Rings of 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 $\phi ({\omega }_R)$ with $\phi \in Ho{m}_R({\omega }_R,R)$. The smallest number $s$ for which there exist ${\phi }_1,\dots ,{\phi }_s\in Ho{m}_R({\omega }_R,R)$ with $tr({\omega }_R)={\phi }_1({\omega }_R)+\cdots +{\phi }_s({\omega }_R)$ 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.