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- W3038987990 abstract "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 Ap'ery 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-gamma} z^gamma, z^c, x^{a-b+gamma}y^{b-beta}, y^{b-beta}z^{c-gamma}), end{align*} where $ 1leq betaleq b-1,; max{1, b-a+1 }leq gammaleq min {b-1,c-1}$ and $ageq cgeq 2$. We prove that $A$ has the weak Lefschetz property in the following cases: (a) $ max{1,b-a+c-1}leq betaleq b-1$ and $gammageq lfloorfrac{beta-a+b+c-2}{2}rfloor$; (b) $ aleq 2b-c$ and $| a-b| +c-1leq betaleq b-1$; (c) one of $a,b,c$ is at most five." @default.
- W3038987990 created "2020-07-10" @default.
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- W3038987990 date "2020-11-01" @default.
- W3038987990 modified "2023-09-24" @default.
- W3038987990 title "The weak Lefschetz property of Gorenstein algebras of codimension three associated to the Apéry sets" @default.
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- W3038987990 doi "https://doi.org/10.1016/j.laa.2020.07.008" @default.
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