交换代数

• 交叉列表
• 替换

查看 最近的 文章

显示 2025年06月06日， 星期五 新的列表

交叉提交 (展示 3 之 3 条目 )

[1] arXiv:2506.04899 (交叉列表自 math.AC) [中文pdf, pdf, html, 其他]

标题： 纤维乘积的典范迹及其应用 显示英文标题

标题： Canonical traces of fiber products and their applications

Shinya Kumashiro, Sora Miyashita

评论： 25页，欢迎评论！

主题： 交换代数 (math.AC)

我们研究了诺特环纤维积的典范迹。此外，我们将关于称为泰特类型的科恩-麦考利环的结果推广到诺特环上。作为我们关于典范迹和泰特类型诺特环的研究应用，我们计算了由非连通单纯复形产生的斯坦利-里斯纳环的典范迹。特别是，即使底层单纯复形不连通，我们也给出了一个刻画，表明对于哪些斯坦利-里斯纳环，其典范迹包含分级极大理想。 显示英文摘要

We study the canonical trace of the fiber product of Noetherian rings. Furthermore, we extend results on the class of Cohen-Macaulay rings called Teter type to Noetherian rings. As an application of our study on canonical traces and Noetherian rings of Teter type, we compute the canonical trace of the Stanley-Reisner ring arising from a non-connected simplicial complex. In particular, we provide a characterization of Stanley-Reisner rings for which the canonical trace contains the graded maximal ideal, even when the underlying simplicial complex is not connected.

[2] arXiv:2506.05193 (交叉列表自 math.AC) [中文pdf, pdf, html, 其他]

标题： 关于确定性理想初始理想在对角单项式序下的弱和强Lefschetz性质 显示英文标题

标题： On the weak and strong Lefschetz properties for initial ideals of determinantal ideals with respect to diagonal monomial orders

Hongmiao Yu

评论： 30页，12幅图

主题： 交换代数 (math.AC)

我们研究了对于$R/\mathrm{in}(I_t)$的弱和强勒夫谢茨性质，其中$I_t$是由不定元的$m\times n$矩阵的$t$子式生成的多项式环$R$中的理想，而$\mathrm{in}(I_t)$表示关于对角单项式序的$I_t$的初始理想。 我们证明当 $I_t$ 由极大子式生成（即 $t=\mathrm{min}\{m,n\}$），环 $R/\mathrm{in}(I_t)$ 对所有 $m$，$n$ 都具有强 Lefschetz 性质。 相反，对于$t<\mathrm{min}\{m,n\}$，我们给出一个界值，只要乘积$mn$超过这个界值，$R/\mathrm{in}(I_t)$就无法满足弱勒夫谢茨性质。 作为一个应用，我们给出了反例，对 Murai 提出的关于自由单项式 Gröbner 变形下勒夫谢茨性质保持问题给出了否定回答。 显示英文摘要

We study the weak and strong Lefschetz properties for $R/\mathrm{in}(I_t)$, where $I_t$ is the ideal of a polynomial ring $R$ generated by the $t$-minors of an $m\times n$ matrix of indeterminates, and $\mathrm{in}(I_t)$ denotes the initial ideal of $I_t$ with respect to a diagonal monomial order. We show that when $I_t$ is generated by maximal minors (that is, $t=\mathrm{min}\{m,n\}$), the ring $R/\mathrm{in}(I_t)$ has the strong Lefschetz property for all $m$, $n$. In contrast, for $t<\mathrm{min}\{m,n\}$, we provide a bound such that $R/\mathrm{in}(I_t)$ fails to satisfy the weak Lefschetz property whenever the product $mn$ exceeds this bound. As an application, we present counterexamples that provide a negative answer to a question posed by Murai regarding the preservation of Lefschetz properties under square-free Gr\"obner degenerations.

