Vasconcelos’ conjecture on the conormal module

Abstract

For any ideal $I$ of finite projective dimension in a commutative noetherian ring $R$, we prove that if the conormal module $I/I^2$ has finite projective dimension over $R/I$, then $I$ must be locally generated by a regular sequence. This resolves a conjecture of Vasconcelos. We prove similar results for the first Koszul homology of $I$, and (with certain caveats) for the module of Kahler differentials of $R/I$. The arguments exploit the structure of the homotopy Lie algebra in an essential way.