Representation Theory of Artin Algebras

封面
Cambridge University Press, 1997年8月21日 - 425 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
 

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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
Bibliography
413
Relevant conference proceedings
421
Index
423
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