GTM283-Basic Representation Theory of Algebras2020

"This textbook introduces the representation theory of algebras by focusing on two of its most important aspects: the Auslander-Reiten theory and the study of the radical of a module category. It starts by introducing and describing several characterisations of the radical of a module category, then presents the central concepts of irreducible morphisms and almost split sequences, before providing the definition of the Auslander-Reiten quiver, which encodes much of the information on the module category. It then turns to the study of endomorphism algebras, leading on one hand to the definition of the Auslander algebra and on the other to tilting theory. The book ends with selected properties of representation-finite algebras, which are now the best understood class of algebras. Intended for graduate students in representation theory, this book is also of interest to any mathematician wanting to learn the fundamentals of this rapidly growing field. A graduate course in non-commutative or homological algebra, which is standard in most universities, is a prerequisite for readers of this book"--Provided by publisher 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