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arXiv · 2611.05788

Betti Sequences of Four-Generated Non-Almost-Symmetric Nearly Gorenstein Numerical Semigroup Rings

Zhi-Lin Zhang
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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 $$(1,4,5,2),\quad (1,5,6,2),\quad (1,6,8,3),$$ and all three possibilities occur. The type-three argument uses a canonical-ideal exact sequence, a chain-level comparison on $\operatorname{Tor}$, 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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Zhi-Lin Zhang. 2026-11-04. Betti Sequences of Four-Generated Non-Almost-Symmetric Nearly Gorenstein Numerical Semigroup Rings. https://arxiv.org/abs/2611.05788

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