When do pseudo-Gorenstein rings become Gorenstein?

TL;DR

This paper explores conditions under which pseudo-Gorenstein rings are actually Gorenstein, establishing new links between various generalizations of Gorenstein rings and providing a method to construct quasi-Gorenstein rings.

Contribution

It proves that under mild conditions, pseudo-Gorenstein nearly Gorenstein graded domains are Gorenstein and introduces a construction method for quasi-Gorenstein rings via Veronese subalgebras.

Findings

Pseudo-Gorenstein nearly Gorenstein graded domains are Gorenstein under certain assumptions.

Relationships among nearly Gorensteinness, almost Gorensteinness, and levelness are clarified.

A new method for constructing quasi-Gorenstein rings using Veronese subalgebras is provided.

Abstract

We discuss the relationship between the trace ideal of the canonical module and pseudo-Gorensteinness. In particular, under certain mild assumptions, we show that every pseudo-Gorenstein nearly Gorenstein graded domain is Gorenstein. As an application, we clarify the relationships among nearly Gorensteinness, almost Gorensteinness, and levelness -- notions that generalize Gorensteinness -- in the context of standard graded domains. Moreover, we give a method for constructing quasi-Gorenstein rings by taking a Veronese subalgebra of certain Noetherian graded rings.