Homological Aspects of Noetherian PI Hopf Algebras and Irreducible Modules of Maximal Dimension, Original Research Article

We prove that a Noetherian Hopf algebra of finite global dimension possesses further attractive homological properties, at least when it satisfies a polynomial identity. This applies in particular to quantized enveloping algebras and to quantized function algebras at a root of unity, as well as to classical enveloping algebras in positive characteristic. In all three cases we show that these algebras are Auslander-regular and Macaulay. We derive representation theoretic consequences concerning the coincidence of the non-Azumaya and singular loci for each of the above three classes of algebras.