2024/2025
Grothendieck Duality
Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
1, 2 module
Open to:
students of one campus
Instructors:
Alexander Pavlov
Language:
English
ECTS credits:
6
Course Syllabus
Abstract
Grothendieck duality is a deep generalization of the Serre duality in algebraic geometry, which holds for proper morphisms of schemes and complexes of quasi-coherent sheaves. The goal of this course is to give a complete modern proof of the Grothendieck duality following ideas of A. Neeman. The proof is based on the Brown’s representability theorem, which is a general theorem on compactly generated triangulated categories. The course will consists of two part. In the first part we’ll develop general theory of triangulated categories from the definition to the Brown’s representability theorem with applications to derived categories of abelian categories. This part can be considered as a second course in homological algebra based on the notion of triangulated categories. In the second part of the course we apply those techniques to the derived categories of the quasi-coherent sheaves on schemes. We’ll show that derived categories of the quasi-coherent sheaves on quasi-compact quasi-separated schemes are compactly generated, and that compact objects are the same as perfect complexes. Derive existence and general properties of the functor f^\times and its relations to the dualizing objects on schemes.
Course Contents
- Definition and basic properties of triangulated categories
- Verdier localizations. Derived categories
- Compact objects, compactly generated triangulated categories and Thomasons localizations
- Brown’s representability theorem
- 𝐾-injective and 𝐾-projective resolutions. Derived functors
- Derived functors in algebraic geometry
- Derived categories of the quasi-coherent sheaves on quasi-compact quasi-separated schemes
- Functor 𝑓× and its special cases
- Dualizing objects.
