Transcription of THE RISING SEA Foundations of Algebraic Geometry
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THE RISING SEA Foundations of Algebraic Geometry
THE RISING SEA. Foundations of Algebraic Geometry November 18, 2017 draft . c 2010 2017 by Ravi Vakil. Note to reader: the index and formatting have yet to be properly dealt with. There remain many issues still to be dealt with in the main part of the notes (including many of your corrections and suggestions). Contents Preface 11. For the reader 12. For the expert 16. Background and conventions 17. The goals of this book 18. Part I. Preliminaries 21. Chapter 1. Some category theory 23. Motivation 23. Categories and functors 25. Universal properties determine an object up to unique isomorphism 31. Limits and colimits 39. Adjoints 43. An introduction to abelian categories 47. Spectral sequences 57. Chapter 2. Sheaves 71. Motivating example: The sheaf of differentiable functions 71. Definition of sheaf and presheaf 73. Morphisms of presheaves and sheaves 78. Properties determined at the level of stalks, and sheafification 82. Recovering sheaves from a sheaf on a base 86. Sheaves of abelian groups, and OX -modules, form abelian categories 89.
0.4. ⋆⋆ The goals of this book 18 Part I. Preliminaries 21 Chapter 1. Some category theory 23 1.1. Motivation 23 1.2. Categories and functors 25 1.3. Universal properties determine an object up to unique isomorphism 31 1.4. Limits and colimits 39 1.5. Adjoints 43 1.6. An introduction to abelian categories 47 1.7. ⋆ Spectral sequences 57 ...
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