Rognes J. Galois extensions of structured ring spectra stably dualizable groups. - Providence: AMS, 2008. - 137 p. - (Memoirs of the American mathematical society; N 898). - ISSN 0065-9266; ISBN 0821840762  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

Galois Extensions of Structured Ring Spectra .................... 1 Abstract ........................................................ 2 Chapter 1. Introduction ......................................... 3 Chapter 2. Galois extensions in algebra ........................ 10 §2.1. Galois extensions of fields ............................ 10 §2.2. Regular covering spaces ................................ 10 §2.3. Galois extensions of commutative rings ................. 11 Chapter 3. Closed categories of structured module spectra ...... 14 §3.1. Structured spectra ..................................... 14 §3.2. Localized categories ................................... 15 §3.3. Dualizable spectra ..................................... 17 §3.4. Stably dualizable groups ............................... 18 §3.5. The dualizing spectrum ................................. 19 §3.6. The norm map ........................................... 20 Chapter 4. Galois extensions in topology ....................... 21 §4.1. Galois extensions of E-local commutative S-algebras .... 21 §4.2. The Eilenberg-Mac Lane embedding ....................... 22 §4.3. Faithful extensions .................................... 23 Chapter 5. Examples of Galois extensions ....................... 25 §5.1. Trivial extensions ..................................... 25 §5.2. Eilenberg-Mac Lane spectra ............................. 25 §5.3. Real and complex topological K-theory .................. 25 §5.4. The Morava change-of-rings theorem ..................... 27 §5.5. The K(1)-local case .................................... 34 §5.6. Cochain S-algebras ..................................... 37 Chapter 6. Dualizability and alternate characterizations ....... 40 §6.1. Extended equivalences .................................. 40 §6.2. Dualizability .......................................... 41 §6.3. Alternate characterizations ............................ 45 §6.4. The trace map and self-duality ......................... 46 §6.5. Smash invertible modules ............................... 49 Chapter 7. Galois theory I ..................................... 51 §7.1. Base change for Galois extensions ...................... 51 §7.2. Fixed S-algebras ....................................... 52 Chapter 8. Pro-Galois extensions and the Amitsur complex ....... 56 §8.1. Pro-Galois extensions .................................. 56 §8.2. The Amitsur complex .................................... 57 Chapter 9. Separable and étale extensions ...................... 61 §9.1. Separable extensions ................................... 61 §9.2. Symmetrically étale extensions ......................... 63 §9.3. Smashing maps .......................................... 65 §9.4. Étale extensions ....................................... 66 §9.5. Henselian maps ......................................... 68 §9.6. I-adic towers .......................................... 72 Chapter 10. Mapping spaces of commutative S-algebras ........... 77 §10.1. Obstruction theory .................................... 77 §10.2. Idempotents and connected S-algebras .................. 80 §10.3. Separable closure ..................................... 82 Chapter 11. Galois theory II ................................... 85 §11.1. Recovering the Galois group ........................... 85 §11.2. The brave new Galois correspondence ................... 86 Chapter 12. Hopf-Galois extensions in topology ................. 89 §12.1. Hopf-Galois extensions of commutative S-algebras ...... 89 §12.2. Complex cobordism ..................................... 91 References ..................................................... 94 Stably Dualizable Groups ....................................... 99 Abstract ...................................................... 100 Chapter 1. Introduction ....................................... 101 §1.1. The symmetry groups of stable homotopy theory ......... 101 §1.2. Algebraic localizations and completions ............... 101 §1.3. Chromatic localizations and completions ............... 103 §1.4. Applications .......................................... 104 Chapter 2. The dualizing spectrum ............................. 106 §2.1. The E-local stable category ........................... 106 §2.2. Dualizable spectra .................................... 107 §2.3. Stably dualizable groups .............................. 108 §2.4. E-compact groups ...................................... 109 §2.5. The dualizing and inverse dualizing spectra ........... 111 Chapter 3. Duality theory ..................................... 114 §3.1. Poincaré duality ...................................... 114 §3.2. Inverse Poincare duality .............................. 116 §3.3. The Picard group ...................................... 119 Chapter 4. Computations ....................................... 121 §4.1. A spectral sequence for E-homology .................... 121 §4.2. Morava K-theories ..................................... 121 §4.3. Eilenberg-Mac Lane spaces ............................. 124 Chapter 5. Norm and transfer maps ............................. 126 §5.1. Thom spectra .......................................... 126 §5.2. The norm map and Tate cohomology ...................... 126 §5.3. The G-transfer map .................................... 129 §5.4. E-local homotopy classes .............................. 129 References .................................................... 131 Index ......................................................... 133