Projective freeness over Tambara fields

Determine whether every projective module over a Tambara field is free, where a Tambara field is a Tambara functor having no nonzero proper ideals.

Background

The paper defines a Tambara field as a Tambara functor with no nonzero proper ideals. In its application of Nakayama’s lemma, the paper reduces freeness questions for finitely generated projective modules over local Tambara functors to the corresponding question for their residue Tambara fields.

The authors note that sufficient conditions are known in special cases, including constant C_{p^n}-Tambara functors at a field, but the general implication from being a Tambara field to projective freeness remains unresolved. Establishing this would broaden the class of local Tambara functors for which finitely generated projective modules are free.