Square-free Groebner degenerations

TL;DR

This paper proves that for homogeneous ideals with square-free initial ideals, key algebraic invariants like Betti numbers, depth, and regularity are preserved between the ideal and its initial ideal, highlighting structural stability.

Contribution

It establishes that square-free initial ideals preserve extremal Betti numbers, depth, and regularity of the original ideal, providing new insights into their algebraic properties.

Findings

Betti numbers of S/I and S/J coincide

Depth of S/I and S/J are equal

Regularity of S/I and S/J are equal

Abstract

Let I be a homogeneous ideal of a polynomial ring S. We prove that if the initial ideal J of I, w.r.t. a term order on S, is square-free, then the extremal Betti numbers of S/I and of S/J coincide. In particular, depth(S/I)=depth(S/J) and reg(S/I)=reg(S/J).