论文标题
gorenstein代数的弱的lefschetz属性与Apéry集有关
The weak Lefschetz property of Gorenstein algebras of codimension three associated to the Apéry sets
论文作者
论文摘要
据推测，{\ it as as as and}渐变的Artinian Gorenstein代数三，具有弱的lefschetz属性，其特性在特征零的领域上。在本文中，我们研究了相关分级代数$ a $ aapéry套件的弱级属性属性的$ m $ pure对称数值半群，该属性由四个自然数产生。 2010年，科比证明了这些代数为Codimension三分之一的Artinian Gorenstein代数。在最近的一篇文章中，Guerrieri表明，如果$ a $不是完整的交叉点，那么$ a $是$ a = r = r = k [x，x，y，z] $和\ begin {align*} i =（x^a，x^a，y^b-x^a，y^b-g-γ} z^q^γ，z^γ，z^γ，z^c，x^c，x^c，x^y-b^y^y} y^{b-β} z^{c-γ}），\ end {align*}其中$ 1 \ leqβ\ leq b-1，\; \ max \ {1，b-a+1 \} \ leqγ\ leq \ min \ {b-1，c-1 \} $和$ a \ geq c \ geq 2 $。我们证明，$ a $在以下情况下具有较弱的lefschetz属性：（a）$ \ max \ {1，b-a+c-1 \} \ leqβ\ leq b-1 $和$γ\ geq \ geq \ geq \ lfloor \ lfloor \ frac {β-a+b+b+b+b+c-2} （b）$ a \ leq 2b-c $和$ | A-B | +c-1 \ leqβ\ leq b-1 $; （c）$ a，b，c $之一最多是五个。
It has been conjectured that {\it all} graded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded algebras $A$ of the Apéry set of $M$-pure symmetric numerical semigroups generated by four natural numbers. In 2010, Bryant proved that these algebras are graded Artinian Gorenstein algebras of codimension three. In a recent article, Guerrieri showed that if $A$ is not a complete intersection, then $A$ is of form $A=R/I$ with $R=K[x,y,z]$ and \begin{align*} I=(x^a, y^b-x^{b-γ} z^γ, z^c, x^{a-b+γ}y^{b-β}, y^{b-β}z^{c-γ}), \end{align*} where $ 1\leq β\leq b-1,\; \max\{1, b-a+1 \}\leq γ\leq \min \{b-1,c-1\}$ and $a\geq c\geq 2$. We prove that $A$ has the weak Lefschetz property in the following cases: (a) $ \max\{1,b-a+c-1\}\leq β\leq b-1$ and $γ\geq \lfloor\frac{β-a+b+c-2}{2}\rfloor$; (b) $ a\leq 2b-c$ and $| a-b| +c-1\leq β\leq b-1$; (c) one of $a,b,c$ is at most five.
