Compatibility of (co)actions and localizations

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.

Some Equivariant Constructions in Noncommutative Algebraic Geometry

“…In [34], we prove an interesting generalization of Theorem 5.8 for ρ-compatible localizations to localization-compatible pairs of entwining structures introduced therein.…”

“…The notion of the compatibility of (co)monads and localization functors can easily be extended to the compatibility of actions of monoidal groups and localizations. For simplicity, we leave out this generalization (and present it in [34]). We implicitly (verbally in a definition) use it in the next section though.…”