Torsion structures in triangulated categories

我们主要研究三角范畴中的扭偶。扭偶是表示论的基本框架。它们使我们能够将给定的三角范畴分解为子范畴。我们的目的是研究现有的“好的”扭转副的机理并加以应用。我们得到了以下结果：(1)发现了一个具有高度对称性的“回忆三角形”，并作为应用给出了一个新的三角形等价。(2)在适当的扩张环上，引入n-复形的同伦范畴和派生范畴以及它们与正规复形的同伦范畴和派生范畴的三角等价关系。(3)商范畴中弱扭对与扭对的一一对应。

We focused on torsion pairs in triangulated categories. Torsion pairs are basic framework in representation theory. They enable us to decompose a given triangulated category into subcategories. We aimed at studying mechanism of existing "good" torsion pairs and applying them. We have obtained the following results: (1) discovery of a "triangle of recollements" which induces high symmetry and a new triangle equivalence as an application. (2) introducing homotopy and derived categories of "n-complexes" and triangle equivalence of these categories to homotopy and derived categories of normal complexes over a suitable extension ring. (3) one to one correspondence of weak torsion pairs to torsion pairs in a quotient category.