On geometrically transitive Hopf algebroids

TL;DR

This paper characterizes geometrically transitive commutative Hopf algebroids, showing their equivalence to several conditions involving base change morphisms, principal bi-bundles, and local isomorphisms, and explores their relation to transitive groupoids.

Contribution

It provides new characterizations of geometrically transitive Hopf algebroids, linking algebraic properties with geometric and groupoid-theoretic conditions.

Findings

Base change morphisms are weak equivalences in geometrically transitive cases.

Any two isotropy Hopf algebras are weakly equivalent and conjugated.

The character groupoid is transitive, with local isomorphisms between objects.

Abstract

This paper contributes to the characterization of a certain class of commutative Hopf algebroids. It is shown that a commutative flat Hopf algebroid with a non zero base ring and a nonempty character groupoid is geometrically transitive if and only if any base change morphism is a weak equivalence (in particular, if any extension of the base ring is Landweber exact), if and only if any trivial bundle is a principal bi-bundle, and if and only if any two objects are fpqc locally isomorphic. As a consequence, any two isotropy Hopf algebras of a geometrically transitive Hopf algebroid (as above) are weakly equivalent. Furthermore, the character groupoid is transitive and any two isotropy Hopf algebras are conjugated. Several other characterizations of these Hopf algebroids in relation to transitive groupoids are also given.