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 34pm, 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.1.11.5.

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.1.61.9.

3. Tilting modules, Julia Sauter, Friday 26 March.
Extprojective and Extinjective modules;
criteria for being Extprojective or Extinjective in a torsion class
or torsionfree 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 APRtilting 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 APRtilting module corresponding
to simple projective and the corresponding reflection functor coincide.
[ASS] VI.1.102.1, 2.52.8. [K], Section 3.3.

4. Properties of partial tilting modules, Andrew Reeves, Friday 30 April.
Equivalent conditions for a faithful module;
Extorthogonal subcategory T(T);
relationship between T(T) and Gen(T);
Bongartz's Lemma.
[ASS] VI.2.22.4.

7. BrennerButler 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.3.13.4.

8. BrennerButler 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;
BrennerButler tilting theorem.
[ASS] VI.3.53.8.

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.