小Sperner数的余维数为3的Gorenstein代数的Jordan次数型
Jordan degree type for codimension three Gorenstein algebras of small Sperner number
AI总结：
该研究将余维数2分次代数的Jordan次数型研究拓展至小Sperner数的余维数3分次Artinian Gorenstein代数，完全确定了s=3,4,5时对应Gorenstein序列的所有可能JDT，并划定了s=6时的潜在JDT范围。
摘要：
域$\textsf{k}$上的分次Artinian代数$A$上作用的线性形式$\bullet$的Jordan型$P_{A,\bullet}$，是描述幂零乘法映射$m_\bullet$的Jordan块分解的分划。Jordan次数型$\boldsymbol{\text{JDT}}$是更精细的不变量，它还描述了$A$作为$\textsf{k}[\bullet]$-模分解时，$A$的单子模的初始次数。当$\bullet$变化时，$A$的Jordan型集合或Jordan次数型集合是代数的不变量。该不变量已在余维数为2的分次代数中得到研究。本文将以往结果推广到一类余维数为3的分次Artinian Gorenstein（AG）代数——即小Sperner数的这类代数。给定一个Gorenstein序列$T$（余维数为3的AG代数的Hilbert函数的一种可能序列），不可约簇$\text{Gor}(T)$参数化了所有具有Hilbert函数$T$的Gorenstein代数。本文针对形如$T=(1,3,s^k,3,1)$、Sperner数$s=3,4,5$且重数$k$任意的Gorenstein序列$T$，完全确定了所有对$(A,\bullet)$（$A\text{Gor}(T)$）的可能JDT。对于$s=6$，我们划定了潜在的JDT范围，但未验证每一种是否都能出现。
英文摘要：
The Jordan type $P_{A,\ell}$ of a linear form $\ell$ acting on a graded Artinian algebra $A$ over a field $\sf k$ is the partition describing the Jordan block decomposition of the multiplication map $m_\ell$, which is nilpotent. The Jordan degree type $\mathcal S_{A,\ell}$ is a finer invariant, describing also the initial degrees of the simple submodules of $A$ in a decomposition of $A$ as ${\sf k}[\ell]$-modules. The set of Jordan types of $A$ or Jordan degree types (JDT) of $A$ as $\ell$ varies, is an invariant of the algebra. This invariant has been studied for codimension two graded algebras. We here extend the previous results to certain codimension three graded Artinian Gorenstein (AG) algebras - those of small Sperner number. Given a Gorenstein sequence $T$ - one possible for the Hilbert function of a codimension three AG algebra - the irreducible variety $\mathrm{Gor}(T)$ parametrizes all Gorenstein algebras of Hilbert function $T$. We here completely determine the JDT possible for all pairs $(A,\ell), A\in \mathrm{Gor}(T)$, for Gorenstein sequences $T$ of the form $T=(1,3,s^k,3,1)$ for Sperner number $s=3,4,5$ and arbitrary multiplicity $k$. For $s=6$ we delimit the prospective JDT, without verifying that each occurs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
