# nLab coseparable coring

Coseparability of corings is a dual notion to separability of rings? (where one requires that the multiplication map is split).

An $A$-coring $\left(C,\Delta ,ϵ\right)$ is coseparable if the comultiplication $\Delta :C\to C{\otimes }_{A}C$ splits as a $C$-$C$-bicomodule morphism. In other words, there is a morphism of $A$-$A$-bimodules $p:C{\otimes }_{A}C\to C$ such that

$\Delta \circ p=\left(C{\otimes }_{A}p\right)\circ \left(\Delta {\otimes }_{A}C\right)=\left(p{\otimes }_{A}C\right)\circ \left(C{\otimes }_{A}\Delta \right)$\Delta\circ p = (C \otimes_A p)\circ(\Delta\otimes_A C) = (p\otimes_A C)\circ(C\otimes_A\Delta)
$p\circ \Delta =C$p\circ\Delta = C

where $C={\mathrm{Id}}_{C}$.

Revised on July 2, 2009 20:42:45 by Toby Bartels (71.104.230.172)