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On a minimal free resolution of the residue field over a local ring of codepth 3 of class $T$

Van C. Nguyen, Oana Veliche

TL;DR

This paper develops a general framework to compute the minimal free resolution of the residue field $\mathsf{k}$ over a noetherian local ring $R$ from a graded minimal resolution of $\mathsf{k}$ over the Koszul homology algebra $A=\mathrm{H}(K)$. Central to the approach is an iterated mapping cone construction that uses a graded resolution of $\mathsf{k}$ over $A$ and a sequence of complexes $\{\mathsf{C}^{(k)}\}$ with maps $\{\varphi^{(k)}\}$, yielding a minimal free resolution over $R$ with $P^{R}_{\mathsf{k}}(t)=(1+t)^n P^{A}_{\mathsf{k}}(t,t)$. The authors first reformulate complete intersections (class $\mathbf{C}(c)$) within this framework, recovering known CI resolutions and Poincaré series, and then focus on codepth $3$ class $\mathbf{T}$, providing a detailed construction of the graded $A$-resolution and the corresponding $R$-resolution via a tree-like combinatorial scheme. They also present a concrete class $\mathbf{T}$ example to illustrate the explicit Koszul-block description up to degree seven, including the full Betti table and differential maps. The work fills a gap identified in previous classifications by supplying an explicit, implementable description of minimal free resolutions for class $\mathbf{T}$ rings and demonstrates how the $A$-structure governs the overall $R$-resolution.

Abstract

Let $R$ be any noetherian local ring with residue field $k$, and $A$ the homology of the Koszul complex on a minimal set of generators of the maximal ideal of $R$. In this paper, we show that a minimal free resolution of $k$ over $R$ can be obtained from a graded minimal free resolution of $k$ over $A$. More precisely, this is done by the iterated mapping cone construction, introduced by the authors in a previous work, using specific choices of ingredients. As applications, using this general perspective, we exhibit a minimal free resolution of $k$ over a complete intersection ring of any codepth, and explicitly construct a minimal free resolution of $k$ over a noetherian local ring of codepth 3 of class $T$ in terms of Koszul blocks.

On a minimal free resolution of the residue field over a local ring of codepth 3 of class $T$

TL;DR

This paper develops a general framework to compute the minimal free resolution of the residue field over a noetherian local ring from a graded minimal resolution of over the Koszul homology algebra . Central to the approach is an iterated mapping cone construction that uses a graded resolution of over and a sequence of complexes with maps , yielding a minimal free resolution over with . The authors first reformulate complete intersections (class ) within this framework, recovering known CI resolutions and Poincaré series, and then focus on codepth class , providing a detailed construction of the graded -resolution and the corresponding -resolution via a tree-like combinatorial scheme. They also present a concrete class example to illustrate the explicit Koszul-block description up to degree seven, including the full Betti table and differential maps. The work fills a gap identified in previous classifications by supplying an explicit, implementable description of minimal free resolutions for class rings and demonstrates how the -structure governs the overall -resolution.

Abstract

Let be any noetherian local ring with residue field , and the homology of the Koszul complex on a minimal set of generators of the maximal ideal of . In this paper, we show that a minimal free resolution of over can be obtained from a graded minimal free resolution of over . More precisely, this is done by the iterated mapping cone construction, introduced by the authors in a previous work, using specific choices of ingredients. As applications, using this general perspective, we exhibit a minimal free resolution of over a complete intersection ring of any codepth, and explicitly construct a minimal free resolution of over a noetherian local ring of codepth 3 of class in terms of Koszul blocks.

Paper Structure

This paper contains 9 sections, 14 theorems, 114 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The minimal free resolution of the residue field
  3. Application on class $\mathbf{C}(c)$
  4. Application on class $\mathbf{T}$
  5. Exactness of sequences on Koszul homologies
  6. Constructing the sequences of complexes ${\mathsf C}^{(k)}$ and of chain maps $\varphi^{(k)}$
  7. A graded minimal resolution of $\mathsf{k}$ over $A$
  8. A minimal free resolution of $\mathsf{k}$ over $R$
  9. A class $\mathbf{T}$ example

Key Result

Theorem 2.2

Let $(R,{\mathfrak m},\mathsf{k})$ be a noetherian local ring of codepth $c \operatorname{\geqslant} 1$, $(K, \partial)$ the Koszul complex of $R$ on a minimal set of generators of ${\mathfrak m}$, and $A= \bigoplus_{0 \operatorname{\leqslant} i \operatorname{\leqslant} c} A_i$ its Koszul homology w with differentials $\partial_k^\mathbb{F}=[\varphi^{(k)}]$ for all integers $k \operatorname{\geqsl

Figures (2)

  • Figure 1: Long exact sequences of homology via the iterated mapping cone construction
  • Figure 2: A directed rooted tree associated to the class $\mathbf{T}$

Theorems & Definitions (40)

  • Theorem 2.2
  • proof
  • Remark 2.3
  • Corollary 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Corollary 2.8
  • proof
  • Definition 3.1
  • ...and 30 more