a morphism of rings and $C$ an $A$-coring. Then the $B$-$B$-bimodule $B\otimes_A C\otimes_A B$ has an induced structure of a $B$-coring with comultiplication

\begin{displaymath}
B\otimes_A C\otimes_A B \stackrel{B\otimes \Delta_C\otimes B}\longrightarrow
 B\otimes_A C\otimes_A C\otimes_A B
\stackrel{B\otimes C\otimes 1_B\otimes C\otimes B}\longrightarrow
B\otimes_A C\otimes_A B\otimes_A C\otimes_A B \cong
(B\otimes_A C\otimes_A B)\otimes_B (B \otimes_A C\otimes_A B)
\end{displaymath}
and the counit

\begin{displaymath}
B\otimes_A C\otimes_A B \stackrel{B\otimes\epsilon_C\otimes B}\longrightarrow B\otimes_A A\otimes_A B \stackrel{B\otimes\phi\otimes B}\longrightarrow 
B\otimes_A B\otimes_A B \stackrel{mult}\longrightarrow B
\end{displaymath}
\hypertarget{morphisms_of_corings_over_different_bases}{}\subsection*{{Morphisms of corings over different bases}}\label{morphisms_of_corings_over_different_bases}

A morphism $(A,C)\to (B,D)$ is a pair $(\alpha,\gamma)$ where

\begin{itemize}%
\item $\alpha : A\to B$ is an $R$-algebra morphism; by restriction this makes $D$ an $A$-$A$-bimodule by restriction. Denote also by $p:D\otimes_A D\to D\otimes_B D$ the canonical projection of bimodules induced by $\alpha$.
\item $\gamma : C\to D$ is a map of $A$-$A$-bimodules
\item $\gamma$ commutes with counit $\alpha \circ \epsilon_C = \epsilon_D\circ \gamma$
\item $p\circ (\gamma\otimes_A\gamma)\circ \Delta_C = \Delta_D\circ \gamma$

\end{itemize}
The last two conditions can be said that the base ring extension coring $B\otimes_A C\otimes_A B$ of $C$ maps to $D$ (via map induced by $\gamma$) as a morphism of $B$-corings.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{sweedler_corings}{}\subsubsection*{{Sweedler corings}}\label{sweedler_corings}

The classical example of a coring is the [[Sweedler coring]] corresponding to an extension $R\hookrightarrow S$ of unital rings. The category of [[descent]] data for this extension is equivalent to the category of [[comodule]]s over the Sweedler coring.

Corings are in general useful for the treatment of [[descent in noncommutative algebraic geometry]].

\hypertarget{matrix_corings}{}\subsubsection*{{Matrix corings}}\label{matrix_corings}

Another major class of examples are the so-called [[matrix coring]]s.

\hypertarget{references}{}\subsection*{{References}}\label{references}

The notion of an $A$-coring is introduced by M. Sweedler and recently lived through a renaissance in works of [[T. Brzeziński]], R. Wisbauer, [[G. Böhm]], L. Kaoutit, G\'o{}mez-Torrecillas, S. Caenepeel, J. Y. Abuhlail, J. Vercruysse and others, including the creation of Galois theory for corings. Some prefer to speak about $A$-[[cocategory|cocategories]].

There is already a monograph:

\begin{itemize}%
\item [[T. Brzeziński]], R. Wisbauer, \emph{Corings and comodules}, London Math. Soc. Lec. Note Series \textbf{309}, Cambridge 2003.

\end{itemize}
Special topics:

\begin{itemize}%
\item [[T. Brzeziński]], \emph{Descent cohomology and corings}, Comm. Algebra 36:1894-1900, 2008, \href{http://arxiv.org/abs/math.RA/0601491}{arxiv:math.RA/0601491}


\item L. El Kaoutit, J. Gomez-Torrecillas, \emph{On the set of grouplikes of a coring}, \href{http://arxiv.org/abs/0901.4291}{arxiv/0901.4291}


\item [[T. Brzeziński]], \emph{Flat connections and (co)modules}, in: New Techniques in Hopf Algebras and Graded Ring Theory, S Caenepeel and F Van Oystaeyen (eds), Universa Press, Wetteren, 2007 pp. 35-52 \href{http://arxiv.org/abs/math.QA/0608170}{arxiv:math.QA/0608170}


\item [[T. Brzeziński]], \emph{The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties}, Algebras and Representation Theory \textbf{5} (2002) 389-410, \href{http://arxiv.org/abs/math.QA/0002105}{math.QA/0002105}


\item Lars Kadison, \emph{Depth two and Galois coring}, \href{http://arxiv.org/abs/math.RA/0408155}{math.RA/0408155}


\item [[George M. Bergman]], Adam O. Hausknecht, \emph{Cogroups and co-rings in categories of associative rings}, A.M.S. Math. Surveys and Monographs \textbf{45}, ix+388 pp., 1996; ISBN 0-8218-0495-2. \href{http://www.ams.org/mathscinet-getitem?mr=97k:16001}{MR 97k:16001} \href{http://math.berkeley.edu/~gbergman/papers/updates/coalg.html}{errata and updates}.


\item [[T. Brzeziński]], L. Kadison, [[R. Wisbauer]], \emph{On coseparable and biseparable corings}, Hopf algebras in noncommutative geometry and physics, 71--87, Lecture Notes in Pure and Appl. Math., 239, Dekker, New York, 2005.



\end{itemize}
There is a generalization of corings:

\begin{itemize}%
\item Jawad Y. Abuhlail, \emph{Semicorings and semicomodules}, \href{http://arxiv.org/abs/1303.3924}{arxiv/1303.3924}

\end{itemize}
category: algebra [[!redirects corings]] [[!redirects semicoring]]



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