The Generalization Of Taft-Wilson Theorem For Pointed Coalgebras

The quivers have become an important foundation for algebra and played an important role in algebra research. Since the the seventies of lastcentury,many people have started to study algebras using quivers, for example in the studying elementary algebra and pointed algebra, The quivers play an important role both in the structure of algebras and the representations of algebras.In chapter 1 of this dissertation,we give an introduction,in eluding the back-ground and recent developments related to this dissertation,and make a systemic exposition of study background and my main Jesuits.In chapter 2,we give an introduction to the defintions and the main theorem which we needed in this dissertation.There are two sections in this chapter.in section 1,we give the defition and basic properties of pointed coalgebras, the defition and basic properties of coradical filtra-tion.In section 2,we give the defitions of quivers and path coalgebras,the gradations to the path coalgebras.We also give K1 the bi-KQ0 comodule structure.We then give the defintions of superquivers and path supercoalgebras.In the last,we give the defition of Gabriel quiver for pointed coalgebras.In chapter 3 we prove the Gabriel theorem for pointed coalgebras with the universal property for path coalgebras.We also give two important lemmas which have proved in the refered papers.In chapter 4 we generalize the Taft-Wilson thoerem with the Gabriel theorem for pointed coalgebras and the important property for coradical filtration which have been proved in chapter 3.Then we apply the the generalized Taft-Wilson theorem to pointed supercoalgebra.