Ascension of G-regularity under flat local homomorphisms with regular closed fiber

Determine whether G-regularity ascends along flat local homomorphisms with regular closed fiber; specifically, given a flat local homomorphism f: (R,m) → (R′,m′) of noetherian local rings such that the closed fiber R′/mR′ is regular, ascertain whether R′ is G-regular whenever R is G-regular.

Background

G-regular local rings are those in which every totally reflexive module is free. This property plays a central role in refining homological dimensions, as in such rings the Gorenstein (totally reflexive) dimension coincides with projective and flat dimensions for finitely generated modules.

The paper’s main results (e.g., Theorem 3.5) detect exceptional complete intersection maps via bounds on Gorenstein flat dimension under the assumption that the map has finite flat dimension. Remark 3.9 raises whether these bounds and conclusions can be derived under a weaker assumption: replacing finite flat dimension by G-regularity of the source ring in the presence of a flat local homomorphism with regular closed fiber.

An affirmative answer would strengthen the main theorem by allowing the finite flat dimension hypothesis to be replaced by G-regularity and, moreover, would imply that the target module is free over S, thus sharpening the conclusions.