Higher-dimensional Teter rings via the canonical trace
Sora Miyashita, Taiga Ozaki
TL;DR
This work develops a graded framework for Teter rings via the canonical trace in Cohen–Macaulay graded rings, unifying Artinian and local perspectives. The authors define graded Teter rings and prove a central theorem: a sequence of conditions involving the canonical trace $tr_R(\omega_R)$ and invariants $r(R)$, $r_{min}(R)$, and $codim(R)$ are equivalent in the standard graded case, yielding a precise characterization of Teterness. They show that many nearly Gorenstein families are Teter and obtain codimension-based bounds on the CM type under suitable hypotheses. The paper also analyzes how Teterness behaves under fiber products and Veronese subalgebras, and provides a complete description for Teter numerical semigroup rings, including explicit criteria in terms of pseudo-Frobenius numbers and $T$-sets, with several sharp exceptions highlighted.
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.