General Heart Construction on a Triangulated Category (I): Unifying t-Structures and Cluster Tilting Subcategories
General Heart Construction on a Triangulated Category (I): Unifying t-Structures and Cluster Tilting Subcategories
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DOI：
10.1007/s10485-010-9223-2
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发表时间：
2009-07
影响因子：
0.6
通讯作者：
H. Nakaoka
中科院分区：
文献类型：
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作者：
H. Nakaoka
Keller和Reiten的论文证明了一个三角范畴（在一定条件下）与一个簇倾斜子范畴的商成为一个阿贝尔范畴。在此之后，Koenig和Zhu详细地展示了如何在更抽象的背景下给出这个商范畴的阿贝尔结构。另一方面，正如20世纪80年代以来所熟知的，任何结构的核心都是阿贝尔的。我们统一这两个建设使用的概念，一个cotorsion对。对于一个三角范畴中的任一余挠对，当余挠对分别来自一个簇倾斜子范畴或at-结构时，我们可以自然地将一个阿贝尔范畴与之联系起来，它分别给出上述两个阿贝尔范畴中的每一个。
In the paper of Keller and Reiten, it was shown that the quotient of a triangulated category (with some conditions) by a cluster tilting subcategory becomes an abelian category. After that, Koenig and Zhu showed in detail, how the abelian structure is given on this quotient category, in a more abstract setting. On the other hand, as is well known since 1980s, the heart of anyt-structure is abelian. We unify these two constructions by using the notion of a cotorsion pair. To any cotorsion pair in a triangulated category, we can naturally associate an abelian category, which gives back each of the above two abelian categories, when the cotorsion pair comes from a cluster tilting subcategory, or at-structure, respectively.