Torsion Theories And Their Cluster Structures In Calabi-Yau Categories

The main object in this thesis is to study cotorsion pairs and their cluster structuresand in particular, to study two special classes of cotorsion pairs: higher cluster tiltingobjects and maximal rigid objects.We define the notion of mutations of cotorsion pairs in a triangulated category andprove that the mutation of every cotorsion pair is also a cotorsion pair. We give a corre-spondence between cotorsion pairs in the subfactor category of a rigid subcategory andcotorsion pairs whose cores contain this rigid subcategory. It induces a bijection betweencotorsion pairs with the same core and t-structures in the subfactor category of the core.Using the fact that the decompositions of categories are determined by the decomposi-tions of their cluster tilting objects, we prove that there are no non-trivial t-structures in atriangulated2-Calabi-Yau category with cluster tilting objects. Based on this, we classifycotorsion pairs in this case. As an application, we give a geometry model of cotorsionpairs and their mutations via Riemann surfaces.Higher cluster tilting objects and maximal rigid objects as two special cases of co-torsion pairs have rich structures. The equivalence between higher cluster tilting objectsand higher maximal rigid objects in a higher cluster categories is proven in this thesis.We prove that any almost complete (d+1) cluster tilting object has exactly d+1com-plements in a (d+1) cluster category. A necessary and sufcient condition of a set ofobjects being the set of complements of one is given. We prove that there are no commonsummands between the middle items in the connecting triangles. Some basic propertiesof higher cluster complexes are studied.In a triangulated2-Calabi-Yau category, we prove that the extension category of amaximal rigid object and its shift contains any rigid objects and that any maximal rigidobject objects are cluster tilting if a cluster tilting object exists. Comparing the relativeresults about cluster tilting objects, we prove that the numbers of direct summands ofmaximal rigid objects are equal and that the endomorphism algebra of a maximal rigid isa Gorenstein algebra with the Gorenstein dimension at most1. In cluster tube, where thereare only maximal rigid objects and no cluster tilting objects, we construct an analogouscluster map which is compatible with the exchange relations of the mutations of maximalrigid objects. Then This map gives a categorification of cluster algebras of type B and type C.