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基本説明
A textbook, for an undergraduate course in modern algebraic geometry. Focuses on Serre's GAGA theorem, which perhaps best encapsulates the link between algebra and analysis.
Full Description
This textbook, for an undergraduate course in modern algebraic geometry, recognizes that the typical undergraduate curriculum contains a great deal of analysis and, by contrast, little algebra. Because of this imbalance, it seems most natural to present algebraic geometry by highlighting the way it connects algebra and analysis; the average student will probably be more familiar and more comfortable with the analytic component. The book therefore focuses on Serre's GAGA theorem, which perhaps best encapsulates the link between algebra and analysis. GAGA provides the unifying theme of the book: we develop enough of the modern machinery of algebraic geometry to be able to give an essentially complete proof, at a level accessible to undergraduates throughout. The book is based on a course which the author has taught, twice, at the Australian National University.
Contents
Foreword; 1. Introduction; 2. Manifolds; 3. Schemes; 4. The complex topology; 5. The analytification of a scheme; 6. The high road to analytification; 7. Coherent sheaves; 8. Projective space - the statements; 9. Projective space - the proofs; 10. The proof of GAGA; Appendix. The proofs concerning analytification; Bibliography; Glossary; Index.
