## Representation Theory of Artin AlgebrasThis book is an introduction to the contemporary representation theory of Artin algebras, by three very distinguished practitioners in the field. Beyond assuming some first-year graduate algebra and basic homological algebra, the presentation is entirely self-contained, so the book is suitable for any mathematicians (especially graduate students) wanting an introduction to this active field.'...written in a clear comprehensive style with full proofs. It can very well serve as an excellent reference as well as a textbook for graduate students.' EMS Newletter |

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### Contents

Artin rings | 1 |

2 Right and left minimal morphisms | 6 |

3 Radical of rings and modules | 8 |

4 Structure of projective modules | 12 |

5 Some homological facts | 16 |

Exercises | 23 |

Notes | 25 |

Artin algebras | 26 |

The AuslanderReitenquiver | 224 |

2 AuslanderReitenquivers and finite type | 232 |

3 Cartan matrices | 241 |

4 Translation quivers | 248 |

Exercises | 253 |

Notes | 256 |

Hereditary algebras | 257 |

1 Preprojective and preinjective modules | 258 |

2 Projectivization | 32 |

3 Duality | 37 |

4 Structure of injective modules | 39 |

5 Blocks | 43 |

Exercises | 45 |

Notes | 47 |

Examples of algebras and modules | 49 |

2 Triangular matrix rings | 70 |

3 Group algebras | 79 |

4 Skew group algebras | 83 |

Exercises | 94 |

Notes | 99 |

The transpose and the dual | 100 |

2 Nakayama algebras | 111 |

3 Selfinjective algebras | 122 |

4 Defect of exact sequences | 128 |

Exercises | 133 |

Notes | 135 |

Almost split sequences | 136 |

2 Interpretation and examples | 147 |

3 Projective or injective middle terms | 153 |

4 Group a1gebras | 158 |

5 Irreducible morphisms | 166 |

6 The middle term | 173 |

7 The radical | 178 |

Exercises | 185 |

Notes | 189 |

Finite representation type | 191 |

2 Nakayama algebras | 197 |

3 Group algebras | 200 |

4 Grothendieck groups | 206 |

5 Auslander algebras | 209 |

Exercises | 219 |

Notes | 221 |

2 The Coxeter transformation | 269 |

3 The homological quadratic form | 272 |

4 Regular components | 277 |

5 Finite representation type | 288 |

6 Quadratic forms and roots | 294 |

7 Kronecker algebras | 302 |

Exercises | 309 |

Notes | 311 |

Short chains and cycles | 313 |

2 Modules determined by composition factors | 320 |

3 Sincere modules and short cycles | 323 |

4 Modules determined by their top and socle | 326 |

Exercises | 332 |

Notes | 333 |

Stable equivalence | 335 |

2 Artin algebras with radical square zero | 344 |

3 Symmetric Nakayama algebras | 352 |

Exercises | 362 |

Notes | 364 |

Modules determining morphisms | 365 |

2 Modules determining a morphism | 370 |

3 Classification of morphisms | 379 |

4 Rigid exact sequences | 385 |

5 Indecomposable middle terms | 389 |

Exercises | 399 |

Notes | 405 |

Notation | 406 |

Conjectures | 409 |

Open problems | 411 |

413 | |

Relevant conference proceedings | 421 |

423 | |

### Other editions - View all

Representation Theory of Artin Algebras Maurice Auslander,Smalø. Sverre O. No preview available - 1995 |

### Common terms and phrases

A-module abelian group algebras of finite AR-quiver arrow Assume c-basis Cartan matrix cG-module commutative diagram component composition factors consequence Corollary corresponding decomposition define denote DTrC duality Dynkin diagram equivalence of categories exact sequence finite dimensional finite length finite representation type following are equivalent functor give given gl.dim Grothendieck group group algebra Hence hereditary algebras hereditary artin algebra HomA(/l HomA(X idempotents indecomposable injective indecomposable modules indecomposable projective modules induced injective envelope injective module integer irreducible morphisms isomorphism left almost split left artin ring Lemma minimal projective presentation minimal right mod(Aop monomorphism morphism in mod Nakayama algebra phism preinjective preprojective projective cover Proof Let Proposition prove quadratic form R-algebra R-functor radical result right almost split semisimple short cycle simple modules socle split epimorphism split monomorphism split morphism split sequence stable equivalence subadditive function submodule Suppose Theorem translation quiver uniserial modules vector space vertex zero