The Lipman-Zariski conjecture in characteristic zero

Theorem 1.1: there are a finitely generated normal integral $\mathbb C$-algebra $A$ of dimension two and a maximal ideal $\mathfrak m$ with $\mathrm{Der}_{\mathbb C}(A)\cong A^{\oplus2}$ and $A_{\mathfrak m}$ not regular; the surface can be smooth away from that point. By Jorder's theorem no local tangent frame there commutes. The paper also answers negatively, in characteristic zero, the derivation-blowup question of Barajas, Chavez-Martinez and Romano-Velazquez (2026, Question 1). It does not address higher-dimensional non-isolated cases left open by earlier results or give a counterexample with a commuting frame (impossible by Jorder).

Theorem 1.1: there are a finitely generated normal integral $\mathbb C$-algebra $A$ of dimension two and a maximal ideal $\mathfrak m$ with $\mathrm{Der}_{\mathbb C}(A)\cong A^{\oplus2}$ and $A_{\mathfrak m}$ not regular; the surface can be smooth away from that point. By Jorder's theorem no local tangent frame there commutes. The paper also answers negatively, in characteristic zero, the derivation-blowup question of Barajas, Chavez-Martinez and Romano-Velazquez (2026, Question 1). It does not address higher-dimensional non-isolated cases left open by earlier results or give a counterexample with a commuting frame (impossible by Jorder).