| The main object in this thesis is to study ghost-tilting objects in a triangulated cate-gory with Serre duality and in particular, to study cluster-tilting objects. We also investi-gate connections between objects in triangulated categories, modules over endomorphis-m algebras of cluster-tilting objects and complexes in homotopy categories of bounded complexes of finitely generated projective modules.Assume that D is a Krull-Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object T. We introduce the notion of ghost-tilting objects and T[1]-tilting objects in D, which are a generalization of cluster-tilting objects. When D is 2-Calabi-Yau, the ghost-tilting objects are cluster-tilting. We show that any basic almost T[1]-tilting object in D has exactly two non-isomorphic indecomposable complements. We first introduce a natural partial order on T[1]-tilting objects and mutation of T[1]-tilting objects. Then we show that starting with a complement, one can calculate the other one by an exchange triangle, which is constructed from a left approximation or a right approximation.Let A= EndDop(T) be the endomorphism algebra of T. We show that there is an order-preserving bijection between T[1]-tilting objects in D and support Ï„-tilting A-modules, which generalizes some results [1-7]. Using this bijection and mutation of T[1]-tilting objects,we give a partial answer to a question raised by Adachi, Iyama and Reit-en.As a special case of ghost-tilting objects, cluster-tilting objects have a rich theory. We investigate connections between cluster-tilting objects in D, functorially finite torsion classes in modA and two-term silting complexes in Kb(projA). And we show that all cluster-tilting objects in D have the same number of non-isomorphic indecomposable direct summands, which generalizes some results [8,9]. Finally, we show the compatibility between Iyama-Yoshino reduction and Ï„-tilting reduction. |
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