# Differential graded algebra over quotients of skew polynomial rings by normal elements

## TL;DR

This paper extends differential graded algebra techniques to analyze the Ext algebra of quotients of skew polynomial rings by normal elements, providing new structural insights and generalizations of existing theorems in homological algebra.

## Contribution

It introduces a generalized construction of the Koszul complex and acyclic closure for skew polynomial rings, and characterizes the Ext algebra in this broader context.

## Key findings

- Presented a new description of the Ext algebra for quotients by normal elements.
- Showed the Ext algebra is noetherian when generated by a regular sequence.
- Generalized a theorem of Bergh and Oppermann to a wider class of rings.

## Abstract

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.