Some equivariant constructions in noncommutative algebraic geometry

TL;DR

This paper develops a framework for geometric actions in noncommutative algebraic geometry using monoidal categories, generalizing classical (co)module algebra approaches and introducing new notions of equivariance and quotients.

Contribution

It introduces a novel approach to geometric actions via monoidal categories, extending classical concepts and defining compatibility with localizations in noncommutative settings.

Findings

Established a distributive law for monoidal actions and localizations.

Defined notions of equivariant objects and noncommutative fiber bundles.

Provided a framework for quotients in noncommutative algebraic geometry.

Abstract

We here present rudiments of an approach to geometric actions in noncommutative algebraic geometry, based on geometrically admissible actions of monoidal categories. This generalizes the usual (co)module algebras over Hopf algebras which provide affine examples. We introduce a compatibility of monoidal actions and localizations which is a distributive law. There are satisfactory notions of equivariant objects, noncommutative fiber bundles and quotients in this setup.