, Volume 19, Issue 6, pp 879-899
Date: 04 Feb 2010

General Heart Construction on a Triangulated Category (I): Unifying t-Structures and Cluster Tilting Subcategories

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Abstract

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 any t-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 a t-structure, respectively.

The author wishes to thank Professor Toshiyuki Katsura for his encouragement. The author wishes to thank his colleague Professor Noriyuki Abe. This work was never possible without his advices. The author wishes to thank Professor Osamu Iyama and Professor Bernhard Keller for their useful comments and advices, especially on the terminology.