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=Bbbk[S] over an arbitrary field Bbbk. 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
$$
(1,4,5,2),\quad (1,5,6,2),\quad (1,6,8,3),
$$
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 = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23538799},
url = {https://zenodo.org/records/23538799},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Betti sequences of four-generated non-almost-symmetric nearly Gorenstein numerical semigroup rings
AU - Zhi-Lin Zhang
PY - 2026
DA - 2026/10/21
PB - Zenodo
DO - 10.5281/zenodo.23538799
UR - https://zenodo.org/records/23538799
AB - Let S be a minimally four-generated numerical semigroup that is nearly Gorenstein but not almost symmetric, and let R=Bbbk[S] over an arbitrary field Bbbk. 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 -