Local Analytic Geometry.
Title:
Local Analytic Geometry.
Author:
Abhyankar, Shreeram Shankar.
ISBN:
9789812810342
Personal Author:
Physical Description:
1 online resource (506 pages)
Contents:
Contents -- PREFACE -- INSTRUCTIONS TO THE READER -- CHAPTER I Elementary Theory in Cn -- 1. Notation and Terminology -- 2. Convergent Power Series -- 3. Laurent Series -- 4. Cauchy Theory -- 5. Convexity in Rn1 -- 6. Laurent Expansion in Cn -- 7. Domains of Holomorphy -- 8. A Theorem of Rado -- 9. Comments on Totally Disconnected Fields -- CHAPTER II Weierstrass Preparation Theorem -- 10. Weierstrass Preparation Theorem. Identity Theorem. Finite Ideal Bases and Unique Factorization in Power Series Rings. Implicit Function Theorem -- 11. Continuity of Roots and Open Map Theorem -- 12. Hensel's Lemma. Continuity of Algebroid Functions -- 13. Complex Weierstrass Preparation Theorem -- 14. Riemann Extension Theorem and Connectivity of Algebroid Hypersurfaces -- 15. Oka Coherence -- 16. Cartan Module Bases -- CHAPTER III Review from Local Algebra -- 17. Depth Height and Dimension. Completions. Direct Sums. Resultants and Discriminants -- 18. Quotient Rings -- 19. Integral Dependence and Finite Generation -- 20. Henselian Rings -- 21. Order and Rank in Local Rings. Regular Local Rings -- 22. Another Proof that a Formal Power Series Rings is Noetherian -- CHAPTER IV Parameters in Power Series Rings -- 23. Parameters for Ideals -- 24. Perfect Fields -- 25. Regularity of Quotient Rings -- 26. Translates of Ideals -- 27. Dimension of an Intersection -- 28. Algebraic Lemmas on Algebroid Functions -- CHAPTER V Analytic Sets -- 29. The Language of Germs -- 30. Decomposition of an Analytic Set Germ -- 31. Ruckert-Weierstrass Parametrization of an Irreducible Analytic Set Germ -- 32. Ruckert-Weierstrass Parametrization of an Irreducible Analytic Set Germ (Summary) -- 33. Local Properties of Analytic Sets -- 34. Connectivity Properties of Complex Analytic Sets.
35. Parametrization of a Pure Dimensional Analytic Set -- 36. Normal Points of Complex Analytic Sets. Remarks on Algebraic Varieties -- 37. Remmert-Stein-Thullen Theorem on Essential Singularities of Complex Analytic Sets. Theorem of Chow -- 38. Topological Dimension -- 39. Remarks on the Fundamental Group -- CHAPTER VI Language of Sheaves -- 40. Inductive Systems and Presheaves -- 41. Sheaves -- 42. Coherent Sheaves -- CHAPTER VII Analytic Spaces -- 43. Definitions -- 44. Recapitulation of Properties of Analytic Spaces -- 45. Invariance of Order and Rank -- 46. Bimeromorphic Maps and Normalizations -- BIBLIOGRAPHY -- INDEX OF NOTATION -- SUBJECT INDEX.
Abstract:
This book provides, for use in a graduate course or for self-study by graduate students, a well-motivated treatment of several topics, especially the following: (1) algebraic treatment of several complex variables; (2) geometric approach to algebraic geometry via analytic sets; (3) survey of local algebra; (4) survey of sheaf theory. The book has been written in the spirit of Weierstrass. Power series play the dominant role. The treatment, being algebraic, is not restricted to complex numbers, but remains valid over any complete-valued field. This makes it applicable to situations arising from number theory. When it is specialized to the complex case, connectivity and other topological properties come to the fore. In particular, via singularities of analytic sets, topological fundamental groups can be studied. In the transition from punctual to local, i.e. from properties at a point to properties near a point, the classical work of Osgood plays an important role. This gives rise to normic forms and the concept of the Osgoodian. Following Serre, the passage from local to global properties of analytic spaces is facilitated by introducing sheaf theory. Here the fundamental results are the coherence theorems of Oka and Cartan. They are followed by theory normalization due to Oka and Zariski in the analytic and algebraic cases, respectively. Contents: Elementary Theory in Cn; Weierstrass Preparation Theorem; Review from Local Algebra; Parameters in Power Series Rings; Analytic Sets; Language of Sheaves; Analytic Spaces. Readership: Graduate students and researchers in pure mathematics.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
Genre:
Electronic Access:
Click to View