Generalized coderivations of bicomodules

Abstract

We introduce a generalized coderivation from a bicomodule to a bicomodule over corings, which is a generalization of a coderivation. For each $({\mathcal D},{\mathcal C})$-bicomodule $N$ over corings ${\mathcal C}$ and ${\mathcal D}$, we construct the universal generalized coderivation $\upsilon_N:{\mathcal U}(N)\to N$ such that every generalized coderivation from a $({\mathcal D},{\mathcal C})$-bicomodule $M$ to $N$ is uniquely expressed as $\upsilon_N\circ f$ with some $({\mathcal D},{\mathcal C})$-bicomodule map $f:M\to{\mathcal U}(N)$. ${\mathcal U}(N)$ is isomorphic to the cotensor product of $N$ and ${\mathcal U}({\mathcal D}\otimes_R{\mathcal C})$. We show that a coring ${\mathcal C}$ is coseparable if and only if, for any coring ${\mathcal D}$, all generalized coderivations from a $({\mathcal D},{\mathcal C})$-bicomodule to a $({\mathcal D},{\mathcal C})$-bicomodule are inner.