Syzygies of the residue field over Golod rings

TL;DR

This work shows that over Golod local rings, the entire syzygy structure of the residue field $k$ is governed by a fixed, small set of indecomposable modules and a universal recursion driven by Koszul homology. In embedding dimension two, the authors give detailed, explicit decompositions of all syzygies of $k$, reducing the problem to the low‑syzygy data ${\rm syz}_{1}^{R}(k)=\mathfrak{m}$ and ${\rm syz}_{2}^{R}(k)={\rm syz}_{1}^{R}(\mathfrak{m})$, with ${\rm syz}_{1}^{R}(\mathfrak{m})\cong \mathfrak{m}^{*}$ when $R$ is not a DVR or zero‑dimensional CI. The core mechanism is a recursive formula for ${\rm syz}_i^R(k)$ in terms of earlier syzygies, via the Koszul complex $K(\mathfrak{m};R)$ and the tensor algebra on higher syzygies, which explains Golod’s classical rank formulas and yields tight indecomposability criteria. The results persist under completion and yield a finite, small set of indecomposables from which all syzygies of $k$ can be assembled, at least in codimension two, with precise combinatorial data encoded by $a=\dim_k((\mathfrak{n}(I:\mathfrak{n}))/ (\mathfrak{n}I))$ and related invariants. This provides a concrete, structural understanding of infinite free resolutions in Golod settings and highlights stark contrasts with zero‑dimensional Gorenstein cases.

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 $k$.

Theorems & Definitions (32)

• Theorem 1.1
• Proposition 1
• Theorem 1.2
• Corollary 1
• Conjecture 1.3
• Theorem 2.1
• Theorem 2.2
• Example 2.3
• proof : Proof of \ref{['3 equi']}
• Proposition 2