The Herzog-Takayama resolution over a skew polynomial ring

TL;DR

This paper extends the Herzog-Takayama minimal free resolution, originally for monomial ideals in polynomial rings, to the setting of skew polynomial rings, broadening its applicability.

Contribution

It generalizes the Herzog-Takayama resolution to monomial ideals in skew polynomial rings, expanding the class of ideals with known minimal free resolutions.

Findings

Successfully constructed a minimal free resolution in skew polynomial rings.

The resolution reduces to the classical Herzog-Takayama resolution in the commutative case.

Provides a framework for resolving monomial ideals in non-commutative settings.

Abstract

Let $\Bbbk$ be a field, and let $I$ be a monomial ideal in the polynomial ring $R=\Bbbk[x_1,\ldots,x_n]$. In her thesis, Taylor introduced a complex that provides a finite free resolution of $R/I$ as an $R$-module. Building on this, Ferraro, Martin and Moore extended this construction to monomial ideals in skew polynomial rings. Since the Taylor resolution is generally not minimal, significant effort has been devoted to identifying classes of ideals with minimal free resolutions that are relatively straightforward to construct. In a 1987 paper, Eliahou and Kervaire developed a minimal free resolution for a class of monomial ideals in $R$ known as stable ideals. This result was later generalized to stable ideals in skew polynomial rings by Ferraro and Hardesty. In a 2002 paper, Herzog and Takayama constructed a minimal free resolution for monomial ideals with linear quotients, a broader class of ideals containing stable ideals. Their resolution reduces to the Eliahou-Kervaire resolution in the stable case. In this paper, we generalize the Herzog-Takayama resolution to skew polynomial rings.