Geometrically Partial actions

TL;DR

This paper introduces geometric partial comodules over coalgebras, providing a new framework that better describes partial actions in algebraic geometry and extends Hopf-Galois theory.

Contribution

It proposes a novel notion of geometric partial comodules, demonstrating their categorical properties and potential applications in Hopf algebra theory.

Findings

Category of geometric partial comodules is complete and cocomplete

Partial comodules over a Hopf algebra form a lax monoidal category

Hopf-Galois theory is extended to geometric partial coactions

Abstract

We introduce "geometric" partial comodules over coalgebras in monoidal categories, as an alternative notion to the notion of partial action and coaction of a Hopf algebra introduced by Caenepeel and Janssen. The name is motivated by the fact that our new notion suits better if one wants to describe phenomena of partial actions in algebraic geometry. Under mild conditions, the category of geometric partial comodules is shown to be complete and cocomplete and the category of partial comodules over a Hopf algebra is lax monoidal. We develop a Hopf-Galois theory for geometric partial coactions to illustrate that our new notion might be a useful additional tool in Hopf algebra theory.