论文标题
圆锥分区的理想和完全多部分图的边缘环的非交流性毛茸茸的分辨率
Conic divisorial ideals and non-commutative crepant resolutions of edge rings of complete multipartite graphs
论文作者
论文摘要
本文的第一个目的是研究完整多部分图的边缘环的班级组，由$ \ bbbk [k_ {r_1，\ ldots，r_n}] $表示，其中$ 1 \ leq r_1 \ leq r_1 \ leq \ leq \ cdots \ cdots \ leq r_n $。更具体地说，我们证明了$ \ bbbk [k_ {r_1，\ ldots，r_n}]的类组$是同构为$ \ \ m m i \ m m mathbb {z}^n $，如果$ n = 3 $，如果$ r_1 \ geq 2 $ n \ geq 2 $或$ n \ geq 4 $，同时又可以分配为hib code code to n of def to hib。第二个目标是研究$ \ bbbk [k_ {r_1，\ ldots，r_n}] $的特殊类别理想的特殊类别，称为锥形分区理想。我们描述了某些$ k_ {r_1，\ ldots，r_n} $的圆锥分区理想，包括$ \ bbbk [k_ {r_1，\ ldots，r_n}] $ is gorenstein。最后，在Gorenstein的情况下，我们给出了$ \ bbbk [k_ {r_1，\ ldots，r_n}] $的非交换性毛皮分辨率（NCCR）。
The first goal of the present paper is to study the class groups of the edge rings of complete multipartite graphs, denoted by $\Bbbk[K_{r_1,\ldots,r_n}]$, where $1 \leq r_1 \leq \cdots \leq r_n$. More concretely, we prove that the class group of $\Bbbk[K_{r_1,\ldots,r_n}]$ is isomorphic to $\mathbb{Z}^n$ if $n =3$ with $r_1 \geq 2$ or $n \geq 4$, while it turns out that the excluded cases can be deduced into Hibi rings. The second goal is to investigate the special class of divisorial ideals of $\Bbbk[K_{r_1,\ldots,r_n}]$, called conic divisorial ideals. We describe conic divisorial ideals for certain $K_{r_1,\ldots,r_n}$ including all cases where $\Bbbk[K_{r_1,\ldots,r_n}]$ is Gorenstein. Finally, we give a non-commutative crepant resolution (NCCR) of $\Bbbk[K_{r_1,\ldots,r_n}]$ in the case where it is Gorenstein.
