【作者】 郑敏；

【导师】 陈清华；

【作者基本信息】 福建师范大学 ， 基础数学， 2022， 博士

【摘要】 代数表示论兴起于上世纪70年代,是当前国际数学研究的前沿分支,在数学的其它分支及相关学科中有着深刻广泛的应用.2022年,数学家Lusztig更是被授予沃尔夫数学奖,以表彰其对表示论及相关领域的开创性贡献.范畴理论是表示论研究的重要工具之一,而范畴的Recollement与代数K-理论是其两大热点研究方向.本学位论文致力于以范畴理论为工具,围绕范畴的Recollement与代数K-理论的若干问题展开相关研究,共分为七章.绪论在介绍外部范畴(Extriangulated category)、代数K-理论和范畴的Rec-ollement的研究背景和研究现状的基础上,简要阐述了本学位论文的主要结论和论文框架:第一章回顾本学位论文所涉及的主要概念和相关结论;第二、三章研究相关范畴的低阶K-群;第四、五章研究相关范畴的高阶K-群;第六章总结与展望.第二章研究外部范畴Recollement中三项范畴间低阶K-群的关系,利用范畴对象和自同构态射定义了外部范畴的低阶K-群,在相关函子正合的情形,证明了外部范畴Recollement的中间项范畴的低阶K-群同构于剩余两项范畴低阶K-群的直和.它推广了三角范畴Recollement低阶K-群的已有成果.作为应用,得到关于幂等完备化范畴Recollement低阶K-群的结论.第三章讨论n-同调对象—预n-角范畴的K1-群,定义了预n-角范畴的K1-群,研究了其性质,得到预n-角范畴K1-群的等价刻画.作为应用,利用n-角给出其元素相等的充分必要条件.第四章在构造外部范畴的Q-范畴的基础上,利用代数拓扑定义了外部范畴的高阶K-群(Higher K-groups),研究外部范畴Recollement中三项范畴高阶K-群的关系.所得结论不仅推广了 Abel范畴Recollement高阶K-群的已有成果,也得到关于三角范畴Recollement高阶K-群的结论.作为应用,刻画了正合范畴Recollement中三项范畴的幂等完备化范畴的高阶K-群关系.第五章研究平凡扩张范畴高阶K-群的不变性,证明了范畴的平凡扩张前后其有限生成投射满子范畴的高阶K-群均同构,应用到Comma范畴得到对应的满子范畴高阶K-群的结论.进而,给出一类平凡扩张环高阶K-群的不变性.最后,在总结本学位论文研究成果的基础上展望了后续研究的主要努力方向.更多还原

【Abstract】 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.更多还原