The Monomial Conjecture and order ideals II

TL;DR

This paper establishes the equivalence between key conjectures in commutative algebra related to regular local rings and proves several special cases, advancing understanding of the monomial and order ideal conjectures.

Contribution

It proves the equivalence of the monomial conjecture, order ideal conjecture, and a specific edge homomorphism non-vanishing condition, and verifies several special cases.

Findings

Equivalence of conjectures established

Special cases of the conjectures proved

Edge homomorphism non-vanishing linked to conjecture validity

Abstract

Let $I$ be an ideal of height $d$ in a regular local ring $(R,m,k=R/m)$ of dimension $n$ and let $\Omega$ denote the canonical module of $R/I$. In this paper we first prove the equivalence of the following: the non-vanishing of the edge homomorphism $\eta_d: \ext{R}{n-d}{k,\Omega} \rightarrow \ext{R}{n}{k,R}$, the validity of the order ideal conjecture for regular local rings, and the validity of the monomial conjecture for all local rings. Next we prove several special cases of the order ideal conjecture/monomial conjecture.