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.

Let H be a semisimple Hopf algebra acting on an Artin-Schelter regular algebra A, homogeneously, inner-faithfully, preserving the grading on A, and so that A is an H-module algebra. When the fixed subring A H is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that H is a reflection Hopf algebra for A. We show that each of the semisimple Hopf algebras H 2n 2 of Pansera, and A 4m and B 4m of Masuoka is a reflection Hopf algebra for an AS regular algebra of dimension 2 or 3.2010 Mathematics Subject Classification. Primary 16T05, 16E65, 16G10.The Hopf algebras A 4m and B 4m can be viewed as deformations of kQ 4m (see [2]), and are extensions of the form:The examples of reflection Hopf algebras that we have computed indicate that there are an abundance of examples. The properties that characterize such a pair (A, H) are not clear, and invite further investigation. One obvious question is:Question 0.1. When is a bicrossed product H = K# τ σ H a reflection Hopf algebra for some AS regular algebra A?The method that is used in this paper is as follows. First, we compute the Grothendieck ring of finite-dimensional H-modules for each Hopf algebra H. The results are summarized in the following table.

For each nontrivial semisimple Hopf algebra H of dimension sixteen over C, the smallest dimension inner-faithful representation of H acting on a quadratic AS regular algebra A of dimension 2 or 3, homogeneously and preserving the grading, is determined. Each invariant subring A H is determined. When A H is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that H is a reflection Hopf algebra for A.

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