导出范畴与同调猜测

Finitistic dimension conjecture and related homological conjectures are important in the study of homological algebra, algebra representation theory and commutative algebra etc., while tilting theory, derived equivalences, cluster categories and cluster tilting theory are also hot in these fields. In the previous work, the applicant extended results of Igusa,Todorov, Xi,Huard and others on the finitistic dimension conjecture by studying Igusa-Todorov algebras, provided the interesting methods to judge algebras satisfying the finitistic dimension conjecture and proved that Auslander-Reiten conjecture and some other conjectures are stable under tilting equivalences and derived equivalences. The project will further study derived categories and derived equivalences, develop new methods to study the above homological conjectures, provide new classes of algebras satisfying these homological conjectures and new methods to judge them, and extend the important results in tilting theory to the context of derived categeries and other triangulated categories.

有限维数猜测以及相关的其他同调猜测是同调代数、代数表示论以及交换代数等领域研究的重要内容，倾斜理论、导出等价、从范畴和丛倾斜理论也是这些领域中的研究热点。在之前的工作中，申请者通过研究Igusa-Todorov 代数推广了Igusa 和Todorov,惠昌常及Huard等人关于有限维数猜测的工作，给出了满足有限维数猜测的代数的有意义的判定条件，证明了Auslander-Reiten 猜测等在倾斜等价及导出等价下的稳定性。本项目将进一步研究导出范畴和导出等价，发展新的相关同调猜测的研究方法，给出新的的相关同调猜测成立的代数类和判定条件，并将把经典倾斜理论中的一些重要结果发展到导出范畴等三角范畴上。

导出范畴是当今同调代数的重点研究领域，同调猜测是推动同调代数发展的强大动力，倾斜理论是研究导出范畴和同调猜测的有力工具。本项目主要研究内容即为这三个主题。项目把一些经典的倾斜理论的结果推广到了导出范畴上，并由此给出了部分同调猜测的导出范畴层次上的刻画，得到了这些同调猜测的重要性质，如导出不变性等。项目还给出了部分同调猜测成立的判断条件，并给出了适用这些条件的例子。