Correspondence theorems for Hopf algebroids with applications to affine groupoids

TL;DR

This paper establishes a correspondence between certain substructures in bialgebroids and Hopf algebroids, with applications to affine groupoids, enhancing understanding of their ideal and coideal relationships.

Contribution

It introduces a bijective correspondence between coideal subrings and coideals in Hopf algebroids under specific conditions, extending prior algebraic frameworks.

Findings

Established a correspondence between coideal subrings and coideals in bialgebroids.

Proved bijectivity of this correspondence for Hopf algebroids under additional conditions.

Applied results to classify normal Hopf ideals in affine groupoid schemes.

Abstract

We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for Hopf algebroids. As an application, we investigate normal Hopf ideals in commutative Hopf algebroids (affine groupoid schemes) in connection with the study of normal affine subgroupoids.