Local Analytic Geometry (eBook)

Front Cover 1
Local Analytic Geometry 4
Copyright Page 5
Contents 14
Preface 8
Instructions to the Reader 12
Chapter I. Elementary Theory in Cn 20
1. Notation and Terminology 20
2. Convergent Power Series 25
3. Laurent Series 36
4. Cauchy Theory 42
5. Convexity in Rn 54
6. Laurent Expansion in Cn 67
7. Domains of Holomorphy 74
8. A Theorem of Rad. 78
9. Comments on Totally Disconnected Fields 84
Chapter II. Weierstrass Preparation Theorem 90
10. Weierstrass Preparation Theorem. Identity Theorem. Finite Ideal Bases and Unique Factorization in Power Series Rings. Implicit Function Theorem 90
11. Continuity of Roots and Open Map Theorem 109
12. Hensel’s Lemma. Continuity of Algebroid Functions 112
13. Complex Weierstrass Preparation Theorem 119
14. Riemann Extension Theorem and Connectivity of Algebroid Hypersurfaces 126
15. Oka Coherence 136
16. Cartan Module Bases 144
Chapter III. Review from Local Algebra 160
17. Depth, Height, and Dimension. Completions. Direct Sums. Resultants and Discriminants 160
18. Quotient Rings 168
19. Integral Dependence and Finite Generation 177
20. Henselian Rings 192
21. Order and Rank in Local Rings. Regular Local Rings 197
22. Another Proof that a Formal Power Series Rings is Noetherian 203
Chapter IV. Parameters in Power Series Rings 208
23. Parameters for Ideals 208
24. Perfect Fields 217
25. Regularity of Quotient Rings 223
26. Translates of Ideals 230
27. Dimension of an Intersection 234
28. Algebraic Lemmas on Algebroid Functions 242
Chapter V. Analytic Sets 249
29. The Language of Germs 249
30. Decomposition of an Analytic Set Germ 252
31. Rückert–Weierstrass Parametrization of an Irreducible Analytic Set Germ 265
32. Rückert–Weierstrass Parametrization of an Irreducible Analytic Set Germ (Summary) 283
33. Local Properties of Analytic Sets 290
34. Connectivity Properties of Complex Analytic Sets 308
35. Parametrization of a Pure Dimensional Analytic Set 324
36. Normal Points of Complex Analytic Sets. Remarks on Algebraic Varieties 342
37. Remmert–Stein–Thullen Theorem on Essential Singularities of Complex Analytic Sets. Theorem of Chow 352
38. Topological Dimension 365
39. Remarks on the Fundamental Group 368
Chapter VI. Language of Sheaves 376
40. Inductive Systems and Presheaves 376
41. Sheaves 388
42. Coherent Sheaves 403
Chapter VII. Analytic Spaces 413
43. Definitions 413
44. Recapitulation of Properties of Analytic Spaces 421
45. Invariance of Order and Rank 443
46. Bimeromorphic Maps and Normalizations 461
Bibliography 490
Index of Notation 494
Subject Index 498