On the gradient of a monomial ideal
Antonino Ficarra
TL;DR
This work analyzes how differentiation via the gradient ∂(I) interacts with the Castelnuovo–Mumford regularity of homogeneous ideals. It shows that reg ∂(I) can exceed or fall behind reg I by any integer, even when I has a linear resolution, and introduces differential linear resolution to track the entire gradient chain. The authors identify several gradient-closed classes, notably polymatroidal and equigenerated stable ideals, for which reg ∂ ≤ reg and many instances satisfy reg ∂ = reg I − 1; they also develop results for squarefree Stanley–Reisner ideals with many generators and for powers of squarefree quadratic ideals, demonstrating extensive preservation of linearity under gradient and taking steps toward a broader differential-linear framework. These findings clarify how the gradient operation shapes regularity and establish broad families with predictable gradient behavior, with implications for both combinatorial and algebraic properties of monomial ideals.
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 Busé, 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.