Some Realted Studies On Weakly λ-koszul Modules

Koszul algebras, first introduced by Priddy in1970, are a class of quadratic alge-bras with linear resolutions and play an important role in different branches in mathe-matics, such as Lie algebra and Quantum groups. In the past of over40years, Koszul theory was developed rapidly. More precisely, motivated by the cubic Artin-Schelter regular algebras, Berger first introduced the notion of nonquadratic Koszul algebra in2001. And motivated by the quiver theory, Green and Zhang Pu et al first generalized the nonquadratic Koszul algebras to the nonlocal case. Then in2007Lu Diming et al introduced the notion of piecewise-Koszul algebra, and bi-Koszul algebra in2008. Note that Koszul algebras,d-Koszul algebras and piecewise-Koszul algebras are both have only one jump-degree, in order to breakthrough this restrict, Lu Jiafeng introduced the notion of λ-Koszul algebra (also called d-Koszul-type algebra) in2009, which have arbitrary finite jump-degrees.In this thesis, we start from the minimal graded projective resolution and discuss some basic properties of minimal graded projective resolutions of weakly λ-Koszul modules in terms of minimal Horseshoe Lemma. Meanwhile, we study the associated graded module of a λ-Koszul module and discuss the relationships between the minimal graded projective resolutions of a λ-Koszul module and that of its associated graded module.The whole thesis is arranged as follows:In Chapter1, we give some basic definitions and the main results. In Chapter2, we will study the relationship between P*i and P*, where P*i�'Ui/Ui-i�'0and P*�'M�'0are the corresponding minimal graded projective resolutions, M a weakly A-Koszul module and Ui are the fixed graded submodules. Moreover, we will give the definition of block λ-Koszul module, and study the rela-tionship between block λ-Koszul modules and λ-Koszul modules.In Chapter3, we will begin with associated graded functor, and study the associ-ated graded module of a weakly λ-Koszul module. We prove that a finitely generated graded module is a weakly A-Koszul module if and only if its associated graded mod-ule is a λ-Koszul module. Further, the relationship of the minimal graded projective resolutions between a weakly λ-Koszul and its associated graded module is discussed.