[3] arXiv:2506.05248 (交叉列表自 math.AC) [中文pdf, pdf, 其他]

标题： 齐次理想族的度函数 显示英文标题

标题： Degree functions of graded families of ideals

Steven Dale Cutkosky, Jonathan Montaño

主题： 交换代数 (math.AC)

We express multiplicities and degree functions of graded families of $\mathfrak{m}_R$-primary ideals in an excellent normal local ring $(R,\mathfrak{m}_R)$ as limits of intersection products. Moreover, in dimension 2, we show more refined results for divisorial filtrations. Finally, also in dimension 2, we give an example of a non-Noetherian divisorial filtration $\{I_n\}_{n\geqslant 0}$ of $\mathfrak{m}_R$-primary ideals such that the union of all the sets of Rees valuations of all the $I_n$ is a finite set, and another example of a (necessarily non-Noetherian) divisorial filtration of $\mathfrak{m}_R$-primary ideals such that the set of all Rees valuations is infinite. 显示英文摘要

We express multiplicities and degree functions of graded families of $\mathfrak{m}_R$-primary ideals in an excellent normal local ring $(R,\mathfrak{m}_R)$ as limits of intersection products. Moreover, in dimension 2, we show more refined results for divisorial filtrations. Finally, also in dimension 2, we give an example of a non-Noetherian divisorial filtration $\{I_n\}_{n\geqslant 0}$ of $\mathfrak{m}_R$-primary ideals such that the union of all the sets of Rees valuations of all the $I_n$ is a finite set, and another example of a (necessarily non-Noetherian) divisorial filtration of $\mathfrak{m}_R$-primary ideals such that the set of all Rees valuations is infinite.

替换提交 (展示 2 之 2 条目 )

[4] arXiv:2502.13276 (替换) [中文pdf, pdf, html, 其他]

标题： CW-复形与分级Artinian Gorenstein代数的极小Hilbert向量 显示英文标题

标题： CW-complexes and minimal Hilbert vector of graded Artinian Gorenstein algebras

Armando Capasso

评论： 新章节和结果

主题： 交换代数 (math.AC) ; 组合数学 (math.CO) ; 环与代数 (math.RA)

我引入了标准分次Artinian Gorenstein代数集合的一种几何解释，其协维度为$n$，度数为$d$：称为标准点集，它是度数为$d$的多项式在$n$变量的射影空间的一个子集，并对其进行了刻画。 在适当的假设下，我证明了全Perazzo多项式的点集是标准点集中最小维不可约分支的并集，且是一个纯维数的子集。 另一方面，我将任何齐次多项式与一个拓扑空间关联起来，该拓扑空间是一个CW-复形。 利用所有这些集合，我证明了Hilbert函数在定义域的任意不可约分支上在标准点集上取到最小值。 我将所有这些应用于Full Perazzo猜想，并证明了该猜想。 显示英文摘要

I introduce a geometric interpretation of the set of standard graded Artinian Gorenstein algebras of codimension $n$ and degree $d$: the standard locus, which is a subset of the projective space of degree $d$ polynomials in $n$ variables, and I characterize it. Under opportune hypothesis, I prove that the locus of full Perazzo polynomials is the union of the minimal dimensional irreducible components of the standard locus and it is pure dimensional subset. On the other hand, I associate to any homogeneous polynomial a topological space, which is a CW-complex. Using all these sets, I prove that the Hilbert function restricted to the standard locus has minimal values on any irreducible component of the domain. I apply all this to the Full Perazzo Conjecture and I prove it.

[5] arXiv:2503.17464 (替换) [中文pdf, pdf, html, 其他]

标题： 主理想域的其他例子，它们不是欧几里得域 显示英文标题

标题： Other Examples of Principal Ideal Domains that are not Euclidean Domains

Nicolás Allo-Gómez

评论： 9页。已提交。

主题： 交换代数 (math.AC) ; 环与代数 (math.RA)

众所周知且容易证明的是，每个欧几里得整环也是一个主理想整环。 然而，其逆命题并不成立，通常通过展示某个特定的二次域中的代数整数环作为反例来证明这一点，而这种证明方法显得相当不自然且技术性很强。 本文将介绍一类利用实闭域构造的反例。 显示英文摘要

It is a well-known and easily established fact that every Euclidean domain is also a principal ideal domain. However, the converse statement is not true, and this is usually shown by exhibiting as a counterexample the ring of algebraic integers in a certain, very specific quadratic field, and the proof that this works is quite unnatural and technical. In this article, we will present a family of counterexamples constructed using real closed fields.