Representation Theory of Algebras I: Modules (preliminary version Feb 10, 2009)

283 Pages·2009·1.685 MB·English

by  Claus Michael Ringel, Jan Schröer

February 10, 2009
REPRESENTATION THEORY OF ALGEBRAS I: MODULES
CLAUS MICHAEL RINGEL AND JAN SCHRO¨ER
Abstract. This is the firstpartofa plannedbook “Introductionto Representa-
tion Theory of Algebras”. These notes are based on a 3 Semester Lecture Course
on Representation Theory held by the first author at the University of Bielefeld
fromAutumn 1993andonan(ongoing)4SemesterLecture Courseby the second
author at the University of Bonn from Autumn 2006.
Preliminary version
Contents
1. Introduction 10
1.1. About this course 10
1.2. Notation and conventions 11
1.3. Acknowledgements 11
Part 1. Modules I: J-Modules 12
2. Basic terminology 12
2.1. J-modules 12
2.2. Isomorphisms of J-modules 12
2.3. Submodules 12
2.4. Factor modules 14
2.5. The lattice of submodules 15
2.6. Examples 16
2.7. Decompositons and direct sums of modules 18
2.8. Products of modules 19
2.9. Examples: Nilpotent endomorphisms 19
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2 CLAUS MICHAEL RINGEL ANDJAN SCHRO¨ER
2.10. Exercises 22
3. Homomorphisms between modules 23
3.1. Homomorphisms 23
3.2. Definition of a ring 24
3.3. Definition of an algebra 25
3.4. Homomorphism Theorems 25
3.5. Homomorphisms between direct sums 27
3.6. Idempotents and direct decompositions 29
3.7. Split monomorphisms and split epimorphisms 30
3.8. Short exact sequences and Hom-functors 30
3.9. Exercises 35
4. Digression: Categories 37
4.1. Categories 37
4.2. Functors 38
4.3. Equivalences of categories 39
4.4. Module categories 39
4.5. Hom-functors 39
4.6. Exercises 40
5. Examples of infinite dimensional 1-modules 40
5.1. The module N( ) 40
∞
5.2. Polynomial rings 41
5.3. The module (K[T], d ) 42
dT
5.4. The module (K[T],T ) 43
·
5.5. The module (K(T),T ) 46
·
5.6. Exercises 46
6. Semisimple modules and their endomorphism rings 48
6.1. Semisimple modules 48
REPRESENTATION THEORY OF ALGEBRAS I: MODULES 3
6.2. Endomorphism rings of semisimple modules 52
6.3. Exercises 54
7. Socle and radical of a module 54
7.1. Socle of a module 54
7.2. Radical of a module 55
7.3. Large and small submodules 57
7.4. Exercises 61
Part 2. Modules of finite length 63
8. Filtrations of modules 63
8.1. Schreier’s Theorem 63
8.2. The Jordan-H¨older Theorem 66
8.3. Exercises 67
9. Digression: Local rings 67
9.1. Local rings 68
9.2. Exercises 71
10. Modules of finite length 72
10.1. Some length formulas for modules of finite length 72
10.2. The Fitting Lemma 75
10.3. The Harada-Sai Lemma 76
References 78
10.4. Exercises 78
11. Direct summands of finite direct sums 79
11.1. The Exchange Theorem 79
11.2. Consequences of the Exchange Theorem 80
References 82
11.3. Examples 82
11.4. Exercises 83
4 CLAUS MICHAEL RINGEL ANDJAN SCHRO¨ER
Part 3. Modules II: A-Modules 85
12. Modules over algebras 85
12.1. Representations of an algebra 85
12.2. Modules over an algebra 86
12.3. Modules and representations 86
12.4. A-modules and A -modules 87
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12.5. Free modules 88
12.6. The opposite algebra 90
12.7. Right A-modules 90
12.8. Examples 91
12.9. Direct decompositions of the regular representation 91
12.10. Modules defined by idempotents 93
12.11. Modules over factor algebras 96
12.12. Modules over products of algebras 96
12.13. Bimodules 97
12.14. Modules over tensor products of algebras 99
12.15. Exercises 99
13. Semisimple algebras 99
13.1. Semisimple algebras and their modules 99
13.2. Examples: Group algebras 101
13.3. Remarks 102
13.4. Exercises 103
