I Algebra Problem

This dissertation consists of five sections.We shall investigate the rcsprcscntation of coalgcbra.The first and second sections are introduction and preliminaries respectively.In the third section,we introduce the concept of irreducible homomorphism,almost split sequence and so on,study their properties,find the relation between irreducible homomorphism and minimal split homomorphism.So that the research of irreducible homomorphism can be converted to the study of the almost split sequence.Then we get some cquvilant conditions of ahnost split sequence.In the forth seetion,in the quasi-finit comodule category,through the foundation of AR transpose and the construction of the coNakayana functor,we study their properties in indccomposble injeetive,projective comodule and the ahnost split sequence derived from them,combine the relation between copresentation and comodule,caculate a conerete quiver's AR value.In the fifth seetion,through the existence of the colnodule almost split sequence, combined the properties of quasi-finite comodule and the former related conclusions,we result that there is irrducible homomorphism start or end at paticular comodule,make clear the concrete form of irreducible homomorphism further.