Time and place: Tuesdays 16.00h-18.00h in CH12.0.17 (in Chemistry) and Fridays 9.00h-10.00h in M101
Teacher: Oriol Raventós
Office: bio 2.2.07 (Building Bio 2, 2nd floor)
Telephone: +94 (0)941 943 5812
E-mail: oriol.raventos-morera(at)ur.de
Office hours: By appointment
Credits: 4.5 ECTS
Link to the official webpage of the course "vorlesung"
Link to the official webpage of the course "übungen"
Course description:
The aim of this course is to expose the theory of triangulated categories as a privileged setting for the study of the interactions between topology and algebra. The core of the course will present the abstract theory as exposed in Neeman's monograph. Before that, we will describe two examples: Derived categories of abelian categories and spectra. In the final part of the course we will see how the abstract setting enables us to transport results between algebra and topology.
Course structure:
Chapter 1: Preliminaries and examples
1.1. Basic theory of additive and abelian categories
1.2. The derived category of an abelian category
1.3. Spectra
Chapter 2: Triangulated categories
2.1. Definition and basic properties
2.2. Examples
2.3. Small objects and generation
2.4. Triangulated subcategories and Verdier quotients
2.5. Thomason localization theorem
2.6. Brown representability
Chapter 3: Interactions between topology and algebra.
Depending on remaining time and interest of the students we could discuss:
- Classification of thick subcategories
- Derived Morita theory
- Presentability of triangulated categories
- Adams representability
- Grothendieck's 6 operations formalism
Main bliography:
A. Neeman, Triangulated Categories. This will be the main source for Chapter 2 of this course.
C. A. Weibel, An introduction to homological algebra. We will follow this book for the introduction to derived categories.
H. R. Margolis, Spectra and Steenrod algebra. We will follow this book to introduce spectra.
Useful notes:
D. Murfet, Triangulated Categories I, II and III. Nicely written notes available at the web of the author. He also have notes about other topics of interest to this course.
Extended bibliography:
J. L. Verdier, Catégories derivées (état 0) in SGA 4 1/2. The original published source for triangulated categories.
J. L. Verdier, Des catégories derivées des catégories abélinnes. The thesis of the author published afterwards in Astérisque.
S. I. Gelfand and Y. Manin, Methods of homological algebra. In contains an introduction to triangulated categories.
R. Hartshorne, Residues and duality. An exposition of the application of triangulated categories to algebraic geometry that motivated the definition of triangulated categories.
Y. B. Rudyak, On Thom Spectra, orientability and cobordism. Contains an introduction to spectra.
J. F. Adams, Stable homotopy theory and generalised homology. The original source for the theory of spectra.
J. Lurie, Higher Algebra. A treatment of the topics of this course using the formalism of infinity categories.
P. Freyd, Abelian categories.
S. MacLane, Cateogries for the working mathematician.
F. Borceaux, Handbook of categorical algebra I, II and III.
J. Adámek and J. Rosický, Locally presentable categories.
B. Keller, Introduction to abelian and derived categories.
J. Lurie, A Theorem of Gabriel-Kuhn-Popesco (Lecture 8).
M. Kashiwara and P. Schapira, Sheaves on Manifolds. Very nice introduction to derived categories and derived functors.
Summary of the lectures:
Lecture 1 (10th of October): We have given a brief description of the course. Including a quick definition of triangulated categories, derived categories and spectra. We have pointed out how form Brown representability for abstract triangulated categories we will infer the classical Brown representability for (co)homology theories and the existence of certain adjoints used to develop Grothendieck duality. We will elaborate on all that during the course.
Lecture 2 (14th of October): We review some general facts about category theory: Isomorphism, monomorphism, epimorphism, set of generator, split idempotents, subojects and quotient objects, limits and colimits, zero object, kernel and cokernel. We state two equivalent definitions of adjoint functors, four equivalent definitions of category equivalences and Freyd's adjoint functor theorem (the general and the special one). In the last part of the class we introduce pre-additive and additive categories. We follow Freyd's book and used some things from Borceaux's too.
Exercise 2 (17th of October): Proofs of Freyd's Adjoint Functor Theorems.
Lecture 3 (21st of October): We study some examples of additive categories, like the category of additive functors, and define abelian categories. In the second part we see applications of the adjoint functor theorems.
Exercise 3 (24th of Obtober): Exercises about additive categories and (co)generators of categories.
Lecture 4 (28th of October): We state the adjoint functor theorems for locally presentable categories following the book of Adámek and Rosický. We list the main properties of abelian categories.
Exercise 4 (31st of October): Exercises about abelian categories.
Lecture 5 (4th of November): We finished stating some basic properties of abelian categories, exact functors and derived functors. Localizations and Gabriel-Zisman factorization systems. Also reflective subcategories, and orthogonality (local objects, local equivalences) and Serre subcategories.
Exercise 5 (7th of November): Exercises about localizations.
Lecture 6 (11th of November): Gabriel-Popescu Theorem, Morita equivalences and Mitchell embedding theorem (following Weibel's book and some notes by Lurie and Keller).
Exercise 6 (14th of November): Exercises related to the proofs of the theorems in the previous class and Serre subcategories.
Lecture 7 (18th of November): Derived categories. Examples: Derived categories of rings and derived categories of Sheaves (following Weibel's book).
Exercise 7 (21th of November): Exercises related to the proofs of the theorems in the previous class (construction of the derived category, cones and cylinders).
Lecture 8 (25th of November): Derived functors. Examples: Derived Hom, derived tensor product, sheaf cohomology and Grothendieck six operations.
Exercise 8 (28th of November): Exercises related to the proofs of the theorems in the previous class (existence of derived functors).
Exercise 9 (5th of December): Finish the proof of the existence of derived functors and start with triangulated categories.
Lecture 9 (9th of December): Definition of triangulated categories, main properties. We proved that a triangulated category is abelian if and only if it is semisimple and that the derived category of an abelian semisimple category is formal.
Exercise 10 (12th December): Prove some basic properties of triangulated categories.
Lecture 10 (16th December): More properties of triangulated categories and basic constructions. Prove that the homotopy category of an abelian category is triangulated.
Exercise 11 (19 December): Exercises on basic properties of triangulated categories.
Exercise 12 (9th January): Neeman's octahedral axiom. Homotopy colimits.
Lecture 11 (13th January): Triangulated subcategories and Verdier quotients. Prove that the derived category of a Grothendieck category is triangulated. Bousfiled localizations.
Exercise 13 (16th January): Exercises about localizations. Recollements and t-structures. We see how an open immersion of schemes gives rise to a recollement between the derived categories (using Grothendieck six operations) and explain how Beilinson, Bersntein and Deligne use this to create new (non-canonical) t-structure on the starting derived category (whose heart is the abelian category of perverse sheaves).
Lecture 12 (20th January): Algebraic triangulated triangulated categories. Small and compact objects. Well generated triangulated categories. Characterization of algebraic well generated triangulated categories after Keller and Porta. Thomason localization theorem and Brown representability for well generated triangulated categories. Discuss applications to K_0 and existence of Bousfield localizations, respectively.
Exercise 14 (23rd January): Exercises about well generated triangulated categories.
Lecture 13 (27th January): Spectra. Topological triangulated categories. Characterization of topological well generated triangulated categories after Schwede-Schipley and Heider. Relation to algebraic ones. Adams representability of homology theories. Canonical t-structure.
Exercise 15 (30th January): Exercises about spectra.