Presentations of rings with a chain of semidualizing modules

TL;DR

This paper generalizes previous results by showing that Cohen-Macaulay local rings with a chain of semidualizing modules can be represented as quotients of Gorenstein rings, revealing new cohomological properties.

Contribution

It extends prior work by characterizing rings with chains of semidualizing modules as quotients of Gorenstein rings and analyzing their cohomological properties.

Findings

Rings with chains of semidualizing modules are quotients of Gorenstein rings.

Subquotients of these rings have notable cohomological properties.

The results generalize earlier theorems by Jorgensen, Foxby, and Reiten.

Abstract

Inspired by Jorgensen et. al., it is proved that if a Cohen--Macaulay local ring $R$ with dualizing module admits a suitable chain of semidualizing $R$--modules of length $n$, then $R\cong Q/(I_1+\cdots+I_n)$ for some Gorenstein ring $Q$ and ideals $I_1,\cdots, I_n$ of $Q$; and, for each $\Lambda\subseteq [n]$, the ring $Q/(\Sigma_{l\in \Lambda} I_l)$ has some interesting cohomological properties . This extends the result of Jorgensen et. al., and also of Foxby and Reiten.