Homological Properties of the Homology Algebra of the Koszul Complex of a Local Ring. Examples and Questions

TL;DR

This paper investigates the homological properties of the homology algebra of the Koszul complex over local rings, focusing on spectral sequence degeneration and Hilbert series calculations, using computational algebra tools.

Contribution

It provides a detailed analysis of spectral sequence degeneration cases and computes Hilbert series for the homology algebra in specific local ring settings, expanding understanding of their homological structure.

Findings

102 cases of spectral sequence degeneration identified

Complete Hilbert series determined for certain cases

Only 3 cases show non-degeneration of the spectral sequence

Abstract

Let $R$ be a local commutative noetherian ring and $HKR$ the homology ring of the corresponding Koszul complex. We study the homological properties of $HKR$ in particular the so-called Avramov spectral sequence. When the embedding dimension of $R$ is four and when $R$ can be presented with quadratic relations we have found 102 cases where this spectral sequence degenerates and only three cases where it does not degenerate. We also determine completely the Hilbert series of the bigraded Tor of these $HKR$ in tables A-D of section 5. We also study some higher embedding dimensions. Among the methods used are the programme {\tt BERGMAN} by J\"orgen Backelin et al, the {\tt Macaulay2}-package {\tt DGAlgebras} by Frank Moore, combined with results by Govorov, Clas L\"ofwall, Victor Ufnarovski and others.