TWIST Equivalent To CALABI-YAU Algebra

This Master thesis is mainly about the twist equivalence and Calabi-Yau algebras. Recently, M. Reyes, D. Rogalski and J. J. Zhang have described the re-lation between the Nakayama automorphisms of a Noetherian AS-regular algebra and its twisted algebra while the twist system is generated by a graded algebra automorphism by using group algebra and Hopf action [RRZ]. This method is not workable for general twist systems. We give a possible way of constructing gen-eral twist systems firstly. Actually, for any finitely generated connected graded algebra, its arbitrary twist system can be lifted to a twist system of the tensor algebra, and the transpose of the inverse of the lifted twist system restricts to a twist system of the dual algebra of the original algebra.Based on above facts, we describe the relation between the Nakayama au-tomorphism of a p-Koszul AS-regular algebra and its twisted algebra for general twist systems. Especially, when those twist systems are generated by algebra automorphisms, our result coincides with the conclusion in [RRZ]. On the other hand, we prove that skew polynomial algebras and generic AS-regular algebras of dimension3except Type E are all twist equivalent to Calabi-Yau algebras.M. Reyes, D. Rogalski and J. J. Zhang raised a question in [RRZ2]:for any AS-regular algebra A, is there a Calabi-Yau algebra B such that GrMod(A) is equivalent to GrMod(B)? Or equivalently, can A be twist equivalent to a Calabi-Yau algebra? In [RRZ2], the authors found an algebra which can not be twist equivalent to any Calabi-Yau algebra by twist systems generated by algebra au-tomorphisms. We prove that for any general twist systems, it can not be twist equivalent to any Calabi-Yau algebra either. Thus we answer the question.