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 $$(1,4,5,2),\quad (1,5,6,2),\quad (1,6,8,3),$$ 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.