导出范畴源自代数几何，在表示论、数学物理等多个数学分支有着广泛、重要的应用。阿贝尔范畴的导出等价是代数学中的重要研究课题。本项目旨在研究表示论中的标准等价猜想：模范畴间的导出等价函子自然同构于一个标准等价函子。利用该猜想的两个等价形式，问题可归结为导出范畴中相应函子的延拓和dg 提升问题。本项目计划对这两个问题进行系统研究，并应用到代数导出等价和相关导出不变量的研究中，包括：对gentle代数、拟遗传代数等代数类验证标准等价猜想；研究三角范畴中心与其dg 增强的Hochschild 上同调之间的联系；研究dg 范畴粘合诱导的Hochschild 上同调群的长正合列；研究三角范畴粘合诱导的分次中心间的联系。标准等价猜想有明确的几何背景，本项目从不同角度研究标准等价猜想，对于丰富和完善导出Morita理论有重要的意义，同时也促进了dg 范畴理论在代数表示论中的实质性应用。

Derived category originated in algebraic geometry, and has since found wide and important applications in many research fields, for example in the representation theory and mathematical physics. The study of derived equivalence of abelian categories is one of the most important topics in algebraic geometry as well as in the representation theory of algebras. This project aims to study the well-known conjecture on standard equivalences: any triangle equivalence between bounded derived categories of finite dimensional algebras is naturally isomorphic to a standard equivalence, that is, the derived tensor product by a two-sided tilting complex. Using alternative forms of this conjecture, we may reduce the conjecture to following two main problems, say the extension problem of certain functors and the dg lift problem of certain triangle functors. We aim to study these problems, and apply them to the study of derived equivalences and related derived invariants. In more detail, we will prove the conjecture for gentle algebras and quasi-hereditary algebras; study the connection between the graded center of a triangulated category and the Hochschild cohomology of it dg-enhancement; explore the long exact sequence of Hochschild cohomology groups induced by a gluing of dg categories; study the relation among the graded centers of the triangulated categories in a recollement. In view of the geometric meaning of the conjecture and the importance of derived equivalences, this project will provide us a better understanding for the derived Morita theory, and in the same time promote applications of dg categories into the representation theory of algebras.