On a Consequence of the Order Ideal Conjecture

TL;DR

This paper investigates specific cases where certain algebraic maps vanish in the context of the order ideal conjecture, advancing understanding of the conjecture's implications in regular local rings.

Contribution

It proves new cases where the maps from Koszul homology to Tor groups are zero, linking these results to the order ideal conjecture.

Findings

Identifies cases where the map $K_d(old{x}; R) o ext{Tor}_d^R(R/I, k)$ is zero.

Derives multiple cases where higher-degree maps are also zero, based on the order ideal conjecture.

Provides partial evidence supporting the order ideal conjecture in regular local rings.

Abstract

Given a minimal set of generators $\bold{x}$ of an ideal $I$ of height d in a regular local ring ($R, m, k$) we prove several cases for which the map $K_d(\bold{x}; R) \otimes k \to \Tor_d^R (R/I, k)$ is the 0-map. As a consequence of the order ideal conjecture we derive several cases for which $K_{d+i}(\bold{x}; R) \otimes k \to \Tor_{d+i}^R (R/I, k)$ are 0-maps for $i \ge 0$.