Extremal Betti Numbers and Applications to Monomial Ideals

TL;DR

This paper introduces extremal Betti numbers as a refinement of regularity, explores their invariance under generic initial modules, and relates them to Alexander duality, providing new proofs of classical results in monomial ideal theory.

Contribution

It defines extremal Betti numbers, studies their properties, and connects them to Alexander duality, extending existing theorems and simplifying proofs.

Findings

Extremal Betti numbers are preserved under generic initial modules.

Relations between extremal Betti numbers of ideals and their Alexander duals are established.

New, simpler proofs of classical criteria by Hochster, Reisner, and Stanley are provided.

Abstract

In this short note we introduce a notion of extremality for Betti numbers of a minimal free resolution, which can be seen as a refinement of the notion of Mumford-Castelnuovo regularity. We show that extremal Betti numbers of an arbitrary submodule of a free S-module are preserved when taking the generic initial module. We relate extremal multigraded Betti numbers in the minimal resolution of a square free monomial ideal with those of the monomial ideal corresponding to the Alexander dual simplicial complex and generalize theorems of Eagon-Reiner and Terai. As an application we give easy (alternative) proofs of classical criteria due to Hochster, Reisner, and Stanley.