This book is motivated by the problem of determining the set of rational points on a variety, but its true goal is to equip readers with a broad range of tools essential for current research in algebraic geometry and number theory. The book is unconventional in that it provides concise accounts of many topics instead of a comprehensive account of just one—this is intentionally designed to bring readers up to speed rapidly. Readers who want to learn more about a particular topic can still do so, however, by following the references suggested in each section. Among the topics included are Brauer groups, faithfully flat descent, algebraic groups, torsors, étale and fppf cohomology, the Weil conjectures, and the Brauer–Manin and descent obstructions. A final chapter applies all these to study the arithmetic of surfaces.
The down-to-earth explanations and the over 100 exercises make the book suitable for use as a graduate-level textbook, but even experts will appreciate having a single source covering many aspects of geometry over an unrestricted ground field and containing some material that cannot be found elsewhere.
Bjorn Poonen, Massachusetts Institute of Technology, Cambridge, MA
• Contents 6 • Preface 10 0.1. Prerequisites 10 0.2. What kind of book this is 10 0.3. The nominal goal 11 0.4. The true goal 11 0.5. The content 11 0.6. Anything new in this book? 13 0.7. Standard notation 14 0.8. Acknowledgments 15 • Chapter 1. Fields 18 1.1. Some fields arising in classical number theory 18 1.2. ??? fields 20 1.3. Galois theory 25 1.4. Cohomological dimension 29 1.5. Brauer groups of fields 33 Exercises 44 • Chapter 2. Varieties over arbitrary fields 48 2.1. Varieties 49 2.2. Base extension 49 2.3. Scheme-valued points 54 2.4. Closed points 62 2.5. Curves 64 2.6. Rational points over special fields 66 Exercises 70 • Chapter 3. Properties of morphisms 74 3.1. Finiteness conditions 74 3.2. Spreading out 77 3.3. Flat morphisms 83 3.4. Fppf and fpqc morphisms 85 3.5. Smooth and étale morphisms 86 3.6. Rational maps 109 3.7. Frobenius morphisms 112 3.8. Comparisons 114 Exercises 114 • Chapter 4. Faithfully flat descent 118 4.1. Motivation: Gluing sheaves 118 4.2. Faithfully flat descent for quasi-coherent sheaves 120 4.3. Faithfully flat descent for schemes 121 4.4. Galois descent 123 4.5. Twists 126 4.6. Restriction of scalars 131 Exercises 133 • Chapter 5. Algebraic groups 136 5.1. Group schemes 136 5.2. Fppf group schemes over a field 142 5.3. Affine algebraic groups 147 5.4. Unipotent groups 148 5.5. Tori 151 5.6. Semisimple and reductive algebraic groups 153 5.7. Abelian varieties 159 5.8. Finite étale group schemes 167 5.9. Classification of smooth algebraic groups 168 5.10. Approximation theorems 171 5.11. Inner twists 172 5.12. Torsors 173 Exercises 181 • Chapter 6. Étale and fppf cohomology 186 6.1. The reasons for étale cohomology 186 6.2. Grothendieck topologies 188 6.3. Presheaves and sheaves 190 6.4. Cohomology 195 6.5. Torsors over an arbitrary base 199 6.6. Brauer groups 207 6.7. Spectral sequences 213 6.8. Residue homomorphisms 217 6.9. Examples of Brauer groups 220 Exercises 224 • Chapter 7. The Weil conjectures 226 7.1. Statements 226 7.2. The case of curves 227 7.3. Zeta functions 228 7.4. The Weil conjectures in terms of zeta functions 230 7.5. Cohomological explanation 231 7.6. Cycle class homomorphism 239 7.7. Applications to varieties over global fields 243 Exercises 246 • Chapter 8. Cohomological obstructions to rational points 248 8.1. Obstructions from functors 248 8.2. The Brauer–Manin obstruction 250 8.3. An example of descent 257 8.4. Descent 261 8.5. Comparing the descent and Brauer–Manin obstructions 267 8.6. Insufficiency of the obstructions 270 Exercises 275 • Chapter 9. Surfaces 278 9.1. Kodaira dimension 278 9.2. Varieties that are close to being rational 280 9.3. Classification of surfaces 288 9.4. Del Pezzo surfaces 296 9.5. Rational points on varieties of general type 307 Exercises 310 • Appendix A. Universes 312 A.1. Definition of universe 312 A.2. The universe axiom 313 A.3. Strongly inaccessible cardinals 313 A.4. Universes and categories 314 A.5. Avoiding universes 314 Exercises 315 • Appendix B. Other kinds of fields 316 B.1. Higher-dimensional local fields 316 B.2. Formally real and real closed fields 316 B.3. Henselian fields 317 B.4. Hilbertian fields 318 B.5. Pseudo-algebraically closed fields 319 Exercises 319 • Appendix C. Properties under base extension 320 C.1. Morphisms 320 C.2. Varieties 325 C.3. Algebraic groups 325 • Bibliography 328 • Index 348