Some Related Studies On(Quasi-)λ-Koszui Algebras And(Quasi-)λ-Koszul-modules

Koszul algebras were first introduced by Priddy in1970, which are a class of quadratic algebras with good homological properties and are"as close to semisimplic-ity as possible". Many important algebras are Koszul, such as path algebras, tensor algebras, polynomial algebras, Sklyanin algebras, Steenrod algebras and the quadratic Artin-Schelter regular algebras. Recently, there were a lot of different generalizations on Koszul algebras and the Koszul theory had a rapid development. For examples, mo-tivated by the cubic Artin-Schelter regular algebras, Berger first introduced the notion of nonquadratic Koszul algebra in2001. Motivated by the quiver theory, Zhang Pu et al first generalized the nonquadratic Koszul algebras to the nonlocal case and gave the name "d-Koszul algebras". Lu Diming et al introduced the notion of piecewise-Koszul algebra in2007, and bi-Koszul algebra in2008, and so on. Note that Koszul algebras, d-Koszul algebras and piecewise-Koszul algebras all have only one "jump-degree", in order to breakthrough this restrict, Lu Jiafeng introduced the notion of A-Koszul algebra in2009, which include both Koszul and d-Koszul algebras as special examples and can have any finite jump-degrees.The present thesis is a continuous work on λ-Koszul algebras. More precisely, we discuss the conditions such that the category of λ-Koszul modules coincide with the category of modules with linear presentations; λ-Koszul algebras are generalized to the nongraded case and we introduce the notions quasi-λ-Koszul algebras and quasi-λ-Koszul modules. The whole thesis is arranged as follows:In chapter one, we give some preknowledge, motivations and main results.In chapter two, we discuss the conditions such that the category of λ-Koszul mod-ules coincide with the category of modules with linear presentations.In chapter three, from the definitions of λ-Koszul algebras and λ-Koszul mod-ules, we give the definitions of quasi-λ-Koszul algebras and quasi-λ-Koszul modules, examples and some basic properties, further we discuss the properties of the category of quasi-λ-Koszul modules, gave the conditions such that the category is closed under kernels and cokernels.