Cohen-Macaulay approximations over generically Gorenstein rings

TL;DR

This paper explores the relationships between minimal Cohen-Macaulay approximations and hulls of modules derived from a canonical ideal in generically Gorenstein Cohen-Macaulay rings, revealing new isomorphisms and invariants.

Contribution

It establishes novel isomorphisms connecting various module constructions and analyzes invariants to distinguish Gorenstein from non-Gorenstein rings.

Findings

Isomorphisms relating MCM approximations and hulls of modules from canonical ideals.

Vanishing of invariants $\, ext{delta}_R$ and $\, ext{gamma}_R$ in non-Gorenstein rings.

Abstract

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring with canonical module that is generically Gorenstein. In this paper, I prove isomorphisms relating the minimal MCM approximations and minimal FID hulls of modules constructed from a canonical ideal $\,\omega \subset R$, including $\,\omega/xR$, with $\,x \in \omega\,$ a nonzerodivisor, $\,(\omega/xR)^{\vee}:=\text{Ext}^1_R(\omega/xR,\omega)$, $\,R/\omega^2$, and $\,\omega/\omega^2$. I also prove that if $R$ is not Gorenstein, then $\delta_{R}\left(\omega/xR \right)=\delta_{R}\left(\left(\omega/xR \right)^{\vee} \right)=0\,$ and $\,\gamma_{R}\left(\Omega^{1}_{R}\left(\omega/xR \right) \right)=\gamma_{R}\left(\Omega^{1}_{R}\left(\left(\omega/xR\right)^{\vee}\right) \right)=0$, where $\delta_R$ is Auslander's $\,\delta$-invariant and $\gamma_R$ is the dual $\gamma$-invariant.