14. The Jacobson radical of an algebra 104
14.1. The radical of an algebra 104
14.2. Exercises 107
15. Quivers and path algebras 107
15.1. Quivers and path algebras 107
REPRESENTATION THEORY OF ALGEBRAS I: MODULES 5
15.2. Examples 108
15.3. Idempotents in path algebras 109
15.4. Representations of quivers 110
15.5. Examples 111
15.6. Representations of quivers and modules over path algebras 112
15.7. Exercises 113
16. Digression: Classification problems in Linear Algebra 115
16.1. Classification of endomorphisms 115
16.2. Classification of homomorphisms 115
16.3. The Kronecker problem 116
16.4. The n-subspace problem 116
16.5. Exercises 118
Part 4. Projective modules 119
17. Projective modules 119
17.1. Definition and basic properties 119
17.2. The radical of a projective module 121
17.3. Cyclic projective modules 123
17.4. Projective covers 123
18. Injective modules 127
18.1. Definition and basic properties 127
18.2. Injective envelopes 129
19. Digression: The stable module category 132
20. Projective modules over finite-dimensional algebras 134
21. Projective modules over basic algebras 136
22. Direct summands of infinite direct sums 139
22.1. The General Exchange Theorem 139
22.2. The Krull-Remak-Schmidt-Azumaya Theorem 141
6 CLAUS MICHAEL RINGEL ANDJAN SCHRO¨ER
22.3. The Crawley-Jønsson-Warfield Theorem 143
22.4. Kaplansky’s Theorem 143
23. Projective modules over semiperfect algebras 146
24. Digression: Projective modules in other areas of mathematics 148
Part 5. Homological Algebra I: Resolutions and extension groups 149
25. Pushout and pullback 149
25.1. Pushout 149
25.2. Pullback 150
25.3. Properties of pushout and pullback 151
25.4. Induced exact sequences 153
25.5. Examples 156
25.6. Schanuel’s Lemma 156
25.7. Short exact sequences with projective middle term 157
25.8. Exercises 158
26. Homological Algebra 159
26.1. The Snake Lemma 159
26.2. Complexes 163
26.3. From complexes to modules 165
26.4. Homology of complexes 165
26.5. Homotopy of morphisms of complexes 167
26.6. The long exact homology sequence 168
27. Projective resolutions and extension groups 172
27.1. Projective resolutions 172
27.2. Ext 175
27.3. Induced maps between extension groups 176
27.4. Some properties of extension groups 176
27.5. Long exact Ext-sequences 177
REPRESENTATION THEORY OF ALGEBRAS I: MODULES 7
27.6. Short exact sequences and the first extension group 178
27.7. The vector space structure on the first extension group 181
27.8. Injective resolutions 183
28. Digression: Homological dimensions 183
28.1. Projective, injective and global dimension 183
28.2. Hereditary algebras 184
28.3. Selfinjective algebras 184
28.4. Finitistic dimension 184
28.5. Representation dimension 184
28.6. Dominant dimension 185
28.7. Auslander algebras 185
28.8. Gorenstein algebras 185
29. Tensor products, adjunction formulas and Tor-functors 185
29.1. Tensor products of modules 185
29.2. Adjoint functors 188
29.3. Tor 189
Part 6. Homological Algebra II: Auslander-Reiten Theory 191
30. Irreducible homomorphisms and Auslander-Reiten sequences 191
30.1. Irreducible homomorphisms 191
30.2. Auslander-Reiten sequences and Auslander-Reiten quivers 191
30.3. Properties of irreducible homomorphisms 192
30.4. Dual statements 196
30.5. Examples: Irreducible maps in f.d. 197
N
30.6. Exercises 199
31. Auslander-Reiten Theory 199
31.1. The transpose of a module 199
31.2. The Auslander-Reiten formula 200
8 CLAUS MICHAEL RINGEL ANDJAN SCHRO¨ER
31.3. The Nakayama functor 202
31.4. Proof of the Auslander-Reiten formula 203
31.5. Existence of Auslander-Reiten sequences 205
31.6. Properties of τ, Tr and ν 208
