Vector space summands of lower syzygies
Mohsen Asgharzadeh, Michael DeBellevue, Souvik Dey, Saeed Nasseh, Ryo Takahashi
TL;DR
This work analyzes when the residue field $k$ can be a direct summand of low syzygies $\Omega^n(M)$ for all non-free modules and when the dual $E^*=\operatorname{Hom}_R(E_R(k),R)$ is a $k$-vector space, with a focus on artinian local rings and 1-dimensional cases. It establishes equivalences among low-syzygy summand conditions and a summand of $\mathfrak{m}$, linking these to $\operatorname{soc}(R)\nsubseteq \mathfrak{m}^2$ and the Burch property with $\operatorname{depth}(R)=0$, and then explores $E^*$-vector-space phenomena using Eliahou–Kervaire resolutions and Macaulay inverse systems. The paper further classifies artinian rings with $\mathfrak{m}E^*=0$, showing $\dim_k(E^*)=(\operatorname{type}(R))^2$ when $E^*$ is a $k$-vector space and connecting this to trace ideals and nearly Gorenstein rings. In dimension one, it relates Ulrich properties of the canonical module to minimal multiplicity and provides examples that delineate the boundaries of these phenomena.
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.