标题： Sally型数值半群 显示英文标题

标题： Numerical Semigroups of Sally Type

摘要： Judith Sally在1980年证明了，一维Gorenstein局部环的关联系数环的重次为$e$且嵌入维数为$e-2$时，它们是Cohen-Macaulay的。 她表明，这种环的关联系数环的定义理想由${e-2 \choose 2}$个元素生成。 数值半群环是一类重要的二维Cohen-Macaulay环。 在2014年，Herzog和Stamate证明了数值半群$<e,e+1,e+4,\ldots,2e-1 >$定义了一个满足Sally上述条件的Gorenstein半群环，这样的半群被称为Gorenstein Sally半群。 我们称一个数值半群$S$为Sally型，如果对于某个$2 \leq m <n \leq e-2$，有$<S >= < e,e+1,\ldots,e+m-1, e+m+1,\ldots, e+n-1,e+n+1, \ldots 2e-1>$。 在本文中，我们给出了其Frobenius数的公式，并给出了它为Gorenstein的充要条件。 我们计算了半群环$k[S]$的定义理想的最小生成元数目。 此外，我们提供了一个算法和用于应用 Hochster 的组合公式的 GAP 代码，以计算$k[S]$的第一贝蒂数。 显示英文摘要

摘要： Judith Sally proved in 1980 that the associated graded ring of one-dimensional Gorenstein local rings of multiplicity $e$ and embedding dimension $e-2$ are Cohen-Macaulay. She showed that the defining ideal of the associated graded ring of such rings is generated by ${e-2 \choose 2}$ elements. Numerical semigroup rings are a big class of one-dimensional Cohen-Macaulay rings. In 2014, Herzog and Stamate proved that the numerical semigroup $<e,e+1,e+4,\ldots,2e-1 >$ defines a Gorenstein semigroup ring satisfying Sally's conditions above and such semigroups are called Gorenstein Sally Semigroups. We call a numerical semigroup $S$ as Sally type if $<S >= < e,e+1,\ldots,e+m-1, e+m+1,\ldots, e+n-1,e+n+1, \ldots 2e-1>$ for some $2 \leq m <n \leq e-2$. In this paper, we give a formula for its Frobenius number along with a necessary and sufficient condition for it to be Gorenstein. We compute the minimal number of generators for the defining ideal of the semigroup ring $k[S]$. Additionally, we present an algorithm and a GAP code used in applying Hochster's combinatorial formula to compute the first Betti number of $k[S]$.