Embedded Associated Primes of Powers of Square-free Monomial Ideals

TL;DR

This paper studies the embedded primes of powers of square-free monomial ideals, introducing a new inductive technique and establishing bounds related to the ideal's combinatorial properties, with implications for a conjecture in combinatorial optimization.

Contribution

It develops an inductive method to analyze normally torsion-free square-free monomial ideals and links algebraic properties to hypergraph invariants, advancing understanding of embedded primes.

Findings

Embedded primes of I^t appear after t exceeds the monomial grade beta_1.

If I lacks the packing property, embedded primes occur at t=beta_1+1.

The results relate to a conjecture by Conforti and Cornue9jols.

Abstract

An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/I^t)=Ass(R/I) for all natural numbers t. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a square-free monomial ideal I is minimally not normally torsion-free then the least power t such that I^t has embedded primes is bigger than beta_1, where beta_1 is the monomial grade of I, which is equal to the matching number of the hypergraph H(I) associated to I. If in addition I fails to have the packing property, then embedded primes of I^t do occur when t=beta_1 +1. As an application, we investigate how these results relate to a conjecture of Conforti and Cornu\'ejols.