标题： 赫尔佐格-高山关于斜多项式环的分解 显示英文标题

标题： The Herzog-Takayama resolution over a skew polynomial ring

摘要： 设$\Bbbk$为一个域，令$I$为多项式环$R=\Bbbk[x_1,\ldots,x_n]$中的一个单项式理想。在她的论文中，Taylor 引入了一个复形，该复形为$R/I$提供了作为$R$模的有限自由分解。在此基础上，Ferraro、Martin 和 Moore 将这一构造扩展到了斜多项式环中的单项式理想。由于 Taylor 分解通常不是最小的，因此人们投入了大量努力来识别那些具有相对容易构造的最小自由分解的理想的类别。在 1987 年的一篇论文中，Eliahou 和 Kervaire 为$R$中一类称为稳定理想的单项式理想开发了一个最小自由分解。这一结果后来由 Ferraro 和 Hardesty 推广到了斜多项式环中的稳定理想。在 2002 年的一篇论文中，Herzog 和 Takayama 为具有线性商的单项式理想构造了一个最小自由分解，这是一个包含稳定理想的更广泛的理想类别。他们的分解在稳定情况下退化为 Eliahou-Kervaire 分解。在本文中，我们将 Herzog-Takayama 分解推广到了斜多项式环中。 显示英文摘要

摘要： Let $\Bbbk$ be a field, and let $I$ be a monomial ideal in the polynomial ring $R=\Bbbk[x_1,\ldots,x_n]$. In her thesis, Taylor introduced a complex that provides a finite free resolution of $R/I$ as an $R$-module. Building on this, Ferraro, Martin and Moore extended this construction to monomial ideals in skew polynomial rings. Since the Taylor resolution is generally not minimal, significant effort has been devoted to identifying classes of ideals with minimal free resolutions that are relatively straightforward to construct. In a 1987 paper, Eliahou and Kervaire developed a minimal free resolution for a class of monomial ideals in $R$ known as stable ideals. This result was later generalized to stable ideals in skew polynomial rings by Ferraro and Hardesty. In a 2002 paper, Herzog and Takayama constructed a minimal free resolution for monomial ideals with linear quotients, a broader class of ideals containing stable ideals. Their resolution reduces to the Eliahou-Kervaire resolution in the stable case. In this paper, we generalize the Herzog-Takayama resolution to skew polynomial rings.