Koszul Algebras Of Finite Complexity

The complexity was introduced to study the group representations. It is also a new approach of the representation theory of the hereditary algebras. Using complexity,we can also study the structure of the A-R quiver of a group algebra. The selfinjective Koszul algebra of finite complexity is an important class of algebras,there is some nice relation between the stable category of such algebras with the derived category of coherent sheaves.Koszul algebra was introduced by Priddy in 1970.Roland Berger generalized it to t-Koszul algebras. It palys an important role in noncommutative geometry. Through 0,1-generated Artin-Schelter regular algebras of global dimension 3,one gave an important classification of t-Koszul algebras. It is well known that the hereditary algebra of Koszul algebra and the tensor product of Koszul algebras are Koszul algebras. In chapter 2, we compute the complexity of the tensor product of Koszul algebras; In chapter 3, we study the Koszul uniserial module on Koszul hereditary algebra, moreover we prove that the Koszul composition series of Koszul moduls on Koszul hereditary algebra is unique up to isomorphism.