## 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.

**Problem sheets:**
Problem sheet 1
Problem sheet 2

**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.