Cohen-Macaulay approximations over generically Gorenstein rings

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 $\omega \mathrm{/}xR$, with $x\in \omega$ a nonzerodivisor, $(\omega \mathrm{/}xR{)}^{\vee }:={\text{Ext}}_R^1(\omega \mathrm{/}xR,\omega )$, $R\mathrm{/}{\omega }^2$, and $\omega \mathrm{/}{\omega }^2$. I also prove that if $R$ is not Gorenstein, then ${\delta }_R\left(\omega \mathrm{/}xR\right)={\delta }_R\left({\left(\omega \mathrm{/}xR\right)}^{\vee }\right)=0$ and ${\gamma }_R\left({\mathrm{\Omega }}_R^1\left(\omega \mathrm{/}xR\right)\right)={\gamma }_R\left({\mathrm{\Omega }}_R^1\left({\left(\omega \mathrm{/}xR\right)}^{\vee }\right)\right)=0$, where ${\delta }_R$ is Auslander’s $\delta$-invariant and ${\gamma }_R$ is the dual $\gamma$-invariant.