Syzygies of the residue field over Golod rings

TL;DR

This paper investigates the syzygies of the residue field over Golod rings, providing recursive formulas for high syzygies and complete classifications in low embedding dimensions.

Contribution

It introduces a recurrent formula for high syzygies over Golod rings and fully describes indecomposable summands in low embedding dimensions.

Findings

Recurrent formula for high syzygies of the residue field

Complete description of indecomposable summands in embedding dimension ≤ 2

Advances understanding of syzygy structures over Golod rings

Abstract

Let $(R,m,k)$ be a Golod ring. We show a recurrent formula for high syzygies of $k$ interms of previous ones. In the case of embedding dimension at most $2$, we provided complete descriptions of all indecomposable summands of all syzygies of $k$.