A Five-dimensional Artin - Schelter Regular Algebra - Quantum P ~ 4 Classification

The class of regular algebras, which are also called Artin-Schelter regular algebras in this thesis, was first identified in a joint paper of Artin and Schelter in1987. These regular algebras are viewed as the homogeneous coordinate rings and the noncommuta-tive analogy of the commutative polynomial rings. Since then, the classification of such regular algebras is always an very important question in the field of noncommutative projective geometry.Subsequent papers by Artin, Tate and Van den Bergh completed the classification of regular algebras of dimension three. The methods that they used are essentially geometric. As for the four-dimensional case, if the algebras are domains, then they are of type (12221),(13431) or (14641). Many researchers have studied these types in various views and ways. More explicitly, Lu, Palmieri, Wu and Zhang classified the type (12221). The classification of type (13431) was considered by Rogalski and Zhang. Using the double Ore extension and the pushout methods, Zhang and Zhang studied the type (14641).In this thesis, we focus on the classification of the five-dimensional regular alge-bras which are generated by two generators of degree one. According to the work of Floystad and Vatne, if such regular algebras are domains with Gelfand-Kirillov di-mension larger than or equal to four, then they are of type (123321),(124421) or (125521). On this basis, we specially consider the five-dimensional regular algebras of type (123321) in the thesis. Using the A∞-algebras methods, we complete the clas-sification of such regular algebras with two degree one generators and three relations of degree four.The main result in this thesis is to prove that there are nine families of such regular algebras satisfying the conditions above. We list the algebras as Algebra A, Algebra B, Algebra C, Algebra D, Algebra E, Algebra F, Algebra G, Algebra H, Algebra I. After the detailed analysis for each family, we prove that all these nine families of algebras are Artin-Schelter regular, Auslander regular, strongly noetherian and Cohen-Macaulay. Finally, we study the example called extremal algebra which was introduced by Floystad and Vatne recently. They constructed the minimal resolution of the trivial module, and proved that the extremal algebra is Artin-Schelter regular. More interesting is that its Hilbert series does not occur as the one of the enveloping algebra of a five-dimensional graded Lie algebra generated in degree one. In this thesis, we reconsider this example and prove that the extremal algebra is also Auslander regular, strongly noetherian and Cohen-Macaulay. We also compute the point modules over this algebra and prove that there are nine classes of point modules up to isomorphisms.