Basic Representation Theory of Algebras (Graduate Texts in Mathematics (283), Band 283)

Preface
Contents
I Modules, algebras and quivers
	I.1 Modules over finite dimensional algebras
		I.1.1 Algebras and modules
		I.1.2 The radical and indecomposability
		I.1.3 Idempotents, projectives and injectives
		I.1.4 The Grothendieck group and composition series
	Exercises for Section I.1
	I.2 Quivers and algebras
		I.2.1 Path algebras and their quotients
		I.2.2 Quiver of a finite dimensional algebra
		I.2.3 Projective, injective and simple modules
		I.2.4 Nakayama algebras
		I.2.5 Hereditary algebras
		I.2.6 The Kronecker algebra
	Exercises for Section I.2
II The radical and almost split sequences
	II.1 The radical of a module category
		II.1.1 Categorical framework
		II.1.2 Defining the radical of `3́9`42`\"̇613A``45`47`\"603AmodA
		II.1.3 Characterisations of the radical
	Exercises for Section II.1
	II.2 Irreducible morphisms and almost split morphisms
		II.2.1 Irreducible morphisms
		II.2.2 Almost split and minimal morphisms
		II.2.3 Almost split sequences
	Exercises for Section II.2
	II.3 The existence of almost split sequences
		II.3.1 The functor category `3́9`42`\"̇613A``45`47`\"603AFunA
		II.3.2 Simple objects in `3́9`42`\"̇613A``45`47`\"603AFunA
		II.3.3 Projective resolutions of simple functors
	Exercises for Section II.3
	II.4 Factorising radical morphisms
		II.4.1 Higher powers of the radical
		II.4.2 Factorising radical morphisms
		II.4.3 Paths
	Exercises for Section II.4
III Constructing almost split sequences
	III.1 The Auslander–Reiten translations
		III.1.1 The stable categories
		III.1.2 Morphisms between projectives and injectives
		III.1.3 The Auslander–Reiten translations
		III.1.4 Properties of the Auslander–Reiten translations
	Exercises for Section III.1
	III.2 The Auslander–Reiten formulae
		III.2.1 Preparatory lemmata
		III.2.2 Proof of the formulae
		III.2.3 Application to almost split sequences
		III.2.4 Starting to compute almost split sequences
	Exercises for Section III.2
	III.3 Examples of constructions of almost split sequences
		III.3.1 The general case
		III.3.2 Projective–injective middle term
		III.3.3 Almost split sequences for Nakayama algebras
		III.3.4 Examples of almost split sequences over bound quiver algebras
	Exercises for Section III.3
	III.4 Almost split sequences over quotient algebras
		III.4.1 The change of rings functors
		III.4.2 The embedding of mod B inside mod A
		III.4.3 Split-by-nilpotent extensions
	Exercises for Section III.4
IV The Auslander–Reiten quiver of an algebra
	IV.1 The Auslander–Reiten quiver
		IV.1.1 The space of irreducible morphisms
		IV.1.2 Defining the Auslander–Reiten quiver
		IV.1.3 Examples and construction procedures
		IV.1.4 The combinatorial structure of the Auslander–Reiten quiver
		IV.1.5 The use of Auslander–Reiten quivers
	Exercises for Section IV.1
	IV.2 Postprojective and preinjective components
		IV.2.1 Definitions and characterisations
		IV.2.2 Postprojective and preinjective components for path algebras
		IV.2.3 Indecomposables determined by their composition factors
	Exercises for Section IV.2
	IV.3 The depth of a morphism
		IV.3.1 The depth
		IV.3.2 The depth of a sectional path
		IV.3.3 Composition of two irreducible morphisms
	Exercises for Section IV.3
	IV.4 Modules over the Kronecker algebra
		IV.4.1 Representing Kronecker modules
		IV.4.2 Modules over the Kronecker algebra
		IV.4.3 The Auslander–Reiten quiver of the Kronecker algebra
	Exercises for Section IV.4
V Endomorphism algebras
	V.1 Projectivisation
		V.1.1 The evaluation functor
		V.1.2 Projectivising projectives
	Exercises for Section V.1
	V.2 Tilting theory
		V.2.1 Tilting modules
		V.2.2 A torsion pair in `3́9`42`\"̇613A``45`47`\"603AmodA
		V.2.3 The main theorems
		V.2.4 Consequences of the main results
	Exercises for Section V.2
VI Representation-finite algebras
	VI.1 The Auslander–Reiten quiver and the radical
		VI.1.1 The Harada–Sai lemma
		VI.1.2 The infinite radical and representation-finiteness
		VI.1.3 Auslander\'s theorem
	Exercises for Section VI.1
	VI.2 Representation-finiteness using depths
		VI.2.1 A characterisation using depths
		VI.2.2 The nilpotency index
	Exercises for Section VI.2
	VI.3 The Auslander algebra of a representation-finite algebra
		VI.3.1 The Auslander algebra
		VI.3.2 Characterisation of the Auslander algebra
		VI.3.3 The representation dimension
	Exercises for Section VI.3
	VI.4 The Four Terms in the Middle theorem
		VI.4.1 Preparatory lemmata
		VI.4.2 The theorem
	Exercises for Section VI.4