论文标题
在某些仿射半群代数的代数不变性上
On the algebraic invariants of certain affine semigroup algebras
论文作者
论文摘要
让$ a $ a $ d $是$ \ mathbb {n}^2 $的两个线性独立向量，在有理数的字段上。对于正整数$ k \ geq 2 $，请考虑序列$ a，a+d，\ ldots，a+kd $，以使Aggine semigroup $ s_ {a，d，d，k} = \ langle a，a+d，\ ldots，a+kd \ rangle $最小产生。我们研究了与此半群相关的仿射半群代数$ k [s_ {a，d，k}] $。我们证明$ k [s_ {a，d，k}] $始终是cohen-macaulay，当且仅当$ k = 2 $时，它是Gorenstein。对于$ k = 2,3,4 $，我们明确地计算了$ k [s_ {a，d，k}]的Syzygies，最低分级的免费分辨率和Hilbert系列。$我们还提供了最小的生成集，并给出了一个定义理想的$ k [s_ s_ a，d，k}的定义理想的基础。 $ k [s_ {a，d，k}] $是koszul。最后，我们证明$ k [s_ {a，d，k}] $的Castelnuovo-Mumford规律性是任何$ a，d，k。$的$ 1 $。
Let $a$ and $d$ be two linearly independent vectors in $\mathbb{N}^2$, over the field of rational numbers. For a positive integer $k \geq 2$, consider the sequence $a, a+d, \ldots, a+kd$ such that the affine semigroup $S_{a,d,k} = \langle a, a+d, \ldots, a+kd \rangle$ is minimally generated by this sequence. We study the properties of affine semigroup algebra $k[S_{a,d,k}]$ associated to this semigroup. We prove that $k[S_{a,d,k}]$ is always Cohen-Macaulay and it is Gorenstein if and only if $k=2$. For $k=2,3,4$, we explicitly compute the syzygies, minimal graded free resolution and Hilbert series of $k[S_{a,d,k}].$ We also give a minimal generating set and a Gröbner basis of the defining ideal of $k[S_{a,d,k}].$ Consequently, we prove that $k[S_{a,d,k}]$ is Koszul. Finally, we prove that the Castelnuovo-Mumford regularity of $k[S_{a,d,k}]$ is $1$ for any $a,d,k.$
