On the flatness and the projectivity over Hopf subalgebras of Hopf algebras over discrete valuation rings

TL;DR

This paper investigates conditions under which Hopf algebras over Dedekind rings are flat or projective over their Hopf subalgebras, providing criteria and demonstrating faithful flatness and properties of the module of integral.

Contribution

It establishes new criteria for faithful flatness and projectivity of Hopf algebras over subalgebras, including the role of the module of integral.

Findings

Faithful flatness of flat Hopf algebras over finite normal subalgebras.

Finiteness conditions involving the module of integral for projectivity.

The module of integral has rank one in the studied context.

Abstract

We study the flatness and the projectivity of Hopf algebras, defined over a Dedekind ring, over their Hopf subalgebras. We give a criterion for the faithful flatness and use it to show the faithful flatness of an arbitrary flat Hopf algebra upon its finite normal Hopf subalgebra. For the projectivity of a projective Hopf algebras we need some finiteness condition in terms of the module of integral. In particular we show the the module of integral has rank one.