  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.
