Researches On Graded Algebras With Non-pure Resolutions

The paper mainly discusses a kind of graded algebras with non-pure resolutions, called bi-Koszul algebras.By an algebra with pure resolution,we mean that every projective module,in the minimal projective resolution of its trivial module,is generated in one degree;otherwise,we say an algebra with non-pure resolution.Bi-Koszul algebras,including the non-Koszul Artin-Schelter regular algebras of global dimension 4 as examples,are different from Koszul algebras and d-Koszul algebras, due to their non-purity.In this thesis,we discuss the homological properties of bi-Koszul algebras.Firstly,we give the equivalent description of a bi-Koszul algebra in terms of its Koszul dual,but the later may not be finitely generated.Strongly bi-Koszul algebras are a class of bi-Koszul algebras whose Koszul duals are finitely generated.To solve the generation of the Koszul dual of the bi-Koszul algebra,we introduce the new concept of "finite generation" in the A_∞-setting,which gives a balance between multiplications and elements.We prove that the Koszul dual of the bi-Koszul algebra is[m₂,m₃]-finitely generated,where m₂ and m₃ are two multiplications of the A_∞-algebra.Secondly,we explore how to transform the non-purity of bi-Koszul algebras into purity.The non-purity is presented by the minimal projective resolution or Koszul duality. It is proved that under some condition,from the non-pure resolution of a bi-Koszul module,two pure resolutions are obtained.In the A_∞-language,a special kind of bi-Koszul algebras,called truncated bi-Koszul algebras,can also be decomposed into two pure parts.We study how to construct bi-Koszul algebras from algebras with pure resolutions. The construction methods are various,such as one point extension,normal extension,Ore extension and so on.The notion of(s,t,d)-bi-Koszul algebras is also introduced in the thesis,as a generalization of bi-Koszul algebras.This kind of algebras preserves lots of homological properties of bi-Koszul algebras.Moreover,the(s,t,d)-bi-Koszul algebra can be constructed by the free product of two algebras with pure resolutions.