Differential Algebra & Algebraic Groups (eBook)

Front Cover 1
Differential Algebra and Algebraic Groups 4
Copyright Page 5
Contents 8
Preface 12
Acknowledgments 18
Chapter 0. Algebraic Preliminaries 22
1. Conventions 22
2. Separable dependence 23
3. Quasi-separable field extensions 25
4. Quotients 28
5. Perfect ideals 28
6. Separable, quasi-separable, and regular ideals 29
7. Conservative systems 31
8. Perfect conservative systems 33
9. Noetherian conservative systems 34
10. Morphisms and birational equivalence of ideals 37
11. Polynomial ideals and generic zeros 40
12. Polynomial ideals and ground field extension 41
13. Power series 50
14. Specializations 54
15. Algebraic function fields of one variable 62
16. Dimension of components 64
17. Lattice points 70
18. Shapiro’s lemma 74
19. t-Values 77
Chapter I. Basic Notions of Differential Algebra 79
1. Differential rings 79
2. Homomorphisms and differential ideals 82
3. Differential rings of quotients 84
4. Transformation and restriction of the set of derivation operators 86
5. Differential modules differential algebras
6. Differential polynomial algebras 90
7. Permissible gradings 93
8. Rank 96
9. Autoreduced sets 98
10. Characteristic sets 102
11. Pseudo-leaders 104
12. Differential algebras of power series 105
Chapter II. Differential Fields 107
1. Linear dependence over constants 107
2. Separable extensions 110
3. Differentially perfect and differentially quasi-perfect differential fields 113
4. Separable dependence over constants 114
5. Differential polynomial functions 116
6. Dependence of derivative operators 116
7. Differentially separable dependence 120
8. Differentially separable extensions 121
9. Differential inseparability bases 125
10. Differential transcendence bases 129
11. Finitely generated extensions 130
12. Differential inseparability polynomials 136
13. Differential type typical differential inseparability degree
Chapter III. The Basis Theorem and Some Related Topics 142
1. Differential conservative systems 142
2. Quasi-separable differential ideals 144
3. Differential fields of definition 146
4. The basis theorem 147
5. Differential dimension polynomials 150
6. Extension of the differential field of coefficients 151
7. Universal extensions 154
8. t-Coherent autoreduced sets 156
9. Differential specializations 159
10. Constrained families 163
Chapter IV. Algebraic Differential Equations 166
PART A: CHARACTERISTIC p ARBITRARY 166
1. Differential affine space. The differential Zariski topology 166
2. Generic zeros. The theorem of zeros 167
3. Closed sets and u-separable differential ideals 168
4. The relative topologies differential fields of definition
5. Linear differential ideals 171
6. General components 176
7. General components and differential dimension polynomials 181
8. Multiplicity of zeros 185
PART B: CHARACTERISTIC p = 0 186
9. Finite sets of differential polynomials 187
10. The leading coefficient theorem 192
11. Levi's lemma 197
12. The domination lemma 199
13. Preparations 204
14. The component theorem 206
15. The low power theorem 208
16. The Ritt problem 211
17. Systems of bounded order 215
18. Substitution of powers 223
Bibliography for Chapters I–IV 227
Chapter V. Algebraic Groups 233
1. Introduction 233
2. Pre-K-sets 236
3. K-Groups and homogeneous K-spaces. K-Sets 239
4. Extending the universal field 248
5. Extending the basic field 251
6. Zariski topology K-topology
7. Closed sets 261
8. K-Subgroups 268
9. K-Homomorphisms 270
10. Direct products 278
11. Quotients 288
12. Galois cohomology 294
13. Principal homogeneous K-spaces 302
14. Holomorphicity at a specialization 308
15. K-Mappings 315
16. K-Functions 327
17. K-Cohomology 339
18. Invariant derivations and differentials. The Lie algebra 343
19. Localrings 352
20. Tangent spaces 355
21. Crossed K-homomorphisms 362
22. Logarithmic derivatives 370
23. Linear K-groups 375
24. Abelian K-groups 397
Bibliography for Chapter V 404
Chapter VI. Galois Theory of Differential Fields 406
1. Specializations of isomorphisms 406
2. Strong isomorphisms 409
3. Strongly normal extensions, Galois groups 414
4. The fundamental theorems 419
5. Examples 425
6. Picard–Vessiot extensions 430
7. G-Primitives 439
8. Differential Galois cohomology 442
9. Applications 447
10. V-Primitives 448
Bibliography for Chapter VI 452
Glossary of Notation 456
Index of Definitions 462
Pure and Applied Mathematics 468