Maximal Crossed Product Orders over Discrete Valuation Rings

TL;DR

This paper characterizes when crossed product orders over discrete valuation rings are maximal, extending previous results to residually separable cases and revealing structural conditions involving the inertia subgroup.

Contribution

It generalizes the criteria for maximality of crossed product orders to residually separable extensions, highlighting the role of the inertia subgroup and conductor in this context.

Findings

Maximal order property implies the inertia subgroup is abelian.

The order of the conductor equals the number of maximal two-sided ideals.

A crossed product order is maximal if and only if the conductor subgroup is trivial.

Abstract

The problem of determining when a (classical) crossed product $T=S^f*G$ of a finite group $G$ over a discrete valuation ring $S$ is a maximal order, was answered in the 1960's for the case where $S$ is tamely ramified over the subring of invariants $S^G$. The answer was given in terms of the conductor subgroup (with respect to $f$) of the inertia. In this paper we solve this problem in general when $S/S^G$ is residually separable. We show that the maximal order property entails a restrictive structure on the sub-crossed product graded by the inertia subgroup. In particular, the inertia is abelian. Using this structure, one is able to extend the notion of the conductor. As in the tame case, the order of the conductor is equal to the number of maximal two sided ideals of $T$ and hence to the number of maximal orders containing $T$ in its quotient ring. Consequently, $T$ is a maximal order if and only if the conductor subgroup is trivial.