On the derived category of 1-motives:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Paris
Société Mathématique de France
2016
|
| Schriftenreihe: | Astérisque
Numéro 381 |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xi, 254 Seiten |
| ISBN: | 9782856298374 |
Internformat
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| 245 | 1 | 0 | |a On the derived category of 1-motives |c Luca Barbieri-Viale and Bruno Kahn |
| 264 | 1 | |a Paris |b Société Mathématique de France |c 2016 | |
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Datensatz im Suchindex
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| adam_text | Titel: On the derived category of 1-motives
Autor: Barbieri-Viale, Luca
Jahr: 2016
CONTENTS Introduction ................................................................ 1 General assumptions and notations ...................................... 3 Outline ..................................................................... 5 0.1. The derived category of 1-motives, integrally ........................ 5 0.2. Z[l/p]-integral equivalence .......................................... 5 0.3. Duality .............................................................. 5 0.4. Left adjoint ......................................................... 6 0.5. Smooth schemes .................................................... 6 0.6. LAlb and RPic ...................................................... 7 0.7. Singular schemes .................................................... 7 0.8. Curves .............................................................. 7 0.9. Roitman’s theorem .................................................. 8 0.10. The homotopy i-structure and 1-motivic sheaves ................... 8 0.11. Internal Horn and tensor structure ................................. 8 0.12. A conceptual proof of Deligne’s conjectures ........................ 9 0.13. Hodge structures ................................................... 9 0.14. Mixed realizations ................................................. 10 0.15. Aadic .............................................................. 10 0.16. Going further ...................................................... 10 0.17. Caveat ............................................................. 10 A small reading guide ................................................... 11 Part I. The universal realization functor .................................... 13 1. The derived category of 1-motives ........................................ 15 1.1. Commutative group schemes ........................................ 15 1.2. Deligne 1-motives ...................................................
17 1.3. Weights and cohomological dimension ............................... 19 1.4. Group schemes and sheaves with transfers .......................... 20 1.5. Representable sheaves and exactness ................................ 22 1.6. Local Exts and global Exts ......................................... 24 1.7. Homotopy invariance and strict homotopy invariance ............... 25 1.8. Etale motivic complexes ............................................ 27 SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2016
CONTENTS viii 1.9. 1-motives with torsion and an exact structure on Mi[l/p .......... 29 1.10. The derived category of 1-motives ................................. 31 1.11. Torsion objects in the derived category of 1-motives ............... 31 1.12. Discrete sheaves and permutation modules ......................... 33 1.13. Cartier duality and 1-motives with cotorsion ....................... 34 1.14. How not to invert p ................................................ 37 2. Universal realization ..................................................... 39 2.1. Statement of the theorem ........................................... 39 2.2. Construction of T ................................................... 40 2.3. Full faithfulness of T ................................................ 40 2.4. Gersten’s principle .................................................. 41 2.5. An important computation .......................................... 41 2.6. Essential image ..................................................... 43 2.7. The universal realization functor .................................... 44 3. 1-motivic sheaves and the homotopy ¿-structure ........................... 45 3.1. Some useful lemmas ................................................. 45 3.2. Breen’s method ..................................................... 47 3.3. 1-motivic sheaves ................................................... 48 3.4. Extensions of 1-motivic sheaves ..................................... 52 3.5. A basic example .................................................... 54 3.6. Application: the Néron-Severi group of a smooth scheme ............ 55 3.7. Technical results on 1-motivic sheaves .............................. 56 3.8. Presenting 1-motivic sheaves by group schemes ..................... 57 3.9. The transfer structure on 1-motivic sheaves ......................... 59 3.10. 1-motivic sheaves and DM
