Triangulated Categories and Derived Categories
This course provides an introduction to triangulated categories and derived categories, which are fundamental concepts in modern algebra, algebraic geometry, and representation theory. We will introduce the basic structures and techniques of triangulated categories, including exact triangles, triangulated functors, and derived functors. The course will further discuss the construction and applications of derived categories, with emphasis on their roles in homological algebra and representation theory. The course aims to introduce students to the basic language and techniques of triangulated and derived categories, and to illustrate their importance in modern algebra and representation theory.
讲师
日期
2026年09月07日 至 12月08日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周二,周四 | 15:20 - 16:55 | A3-2-201 | Tencent A | 482 969 7386 | 106457 |
修课要求
Students are expected to have a basic background in abstract algebra, including groups, rings, modules, and linear algebra. A basic knowledge of homological algebra (such as chain complexes, homology, and exact sequences) is helpful but not required; necessary background will be introduced during the course.
课程大纲
1. Basic notions from category theory and homological algebra
1.1 Categories, functors, and natural transformations; additive categories and abelian categories.
1.2 Chain complexes and cochain complexes; homotopy of complexes; homology and exact sequences.
1.3 Basic concepts and constructions in homological algebra.
2. Triangulated categories
2.1 Definition and examples of triangulated categories.
2.2 Distinguished triangles and their basic properties.
2.3 Morphisms of triangles and related constructions.
3. Derived categories
3.1 Construction of derived categories.
3.2 Localization of categories and derived functors.
3.3 Derived equivalences and their applications.
4. Applications and examples
4.1 Derived categories of modules and algebras.
4.2 Applications to representation theory.
4.3 Connections with other areas of modern mathematics.
1.1 Categories, functors, and natural transformations; additive categories and abelian categories.
1.2 Chain complexes and cochain complexes; homotopy of complexes; homology and exact sequences.
1.3 Basic concepts and constructions in homological algebra.
2. Triangulated categories
2.1 Definition and examples of triangulated categories.
2.2 Distinguished triangles and their basic properties.
2.3 Morphisms of triangles and related constructions.
3. Derived categories
3.1 Construction of derived categories.
3.2 Localization of categories and derived functors.
3.3 Derived equivalences and their applications.
4. Applications and examples
4.1 Derived categories of modules and algebras.
4.2 Applications to representation theory.
4.3 Connections with other areas of modern mathematics.
参考资料
Pu Zhang, Triangulated Categories and Derived Categories (章璞，《三角范畴与导出范畴》).
2. Amnon Neeman, Triangulated Categories, Annals of Mathematics Studies, Princeton University Press, 2001.
3. Bernhard Keller, On Differential Graded Categories, International Congress of Mathematicians, 2006.
2. Amnon Neeman, Triangulated Categories, Annals of Mathematics Studies, Princeton University Press, 2001.
3. Bernhard Keller, On Differential Graded Categories, International Congress of Mathematicians, 2006.
听众
Advanced Undergraduate
, Graduate
视频公开
公开
笔记公开
公开
语言
中文