Representation Theory of Artin AlgebrasCambridge University Press, 1997年8月21日 - 440 頁 This 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 |
內容
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 | |
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常見字詞
A-module algebras of finite AR-quiver arrow Cartan matrix commutative diagram component composition factors Corollary define denote duality Dynkin diagram equivalence of categories exact sequence factors through f finite dimensional finite representation type following are equivalent gl.dim Grothendieck group group algebra Hence hereditary algebras hereditary artin algebra HomɅ(X idempotents indecomposable modules indecomposable projective modules induced injective envelope injective module irreducible morphisms isomorphism k-basis kG-module Ko(mod left almost split Lemma Let f Math minimal projective presentation minimal right mod A/C monomorphism morphism f Nakayama algebra nonisomorphic nonprojective nonzero P₁ P₂ phism preinjective preprojective projective cover Proof Let Proposition prove quadratic form R-algebra R-functor result right almost split right minimal semisimple short cycle simple modules socle split epimorphism split monomorphism split morphism split sequence stable equivalence subadditive function submodule summand Suppose Theorem translation quiver vertex zero