Working Seminar: Tilting theory.

SEMESTER 2, 2009/10.

Organiser: Robert Marsh
Office: 10.15 (Maths Satellite), Tel. 0113 343 5164.
Email: marsh@maths.leeds.ac.uk.
Usual Time: Fridays 3-4pm, MALL 2 if available.

List of talks

• 0. Morita equivalences, Alex Collins, Friday 5th March 2010.
  
• 1. Introduction to torsion pairs, Alex Collins, Friday 12 March 2010.
  Definition; example; equivalent conditions for being a torsion class; short exact sequence for any object, with kernel torsion and cokernel torsionfree. [ASS] VI.1.1-1.5.
• 2. Properties of torsion pairs, Raquel Simoes, Friday 19 March 2010.
  Every simple module is torsion or torsion free; Splitting torsion pairs and equivalent conditions Modules Gen T generated by a module T Criterion for being in Gen(T) Criterion for Gen T to be a torsion class [ASS] VI.1.6-1.9.
• 3. Tilting modules, Julia Sauter, Friday 26 March.
  Ext-projective and Ext-injective modules; criteria for being Ext-projective or Ext-injective in a torsion class or torsion-free class; partial tilting modules and tilting modules; faithful modules. Equivalent conditions for a partial tilting module to be a tilting module; existence of add(T)-resolutions by add(T); the torsion pair of a tilting module; criterion to be in T(T) when T is tilting. Description of APR-tilting module, verification that it is tilting; description of the corresponding torsion pair; examples; reflection functors: definition and some simple properties; statement that for a path algebra, the APR-tilting module corresponding to simple projective and the corresponding reflection functor coincide. [ASS] VI.1.10-2.1, 2.5-2.8. [K], Section 3.3.
• 4. Properties of partial tilting modules, Andrew Reeves, Friday 30 April.
  Equivalent conditions for a faithful module; Ext-orthogonal subcategory T(T); relationship between T(T) and Gen(T); Bongartz's Lemma. [ASS] VI.2.2-2.4.
• 7. Brenner-Butler Tilting theorem I, Robert Marsh, Friday 7 May.
  Properties of Hom(T,-), T tilting; functorial isomorphisms for objects in T(T); properties of a tilting module over its endomorphism algebra; centre of the endomorphism algebra of a tilting module. [ASS] VI.3.1-3.4.
• 8. Brenner-Butler Tilting theorem II, Raquel Simoes, Friday 14 May.
  Connectedness of the endomorphism algebra of a tilting module; Torsion pair in the module category of the endomorphism algebra of a tilting module; Brenner-Butler tilting theorem. [ASS] VI.3.5-3.8.
• 9. Grothendieck groups and tilting, Yann Palu, Friday 4 June, 3pm, Miall Lecture Theatre, Baines Wing.
  Dimension vectors of Hom(T,M) and Ext(T,N) for M torsion and N torsionfree; isomorphism of Grothendieck groups; criterion for a partial tilting module to be a tilting module; Euler characteristic of an algebra of finite global dimension. [ASS] VI.3.10, VI.4.3.

References:

• I. Assem, D. Simson and A. Skowronski. Elements of the representation theory of associative algebras. Vol. 1. Techniques of representation theory. London Mathematical Society Student Texts, 65. Cambridge University Press, Cambridge, 2006.
• H. Krause, Representations of quivers via reflection functors. Lecture notes.
• M. Auslander, I. Reiten, S. O. Smal\o. Representation theory of Artin algebras. Corrected reprint of the 1995 original. Cambridge Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge, 1997.