A.H.: What are tilting
modules? -
A very brief introduction to tilting theory.
The main topic will be the Brenner-Butler theorem: If
B = End(_{A}T) is the
endomorphism ring of an A-tilting module
_{A}T, then "certain
subcategories" of the module categories of A and
B are equivalent.
C.M.R.:
If R is a local artinian ring, then
there are no non-trivial tilting complexes. (A tilting complex is trivial if it is
homotopy equivalent to a complex C^{•}
with all but at most one C^{i} zero, and the
only non-zero C^{i} being projective.)
(A stronger assertion:
If T is a bounded complex of free modules of finite rank
and any map f : T → T[n] with n ≠ 0 is homotopic to zero,
then T is homotopy equivalent to a trivial complex.)
C.M.R.: Some tilting complexes for Artin algebras (given by a partition of the vertex set into two disjoint sets
I and I', with the requirement that for x ∈ I any simple
submodule of P(x) is of the form S(i) with i ∈ I)
(Here is the construction: Let P(I) be the projective cover of
the direct sum of all S(i) with i ∈ I, and similarly, P(I') ...
Let
f : M → P(I') be a right (add P(I))-approximation of P(I') such
that M has a direct summand M' isomorphic to P(I) with M' contained
in the kernel of f. Then ... 0 → M → P(I') → 0 ...
is a tilting complex.)
R.F.:
The Burnside theorem: If G is a finite group,
and M is a faithful representation, then any irreducible
representation occurs as a subfactor of some tensor product
of copies of M.
A.H.: Modules with standard filtration
I: Let A be an Artin algebra, mod A the
category of finitely generated A-modules, and
Θ={Θ(1),...,Θ(n)} a sequence of A-modules
with
Ext^{1}_{A}(Θ(j),Θ(i))=0 for all
j≥i. Denote by F(Θ) the full subcategory of
mod A of modules filtered by modules from Θ. Then
F(Θ) is functorially finite in mod A.
Covariantly finite, but not contravariantly finite
full subcategories of a module category
Right F(Δ)-approximations
C.M.R.: The simplicial complex of cluster tilting objects,
corresponding to a hereditary artin algebra A.
We provide a description
in terms of pairs (T,U), where
U is a Serre subcategory of mod(A) and T is a partial
tilting A-module which belongs to U. See
Handbook, Part 3.
C.M.R.: The Auslander problem: Let R be left and right
noetherian
ring. Consider the class W(R) of all (left) R-modules of the form
Ext^{1}(N_{R},R_{R}) where N_{R} is
a finitely generated R-module. Is W(R) closed under submodules?
Huang gave a counter example (A_{3} with 2 sources).
We will show: For a hereditary artin algebra, W(R) is closed under
submodules if and only if R is a Nakayama algebra.
It will be shown that for hereditary artin algebra,
the problem concerns tilting theory: just consider an
injective cogenerator as tilting module.
See
Handbook, Part 2.
C.M.R.: Examples of simplicial complexes of tilting modules
L.H.: The volume of a tilting module This notion allows to show easily that an almost complete partial
tilting module has at most two complements, and at most one in case
the module is not sincere.
A.H.: Torsion theories and tilting
modules I: Let
(T, F) be a torsion theory on mod A
such that D(A_{A})∈T and either
T or F contain only finitely many isomorphism
classes of indecomposable A-modules. Let T be the
sum of all Ext-projectives in T. Then T is a tilting
module such that
(T(T), F(T))=(T, F).
A.H.: The Auslander-Reiten formula
(This talk was based on Chapter IV.4 of ARS, but there is a more
comprehensive reference which also includes the functoriality behind
the Auslander-Reiten formula by H. Krause:
A short proof
for Auslander's defect formula.)
February 8, 2006.
C.M.R.: Pairs (S,T) of orthogonal exceptional modules over tame hereditary algebras with dim Ext^{1}(T,S)=2.
A.H.: Correspondence of dimension vectors of subspace representations with tuples of compositions of a number, and properties of the Tits form for dimension vectors of stars
C.M.R.: Schofield induction.If A is a hereditary
artin algebra with s simple modules, and M is an exceptional sincere
module, then there are precisely s-1 Schofield sequences.
June 14, 2006.
C.M.R.:
Gabriel-Roiter inclusions and irreducible monomorphisms.
If there is a Gabriel-Roiter inclusion X → Y,
there is an irreducible monomorphism X → M and an epimorphism
M → Y such that the composition X → M → Y is a
monomorphism.
L.H.: Slopes for representations of Dynkin quivers (There exists a slope with the property: a module over a path algebra of a quiver is stable precisely if it is indecomposable.)