Compatibility of (co)actions and localizations

TL;DR

This paper explores different notions of compatibility between localizations and algebraic structures, introducing categorical frameworks and distributive laws to unify and extend previous concepts in noncommutative geometry.

Contribution

It presents a new categorical approach to compatibility of localizations with algebraic structures, including Ore localizations and entwining structures, broadening the theoretical understanding.

Findings

Categorical language describes compatibility of Ore localizations with comodule algebra structures.

Distributive laws are key to understanding localization compatibility.

Functoriality properties relate localizations to comonads and monoidal actions.

Abstract

Earlier, Lunts and Rosenberg studied a notion of compatibility of endofunctors with localization functors, with an application to the study of differential operators on noncommutative rings and schemes. Another compatibility -- of Ore localizations of an algebra with a comodule algebra structure over a given bialgebra -- introduced in my earlier work -- is here described also in categorical language, but the appropriate notion differs from that of Lunts and Rosenberg, and it involves a specific kind of distributive laws. Some basic facts about compatible localization follow from more general functoriality properties of associating comonads or even actions of monoidal categories to comodule algebras. We also introduce localization compatible pairs of entwining structures.