Betti Sequences of Four-Generated Non-Almost-Symmetric Nearly Gorenstein Numerical Semigroup Rings
Abstract
Let S be a minimally four-generated numerical semigroup that is nearly Gorenstein but not almost symmetric, and let R=k[S] over an arbitrary field k. We prove that the defining ideal of R has four or five minimal generators when R has Cohen--Macaulay type two, and exactly six when it has type three. Thus the total Betti sequence over the four-variable polynomial presentation ring is one of (1,4,5,2), (1,5,6,2), (1,6,8,3), and all three possibilities occur. The type-three argument uses a canonical-ideal exact sequence, a chain-level comparison on Tor, and a codimension-three Gorenstein reduction. The resulting classification gives an affirmative answer, over every field, to Moscariello and Strazzanti's Question 4.3 on four-generated nearly Gorenstein numerical semigroups that are not almost symmetric.
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BibTeX
@misc{zhang2026betti,
title = {{Betti Sequences of Four-Generated Non-Almost-Symmetric Nearly Gorenstein Numerical Semigroup Rings}},
author = {Zhi-Lin Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05788},
eprint = {2611.05788},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05788},
}
RIS
TY - UNPB
TI - Betti Sequences of Four-Generated Non-Almost-Symmetric Nearly Gorenstein Numerical Semigroup Rings
AU - Zhi-Lin Zhang
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05788
UR - https://arxiv.org/abs/2611.05788
AB - Let S be a minimally four-generated numerical semigroup that is nearly Gorenstein but not almost symmetric, and let R=k[S] over an arbitrary field k. We prove that the defining ideal of R has four or five minimal generators when R has Cohen--Macaulay type two, and exactly six when it has type three. Thus the total Betti sequence over the four-variable polynomial presentation ring is one of (1,4,5,2), (1,5,6,2), (1,6,8,3), and all three possibilities occur. The type-three argument uses a canonical-ideal exact sequence, a chain-level comparison on Tor, and a codimension-three Gorenstein reduction. The resulting classification gives an affirmative answer, over every field, to Moscariello and Strazzanti's Question 4.3 on four-generated nearly Gorenstein numerical semigroups that are not almost symmetric.
ER -