AI 中文总结
本文引入阿贝尔n-截断范畴作为高维构造的统一框架，将余挠对的心构造推广到任意三角范畴上的n-余挠对，证明所得扩张心兼具兼容的预三角化与外三角化结构，且可实现为外三角化局部化。
AI 中文摘要
t-结构的心与由簇倾斜子范畴诱导的理想商是从三角范畴构造阿贝尔范畴的经典构造。它们的高维类似物，即n-扩张心与由(n+1)-簇倾斜子范畴诱导的理想商范畴，通常不再是阿贝尔范畴，但当基础三角范畴是代数三角范畴时，已知其兼具预三角化与外三角化结构。本文引入阿贝尔n-截断范畴的概念，作为这类高维构造的统一框架。我们将余挠对的心构造推广到任意三角范畴上的n-余挠对，证明所得的扩张心自然兼具兼容的预三角化与外三角化结构，构成阿贝尔n-截断范畴。该构造同时推广了t-结构的n-扩张心与由(n+1)-簇倾斜子范畴诱导的理想商，也可视为三角范畴上余挠对一般心构造的高维推广。最后，我们证明该心可实现为环境三角范畴上合适的相对外三角化结构的外三角化局部化。
英文摘要
The heart of a $t$-structure and the ideal quotient by a cluster tilting subcategory are classical constructions that produce abelian categories from triangulated categories. Their higher analogues, namely $n$-extended hearts and ideal quotient categories by $(n+1)$-cluster tilting subcategories, are generally no longer abelian, but are known to carry both pretriangulated and extriangulated structures when the underlying triangulated category is algebraic. In this article, we introduce the notion of an abelian $n$-truncated category as a common framework for such higher constructions. We extend the heart construction for cotorsion pairs to $n$-cotorsion pairs on arbitrary triangulated categories, and prove that the resulting extended heart naturally carries compatible pretriangulated and extriangulated structures forming an abelian $n$-truncated category. This construction simultaneously generalizes the $n$-extended heart of a $t$-structure and the ideal quotient by an $(n+1)$-cluster tilting subcategory. It may also be regarded as a higher-dimensional generalization of the general heart construction for cotorsion pairs on triangulated categories. Finally, we show that the heart can be realized as an extriangulated localization of a suitable relative extriangulated structure on the ambient triangulated category.
Comments51 pages
