Finite projective dimension and a question of Jorgensen

TL;DR

This paper investigates conditions for finite projective dimension over Noetherian local rings, addressing a longstanding question by Jorgensen using spectral sequences and generalized local cohomology.

Contribution

It provides new criteria and results for bounding projective dimension, extending to other homological dimensions and utilizing weakly full ideals.

Findings

Established bounds for projective dimension via Ext vanishing

Extended results to other homological dimensions

Derived criteria using weakly full ideals

Abstract

This paper studies finite projective dimension of finitely generated modules over a Noetherian local ring, by means of spectral sequence methods related to generalized local cohomology. Our main goal is to address a question raised by D. Jorgensen over fifteen years ago, concerning a prescribed bound (via Ext vanishing) for projective dimension over a complete intersection local ring. We obtain similar results involving other homological dimensions as well. Also we make use of weakly full ideals to derive further criteria for prescribed bound on projective dimension.