Some Researches On Recollements And K-theory Of Categories

The representation theory of algebra,which arose in the 1970s,is the frontier branch of mathematics research in the word and has profound and extensive applications in other branches of mathematics and related subjects.In 2022,the mathematician Lusztig was awarded the Wolf Prize in mathematics for his pioneering contributions to the field of representation theory and related fields.Category theory is one of the important tools in the study of the representation theory.The recollement and algebraic K-theory of categories are two hot research directions of the representation theory.By using the category theory,this dissertation is divided into seven chapters to study the recollement and algebraic K-theory of categories.Based on introducing the research background and research status of extriangulated categories,algebraic K-theory and the recollement of categories,the introduction briefly describes the main conclusions and framework of this dissertation:The first chapter reviews the main concepts and conclusions involved in this dissertation.In the second and third chapters,lower K-groups of related categories are studied.In the fourth and fifth chapters,higher K-groups of related categories are studied.In the six chapter,the summary and prospect are introduced.Specifically:In Chapter Two,we study the relationships among lower K-groups of three categories in the recollement of extriangulated categories,and define lower K-groups of extriangulated categories by using objects and automorphisms of categories.It is proved that the lower K-group of the middle category in the recollement of extriangulated categories are isomorphic to the direct sum of lower K-groups of two remaining categories,which generalizes the conclusions of lower K-groups of recollements of triangulated categories.As an application,some conclusions about lower K-groups of the recollement of idempotent completion categories are obtained.In Chapter Three,we discuss the K1-group of the pre-n-angulated category,define the K1-group of the pre-n-angulated category,study its properties,and obtain the equivalent characterization of the K1-group of the pre-n-angulated category.As an application,the necessary and sufficient conditions for the equality of elements in the K1-group are given by using n-angles.In Chapter Four,based on constructing the Q-category of the extriangulated category,higher K-groups of the extriangulated category are defined by applying the algebraic topology.We study the relationships between higher K-groups of three categories in the recollement of extriangulated categories.The results generalizes the conclusion of higher K-groups of the recollement of abelian categories.The result of higher K-groups of the recollement of triangulated categories is also obtained.As an application,we get the relationships among higher K-groups of idempotent completion categories in the recollement of exact categories.In Chapter Five,the invariance of higher K-groups of the trivial extension of a category by a functor is studied.It is proved that there are isomorphisms of higher K-groups of subcategories whose objects are projective and finitely generated object between the original category and its trivial extension category.As an application to comma categories,the conclusion about,higher K-groups of its corresponding full subcategory is obtained.Furthermore,we show that higher K-groups of some kind of the trivial extension ring keep the invariance.Finally,on the basis of summarizing the main results of this dissertation,the main direction of the future research is prospected.