Minimal free resolution of a graded ideal with linear quotients

TL;DR

This paper presents a direct inductive method using the Horseshoe Lemma to construct minimal free resolutions for graded ideals with linear quotients, specifically focusing on $d$-linear resolutions.

Contribution

It introduces a new inductive approach to obtain minimal free resolutions of graded ideals with linear quotients using the Horseshoe Lemma.

Findings

Provides a constructive method for $d$-linear resolutions

Simplifies the process of finding minimal free resolutions

Enhances understanding of ideals with linear quotients

Abstract

Let $I$ be a graded ideal of $K[x_1,\ldots,x_n]$ generated by homogeneous polynomials of a same degree $d$, and assume that $I$ has linear quotients. In this note, we use Horseshoe Lemma to give a relatively direct inductive construction of a minimal free resolution of $I$, which is called a $d$-linear resolution.