Higher dimensional Teter rings

TL;DR

This paper characterizes higher-dimensional Teter rings, especially in the domain case, and explores their properties and examples, including graded Cohen-Macaulay algebras of finite representation type.

Contribution

It provides intrinsic characterizations of Teter and strongly Teter rings, extending the understanding of their structure in higher dimensions and specific algebraic contexts.

Findings

Intrinsic characterization of Teter rings that are domains

Intrinsic characterization of strongly Teter rings that are domains

Graded Cohen-Macaulay algebras of finite representation type have Teter ring completions

Abstract

Let $(A,\mathfrak{m})$ be a complete Cohen-Macaulay local ring. Assume $A$ is not Gorenstein. We say $A$ is a Teter ring if there exists a complete Gorenstein ring $(B,\mathfrak{n})$ with $\dim B = \dim A$ and a surjective map $B \rightarrow A$ with $e(B) - e(A) = 1$ (here $e(A)$ denotes multiplicity of $A$). We give an intrinsic characterization of Teter rings which are domains. We say a Teter ring is a strongly Teter ring if $G(B) = \bigoplus_{i \geq 0}\mathfrak{n}^i/\mathfrak{n}^{i+1}$ is also a Gorenstein ring. We give an intrinsic characterizations of strongly Teter rings which are domains. If $k$ is algebraically closed field of characteristic zero and $R$ is a standard graded Cohen-Macaulay $k$-algebra of finite representation type (and not Gorenstein) then we show that $\widehat{R_\mathfrak{M}}$ is a Teter ring (here $\mathfrak{M}$ is the maximal homogeneous ideal of $R$).