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 $\ddot{I}‰\subset R$, including $\ddot{I}‰/xR$, with $x\in \ddot{I}‰$ a nonzerodivisor, $(\ddot{I}‰/xR{)}^{\vee }:={\text{Ext}}_R^1(\ddot{I}‰/xR,\ddot{I}‰)$, $R/\ddot{I}{‰}^2$, and $\ddot{I}‰/\ddot{I}{‰}^2$. I also prove that if $R$ is not Gorenstein, then $\hat{I}{´}_R\left(\ddot{I}‰/xR\right)=\hat{I}{´}_R\left({\left(\ddot{I}‰/xR\right)}^{\vee }\right)=0$ and ${\hat{I}}_R^3\left(\hat{I}{\text{c◯}}_R^1\left(\ddot{I}‰/xR\right)\right)={\hat{I}}_R^3\left(\hat{I}{\text{c◯}}_R^1\left({\left(\ddot{I}‰/xR\right)}^{\vee }\right)\right)=0$, where $\hat{I}{´}_R$ is Auslander's $\hat{I}´$-invariant and ${\hat{I}}_R^3$ is the dual ${\hat{I}}^3$-invariant.

18 pages