Differential graded algebra techniques have played a crucial role in the development of homological algebra, especially in the study of homological properties of commutative rings carried out by Serre, Tate, Gulliksen, Avramov, and others. In this article, we extend the construction of the Koszul complex and acyclic closure to a more general setting. As an application of our constructions, we shine some light on the structure of the Ext algebra of quotients of skew polynomial rings by ideals generated by normal elements. As a consequence, we give a presentation of the Ext algebra when the elements generating the ideal form a regular sequence, generalizing a theorem of Bergh and Oppermann. It follows that in this case the Ext algebra is noetherian, providing a partial answer to a question of Kirkman, Kuzmanovich and Zhang.

中文翻译：

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正规元素对偏斜多项式环商的微分分级代数

微分分级代数技术在同调代数的发展中发挥了至关重要的作用，特别是在 Serre、Tate、Gulliksen、Avramov 等人进行的交换环同调性质研究中。在本文中，我们将 Koszul 复形和非循环闭包的构造扩展到更一般的设置。作为我们构造的一个应用，我们通过普通元素生成的理想对偏斜多项式环商的 Ext 代数的结构有所了解。因此，当生成理想的元素形成规则序列时，我们给出了 Ext 代数的表示，概括了 Bergh 和 Oppermann 的定理。因此，在这种情况下，Ext 代数是诺特式的，为 Kirkman、Kuzmanovich 和 Zhang 的问题提供了部分答案。