Numbers of Generators of Perfect Ideals

TL;DR

This paper investigates bounds on the number of generators of perfect ideals in regular local rings, establishing a bound based on the ideal's properties and exploring related conjectures with partial results.

Contribution

It introduces new bounds for the number of generators of perfect ideals and proposes conjectures with partial progress in the context of regular local rings.

Findings

Established bounds depending on n and projective dimension

Proposed conjectures with partial results

Analyzed the number of generators modulo powers of m

Abstract

This article is concerned with the number of generators of perfect ideals J in regular local rings (R,m). If J is sufficiently large modulo $m^n$, a bound is established depending only on n and the projective dimension of J. More ambitious conjectures are also introduced with some partial results.