The well-known tame theorem tells that for a given tame bocs and a positive integer n there exist finitely many minimal bocses, such that any representation of the original bocs of dimension at most n is isomorphic to the image of a representation of some minimal bocses under a certain reduction functor. In the present paper we will give an alternative statement of the tame theorem in terms of matrix problem, by constructing a unified minimal matrix problem whose indecomposable matrices cover all the canonical forms of the indecomposable representations of dimension at most n for each non-negative integer n.
The standard Podle′s quantum sphere is Artin-Schelter regular as showed by Kra¨hmer(2012).The non-standard Podle′s quantum spheres are proved to be Auslander-regular,Cohen-Macaulay and Artin-Schelter regular in this paper.
