Zariski–Lipman conjecture for algebraifolds (regularity from finitely generated derivations)

Determine whether, for a finitely generated integral k-algebra A over a characteristic-zero field k, the condition that the A-module of k-derivations Der_k(A) is finitely generated projective (i.e., A is a k-algebraifold) implies that A is regular. Concretely, prove or refute that every finitely generated k-algebra A without zero divisors with Der_k(A) finitely generated projective must be regular in the commutative algebra sense (each localization at a prime is a regular local ring).

Background

Over a characteristic-zero field k, if A is a finitely generated regular k-algebra without zero divisors, then the Kähler differentials \widehat{\Omega}_A are finitely generated projective, which implies A is an algebraifold and \Omega_A = \widehat{\Omega}_A by Theorem 2.7. The converse direction is the Zariski–Lipman conjecture.

The paper frames the conjecture in the algebraifold language and notes partial positive results in the literature, but emphasizes that the general statement remains unresolved. Establishing the implication would tightly connect the algebraifold condition to the classical notion of regularity.