正合三角范畴的代数结构及其高维推广

The notion of n-cluster tilting subcategory in triangulated and exact categories is a general framework to study important structures in representation theory. The notion of an extriangulated category was introduced, which is a simultaneous generalization of exact category and triangulated category. This enables us to generalize the arguments on n-cluster tilting subcategory.. n-angulated category and n-abelian category are analog of triangulated category and abelian category from the point view of higher homological algebra. One interesting example is n-cluster tilting subcategories of an abelian category are also n-abelian. On the other hand, we have n-angulated category which generalizes triangulated category, and n-cluster tilting subcategories on a triangulated category satisfying certain condition are typical examples of such case.. For each positive integer n we introduce the notion of n-exangulated categories as higher dimensional analogues of extriangulated categories defined by Nakaoka-Palu. We characterize which n-exangulated categories are n-exact in the sense of Jasso and which are (n + 2)-angulated in the sense of Geiss-Keller-Oppermann. For extriangulated categories with enough projectives and injectives we introduce the notion of n-cluster tilting subcategories and show that under certain conditions such n-cluster tilting subcategories are n-exangulated.. We will study n-exangulated category and its abelian quotient categories, the n-cluster tilting subcategories on it. Moreover, we will also study cotorsion pairs (a more general setting than n-cluster tilting) and their hearts on the n-exangulated category, try to give a classification of them. Then we consider the application of our results on triangulated and exact categories.

在三角范畴和正合范畴上的n-丛倾斜子范畴和余绕对为我们学习和研究表示论中的许多重要结构提供了一个具有一般性意义的模型。由于其共通性，我们可以在更具一般性的范畴—正合三角范畴上来考察它们。n-角范畴和n-正合范畴是三角范畴和正合范畴的高维推广，一个重要例子是三角范畴上的n-丛倾斜子范畴在满足一定条件时是一个(n+2)-角范畴。对于每个正整数n，导入正合n-角范畴的概念，作为正合三角范畴的高维推广。我们确定在这种定义下何时正合n-角范畴是n-正合范畴，何时是(n+2)-角范畴。对于具有足够投射和内设对象的正合三角范畴，我们引入n-丛倾斜子范畴，并研究在满足何种条件时这些n-丛倾斜子范畴是我们所定义的正合n-角范畴。我们将考察正合n-角范畴及其阿贝尔商范畴，研究其上n-丛倾斜子范畴的性质。进而研究其上的余绕对(更具一般性)及其中心，给出它们的分类。最后考虑结论在三角，正合范畴上的应用。

在三角范畴和正合范畴上的n-丛倾斜子范畴和余绕对为我们学习和研究表示论中的许多重要结构提供了一个具有一般性意义的模型。基于其共通性，我们在更具一般性的范畴—正合三角范畴上考查相关问题。n-角范畴和n-正合范畴是三角范畴和正合范畴的高维推广；另一方面，三角范畴上的n-丛倾斜子范畴在满足一定条件时也是一个(n+2)-角范畴。对于每个正整数n，我们导入正合n-角范畴的概念，作为正合三角范畴的高维推广。我们建立起这种范畴的基本结构，研究了这种范畴的基本性质，并且给出了不同于n-角范畴和n-正合范畴的正合n-角范畴的例子。同时对于具有足够投射和内设对象的正合三角范畴，我们引入n-丛倾斜子范畴，考查了在满足何种条件时这些n-丛倾斜子范畴是正合n-角范畴。我们深入研究了正合三角范畴上的一些重要代数结构：余绕对和相对刚性子范畴。余绕对是应用广泛的代数结构，在许多代数问题的研究中发挥着重要作用，特别是在关于商范畴的研究中。我们的研究还进一步揭示出余绕对在正合三角范畴这样具有一般性的代数结构中仍然具有较好的性质。同时这些结论在各具体的范畴，如模范畴，三角范畴中均能得到有效应用。相对刚性子范畴由于和τ-倾斜理论有着很深的联系而受到关注，我们研究了与之相关的结论在正合三角范畴上的表现，并作出了相应的推广。