Abstract
In the preceding part (I) of this paper, we showed that for any torsion pair (i.e., t-structure without the shift-closedness) in a triangulated category, there is an associated abelian category, which we call the heart. Two extremal cases of torsion pairs are t-structures and cluster tilting subcategories. If the torsion pair comes from a t-structure, then its heart is nothing other than the heart of this t-structure. In this case, as is well known, by composing certain adjoint functors, we obtain a homological functor from the triangulated category to the heart. If the torsion pair comes from a cluster tilting subcategory, then its heart coincides with the quotient category of the triangulated category by this subcategory. In this case, the quotient functor becomes homological. In this paper, we unify these two constructions, to obtain a homological functor from the triangulated category, to the heart of any torsion pair.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers (French) [Perverse sheaves] Analysis and topology on singular spaces, I (Luminy, 1981). Astérisque, vol. 100, pp. 5–171. Soc. Math. France, Paris (1982)
Beligiannis, A., Reiten, I.: Homological and homotopical aspects of torsion theories (English summary). Mem. Amer. Math. Soc. vol. 188, no. 883, viii+207 pp. (2007)
Borceux, F.: Handbook of categorical algebra 1, Basic category theory. Encyclopedia of Mathematics and its Applications, vol. 50, xvi+345 pp. Cambridge University Press, Cambridge (1994)
Iyama, O., Yoshino, Y.: Mutation in triangulated categories and rigid Cohen–Macaulay modules (English summary). Invent. Math. 172(1), 117–168 (2008)
Keller, B., Reiten, I.: Cluster-tilted algebras are Gorenstein and stably Calabi–Yau. Adv. Math. 211(1), 123–151 (2007)
Koenig, S., Zhu, B.: From triangulated categories to abelian categories: cluster tilting in a general framework (English summary). Math. Z. 258(1), 143–160 (2008)
Nakaoka, H.: General heart construction on a triangulated category (I): unifying t-structures and cluster tilting subcategories. arXiv:0907.2080 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abe, N., Nakaoka, H. General Heart Construction on a Triangulated Category (II): Associated Homological Functor. Appl Categor Struct 20, 161–174 (2012). https://doi.org/10.1007/s10485-010-9226-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-010-9226-z