Hypotheses for faithful flatness of Green meadow morphisms for general finite groups

Identify sufficient and verifiable hypotheses on a morphism k → ℓ of Green meadows for an arbitrary finite group G that guarantee the base-change functor ℓ ⊠_k − is faithfully flat, thereby generalizing Theorem thm:fields-are-FF from the case G = C_p with zero transfer to more complicated groups.

Background

Theorem thm:fields-are-FF proves that if G = C_p and the transfer in ℓ vanishes, then any morphism k → ℓ of C_p-Green meadows is faithfully flat. The authors expect analogous results for larger groups but lack precise criteria on the morphism that ensure faithful flatness.

They suggest possible criteria (e.g., preserving isotropy of the Weyl action or matching patterns of nontrivial transfers), but a definitive set of conditions remains to be found.