Vasconcelos’ conjecture on the conormal module

For any ideal I of finite projective dimension in a commutative noetherian local ring R, we prove that if the conormal module $$I/I^2$$ has over R/I, then must be generated by regular sequence. This resolves conjecture Vasconcelos. We similar result for first Koszul homology I. When R is localisation polynomial field K characteristic zero, Vasconcelos conjectured R/I reduced complete intersection $$\Omega _{(R/I)/K}$$ Kähler differentials dimension; this contingent on Eisenbud-Mazur conjecture. The arguments exploit structure homotopy Lie algebra associated to an essential way. By work Avramov and Halperin, every degree 2 element radical, Iyengar shown free summands give rise central elements algebra, establish analogous criterion constructing radical elements, from which deduce our main result.