Higher dimensional Teter rings

TL;DR

This work extends the notion of Teter rings to higher dimensions by introducing $g(A)=\min\{e(B)-e(A)\}$ over Gorenstein approximations and characterizes Teter rings intrinsically for domains via the canonical module $\omega_A$ and a codimension condition. It shows that a Cohen–Macaulay local domain $A$ is Teter iff $\omega_A$ is isomorphic to a proper ideal $J$ with $\mathrm{codim}(A/J)\le 1$, and it constructs a unique Teter Gorenstein approximation using fiber products. The paper defines strongly Teter rings through the associated graded ring $G(B)$ and proves that strongly Teter implies $G(A)$ is Cohen–Macaulay, with dimension-reduction arguments and superficial sequences. It further proves a strong-strong correspondence: a strongly Teter domain has a unique Teter approximation given by a fiber product, and its graded canonical module aligns with $G(A)$. Finally, it demonstrates that completions of standard graded CM algebras of finite representation type over an algebraically closed field of characteristic zero are Teter, with detailed case analyses and numerous explicit examples and counterexamples.

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 $R$).

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• proof : Proof of (i) $\implies$ (ii)
• Theorem 2.1
• proof : Proof of Theorem \ref{['fiber']}
• proof : Proof of Theorem \ref{['main']} continued
• proof : Proof of Theorem \ref{['strong']}
• proof : Proof of Theorem \ref{['main-strong']}.