Norms in motivic homotopy theory:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Paris
Société Mathématique de France
2021
|
| Schriftenreihe: | Astérisque
425 |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | ix, 207 Seiten Diagramme |
| ISBN: | 9782856299395 |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804182673597399040 |
|---|---|
| adam_text | CONTENTS 1. Introduction ........................................................................................................ 1.1. Norm functors ............................................................................................ 1.2. Normed motivic spectra ............................................................................ 1.3. Examples of normed spectra .................................................................... 1.4. Norms in other contexts ........................................................................... 1.5. Norms vs. framed transfers ...................................................................... 1.6. Summary of the construction ................................................................... 1.7. Summary of results .................................................................................... 1.8. Guide for the reader .................................................................................. 1.9. Remarks on oo-categories ......................................................................... 1.10. Standing assumptions .............................................................................. 1.11. Acknowledgments ..................................................................................... 1 1 4 5 7 9 10 11 13 13 14 14 2. Preliminaries ....................................................................................................... 2.1. Nonabelian derived oo-categories ............................................................ 2.2. Unstable motivic homotopy theory
......................................................... 2.3. Weil restriction ........................................................................................... 15 15 18 20 3. Norms of pointed motivic spaces ...................................................................... 3.1. The unstable norm functors ..................................................................... 3.2. Norms of quotients ..................................................................................... 21 22 25 4. Norms of motivic spectra ................................................................................... 4.1. Stable motivic homotopy theory .............................................................. 4.2. The stable norm functors .......................................................................... 29 29 31 5. Properties of norms ............................................................................................ 5.1. Composition and base change .................................................................. 5.2. The distributivity laws .............................................................................. 5.3. The purity equivalence .............................................................................. 5.4. The ambidexterity equivalence ................................................................ 5.5. Polynomial functors ................................................................................... 33 33 35 38 40 43 6. Coherence of norms
............................................................................................ 6.1. Functoriality on the category of spans ................................................... 6.2. Normed oo-categories ................................................................................ 47 47 50 7. Normed motivic spectra ..................................................................................... 7.1. Categories of normed spectra ................................................................... 57 57 SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2021
CONTENTS viii 7.2. Cohomology theories represented by normed spectra .......................... 63 8. The norm-pullback-pushforward adjunctions .................................................. 8.1. The norm-pullback adjunction ................................................................. 8.2. The pullback-pushforward adjunction ..................................................... 69 70 75 9. Spectra over profinite groupoids ........................................................................ 9.1. Profinite groupoids ..................................................................................... 9.2. Norms in stable equivariant homotopy theory ....................................... 77 78 81 10. Norms and Grothendieck’s Galois theory ....................................................... 10.1. The profinite étale fundamental groupoid ............................................ 10.2. Galois-equivariant spectra and motivic spectra ................................... 10.3. The Rost norm on Grothendieck-Witt groups ..................................... 87 87 89 92 11. Norms and Betti realization ............................................................................. 11.1. A topological model for equivariant homotopy theory ....................... 11.2. The real Betti realization functor .......................................................... 95 95 97 12. Norms and localization ................................................ 12.1. Inverting Picard-graded elements .......................................................... 12.2. Inverting elements in
normed spectra ................................................... 12.3. Completion of normed spectra ............................................................... 101 101 104 107 13. Norms and the slice filtration ........................................................................... 13.1. The zeroth slice of a normed spectrum ................................................. 13.2. Applications to motivic cohomology ..................................................... 13.3. Graded normed spectra ........................................................................... 13.4. The graded slices of a normed spectrum .............................................. Ill Ill 114 117 121 14. Norms of cycles ................................................................................................. 14.1. Norms of presheaves with transfers ....................................................... 14.2. The Fulton-MacPherson norm on Chow groups ................................. 14.3. Comparison of norms ............................................................................... 123 123 126 129 15. Norms of linear oo-categories .......................................................................... 15.1. Linear oo-categories ................................................................................. 15.2. Noncommutative motivic spectra and homotopy K-theory ............... 15.3. Nonconnective K-theory .......................................................................... 137 137 141 148 16. Motivic Thom spectra
