Research On Calabi-yau Objects And Torsion Pairs In Triangulated Categories

Triangulated categories were introduced in the middle of twentieth centuryby J.L.Verdier. Originally, they were used in algebraic geometry and algebraictopology, and by now they have become indispensable in many diferent fields ofmathematics. There are important applications of triangulated categories in fieldslike algebraic geometry, algebraic topology(stable homotopy theory), commutativealgebra, diferential geometry(Fukaya categories), microlocal analysis and repre-sentation theory(drived and stable module categories). In this thesis, we studythe Calabi-Yau objects in a triangulated category, the torsion pairs in a triangu-lated category and the Gorenstein homological dimensions of the torsion pairs ina triangulated category.In Chapter2, we study the d th CY objects in a Hom-finite triangulatedk-category. Firstly, we obtain some useful properties on d th CY objects usingthe definitions of d orthogonal subcategory and rigid objects. Secondly, we givethe conditions under which the d th CY objects in a triangulated k-category areprojectively(injectively) Cohen-Macaulay. Finally, we prove that the left orthogo-nal subcategory and right orthogonal subcategory of the triangulated k-categoryare extension closed if this triangulated k-category is Gorenstein with kernels,cokernels and finite global dimension.In Chapter3, we study the properties of torsion pairs in a triangulated cate-gory according to torsion theory in abelian categories. By introducing the notions of d Ext projectivity and d Ext injectivity and giving the definitions ofd cluster tilting torsion pair, rigid torsion pair and maximal rigid torsion pair,we prove under which necessary and sufcient conditions these torsion pairs holdtrue.In Chapter4, we introduce D mutation pairs in a triangulated category, afterendowing subcategory D with some properties, we construct a quotient category.Further more, we study the structure and properties of these torsion pairs in thisquotient category.Chapter5, we study the Gorenstein homological dimensions of torsion pairsin a triangulated category. By using relative homological dimensions, resolutionand coresolution dimensions, we study the relationship of Gorenstein projectivedimension between triangulated category C and its subcategories A and B, where(A, B) is a torsion pair in C.