论文标题
不正常的边缘环满足$（S_2）$ - 条件
Non-normal edge rings satisfying $(S_2)$-condition
论文作者
论文摘要
令$ g $为有限的简单连接图$ v（g）= [d] = \ {1，\ dots，d \} $，带有边缘集$ e（g）= \ {e_ {1}，\ dots，e_ {n} \} $。令$ k [\ mathbf {t}] = k [t_ {1}，\ dots，t_ {d}] $为$ d $变量$ k $中的polyenmial ring。 $ g $的边缘环是由单组$ \ mathbf {t}^{e}生成的semigroup ring $ k [g] $。在本文中，我们将证明，给定整数$ d $和$ n $，其中$ d \ geq 7 $和$ d+1 \ leq n \ leq n \ leq \ frac {d^{2} {2} -7d+24} {2} {2} {2} {2} $，存在有限的简单连接$ g $ g $ g $ g $ g $ n $ n $ | n $ | v $ = g）非正常并满足$（s_ {2}）$ - 条件。
Let $G$ be a finite simple connected graph on the vertex set $V(G)=[d]=\{1,\dots ,d\}$, with edge set $E(G)=\{e_{1},\dots , e_{n}\}$. Let $K[\mathbf{t}]=K[t_{1},\dots , t_{d}]$ be the polynomial ring in $d$ variables over a field $K$. The edge ring of $G$ is the semigroup ring $K[G]$ generated by monomials $\mathbf{t}^{e}:=t_{i}t_{j}$, for $e=\{i,j\} \in E(G)$. In this paper, we will prove that, given integers $d$ and $n$, where $d\geq 7$ and $d+1\leq n\leq \frac{d^{2}-7d+24}{2}$, there exists a finite simple connected graph $G$ with $|V(G)|=d$ and $|E(G)|=n$, such that $K[G]$ is non-normal and satisfies $(S_{2})$-condition.
