Beijing Institute of Mathematical Sciences and Applications

7th September ~ 8th December, 2026

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.

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.

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.

Advanced Undergraduate , Graduate

Yes

Yes

Chinese