【作者】 周宇；

【导师】 肖杰；

【作者基本信息】 清华大学 ， 数学， 2012， 博士

【摘要】 本论文主要研究三角Calabi-Yau范畴中的一般余扭对和其两类特殊情形–高维丛倾斜对象和极大刚性对象，并研究其中的丛结构。我们在一般的三角范畴中定义了余扭对的突变，证明余扭对的突变仍是一个余扭对，并给出了刚性子范畴的子商范畴中的余扭对与核心中包含该刚性子范畴的余扭对的一一对应。这个对应诱导了具有相同核心的余扭对和该核心的子商范畴中的t-结构有一一对应关系。在任意带有丛倾斜对象的三角2-Calabi-Yau范畴中，我们利用范畴的分解由其中丛倾斜对象的分解所决定的事实，证明了非平凡的t-结构的不存在性。在此基础上，我们给出了这类范畴中余扭对的完全分类。作为分类定理的一个应用，我们利用黎曼曲面建立了余扭对及其突变的一个几何模型。作为余扭对的两类特殊情形，高维丛倾斜对象和极大刚性对象都有着丰富的结构。高维丛范畴中的高维丛倾斜对象和高维极大刚性对象的等价性在本论文中得到了证明。我们证明了(d+1)丛范畴中几乎完备丛倾斜对象的补的个数正好是d+1个，并讨论了d+1个不可分解对象的集合是某个几乎完备(d+1)丛倾斜对象的补的集合的充要条件。我们也证明了连接这些补的三角的中间项之间没有公共直和项，且不同的补几乎占有不同的分次。由此我们刻画了高维丛范畴的丛复形的基本性质。在三角2-Calabi-Yau范畴中，我们证明了任意极大刚性对象和它自己平移的扩张范畴中包含了所有刚性对象，如果其中含有丛倾斜对象，则任意极大刚性对象都是丛倾斜的。对比于丛倾斜对象的相关结果，我们证明了同一个范畴中，极大刚性对象的互不同构的不可分解直和项的个数是相等的，极大刚性对象的自同态代数是Gorenstein代数且Gorenstein维数小于等于1。在丛管子中，只存在极大刚性对象而没有丛倾斜对象。我们在这个范畴中构造了类似于带丛倾斜对象的范畴中的丛映射，并证明这个丛映射满足极大刚性对象突变的变换关系。从而这个丛映射给出了B型和C型丛代数的一个范畴化。更多还原

【Abstract】 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.更多还原