Rings of Teter type

TL;DR

This paper investigates rings of Teter type, focusing on 0-dimensional graded and monomial algebras, and establishes conditions under which such rings are of Teter type, especially highlighting that generically Gorenstein rings are not.

Contribution

The paper characterizes Teter type in Cohen-Macaulay and graded algebras, providing classes of 0-dimensional monomial algebras that are of Teter type and analyzing their properties.

Findings

Rings of Teter type are characterized in specific algebra classes.

Generically Gorenstein rings are shown not to be of Teter type.

Various classes of 0-dimensional monomial algebras are identified as Teter type.

Abstract

Let $R$ be a Cohen--Macaulay local $K$-algebra or a standard graded $K$-algebra over a field $K$ with a canonical module $\omega_R$. The trace of $\omega_R$ is the ideal $tr(\omega_R)$ of $R$ which is the sum of those ideals $\varphi(\omega_R)$ with $\varphi\in Hom_R(\omega_R,R)$. The smallest number $s$ for which there exist $\varphi_1, \ldots, \varphi_s \in Hom_R(\omega_R,R)$ with $tr(\omega_R)=\varphi_1(\omega_R) + \cdots + \varphi_s(\omega_R)$ is called the Teter number of $R$. We say that $R$ is of Teter type if $s = 1$. It is shown that $R$ is not of Teter type if $R$ is generically Gorenstein. In the present paper, we focus especially on $0$-dimensional graded and monomial $K$-algebras and present various classes of such algebras which are of Teter type.