N≥3时具有有限整体维数的N-Koszul代数
$N$-Koszul algebras of finite global dimension for $N\geq 3$
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中文总结 AI 辅助
本文针对N≥3的N-Koszul代数，证明若其希尔伯特级数在t=1处的极点阶数满足特定条件，或满足加权多项式环希尔伯特级数与GK维数等于整体维数的条件，则必为已知的整体维数3的3-Koszul AS正则代数。
中文摘要 AI 辅助
设N≥3，N-Koszul AS正则代数类，或更一般地N-Koszul AS Gorenstein代数类，已代数学家广泛关注。然而，除了整体维数为3的N-Koszul AS正则代数外，尚无其他具有有限整体维数的此类代数的已知例子。Kabbaj近期的研究表明，在以下两个假设下：(1)代数A具有加权多项式环的希尔伯特级数；(2)平凡A-模𝕂具有有限自由分解，此类具有有限整体维数的N-Koszul代数A必须具有较大的整体维数，且N必须为素数。所有AS正则代数均满足第二个假设，且被期望也满足第一个假设。本文证明，若N-Koszul代数A的希尔伯特级数h_A(t)在t=1处的极点阶数大于(21d+1)/22（其中d为A的整体维数），则A必为已知的整体维数为3的3-Koszul AS正则代数之一。作为推论，若N-Koszul AS正则代数A具有加权多项式环的希尔伯特级数，且A的GK维数与整体维数相等（这两个条件已被猜想对所有AS正则代数均成立），则A必为已知的整体维数为3的3-Koszul AS正则代数之一。
英文摘要
Let $N\geq 3$. The class of $N$-Koszul AS regular algebras, or more generally, that of $N$-Koszul AS Gorenstein algebras, has attracted much attention from algebraists. Nevertheless, there have been no known examples of $N$-Koszul AS regular algebras of finite global dimension other than the ones of global dimension $3$. A recent work by Kabbaj showed that, such an $N$-Koszul algebra $A$ of finite global dimension has to have a large global dimension and that $N$ has to be prime, under the assumptions that $(1)$ $A$ has a Hilbert series of weighted polynomial rings and that $(2)$ the trivial $A$-module $\mathbb{K}$ has a finite free resolution. All AS regular algebras satisfy the latter assumption and are expected to do the former as well. In this paper, we prove that such an $N$-Koszul algebra $A$ must be one of the known $3$-Koszul AS regular algebras of global dimension $3$ if the order of the pole of its Hilbert series $h_A(t)$ at $t=1$ is greater than $\frac{21d+1}{22}$, where $d$ is the global dimension of $A$. As a corollary, we prove that any $N$-Koszul AS regular algebra $A$ must be one of the known $3$-Koszul AS regular algebras of global dimension $3$ if $A$ has a Hilbert series of weighted polynomial rings and if the GK dimension of $A$ coincides with the global dimension of $A$, both of which have been conjectured to hold for any AS regular algebras.
