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