......................................... 61 3.11. Comparing t-structures ............................................ 62 3.12. Global Ext’ with transfers ......................................... 63 3.13. Local Ext with transfers .......................................... 64 3.14. Ext with and without transfers ................................... 66 3.15. t-cxac.tness ......................................................... 68 4. Comparing two dualities ................................................. 71 1.1. Biextensions of 1-motives ........................................... 71 ?1,2. Biextensions of complexes of 1-motives .............................. 76 ?1.3. A pairing with finite coefficients .................................... 77 -1.4. Comparing two Ext groups ......................................... 78 •1.5. Two Cartier dualities ................................................ 79 Part 11 . The functors LAlb and RPic ....................................... 81 5. Definition of LAlb and RPic ..................... 83 5.1. Motivic Cartier duality .................... 83 as H U ?isgi iK :-iNi
CONTENTS « 5.2. Motivic Albanese ................................................... 84 5.3. Motivic Pic ......................................................... 86 5.4. Motivic 7T 0 .......................................................... 86 5.5. LAlb and Chow motives ............................................ 87 6. The adjunction LAlb - Tot with rational coefficients ...................... 89 6.1. Rational coefficients revisited ....................................... 89 6.2. The functor LAlb^ .................................................. 91 7. A tensor structure on D b {M ® Q) ........................................ 93 7.1. Tensor structure .................................................... 93 7.2. A formula for the internal Horn ..................................... 95 8. The Albanese complexes and their basic properties ........................ 97 8.1. Motives of singular schemes ......................................... 97 8.2. The homological Albanese complex ................................. 98 8.3. The cohomological Picard complex .................................. 101 8.4. Relative LAlb and RPic ............................................. 103 8.5. The Borel-Moore Albanese complex ................................. 103 8.6. Cohomological Albanese complex ................................... 104 8.7. Compactly supported and homological Pic .......................... 105 8.8. Topological invariance ............................................... 105 Part III. Some computations ............................................... 107 9. Computing LAlb(X) and RPic(X) for smooth X ......................... 109 9.1. The Albanese scheme ............................................... 109 9.2. The main theorem .................................................. 110 9.3. Reformulation of Theorem 9.2.2 ..................................... Ill 9.4. Proof of Theorem 9.3.2
.............................................. 112 9.5. An application ...................................................... 113 9.6. RPic(X) ............................................................ 113 10. 1-motivic homology and cohomology of singular schemes ................ 115 10.1. Ux/k for X e Sch(fc) ............................................... 115 10.2. The eh topology ................................................... 116 10.3. Blow-up induction ................................................. 118 10.4. L ; Alb(X) for X singular ........................................... 119 10.5. The cohomological 1-motives R*Pic(A ) ............................ 120 10.6. Borel-Moore variants ............................................... 120 11. 1-motivic homology and cohomology of curves ........................... 121 11.1. “Chow-Kunneth” decomposition for a curve ........................ 121 11.2. L^Alb and R’Pic, of curves ......................................... 121 11.3. Borel-Moore variants ............................................... 123 SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2016
CONTENTS 12. Comparison with Pic + , Pic , Alb + and Alb - ............................ 125 12.1. Torsion sheaves .................................................... 125 12.2. Glueing lemmas .................................................... 125 12.3. Discrete sheaves ................................................... 127 12.4. Normal schemes .................................................... 129 12.5. Some representability results ....................................... 132 12.6. LiAlb(X) and the Albanese schemes ............................... 132 12.7. LiAlb(AT) and Alb - (X) for X normal ............................. 135 12.8. RPic(Ar) and H? h {X,G rn ) ......................................... 136 12.9. H t (X,G m ) and G m ) ...................................... 137 12.10. R 1 Pic(A) and Pic + (Af) for X proper ............................. 137 12.11. The Borel-Moore variant ......................................... 138 12.12. LiAlb* and Alb+ ................................................. 140 13. Generalizations of RoTtman’s theorem ................................... 145 13.1. Motivic and classical Albanese ..................................... 145 13.2. A variant of the Suslin-Voevodsky theorem ........................ 147 13.3. Change of topology and motivic Albanese map .................... 147 13.4. A proof of Roitman’s and Spiefi-Szamuely’s theorems .............. 148 13.5. Generalization to singular schemes ................................. 150 13.6. Borel-Moore Roitman .............................................. 151 13.7. “Cohomological” Roitman .......................................... 152 Part IV. Realizations ....................................................... 155 14. An axiomatic version of Deligne’s conjecture ............................. 157 14.1. A review of base change ........................................... 157 14.2. A weight filtration on M ® Q
