NMAG468: Dualities in triangulated categories

This is the webpage for the course "Dualities in triangulated categories" which I taught in the spring semester of the 2023/2024 academic year.

Lecture notes

These lecture notes will be updated as the course progresses to fix any typos or mistakes. Exercises can be found in the appendix to the lecture notes.

Lectures are Wednesdays 16:30-18:00 in K8 and Thursdays 15:40-17:10 in K9.

Some recommended exercises:

  • Section 5 - Exercises 35, 36
  • Section 6 - Exercises 39, 40, 44
  • Section 7 - Exercises 45, 46, 47
  • Section 8 - Exercises 48, 49, 50.

If you have any questions about the course, or for office hours, please send me an email to arrange a time.

Date What we covered Reference to notes
21/02 Introduction to duality and definition of triangulated categories Sections 1 and 2.1
22/02 The homotopy category of a ring is triangulated Sections 2.2 and 2.3
28/02 Homological functors, the five lemma, and triangulated functors Sections 2.4 and 2.5
29/02 Exercise class Exercises A.1-A.4
06/03 Definition of the derived category of a ring Section 2.6 up to Lemma 2.30
07/03 The derived category is triangulated Rest of Section 2.6
13/03 Calculating maps in the derived category Section 2.7
14/03 Exercise class Exercises A.5-A.7
27/03 Compact objects and Brown representability Sections 3.1 and 3.2
03/04 Homotopy colimits Section 3.3
04/04 Exercise class Exercises A.16-A.18
10/04 The derived category of a commutative ring is tensor-triangulated Section 4.1
11/04 Rigid objects and recap on some algebra Sections 4.2, 5.1 and 5.2
17/04 Local cohomology Section 5.3
18/04 Exercise class Exercises A.24, A.25, A.33, A.34
24/04 Universal property of local cohomology Section 6.1
25/04 Derived completion and localization functors Sections 6.2 and 7.1
09/05 Finite and smashing localizations Section 7.2
22/05 Local duality contexts and recovering Grothendieck local duality Sections 8.1 and 8.2
23/05 The local-to-global principle Section 8.3