Vasconcelos' conjecture on the conormal module

TL;DR

This paper proves Vasconcelos' conjecture that ideals with conormal modules of finite projective dimension are generated by regular sequences, using homotopy Lie algebra techniques and extending to differential modules under certain conditions.

Contribution

It establishes the conjecture relating finite projective dimension of conormal modules to regular sequence generation, and connects this to the structure of the homotopy Lie algebra.

Findings

Ideals with conormal modules of finite projective dimension are generated by regular sequences.

The structure of the homotopy Lie algebra is key to understanding these properties.

Conditional proof that certain modules imply reduced complete intersections.

Abstract

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 $\Omega_{(R/I)/K}$ of differentials has finite projective dimension; we prove this contingent on the Eisenbud-Mazur 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.