Regularity of Sn -invariant monomial ideals
Regularity of Sn -invariant monomial ideals
复制标题
Sn不变单项式理想的正则性
DOI：
10.1016/j.jcta.2020.105307
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Raicu, Claudiu
中科院分区：
文献类型：
--
作者：
Raicu, Claudiu
对于n元多项式环S，通过置换变量来考虑对称群S n对S的自然作用。对于S n-不变单项理想I⊆S和j-≥0，我们给出了计算模Ext S j(S，S)的一个显式公式，并用它来刻画I的投射维数和正则性.我们对具有线性自由分解的S n-不变单项理想I进行了分类，并刻画了它们的性质.然后，我们考虑分析正则性的渐近行为的两种设置：一种是考虑固定理想I的幂，另一种是改变环境多项式环的维度，并考察由I诱导的不变单项理想。在第一种情况下，我们通过求解一个整数线性优化问题来确定由单项S n轨道生成的理想I的渐近正则性。在第二种情况下，我们描述了任意i的正则性行为，恢复了Murai最近的一个结果。
For a polynomial ring S in n variables, we consider the natural action of the symmetric group S n on S by permuting the variables. For an S n-invariant monomial ideal I⊆ S and j≥ 0, we give an explicit recipe for computing the modules Ext S j (S/I, S), and use this to describe the projective dimension and regularity of I. We classify the S n-invariant monomial ideals I that have a linear free resolution, and also characterize those which are Cohen–Macaulay. We then consider two settings for analyzing the asymptotic behavior of regularity: one where we look at powers of a fixed ideal I, and another where we vary the dimension of the ambient polynomial ring and examine the invariant monomial ideals induced by I. In the first case we determine the asymptotic regularity for those ideals I that are generated by the S n-orbit of a single monomial by solving an integer linear optimization problem. In the second case we describe the behavior of regularity for any I, recovering a recent result of Murai.