Vector space summands of lower syzygies

TL;DR

This paper explores when the residue field appears as a summand in syzygy modules of local rings, establishing equivalences and identifying classes of rings with specific syzygy properties.

Contribution

It proves equivalences between conditions for the residue field to be a summand in certain syzygies and introduces a class of artinian rings with duals that are vector spaces.

Findings

Residue field as summand in second and third syzygies are equivalent conditions.

Conditions under which the dual of the injective envelope is a k-vector space.

Introduction of a class of artinian rings satisfying these syzygy conditions.

Abstract

In this paper, we investigate problems concerning when the residue field $k$ of a local ring $(R,\frak m$, $k)$ appears as a direct summand of syzygy modules, from two perspectives. First, we prove that the following conditions are equivalent: (i) $k$ is a direct summand of second syzygies of all non-free finitely generated $R$-modules; (ii) $k$ is a direct summand of third syzygies of all non-free finitely generated $R$-modules; (iii) $k$ is a direct summand of $\frak m$. We also prove various consequences of these conditions. The second point of this article is to investigate for what artinian local rings $R$ the dual $E^*=Hom_R(E_R(k),R)$ of the injective envelope of the residue field, which is also a second syzygy, is a $k$-vector space. Using the notion of Eliahou-Kervaire resolution, we introduce a large class of artinian local rings that satisfy this condition.