Braided Galois objects and Sweedler cohomology of certain Radford biproducts

Abstract

We construct the group of $H$-Galois objects for a flat and cocommutative Hopf algebra in a braided monoidal category with
equalizers provided that a certain assumption on the braiding is fulfilled. We show that it is a subgroup of the group of BiGalois objects
of Schauenburg, and prove that the latter group is isomorphic to the semidirect product of the group of Hopf automorphisms of
$H$ and the group of $H$-Galois objects.
Dropping the assumption on the braiding, we construct the group of $H$-Galois objects with normal basis.
For $H$ cocommutative we construct Sweedler cohomology and prove that the second cohomology group is
isomorphic to the group of $H$-Galois objects with normal basis. We construct the Picard group of invertible $H$-comodules
for a flat and cocommutative Hopf algebra $H$. We show that every $H$-Galois object is an invertible $H$-comodule, yielding
a group morphism from the group of $H$-Galois objects to the Picard group of $H$. A short exact sequence is constructed
relating the second cohomology group and the two latter groups, under the above mentioned assumption on the braiding. We
show how our constructions generalize some results for modules over commutative rings, and some other known for symmetric
monoidal categories. Examples of Hopf algebras are discussed for which we compute the second cohomology group and the group
of Galois objects.