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

TL;DR

This paper presents a method to construct minimal free resolutions of the residue field over certain local rings using graded resolutions over associated homology rings, with explicit examples for rings of codepth 3 of class T.

Contribution

It introduces a novel approach using iterated mapping cones to derive minimal free resolutions from graded resolutions over homology rings, including explicit constructions for specific classes of rings.

Findings

Minimal free resolution of $k$ over a complete intersection ring of any codepth is exhibited.

Explicit minimal free resolution over a noetherian local ring of codepth 3 of class T is constructed.

The approach generalizes previous methods and provides concrete resolutions in new cases.

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.