Handwritten course notes in Hungarian for Lecture 1
Lecture #2 (Febr. 20.) Lifting of mappings to projective resolution; projective resolution as a functor from the module category to the homotopy category. Deived functors of additive fnctors. Exactnes of the projective resolution functor ("Horseshoe lemma"). Long exact sequence of derived functors. Axiomatic characterization of derived functors.
Handwritten course notes in Hungarian for Lecture 2
Lecture #3 (Febr. 27.) 2. The functors Ext and Tor. Extension of modules and the Ext functor. The Yoneda product. The right dervied functors of the covariant and of the contravariant Hom functors coincide. Definition of the functors Extn(-,-). The Torn(-,-) functors, as the left derived functors of the tensor functor. Extension of modules as equivalence classes of short exact sequences. The action of morphisms on equivalence classes: the Ex functor. Baer sum. Abelian group structure on the set of equivalence classes of extensions. Non-split short exact sequences as obstructions to lifting (or extending) morphisms.
Handwritten course notes in Hungarian for Lecture 3
Lecture #4. (March 5.) The isomorphism of Ex and Ext1, the meaning of connecting homomorphism . Equivalence classes of long exact sequences, Yoneda product of long extensions. Eqivalence of Exn and Extn. Extension algebras, Koszul duality (only mentioned).
Handwritten course notes in Hungarian for Lecture 4
Lecture #5. (March 12.) 3. Homological dimensions. Projective and injective dimension of modules, characterization in terms of projective resolutions and the values of the Ext functors. Estimates for the projective dimensions of members of a short exact sequence. Global dimension of rings. Auslander's theorem: the global dimension is the supremum of the projective dimensions of yclic modules. For Artinian rings it is enough to compute the projective dimension of simple modules. Examples. A finite dimensional path algebra is hereditary, i.e. its global dimension is at most 1. Finitistic dimension of algebras, the conjecture (FDC) on the finiteness of the finitistic dimension of finite dimensional algebras. Validity of the FDC for some special classes of algebras.
Handwritten course notes in Hungarian for Lecture 5
Lecture #6. (March 19.) Global dimension and ring constructions. "Change of rings" type theorems. Hilbert's syzygy theorem.
Handwritten course notes in Hungarian for Lecture 6
Lecture #7 (March 26.) Auslander's theorem: if A is a finite
dimensional algebra over a field then there exists a finite dimensional algebra
B and an idempotent element e in B so that
(i) A=eBe and (ii) B is of finite global
dimension. Thus i is the endomorphism ring of a cyclic
projective B-module. The global dimension of Bis bounded
from above by the nilpotency index of the radical of A.
Handwritten course notes in Hungarian for
Lecture 7
Lecture #8 (April 9.) 4. Path alegbra and the graph
(quiver) of an elgebra. Gabriel's theorem. Reminder> basic
properties of path algebras and quotients modulo admissible
ideals. Connected path algebras, the idempotents corrseponding to
vertices are primitive idempotents. The Gabriel graph of an
algebra. Gabriel's theorem: any finite dimensional basic algebra over
an algebraically closed field is isomorphic to a path algebra modulo
some admissible ideal. Examples: the endomorphism ring of the module
in the Auslander construction from last time.
Handwritten course notes in Hungarian for
Lecture 8
Lecture #9 (April 16.) 5. Auslander–Reiten theory:
almost split sequences and irreducible morphisms. (The whole
theory is developed over finite dimensional algebras.) Left and right
minimal, left and right almost split morphisms. The uniqueness of a
left minimal left almost split sequence starting from a module. The
starting module of a left almost split sequence is directly
indecomposable. Irreducible morphisms. An irreducible morphism is
either a monomorphsim or an epimorphism but it is never an
isomorphism. The embedding of the radical of an indecomposable
projective module into the projective module is right almost split and
irerducible. Irreducible morphisms and the radical of the module
category. Charecterization of irreducible monomorphisms and
irreducible epimorphisms. The cokernel of an irreducible morphism is
directly indecomposable.
Handwritten course notes in Hungarian for
Lecture 9
Lecture #10 (April 23.) Characterization of non-split
monomorphisms startng from an indecomposable module. Irreducible
morphisms starting from a module L are precisely those which
are components of a left minimal left almost split morphism starting
from L. Definition of almost split (or Auslander–Reiten, AR)
sequences and characterizations.
Handwritten course notes in Hungarian for
Lecture 10
Lecture #11 (April 30) Construction of almost split
sequences. The transpose of a module, basic properties. The
stable module category. The Auslander–Reiten formulas (without
proof). The existence of almost split sequences; the case of
projective and of injective modules. The Auslander–graph of an
algebra.
Handwritten course notes in Hungarian for
Lecture 11
Lecture #12 (May 7.) 6. The Brauer–Thrall conjectures.
A few words about representation finite algebras. Basic properties of the Auslander–Reiten graph. Short history of the Brauerr–Thrall conjectures.
Long sequences of irreducible morphisms. The Harad–Sai lemma.
The proof of Auslander for the first Brauer–Thrall conjecture.
Handwritten course notes in Hungarian for
Lecture 12
Course notes (in preparation - containing
the first five lectures)