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

IF 1.1 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS Journal of Symbolic Computation 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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来源期刊

Journal of Symbolic Computation 工程技术-计算机：理论方法

CiteScore

2.10

自引率

14.30%

发文量

审稿时长

142 days

期刊介绍： An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects. It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.

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