Algebraic Theory of Locally Nilpotent Derivations (eBook)

Acknowledgments 6
Contents 7
Introduction 10
Historical Overview 12
1 First Principles 17
1.1 Basic Definitions for Derivations 17
1.2 Basic Facts about Derivations 23
1.3 Group Actions 27
1.4 First Principles for Locally Nilpotent Derivations 30
1.5 Ga-Actions 39
2 Further Properties of Locally Nilpotent Derivations 42
2.1 Irreducible Derivations 42
2.2 Minimal Local Slices 44
2.3 Three Lemmas about UFDs 46
2.4 The Defect of a Derivation 47
2.5 Exponential Automorphisms 51
2.6 Wronskians and Kernel Elements 52
2.7 The Star Operator 54
3 Polynomial Rings 55
3.1 Variables, Automorphisms, and Gradings 55
3.2 Derivations of Polynomial Rings 56
3.3 Group Actions on 67
3.4 Locally Nilpotent Derivations of Polynomial Rings 69
3.5 Slices in Polynomial Rings 72
3.6 Triangular Derivations and Automoprhisms 72
3.7 Homogeneous Locally Nilpotent Derivations 76
3.8 Symmetric Locally Nilpotent Derivations 77
3.9 Some Important Early Examples 78
3.10 The Homogeneous Dependence Problem 82
4 Dimension Two 89
4.1 The Polynomial Ring in Two Variables over a Field 92
4.2 Locally Nilpotent R-Derivations of R[ x, y] 97
4.3 Rank-Two Derivations of Polynomial Rings 103
4.4 Automorphisms Preserving Lattice Points 106
4.5 Newton Polygons 107
4.6 Appendix: Newton Polytopes 109
5 Dimension Three 113
5.1 Miyanishi’s Theorem 114
5.2 Other Fundamental Theorems in Dimension Three 121
5.3 Questions of Triangularizability and Tameness 125
5.4 The Homogeneous (2, 5) Derivation 127
5.5 Local Slice Constructions 128
5.6 The Homogeneous Case 133
5.7 Graph of Kernels and Generalized Local Slice Constructions 137
5.8 G2a-Actions 139
5.9 Appendix: An Intersection Condition 140
6 Linear Actions of Unipotent Groups 143
6.1 The Finiteness Theorem 144
6.2 Linear 145
Actions 145
6.3 Linear Counterexamples to the Fourteenth Problem 152
6.4 Linear G2a-Actions 157
6.5 Appendix: Finite Group Actions 161
7 Non-Finitely Generated Kernels 163
7.1 Roberts’ Examples 163
7.2 Counterexample in Dimension Five 166
7.3 Proof for A’Campo-Neuen’s Example 175
7.4 Quotient of a Ga-Module 176
7.5 Proof for the Linear Example in Dimension Eleven 179
7.6 Kuroda’s Examples in Dimensions Three and Four 180
7.7 Locally Trivial Examples 181
7.8 Some Positive Results 182
7.9 Winkelmann’s Theorem 183
7.10 Appendix: Van den Essen’s Proof 184
8 Algorithms 187
8.1 Van den Essen’s Algorithm 189
8.2 Image Membership Algorithm 191
8.3 Criteria for a Derivation to be Locally Nilpotent 192
8.4 Maubach’s Algorithm 194
8.5 Extendibility Algorithm 196
8.6 Examples 196
8.7 Remarks 199
9 The Makar-Limanov and Derksen Invariants 201
9.1 Danielewski Surfaces 203
9.2 A Preliminary Result 205
9.3 The Threefold x + x2y + z2 + t3 = 0 207
9.4 Characterizing k[ x, y] by LNDs 210
9.5 Characterizing Danielewski Surfaces by LNDs 213
9.6 LNDs of Special Danielewski Surfaces 215
9.7 Further Properties of the ML Invariant 218
9.8 Further Results in the Classification of Surfaces 221
10 Slices, Embeddings and Cancellation 224
10.1 Some Positive Results 225
10.2 Torus Action Formula 228
10.3 Asanuma’s Torus Actions 230
10.4 V ´ en ´ ereau Polynomials 235
10.5 Open Questions 238
11 Epilogue 240
11.1 Rigidity of Kernels for Polynomial Rings 240
11.2 The Extension Property 241
11.3 Nilpotency Criterion 241
11.4 Calculating the Makar-Limanov Invariant 241
11.5 Relative Invariants 242
11.6 Structure of LND(B) 242
11.7 Maximal Subalgebras 243
11.8 Invariants of a Sum 243
11.9 Finiteness Problem for Extensions 244
11.10 Geometric Viewpoint 244
11.11 Paragonic Varieties 245
11.12 Stably Triangular Ga-Action 246
11.13 Extending Ga-Actions to Larger Group Actions 246
11.14 Variable Criterion 246
11.15 Bass’s Question on Rational Triangularization 247
11.16 Popov’s Questions 247
11.17 Miyanishi’s Question 247
References 248
Index 262