Continuous Cluster-tilted Categories
نویسندگان
چکیده
We show that the quotients of the continuous cluster category Cπ and the rational subcategory X modulo the additive subcategory generated by any cluster are abelian categories and we show that they are isomorphic to categories of infinite and finite length modules, respectively, over the endomorphism ring of the cluster. These theorems extend the theorems of Caldero-ChapotonSchiffler and Buan-Marsh-Reiten for cluster categories of type An to their continuous and countably infinite limits respective.
منابع مشابه
Continuous Cluster Categories Ii: Continuous Cluster-tilted Categories
We show that the quotients of the continuous cluster category Cπ and the rational subcategory X modulo the additive subcategory generated by any cluster are abelian categories and we show that they are isomorphic to categories of infinite and finite length modules, respectively, over the endomorphism ring of the cluster. These theorems extend the theorems of Caldero-ChapotonSchiffler and Buan-M...
متن کاملCluster-tilted algebras and their intermediate coverings
We construct the intermediate coverings of cluster-tilted algebras by defining the generalized cluster categories. These generalized cluster categories are Calabi-Yau triangulated categories with fraction CY-dimension and have also cluster tilting objects (subcategories). Furthermore we study the representations of these intermediate coverings of cluster-tilted algebras.
متن کاملEquivalences between cluster categories
Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. These results are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object T in a hereditary abelian category H, we verify that the tilting functor HomH(T,−) induces a triangle equivalence from the cluster category C(H) to the cluster category C(A),...
متن کاملTilting Theory and Cluster Algebras
The purpose of this chapter is to give an introduction to the theory of cluster categories and cluster-tilted algebras, with some background on the theory of cluster algebras, which motivated these topics. We will also discuss some of the interplay between cluster algebras on one side and cluster categories/cluster-tilted algebras on the other, as well as feedback from the latter theory to clus...
متن کاملar X iv : m at h / 05 11 38 2 v 2 [ m at h . R T ] 1 9 Ju n 20 06 Equivalences between cluster categories ∗
Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. Some of them are already proved for hereditary abelian categories there. In the present paper, all basic results about tilting theory are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object T in a hereditary abelian category H, we verify that t...
متن کامل