Extriangulated 范畴和丛倾斜理论

We can regard extriangulated categories as a simultaneous generalization of exact categories and triangulated categories, and now it has become a hot topic in representation theory of algebras. This project will study extriangulated categories and its cluster tilting theoy, wihich is divided into the following three parts: . 1. We will construct some examples of extriangulated categories from algebraic topology and algebraic geometry, which are neither exact categories nor triangulated categories.. 2. We will establish a close relationship between cluster tilting theory of extriangulated categories and tau-tilting theory of module categories.. 3. We will explore a basis of Grothendieck groups of cluster tilting subcategories of extriangulated categories.. This project will build the links between extriangulated categories and other fields, and enable extriangulated categories to have new application values and broader research background. Thus this project will enrich and develop the cluster tilting theory in triangulated categories.

Extriangulated范畴作为正合范畴和三角范畴的推广，现已成为代数表示论研究的前沿热点课题。本项目将研究extriangulated范畴和它的丛倾斜理论，具体分为以下三个部分：. 1.从代数拓扑和代数几何中构造一些既不是正合范畴也不是三角范畴的extriangulated范畴的例子。. 2.建立extriangulated范畴的丛倾斜理论与模范畴的tau-倾斜理论之间的联系。. 3.探究extriangulated范畴的丛倾斜子范畴的Grothendieck群的基。.本项目将构建extriangulated范畴与其它领域之间的联系，使得extriangulated范畴有新的应用价值及更广泛的研究背景，从而丰富和发展三角范畴的丛倾斜理论。

Extriangulated范畴的概念由Nakaoka和Palu于2016年提出,其目的是为了发展模型结构与余挠理论之间的联系。它统一了正合范畴,三角范畴以及一些既不是正合也不是三角的加法范畴,在代数,几何以及拓扑上均有重要的应用背景。本项目研究了extriangulated范畴和丛倾斜理论，研究成果主要包含五个方面的内容:.1.找到了一些既不是正合范畴也不是三角范畴的extriangulated范畴例子;.2.建立了extriangulated范畴的丛倾斜理论与tau-倾斜理论之间的联系;.3.给出了Auslander-Reiten理论与Grothendieck群的联系;.4.通过丛倾斜理论实现了从n-exangulated范畴到 n-阿贝尔范畴;.5.研究了extriangulated范畴的相对同调代数和粘合理论。.本项目构建了extriangulated范畴与其它领域之间的联系，使得extriangulated范畴有新的应用价值及更广泛的研究背景，从而丰富和发展了三角范畴的丛倾斜理论，并为一些重要的代数结构的研究提供新的思想与方法。