31.7. Properties of Auslander-Reiten sequences 210
31.8. Digression: The Brauer-Thrall Conjectures 214
31.9. The bimodule of irreducible morphisms 217
31.10. Translation quivers and mesh categories 221
31.11. Examples of Auslander-Reiten quivers 225
31.12. Knitting preprojective components 227
31.13. More examples of Auslander-Reiten quivers 233
32. Grothendieck group and Ringel form 242
32.1. Grothendieck group 242
32.2. The Ringel form 244
33. Reachable and directing modules 245
33.1. Reachable modules 245
33.2. Computations in the mesh category 250
33.3. Directing modules 253
33.4. The quiver of an algebra 256
33.5. Exercises 257
34. Cartan and Coxeter matrix 257
34.1. Coxeter matrix 257
34.2. Cartan matrix 262
34.3. Exercises 266
35. Representation theory of quivers 266
35.1. Bilinear and quadratic forms 267
35.2. Gabriel’s Theorem 271
REPRESENTATION THEORY OF ALGEBRAS I: MODULES 9
36. Cartan matrices and (sub)additive functions 271
Part 7. Extras 274
37. Classes of modules 274
38. Classes of algebras 275
39. Dimensions 277
References 278
Index 279
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10 CLAUS MICHAEL RINGEL ANDJAN SCHRO¨ER
1. Introduction
1.1. About this course. This is the first part of notes for a lecture course “In-
troduction to Representation Theory”. As a prerequisite only a good knowledge of
Linear Algebra is required. We will focus on the representation theory of quivers
and finite-dimensional algebras.
The intersection between the content of this course and a classical Algebra course
just consists of some elementary ring theory. We usually work over a fixed field K.
Field extensions and Galois theory do not play a role.
This part contains an introduction to general module theory. We prove the classical
theorems of Jordan-H¨older and Krull-Remak-Schmidt, and we develop the represen-
tation theory of semisimple algebras. (But let us stress that in this course, semisim-
ple representations carry the label “boring and not very interesting”.) We also start
to investigate short exact sequences of modules, pushouts, pullbacks and properties
of Auslander Reiten sequences. Some first results on the representation theory of
path algebras (or equivalently, the representation theory of quivers) are presented
towards the end of this first part. We study the Jacobson radical of an algebra,
decompositions of the regular representation of an algebra, and also describe the
structure of semisimple algebras (which is again regarded as boring). Furthermore,
we develop the theory of projective modules.
As you will notice, this first part of the script concentrates on modules and algebras.
But what we almost do not study yet are modules over algebras. (An exception are
semisimple modules and projective modules. Projective modules will be important
later on when we begin to study homological properties of algebras and modules.)
Here are some topics we will discuss in this series of lecture courses:
Representation theory of quivers and finite-dimensional algebras
•
Homological algebra
•
Auslander-Reiten theory
•
Knitting of preprojective components
•
Tilting theory
•
Derived and triangulated categories
•
Covering theory
•
Categorifications of cluster algebras
•
Preprojective algebras
•
Ringel-Hall algebras, (dual)(semi) canonical bases of quantized enveloping
•
algebras
Quiver representations and root systems of Kac-Moody Lie algebras
•
Homological conjectures
•
Tame and wild algebras
•
Functorial filtrations and applications to the representation theory of clans
•
and biserial algebras
Gabriel-Roiter measure
•
Degenerations of modules
•