Free resolutions and Lefschetz properties of some Artin Gorenstein rings of codimension four

Pub Date : 2023-08-21 DOI:10.1016/j.jsc.2023.102257

Nancy Abdallah , Hal Schenck

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引用次数: 7

Abstract

In (Stanley, 1978), Stanley constructs an example of an Artinian Gorenstein (AG) ring A with non-unimodal H-vector $(1, 13, 12, 13, 1)$ . Migliore-Zanello show in (Migliore and Zanello, 2017) that for regularity $r = 4$ , Stanley's example has the smallest possible codimension c for an AG ring with non-unimodal H-vector.

The weak Lefschetz property (WLP) has been much studied for AG rings; it is easy to show that an AG ring with non-unimodal H-vector fails to have WLP. In codimension $c = 3$ it is conjectured that all AG rings have WLP. For $c = 4$ , Gondim shows in (Gondim, 2017) that WLP always holds for $r \leq 4$ and gives a family where WLP fails for any $r \geq 7$ , building on Ikeda's example (Ikeda, 1996) of failure for $r = 5$ . In this note we study the minimal free resolution of A and relation to Lefschetz properties (both weak and strong) and Jordan type for $c = 4$ and $r \leq 6$ .

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余维四环的自由分辨率和Lefschetz性质

在（Stanley，1978）中，Stanley构造了具有非单峰H-向量（1,13,12,13,1）的Artinian-Gorenstein（AG）环A的一个例子。Migliore Zanello在（Migliore and Zanello，2017）中证明，对于正则性r＝4，Stanley的例子对于具有非单峰H-向量的AG环具有最小可能余维数c；很容易证明具有非单峰H-向量的AG环不具有WLP。在余维c=3中，推测所有的AG环都具有WLP。对于c=4，Gondim在（Gondim，2017）中表明，对于r≤4，WLP总是成立的，并基于Ikeda关于r=5的失败的例子（Ikeda，1996）给出了一个WLP对于任何r≥7失败的族。在本文中，我们研究了当c=4和r≤6时，A的最小自由分辨率以及与Lefschetz性质（弱和强）和Jordan型的关系。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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