Traces of powers of canonical ideals
Abstract
Let (R,m) be a one-dimensional Cohen--Macaulay local ring with a canonical fractional ideal R⊆ K⊆R̅, and let v denote its embedding dimension. Put σ(R)=sup{j≥0|tr(Kj)⊇m}. By definition, R is nearly Gorenstein if and only if σ(R)>0. We prove that R is almost Gorenstein if and only if σ(R)≥ v, equivalently, σ(R)=∞. Thus every finite value of σ(R) is at most v−1. We then determine the structure of rings for which σ(R) attains its largest finite value v−1, just below the threshold for the almost Gorenstein property. This occurs if and only if Kv−1≅m and Kvnot≅m. When these conditions hold, R has type two, with multiplicity e greater than v and e ≡ 3 pmodv−1. As an application, we characterize the numerical semigroup rings R with σ(R)=v−1 as those generated by an arithmetic progression, of type two and multiplicity greater than v.
Related
- Monomial ideals whose all matching powers are Cohen–Macaulay
- DEPTH OF POWERS OF EDGE IDEALS OF CYCLES AND STARLIKE TREES
- Stabilization index of V-number of powers of edge ideals of graphs
- v-numbers of powers of edge ideals of cubic circulant graphs
- Two-sided bounds for canonical traces of Veronese subalgebras
BibTeX
@misc{hoang2026traces,
title = {{Traces of powers of canonical ideals}},
author = {Le Truong Hoang and Naoyuki Matsuoka},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05150},
eprint = {2610.05150},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05150},
}
RIS
TY - UNPB
TI - Traces of powers of canonical ideals
AU - Le Truong Hoang
AU - Naoyuki Matsuoka
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05150
UR - https://arxiv.org/abs/2610.05150
AB - Let (R,m) be a one-dimensional Cohen--Macaulay local ring with a canonical fractional ideal R⊆ K⊆R̅, and let v denote its embedding dimension. Put σ(R)=sup{j≥0|tr(K^j)⊇m}. By definition, R is nearly Gorenstein if and only if σ(R)>0. We prove that R is almost Gorenstein if and only if σ(R)≥ v, equivalently, σ(R)=∞. Thus every finite value of σ(R) is at most v−1. We then determine the structure of rings for which σ(R) attains its largest finite value v−1, just below the threshold for the almost Gorenstein property. This occurs if and only if K^v−1≅m and K^vnot≅m. When these conditions hold, R has type two, with multiplicity e greater than v and e ≡ 3 pmodv−1. As an application, we characterize the numerical semigroup rings R with σ(R)=v−1 as those generated by an arithmetic progression, of type two and multiplicity greater than v.
ER -