Galois Covers, Smash Products And The Duality Theorem Over The Semigroup Graded Category

In this dissertation,we introduce some definitions ,obtain some basic propertiesand research Galois coverings, smash products, the duality theorem and künnethformula in the semigroup graded category.The dissertation consists of four chapters.In chapter one, we introduce the groundwork of this text, the research directionand the trends of the development, then sum up the main results of this dissertation.In chapter two, we introduce module categories over semigroup graded care-gories,Galois coverings and smash products.We can obtain that in case of a Galoiscovering of the category C to B, the category B-Mod of B-modules coincides withthe full subcategory of the fixed modules over the category C,while to the S-gradedB-Mod ,the category B#S-Mod of B#S-modules coincides with the category of S-graded B-modules.In chapter three, we consider the category C over a field k .In part one,weobtain a coherence result that isα(C)#H(?)α(C#H). As to a S-graded category, weprove that it is also a kS-Mod.In part two,we can obtain a generazation of Cohen-Montgomery duality theorem that is when B is a S-graded category and S is a finitesemigroup, (B#k~S)#kS(?)B.In chapter four, we use the method of member that can be used to prove thekünneth formula over the Abel category C-Mod.