On Modules of Finite Projective Dimension

TL;DR

This paper investigates modules with finite projective dimension over local rings, solving embeddability issues and advancing the order ideal conjecture, with implications for prime ideals and non-zero-divisors in mixed characteristic rings.

Contribution

It provides solutions to the embeddability problem and reduces cases of the order ideal conjecture, especially in mixed characteristic local rings.

Findings

Embeddability problem is solved for modules of finite projective dimension.

In mixed characteristic rings, minimal generators of certain ideals are non-zero-divisors.

Prime ideals of finite projective dimension have minimal generators that are non-zero-divisors.

Abstract

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular we derive that in any local ring R of mixed characteristic p > 0, where p is a non-zero-divisor, if I is an ideal of finite projective dimension over R and p is in I or p is a non-zero-divisor on R/I, then every minimal generator of I is a non-zero-divisor. Hence if P is a prime ideal of finite projective dimension in a local ring R, then every minimal generator of P is a non-zero-divisor in R.