....................................................................................... 16.1. The motivic Thom spectrum functor .................................................... 16.2. Algebraic cobordism and the motivic J-homomorphism .................... 16.3. Multiplicative properties ......................................................................... 16.4. Flee normed spectra ................................................................................ 16.5. Thom isomorphisms ................................................................................. 151 151 155 157 163 165 A. The Nisnevich topology ..................................................................................... 169 B. Detecting effectivity ........................................................................................... 173 ASTÉRISQUE 425
ix CONTENTS C. Categories of spans ............................................................................................ C.l. Spans in extensive oo-categories ............................................................. C.2. Spans and descent ..................................................................................... C.3. Functoriality of spans ............................................................................... 177 177 183 188 D. Relative adjunctions .......................................................................................... 191 Table of notation ..................................................................................................... 195 Bibliography ............................................................................................................ 199 SOCIÉTÉ MATHÉMATIQUE PE FRANCE 2021
|
| adam_txt |
CONTENTS 1. Introduction . 1.1. Norm functors . 1.2. Normed motivic spectra . 1.3. Examples of normed spectra . 1.4. Norms in other contexts . 1.5. Norms vs. framed transfers . 1.6. Summary of the construction . 1.7. Summary of results . 1.8. Guide for the reader . 1.9. Remarks on oo-categories . 1.10. Standing assumptions . 1.11. Acknowledgments . 1 1 4 5 7 9 10 11 13 13 14 14 2. Preliminaries . 2.1. Nonabelian derived oo-categories . 2.2. Unstable motivic homotopy theory
. 2.3. Weil restriction . 15 15 18 20 3. Norms of pointed motivic spaces . 3.1. The unstable norm functors . 3.2. Norms of quotients . 21 22 25 4. Norms of motivic spectra . 4.1. Stable motivic homotopy theory . 4.2. The stable norm functors . 29 29 31 5. Properties of norms . 5.1. Composition and base change . 5.2. The distributivity laws . 5.3. The purity equivalence . 5.4. The ambidexterity equivalence . 5.5. Polynomial functors . 33 33 35 38 40 43 6. Coherence of norms
. 6.1. Functoriality on the category of spans . 6.2. Normed oo-categories . 47 47 50 7. Normed motivic spectra . 7.1. Categories of normed spectra . 57 57 SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2021
CONTENTS viii 7.2. Cohomology theories represented by normed spectra . 63 8. The norm-pullback-pushforward adjunctions . 8.1. The norm-pullback adjunction . 8.2. The pullback-pushforward adjunction . 69 70 75 9. Spectra over profinite groupoids . 9.1. Profinite groupoids . 9.2. Norms in stable equivariant homotopy theory . 77 78 81 10. Norms and Grothendieck’s Galois theory . 10.1. The profinite étale fundamental groupoid . 10.2. Galois-equivariant spectra and motivic spectra . 10.3. The Rost norm on Grothendieck-Witt groups . 87 87 89 92 11. Norms and Betti realization . 11.1. A topological model for equivariant homotopy theory . 11.2. The real Betti realization functor . 95 95 97 12. Norms and localization . 12.1. Inverting Picard-graded elements . 12.2. Inverting elements in
normed spectra . 12.3. Completion of normed spectra . 101 101 104 107 13. Norms and the slice filtration . 13.1. The zeroth slice of a normed spectrum . 13.2. Applications to motivic cohomology . 13.3. Graded normed spectra . 13.4. The graded slices of a normed spectrum . Ill Ill 114 117 121 14. Norms of cycles . 14.1. Norms of presheaves with transfers . 14.2. The Fulton-MacPherson norm on Chow groups . 14.3. Comparison of norms . 123 123 126 129 15. Norms of linear oo-categories . 15.1. Linear oo-categories . 15.2. Noncommutative motivic spectra and homotopy K-theory . 15.3. Nonconnective K-theory . 137 137 141 148 16. Motivic Thom spectra
. 16.1. The motivic Thom spectrum functor . 16.2. Algebraic cobordism and the motivic J-homomorphism . 16.3. Multiplicative properties . 16.4. Flee normed spectra . 16.5. Thom isomorphisms . 151 151 155 157 163 165 A. The Nisnevich topology . 169 B. Detecting effectivity . 173 ASTÉRISQUE 425
ix CONTENTS C. Categories of spans . C.l. Spans in extensive oo-categories . C.2. Spans and descent . C.3. Functoriality of spans . 177 177 183 188 D. Relative adjunctions . 191 Table of notation . 195 Bibliography . 199 SOCIÉTÉ MATHÉMATIQUE PE FRANCE 2021 |
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| spelling | Bachmann, Tom 19XX- Verfasser (DE-588)1237632021 aut Norms in motivic homotopy theory Tom Bachmann & Marc Hoyois Paris Société Mathématique de France 2021 ix, 207 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Astérisque 425 Mit einer Zusammenfassung in englischer und französischer Sprache Hoyois, Marc 1987- Verfasser (DE-588)1237632595 aut Astérisque 425 (DE-604)BV002579439 425 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032809082&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bachmann, Tom 19XX- Hoyois, Marc 1987- Norms in motivic homotopy theory Astérisque |
| title | Norms in motivic homotopy theory |
| title_auth | Norms in motivic homotopy theory |
| title_exact_search | Norms in motivic homotopy theory |
| title_exact_search_txtP | Norms in motivic homotopy theory |
| title_full | Norms in motivic homotopy theory Tom Bachmann & Marc Hoyois |
| title_fullStr | Norms in motivic homotopy theory Tom Bachmann & Marc Hoyois |
| title_full_unstemmed | Norms in motivic homotopy theory Tom Bachmann & Marc Hoyois |
| title_short | Norms in motivic homotopy theory |
| title_sort | norms in motivic homotopy theory |
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