Higher structure maps for free resolutions of length 3 and linkage

TL;DR

This paper develops formulas for higher structure maps in free resolutions of perfect ideals of height 3, linking their properties to linkage and Betti numbers, with implications for classifying licci ideals.

Contribution

It introduces formulas to compute higher structure maps for linked ideals' resolutions, extending the understanding of linkage and free resolutions in Gorenstein rings.

Findings

Formulas for higher structure maps of linked ideals' resolutions

Characterization of licci ideals with specific Betti numbers

Connection between non-vanishing maps and linkage properties

Abstract

Let $I$ be a perfect ideal of height 3 in a Gorenstein local ring $R$. Let $\mathbb{F}$ be the minimal free resolution of $I$. A sequence of linear maps, which generalize the multiplicative structure of $\mathbb{F}$, can be defined using the generic ring associated to the format of $\mathbb{F}$. Let $J$ be an ideal linked to $I$. We provide formulas to compute some of these maps for the free resolution of $J$ in terms of those of the free resolution of $I$. We apply our results to describe classes of licci ideals, showing that a perfect ideal with Betti numbers $(1,5,6,2)$ is licci if and only if at least one of these maps is nonzero modulo the maximal ideal of $R$.