Geometric partial comodules over flat coalgebras in Abelian categories are globalizable

Abstract

The aim of this paper is to prove the statement in the title. As a by-product, we obtain new globalization results in cases never considered before, such as partial corepresentations of Hopf algebras. Moreover, we show that for partial representations of groups and Hopf algebras, our globalization coincides with those described earlier in literature. Finally, we introduce Hopf partial comodules over a bialgebra as geometric partial comodules in the monoidal category of (global) modules. By applying our globalization theorem we obtain an analogue of the fundamental theorem for Hopf modules in this partial setting.

A comonadicity theorem for partial comodules

“…Partial comodules as defined above, should not be confused with the more general notion of 'geometric partial comodule' as introduced in [21]. In [24], it was shown that partial comodules in the sense above are a particular kind of geometric partial comodules.…”

“…In view of this, in order to make partial representation theory applicable to algebraic groups, it is indispensable to consider partial comodules or corepresentations over a Hopf algebra, notions which were recently introduced in [4]. A globalization theorem for partial comodules has been obtained recently in [24], within the more general framework of so-called 'geometric partial comodules' [21]. In spite of this, in [4] the remarkable observation was made that partial comodules do not share the intrinsic finiteness observed in the usual theory of comodules over a coalgebra because, in general, they do not satisfy the fundamental theorem of comodules.…”

On the globalization of geometric partial (co)modules in the categories of topological spaces and algebras

“…The geometric partial (co)modules from [10] provide a general categorical framework to study all sorts of partial actions in a unified way, subsuming partial actions of groups as well as partial (co)representations of Hopf algebras (see [15] for a detailed treatment of the globalization question in these cases). Moreover, geometric partial comodules also allow to treat cases that cannot be described by the Hopf-algebraic partial (co)actions from [4], such as genuine partial actions of algebraic groups on irreducible varieties.…”