If is a number field, arithmetic duality theorems for tori and complexes of tori over are crucial to understand local-global principles for linear algebraic groups over . When is a global field of positive characteristic, we prove similar arithmetic duality theorems, including a Poitou-Tate exact sequence for Galois hypercohomology of complexes of tori. One of the main ingredients is the Artin-Mazur-Milne duality theorem for fppf cohomology of finite flat commutative group schemes.
Sur un corps de nombres , les théorèmes de dualité pour les tores et les complexes de tores sont cruciaux afin de comprendre le principe local-global pour les -groupes algébriques linéaires. Nous démontrons de tels théorèmes de dualité arithmétique quand est un corps global de caractéristique , et en particulier nous établissons une suite de Poitou-Tate pour l’hypercohomologie galoisienne d’un complexe de tores. Un des principaux ingrédients est la dualité d’Artin-Mazur-Milne pour la cohomologie fppf d’un schéma en groupes fini et plat.
Accepted:
Published online:
DOI: 10.5802/jep.129
Keywords: Artin-Mazur-Milne duality, complex of tori, flat cohomology, Poitou-Tate exact sequence
Mot clés : Dualité d’Artin-Mazur-Milne, complexe de tores, cohomologie plate, suite exacte de Poitou-Tate
Cyril Demarche 1; David Harari 2
@article{JEP_2020__7__831_0, author = {Cyril Demarche and David Harari}, title = {Duality for complexes of tori over a~global~field of positive characteristic}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {831--870}, publisher = {\'Ecole polytechnique}, volume = {7}, year = {2020}, doi = {10.5802/jep.129}, zbl = {07154431}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.129/} }
TY - JOUR AU - Cyril Demarche AU - David Harari TI - Duality for complexes of tori over a global field of positive characteristic JO - Journal de l’École polytechnique — Mathématiques PY - 2020 SP - 831 EP - 870 VL - 7 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.129/ DO - 10.5802/jep.129 LA - en ID - JEP_2020__7__831_0 ER -
%0 Journal Article %A Cyril Demarche %A David Harari %T Duality for complexes of tori over a global field of positive characteristic %J Journal de l’École polytechnique — Mathématiques %D 2020 %P 831-870 %V 7 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.129/ %R 10.5802/jep.129 %G en %F JEP_2020__7__831_0
Cyril Demarche; David Harari. Duality for complexes of tori over a global field of positive characteristic. Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 831-870. doi : 10.5802/jep.129. https://jep.centre-mersenne.org/articles/10.5802/jep.129/
[BD13] - “Manin obstruction to strong approximation for homogeneous spaces”, Comment. Math. Helv. 88 (2013) no. 1, p. 1-54 | DOI | MR | Zbl
[Bor96] - “The Brauer-Manin obstructions for homogeneous spaces with connected or abelian stabilizer”, J. reine angew. Math. 473 (1996), p. 181-194 | DOI | MR | Zbl
[Bou07] - Éléments de mathématique. Algèbre commutative. Chapitre 10, Springer-Verlag, Berlin, 2007 | Zbl
[Ces15] - “Topology on cohomology of local fields”, Forum Math. Sigma 3 (2015), article ID e16, 55 pages | DOI | MR | Zbl
[Ces17] - “-Selmer growth in extensions of degree ”, J. London Math. Soc. (2) 95 (2017) no. 3, p. 833-852 | DOI | MR | Zbl
[Dem11a] - “Le défaut d’approximation forte dans les groupes linéaires connexes”, Proc. London Math. Soc. (3) 102 (2011) no. 3, p. 563-597 | DOI | MR | Zbl
[Dem11b] - “Suites de Poitou-Tate pour les complexes de tores à deux termes”, Internat. Math. Res. Notices (2011) no. 1, p. 135-174, short version of “Théorèmes de dualité pour les complexes de tores”, available at https://webusers.imj-prg.fr/~cyril.demarche/ | DOI | Numdam | MR | Zbl
[DH19] - “Artin-Mazur-Milne duality for fppf cohomology”, Algebra Number Theory 13 (2019) no. 10, p. 2323-2357 | DOI | MR | Zbl
[FS02] - “The spectral sequence relating algebraic -theory to motivic cohomology”, Ann. Sci. École Norm. Sup. (4) 35 (2002) no. 6, p. 773-875 | DOI | MR | Zbl
[GA09] - “Arithmetic duality theorems for 1-motives over function fields”, J. reine angew. Math. 632 (2009), p. 203-231 | DOI | Zbl
[GAT09] - “The generalized Cassels-Tate dual exact sequence for 1-motives”, Math. Res. Lett. 16 (2009) no. 5, p. 827-839 | DOI | MR | Zbl
[God73] - Topologie algébrique et théorie des faisceaux, Hermann, Paris, 1973 | DOI | MR | Zbl
[HSS15] - “Weak approximation for tori over -adic function fields”, Internat. Math. Res. Notices (2015) no. 10, p. 2751-2783 | DOI | MR | Zbl
[Lan56] - “Algebraic groups over finite fields”, Amer. J. Math. 78 (1956), p. 555-563 | DOI | Zbl
[Mil80] - Étale cohomology, Princeton Math. Series, vol. 33, Princeton University Press, Princeton, N.J., 1980 | DOI | MR | Zbl
[Mil06] - Arithmetic duality theorems, BookSurge, LLC, Charleston, SC, 2006 | arXiv
[NSW08] - Cohomology of number fields, Grundlehren Math. Wiss., vol. 323, Springer-Verlag, Berlin, 2008 | DOI | MR | Zbl
[Ros18] - “Tamagawa numbers and other invariants of pseudo-reductive groups over global function fields”, 2018, to appear in Algebra Number Theory | arXiv | Zbl
[San81] - “Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres”, J. reine angew. Math. 327 (1981), p. 12-80 | DOI
[Ser68] - Corps locaux, Publications de l’Université de Nancago, vol. VIII, Hermann, Paris, 1968 | Zbl
[SGA4] - Théorie des topos et cohomologie étale des schémas. Tome 3, Lect. Notes in Math., vol. 269, 270 & 305, Springer-Verlag, Berlin-New York, 1973, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4)
[SGA3] - Schémas en groupes (SGA 3). Tome I. Propriétés générales des schémas en groupes, Documents mathématiques, vol. 7, Société Mathématique de France, Paris, 2011, article ID e16, Revised and annotated edition of the 1970 original | DOI | MR | Zbl
[Stacks] - “The Stacks Project”, 2019, https://stacks.math.columbia.edu | DOI | MR | Zbl
Cited by Sources: