Syllabus
The course will give an introduction to triangulated categories, before turning to introducing local cohomology, firstly in the classical algebraic setting, and then in the triangulated realm and explaining how the latter recovers and generalises the former. We will then turn to exploring local duality in thetriangulated setting, which naturally leads us to consider other duality theorems such as Greenlees-May duality, and Warwick duality. We will show how one can recover the classical statement of Grothendieck local duality from this more general triangulated version.
• Introduction to duality (1 lecture):
- philosophy behind duality statements;
- some familiar examples;
- statement of Grothendieck local duality.
• Triangulated categories (6 lectures):
- definition and some examples (derived categories and stable module categories);
- basic properties of triangulated categories;
- tensor-triangulated categories and rigid objects;
- statement of Brown representability and its consequences.
• Local cohomology in algebra (2 lectures):
- recollections on derived functors;
- the classical algebraic definition of local cohomology;
- calculating some examples and proving some key properties.
• Local cohomology in triangulated categories (4 lectures):
- local cohomology as a colocalization;
- generalization from the algebraic setting to more general triangulated categories.
• Duality theorems in triangulated categories (3 lectures):
- triangulated duality theorems (Greenlees-May duality and Warwick duality);
- deduction of Grothendieck’s local duality theorem from Greenlees-May duality.
Annotation
The goal of this lecture course is to give a modern point of view on some important duality theorems in algebra, from the point of view of triangulated categories. This perspective also enables one to view these dualities not just in an algebraic setting, but to transport them into other realms, such as geometry and topology.
The main focus will be on Grothendieck’s local duality theorem, which relates the Matlis dual of local cohomology to the ordinary functional dual.