Textbook for a graduate course or for self-study by graduate students on a number of topics related to local analytical geometry. Includes an algebraic treatment of several complex variables, a geometric approach to algebraic geometry versus analytic sets, a survey of local algebra, and a survey of sheaf theory.
Author(s): Shreeram Shankar Abhyankar
Publisher: World Scientific
Year: 2001
Language: English
Pages: 506
City: Singapore; River Edge, NJ
PREFACE......Page 8
INSTRUCTIONS TO THE READER......Page 12
Contents......Page 14
1. Notation and Terminology......Page 18
2. Convergent Power Series......Page 23
3. Laurent Series......Page 34
4. Cauchy Theory......Page 40
5. Convexity in Rn1......Page 52
6. Laurent Expansion in Cn......Page 65
7. Domains of Holomorphy......Page 72
8. A Theorem of Rado......Page 76
9. Comments on Totally Disconnected Fields......Page 82
10. Weierstrass Preparation Theorem. Identity Theorem. Finite Ideal Bases and Unique Factorization in Power Series Rings. Implicit Function Theorem......Page 88
11. Continuity of Roots and Open Map Theorem......Page 107
12. Hensel's Lemma. Continuity of Algebroid Functions......Page 110
13. Complex Weierstrass Preparation Theorem......Page 117
14. Riemann Extension Theorem and Connectivity of Algebroid Hypersurfaces......Page 124
15. Oka Coherence......Page 134
16. Cartan Module Bases......Page 142
17. Depth Height and Dimension. Completions. Direct Sums. Resultants and Discriminants......Page 158
18. Quotient Rings......Page 166
19. Integral Dependence and Finite Generation......Page 175
20. Henselian Rings......Page 190
21. Order and Rank in Local Rings. Regular Local Rings......Page 195
22. Another Proof that a Formal Power Series Rings is Noetherian......Page 201
23. Parameters for Ideals......Page 206
24. Perfect Fields......Page 215
25. Regularity of Quotient Rings......Page 221
26. Translates of Ideals......Page 228
27. Dimension of an Intersection......Page 232
28. Algebraic Lemmas on Algebroid Functions......Page 240
29. The Language of Germs......Page 247
30. Decomposition of an Analytic Set Germ......Page 250
31. Ruckert-Weierstrass Parametrization of an Irreducible Analytic Set Germ......Page 263
32. Ruckert-Weierstrass Parametrization of an Irreducible Analytic Set Germ (Summary)......Page 281
33. Local Properties of Analytic Sets......Page 288
34. Connectivity Properties of Complex Analytic Sets......Page 306
35. Parametrization of a Pure Dimensional Analytic Set......Page 322
36. Normal Points of Complex Analytic Sets. Remarks on Algebraic Varieties......Page 340
37. Remmert-Stein-Thullen Theorem on Essential Singularities of Complex Analytic Sets. Theorem of Chow......Page 350
38. Topological Dimension......Page 363
39. Remarks on the Fundamental Group......Page 366
40. Inductive Systems and Presheaves......Page 374
41. Sheaves......Page 386
42. Coherent Sheaves......Page 401
43. Definitions......Page 411
44. Recapitulation of Properties of Analytic Spaces......Page 419
45. Invariance of Order and Rank......Page 441
46. Bimeromorphic Maps and Normalizations......Page 459
BIBLIOGRAPHY......Page 488
INDEX OF NOTATION......Page 492
SUBJECT INDEX......Page 496
