Inhaltsverzeichnis
Contents vii
Preface xi
Acknowledgements xv
Conventions xvii
Chapter 0 Differential Algebraic Preliminaries
1 Compatible homomorphisms 1
2 Numerical polynomials 5
3 Finitely generated extensions 7
4 Differential derivations 8
5 The derivation operators of a differential field 12
6 Varying the set of derivation operators 16
7 A kind of disjoitness 19
8 Some groups and rings 24
9 Integrability conditions 26
Chapter I The Axioms and First Consequences
1 Pre differential algebraic sets 29
2 Differential algebraic groups and homogenous differential algebraic spaces. Differential algebraic sets 32
3 The component theorem and its corollaries 39
Chapter II Variations in the Data
1 Varying the universal differential field 44
2 Extending the basic differential field 48
3 Varying the set of derivation operators 56
Chapter III Topology
1 The differential Zariski topologies 68
2 Differentially closed sets 73
3 Constrained elements 79
Chapter IV Subgroups and Homomorphisms
1 Differential algebraic subgroups 87
2 Differential rational homomorphisms 89
3 Direct products 97
4 Quotients 103
5 Solvable and nilpotent differential algebraic groups 110
Chapter V Mappings and Functions
1 Holomorphicity at a differential specialization 113
2 Differential rational mappings 120
3 Differential rational functions 136
Chapter VI Other Structures
1 Two lemmas 145
2 Some other differential algebraic structures 147
Chapter VII Cohomology
1 The differential algebraic set of a finitely generated constrained extension 157
2 Constrained cohomology 159
3 Differential rational cohomology 170
4 Differential algebraic vector spaces over U 178
5 Principal homogeneous differential algebraic spaces 180
6 Examples 191
Chapter VIII Lie Theory
1 Local differential rings 194
2 Tangent spaces 197
3 Differential derivations 205
4 Invariant differential derivations 208
5 The Lie algebra 210
6 Examples 221
7 Differential algebraic spaces and crossed differential rational homomorphisms 227
8 Logarithmic derivations 235
9 Further properties of logarithmic derivations 240
10 The Lie-Cassidy-Kovacic method 247
11 Images and kernels 249
12 The Lie algebra of the connected differential algebraic subgroups of a connected differential algebraic group 251
References 263
Glossary of Notation 265
Index of Definitions 269 - 271