Higher-dimensional Teter rings via the canonical trace

TL;DR

This paper characterizes higher-dimensional Teter rings using the canonical trace ideal, providing criteria for Teterness, and explores its implications for nearly Gorenstein rings, fiber products, Veronese subrings, and numerical semigroup rings.

Contribution

It offers a complete characterization of Teter rings in the standard graded case and analyzes Teterness in various ring constructions.

Findings

Nearly Gorenstein families are Teter rings.

Cohen--Macaulay type is bounded by codimension under certain conditions.

Criteria for Teterness are both sufficient and necessary in the standard graded case.

Abstract

We study Puthenpurakal's higher-dimensional Teter rings via the canonical trace ideal. We give a sufficient criterion for Teterness and show that, in the standard graded case, it is also necessary, yielding a characterization. Consequently, several nearly Gorenstein families are Teter; moreover, under certain hypotheses, the Cohen--Macaulay type of nearly Gorenstein rings is bounded by the codimension. We also analyze Teterness for fiber products, Veronese subrings, and numerical semigroup rings.