..................................... 157 14.3. A left adjoint in the category of realizations ....................... 158 14.4. Realization functor ................................................ 159 14.5. The base change theorem .......................................... 161 15. The Hodge realization ................................................... 165 15.1. LAIN 7 for mixed Hodge structures ................................. 165 15.2. Huber’s Hodge realization functor ................................. 166 15.3. Deligne’s conjecture ................................................ 168 15.4. Deligne’s Hodge realization functor ................................ 169 16. The mixed realization ................................................... 171 16.1. LAlb 7 for mixed realizations ...................................... 171 16.2. Huber’s mixed realization functor .................................. 174 16.3. Deligne’s conjecture ................................................ 174 17. The Gadic realization in positive characteristic ........................... 177 ASTÉRISQUE . 381
CONTENTS xi 17.1. The tame derived category of l -adic sheaves ....................... 177 17.2. DM and DA over a base ........................................... 178 17.3. ¿-adic realizations for DA and DM ................................. 179 17.4. Weights and effectivity ............................................. 180 17.5. Sheaves of level 1 ................................................ 183 17.6. Derived realization for 1-motives ................................... 184 17.7. An ¿-adic version of Deligne’s conjecture ........................... 184 Appendices ................................................................. 187 A. Homological algebra ..................................................... 189 A.l. Some comparison lemmas .......................................... 189 A. 2. The Tot construction ............................................... 190 B. Torsion objects in additive categories ..................................... 193 B. l. Additive categories ................................................. 193 B.2. Triangulated categories ............................................. 194 B.3. Torsion objects in an abelian category .............................. 197 B. 4. Abelian and derived categories ..................................... 197 C. 1-motives with torsion ................................................... 199 C. l. Effective 1-motives ................................................. 199 C.2. Quasi-isomorphisms ................................................ 201 C.3. 1-motives ........................................................... 203 C.4. Strict morphisms ................................................... 204 C.5. Exact sequences of 1-motives ....................................... 205 C.6. ¿-adic realization ................................................... 208 C.7. Deligne 1-motives ................................................... 209 C.8. Homs and Extensions
............................................... 210 C.9. Projective objects in t Mi [1/pj ...................................... 212 C.10. Weights ........................................................... 213 C. ll. l-motives over a base .............................................. 214 D. Weight filtrations ........................................................ 215 D. l. Filtrations of abelian categories .................................... 215 D.2. Morphisms of filtered categories .................................... 227 D.3. Glueing natural transformations .................................... 228 D.4. Glueing equivalences of abelian categories .......................... 231 D.5. The case of triangulated categories ................................. 235 D.6. The case of f-categories ............................................. 240 Index ....................................................................... 245 Bibliography ................................................................ 247 SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2016
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| author | Barbieri-Viale, Luca 1960- Kahn, Bruno |
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| indexdate | 2024-07-10T07:34:41Z |
| institution | BVB |
| isbn | 9782856298374 |
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| spelling | Barbieri-Viale, Luca 1960- Verfasser (DE-588)1114152986 aut On the derived category of 1-motives Luca Barbieri-Viale and Bruno Kahn Paris Société Mathématique de France 2016 xi, 254 Seiten txt rdacontent n rdamedia nc rdacarrier Astérisque Numéro 381 Kahn, Bruno Verfasser aut Astérisque Numéro 381 (DE-604)BV002579439 381 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029183718&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Barbieri-Viale, Luca 1960- Kahn, Bruno On the derived category of 1-motives Astérisque |
| title | On the derived category of 1-motives |
| title_auth | On the derived category of 1-motives |
| title_exact_search | On the derived category of 1-motives |
| title_full | On the derived category of 1-motives Luca Barbieri-Viale and Bruno Kahn |
| title_fullStr | On the derived category of 1-motives Luca Barbieri-Viale and Bruno Kahn |
| title_full_unstemmed | On the derived category of 1-motives Luca Barbieri-Viale and Bruno Kahn |
| title_short | On the derived category of 1-motives |
| title_sort | on the derived category of 1 motives |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029183718&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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