Tensor Product Of Flat Exact Sequence

This master’s thesis focuses on the research of n-abelian category and n-exact category on the basis of Gustavo Jasso,in the same time,we introduce the concept of flat exact sequence in the module category,we give and prove the properties of the tensor product of flat exact sequence.This thesis consists of four chapters:Chapter 1,We state historical background and summarize the content structure arrangement.Chapter 2,we give some basic concepts and related properties,such as additive category,modular category,homotopy category etc.And some results of n-kernel,n-exact sequence and n-pullback graph are given.Chapter 3,First of all,we introduce the concept of n-abelian category,and discuss the related issues of kernel and weak kernel,n-kernel and n-cokernel.At the same time,we prove that there are close relationship between n-pullback graph and n-kernel in n-abelian category;Then,We give the definition of n-exact category and admissible n-exact sequence,and prove that the relation between admissible epimorphism,n-kernel and admissible n-exact sequence.We also research the basic properties of the tensor product of admissible n-exact sequence.Chapter 4,On the basis of the n-exact sequence of the module category,we introduce the concept of flat exact sequence.By using two flat exact columns X,Y,as a tensor product X(?)Y which leads to a flat double complex.We prove that all the rows and generated complex are flat exact sequence in the flat double complex,and further proof that the tensor product of two flat exact sequence is still flat exact sequence.