Field independence of algebra and module amenability

Does amenability depend on the ground field? More precisely, for a field extension K⊂L, an associative L-algebra A and an A-module M, is amenability over A viewed as a K-algebra equivalent to amenability over L?

Why it matters and what counts as progress

Why it matters: The ground-field question asks whether amenability is intrinsic to an algebra or module rather than dependent on its scalar description.

What counts as progress: Prove field independence for arbitrary associative algebras and modules, or exhibit an extension where amenability changes.

How the problem developed

2018 · Open question [1]
Yves Cornulier Raised whether algebraic amenability can depend on the base field, motivating the field-extension question for algebras and modules.
8 Aug 2026 · Field invariance result [2]
Be’eri Greenfeld; ChatGPT 5.6 Sol contributed significant proof ideas, and Greenfeld wrote and checked the paper Proved that the paper’s algebra- and module-amenability notions are invariant under extension of the ground field.

AI contribution

Amenability is not sensitive to the base field

Be'eri Greenfeld

For any field extension K⊂L, associative L-algebra A and A-module M, Greenfeld’s manuscript states that amenability of M over A viewed as a K-algebra is equivalent to amenability over L. This also applies to amenable and fully amenable algebras.

AI involvementAI-centered

ImportanceNotable

Technical verificationReported

Human understandingFull