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 υ_N: ${\mathcal U}$(N) → N such that every generalized coderivation from a $({\mathcal D},{\mathcal C})$-bicomodule M to N is uniquely expressed as υ_N ∘ f with some $({\mathcal D},{\mathcal C})$-bicomodule map f: M → ${\mathcal U}$(N). ${\mathcal U}$(N) is isomorphic to the cotensor product of N and ${\mathcal U}$(${\mathcal D}$ ⊗_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.