On the gradient of a monomial ideal

TL;DR

This paper investigates the relationship between the regularity of monomial ideals and their gradient ideals, demonstrating that these regularities can differ arbitrarily even for ideals with linear resolutions, and introduces the concept of monomial ideals with differential linear resolution.

Contribution

It proves the regularity difference can be arbitrarily large for monomial ideals with linear resolution and introduces monomial ideals with differential linear resolution, expanding understanding of their properties.

Findings

Regularity differences can be arbitrarily large for monomial ideals with linear resolution.

Constructed examples of monomial ideals with prescribed regularity differences.

Identified classes of monomial ideals with differential linear resolution.

Abstract

Let $K$ be a field of characteristic zero, let $I \subset S = K[x_1,\dots,x_n]$ be a homogeneous ideal, and let $\partial(I)$ be its gradient ideal. We study the relationship between $\mathrm{reg}\,I$ and $\mathrm{reg}\,\partial(I)$. While earlier work by Bus\'e, Dimca, Schenck, and Sticlaru showed these regularities are generally incomparable for hypersurface ideals, we prove they remain incomparable even for monomial ideals with linear resolution, answering a question of J. Herzog. In fact, for any integers $a \in \mathbb{Z}$ and $b \ge - 1$, we construct monomial ideals $I$ and $J$ such that $\mathrm{reg}\,I - \mathrm{reg}\,\partial(I) = a$, $\mathrm{reg}\,\partial(J) - \mathrm{reg}\,J = b$ and $J$ has linear resolution. We introduce monomial ideals with differential linear resolution as those monomial ideals whose all iterated gradient ideals have linear resolution. We prove that polymatroidal ideals, equigenerated (strongly) stable ideals, powers of edge ideals with linear resolution, complementary edge ideals with linear resolution, and certain equigenerated squarefree monomial ideals with many generators satisfy this property.