Applications Of Quivers In Coalgebras And Graded Comodules

This Ph.D thesis mainly deals with graded comodules,serial coalgebras and co-Frobenius coalgebras.Our ideas and methods come from the representation theory of finite dimensional algebras,especially the quiver-theoretic method.It consists of the following three parts:1.The second chapter starts with a discussion of the general theory of graded coalgebras and graded cornodules,and we point out the uniqueness of gradation on certain comodules up to the degree shift.We are interested in the(graded)cornodules on path coalgebras.For this,we introduce the notion of arrow function and vertex-gradation.We prove that if the quiver Q does not contain basic cycles,all kQ^c-comodules are gradable if and only if Q admits an arrow function,and in this case,all comodules are vertex-graded. The case where Q is a basic cycle is subtle.We classify all the indecomposable comodules on basic cycles,and show that all of them are Z-graded.Combining these results,we prove that all the kQ^c-comodules are gradable if and only if Q is a basic cycle or does not have nonsymmetric cycles.2.The third chapter starts with a survey of serial coalgebras and co-Frobenius coalgebras.Based on[CT2],Theorem 2.10,we focus on the pointed case and then the quiver-theoretic method can be applied.We prove that a pointed coalgebra C is serial and co-Frobenius if and only if C is isomorphic to(?)_i^m=0^kQ_i,m≥1,where Q is the linear quiver A_∞^∞or the basic cycle Z_n.3.In the last chapter,by applying the Warfield Lemma we observe three new classes of Krull-Schmidt categories.As an application,we obtain the uniqueness of the gradation of certain modules(and comodules).