Maximal Crossed Product Orders Over Discrete Valuation Rings

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 1960s 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 article 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 subcrossed 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.

Key Words:

• Conductor
• Crossed product
• Discrete valuation ring
• Hereditary order
• Maximal order

2000 Mathematics Subject Classification:

• 16H05
• 16E60
• 20J06

ACKNOWLEDGMENT

I am grateful to my colleague Amiram Braun and to Mota Frances for their help.

Notes

Communicated by M. Cohen.