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Triangulated Categories

Winter semester 2017/18

Dates:
Lecture Monday 9.45-11.15 in 7.527
Lecture / problem class Thursday 14.00-15.30 in 7.527
First lecture Monday 16th of October.

Contents:

Chapter one: Brief introduction.

Monday, October 16.


Chapter two: Stable module categories

Monday, October 16: Stable categories. Stable equivalences, Examples.

Thursday, October 19: More examples. Syzygies. Auslander-Reiten conjecture. Nodes.

Monday, October 23: More on nodes. Martinez-Villa's results on invariants of stable equivalences. Comparing exact sequences. Yoneda's lemma and functor categories.

Thursday, October 26: Almost split sequences and projective resolutions of simple functors.

Monday, October 30: From exact sequences to projective resolutions and back. Why nodes cause problems. Projective dimensions of simple functors in the stable functor category, and applications.

Thursday, November 2: Exact sequences, injective functors and stable equivalences.

Monday, November 6: Martinez-Villa's technical main theorem. Extensions, global dimension and dominant dimension under stable equivalences, when there are no nodes.

Thursday, November 8: Problem class.


Chapter three: Triangulated categories: Definition and basic properties

Monday, November 6. (TR1), (TR2), (TR3), (TR4).

Monday, November 13. Basic properties. Connections between the axioms.

Thursday, November 16. Long exact sequences and applications.

Monday, November 20. Another formulation of the octahedral axiom, related to pullback and pushout.


Chapter four: Triangulated categories: Examples

Monday, November 20. Stable categories.

Thursday, November 23. (TR4) for stable categories. Homotopy categories.

Monday, November 27. Proof, continued. Frobenius categories.

Thursday, November 30. Problem class.

Monday, December 4. Algebraic triangulated categories.


Chapter five: Localisations and quotients

Monday, December 4. Quasi-isomorphisms and derived categories.

Thursday, December 7. Some derived categories as homotopy categories. Rickard's theorem on stable categories as quotients of derived categories.

Monday, December 11. More on Rickard's theorem. Quotient functors. Calculus of fractions. Compatibility with triangulated structures.

Thursday, December 14. Problem class.

Monday, December 18. Two quotients by calculus of fractions: derived categories, Verdier quotients.

Thursday, December 21. Adjoint functors.




Problem sheets:

Problem sheet 1
Problem sheet 2
Problem sheet 3



References:

Chapter two:
Auslander, Reiten and Smalø, Representation theory of Artin algebras.
Martinez-Villa, Properties that are left invariant under stable equivalence. Comm. Alg. 18 (1990), 4141-4169.
Auslander and Reiten, Stable equivalence of dualizing R-varieties, Advances Math. 12 (1974), 306-366.
Auslander and Reiten, Representation theory of Artin algebras. VI. A functorial approach to almost split sequences. Comm. Alg. 6 (1978), 257-300.

Chapter three:
Weibel, An introduction to homological algebra.
Neeman, Triangulated categories.

Chapter four:
Happel, Triangulated categories in the representation theory of finite-dimensional algebras.
Zimmermann, Representation theory. A homological algebra point of view.
Schwede, Algebraic versus topological triangulated categories. Triangulated categories, 389-407, London Math. Soc. Lecture Note Ser., 375, Cambridge Univ. Press, Cambridge, 2010.

Chapter five:
Gabriel and Zisman, Calculus of fractions and homotopy theory.
Krause, Localization theory for triangulated categories. Triangulated categories, 161-235, London Math. Soc. Lecture Note Ser., 375, Cambridge Univ. Press, Cambridge, 2010.