On partial (Co)actions on coalgebras : globalizations and some galois theory

Abstract

Partial module coalgebra and partial comodule coalgebra are the dual notions of partial module algebra, and all these structures are close related. Partial module algebra was defined by Caenepeel and Janssen in [11] and developed in a certain direction by Alves and Batista in [1–3]. We are interested in some constructions made by Alves and Batista, namely: globalization for partial module algebras, a Morita context relating the invariant subalgebra and the partial smash product, and Galois theory. In this work, we introduce the notion of globalization for partial module coalgebra and for partial comodule coalgebra. We show that every partial module coalgebra is globalizable constructing a globalization, named standard. For the case of partial comodule coalgebra we need assume some kind of rationality condition to obtain a correspondent globalization. Moreover, for a partial comodule coalgebra, we construct a Morita-Takeuchi context relating the coinvariant coalgebra and the partial smash coproduct, and we define a Galois coextension and show some properties, relating the Galois coextension for partial comodule coalgebra with the Galois extension for partial coaction on algebras, extending results of Dascalescu, Raianu, and Zhang obtained in [20].