Equivalences of comodule categories for coalgebras over rings

Abstract

In this article we defined and studied the quasi-finite comodules, cohom functors for coalgebras over rings. Linear functors between categories of comodules are also investigated and it is proved that good enough linear functors are nothing but a cotensor functor. Our main result of this work characterizes equivalences between comodule categories generalizing the Morita–Takeuchi theory to coalgebras over rings. Morita–Takeuchi contexts in our setting is de2ned and investigated, a correspondence between strict Morita–Takeuchi contexts and equivalences of comodule categories over the involved coalgebras is obtained. Finally, we proved that for coalgebras over QF-rings Takeuchi’s representation of the cohomfunctor is also valid.