Cohen-Macaulay approximations over generically 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 $\,ω\subset R$, including $\,ω/xR$, with $\,x \in ω\,$ a nonzerodivisor, $\,(ω/xR)^{\vee}:=\text{Ext}^1_R(ω/xR,ω)$, $\,R/ω^2$, and $\,ω/ω^2$. I also prove that if $R$ is not Gorenstein, then $δ_{R}\left(ω/xR \right)=δ_{R}\left(\left(ω/xR \right)^{\vee} \right)=0\,$ and $\,γ_{R}\left(Ω^{1}_{R}\left(ω/xR \right) \right)=γ_{R}\left(Ω^{1}_{R}\left(\left(ω/xR\right)^{\vee}\right) \right)=0$, where $δ_R$ is Auslander's $\,δ$-invariant and $γ_R$ is the dual $γ$-invariant.

Theorems & Definitions (20)

• Proposition 1.1
• Definition 1.2
• Lemma 1.3
• Definition 2.1
• Remark 2.2
• Proposition 2.3
• Proposition 2.4
• proof
• Corollary 2.5
• proof