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THE RIGID SYNTOMIC RING SPECTRUM

Part of: (Co)homology theory

Published online by Cambridge University Press:  13 June 2014

F. Déglise  and
N. Mazzari
F. Déglise
Affiliation:
ENS de Lyon, France (frederic.deglise@ens-lyon.fr)
N. Mazzari
Affiliation:
Université de Bordeaux, France (nicola.mazzari@math.u-bordeaux1.fr)
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Abstract

The aim of this paper is to show that rigid syntomic cohomology – defined by Besser – is representable by a rational ring spectrum in the motivic homotopical sense. In fact, extending previous constructions, we exhibit a simple representability criterion and we apply it to several cohomologies in order to get our central result. This theorem gives new results for rigid syntomic cohomology such as h-descent and the compatibility of cycle classes with Gysin morphisms. Along the way, we prove that motivic ring spectra induce a complete Bloch–Ogus cohomological formalism and even more. Finally, following a general motivic homotopical philosophy, we exhibit a natural notion of rigid syntomic coefficients.

Keywords

Rigid syntomic cohomologyBeilinson motivesBloch–Ogus

MSC classification

Primary: 14F42: Motivic cohomology; motivic homotopy theory 14F30: $p$-adic cohomology, crystalline cohomology
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 4 , October 2015 , pp. 753 - 799
Copyright
© Cambridge University Press 2014 

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References

Ayoub, Joseph, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (I, II), Astérisque, Volume 314, 315, (Soc Math, France, 2007).Google Scholar
Bannai, Kenichi, Syntomic cohomology as a p-adic absolute Hodge cohomology, Math. Z. 242(3) (2002), 443–480MR 1985460 (2004e:14037).Google Scholar
Beĭlinson, Alexander, Higher regulators and values of L-functions, Current problems in mathematics, Volume 24, pp. 181–238 (Itogi Nauki i Tekhniki, Akad Nauk SSSR Vsesoyuz Inst Nauchn i Tekhn Inform, Moscow, 1984) MR 760999 (86h:11103).Google Scholar
Beĭlinson, Alexander, Higher regulators of modular curves, in Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo, 1983), Contemp. Math., Volume 55, pp. 1–34 (Amer. Math. Soc., Providence, RI, 1986) MR 862627 (88f:11060).Google Scholar
Beĭlinson, Alexander, Notes on absolute Hodge cohomology, in Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo, 1983), Contemp. Math., Volume 55, pp. 35–68 (Amer. Math. Soc., Providence, RI, 1986).Google Scholar
Besser, Amnon, Syntomic regulators and p-adic integration I Rigid syntomic regulators, Proceedings of the Conference on p-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998), Volume 120, pp. 291–334. (2000).Google Scholar
Besser, Amnon, On the syntomic regulator for K 1 of a surface, Israel J. Math. 190 (2012), 29–66MR 2956231.CrossRefGoogle Scholar
Bloch, Spencer and Ogus, Arthur, Gersten’s conjecture and the homology of schemes, Ann. Sci. École Norm. Sup. (4) 7 (1975), 181–201.CrossRefGoogle Scholar
Cisinski, Denis-Charles, Images directes cohomologiques dans les catégories de modèles, Ann. Math. Blaise Pascal 10(2) (2003), 195–244MR 2031269 (2004k:18009).Google Scholar
Cisinski, Denis-Charles and Déglise, Frédéric, Local and stable homological algebra in Grothendieck abelian categories, Homology, Homotopy and Applications 11(1) (2009), 219–260.CrossRefGoogle Scholar
Cisinski, Denis-Charles and Déglise, Frédéric, Mixed weil cohomologies, Adv. Math. 230 (2012), 55–130.Google Scholar
Cisinski, Denis-Charles and Déglise, Frédéric,Triangulated categories of mixed motives, arXiv:0912.2110v3 (2012).Google Scholar
Chiarellotto, Bruno, Ciccioni, Alicia and Mazzari, Nicola, Cycle classes and the syntomic regulator, Algebra Number Theory 7(3) (2013), 533–566.Google Scholar
Déglise, Frédéric, Around the Gysin triangle II, Doc. Math. 13 (2008), 613–675.Google Scholar
Déglise, Frédéric,Orientation theory in arithmetic geometry, arXiv:1111.4203, 2011.Google Scholar
Deligne, Pierre, Théorie de Hodge III, Inst. Hautes Études Sci. Publ. Math. (44) (1974), 5–77MR MR0498552 (58 #16653b).Google Scholar
Deligne, Pierre, Cohomologie étale, Lecture Notes in Mathematics, Volume 569, (Springer-Verlag, Berlin, 1977) Séminaire de Géométrie Algébrique du Bois-Marie SGA 4 1/2, Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier MR 0463174 (57 #3132).Google Scholar
Fontaine, Jean-Marc and Messing, William, p-adic periods and p-adic étale cohomology, in Current trends in arithmetical algebraic geometry (Arcata, Calif, 1985), Contemp Math, Volume 67, pp. 179–207 (Amer Math Soc, Providence, RI, 1987) MR 902593 (89g:14009).Google Scholar
Fulton, William, Intersection theory. second ed, (Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Folge A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas 3rd Series A Series of Modern Surveys in Mathematics], Volume 2), (Springer-Verlag, Berlin, 1998) MR 1644323 (99d:14003).Google Scholar
Gillet, Henri, Riemann-Roch theorems for higher algebraic K-theory, Adv. Math. 40(3) (1981), 203–289MR 624666 (83m:14013).Google Scholar
Grosse-Klönne, Elmar, Rigid analytic spaces with overconvergent structure sheaf, J. Reine Angew. Math. 519 (2000), 73–95MR 1739729 (2001b:14033).Google Scholar
Gros, Michel, Régulateurs syntomiques et valeurs de fonctions L p-adiques I, Invent. Math. 99(2) (1990), 293–320 With an appendix by Masato Kurihara MR 1031903 (91e:11070).Google Scholar
Holmstrom, Andreas and Scholbach, Jacob, Arakelov motivic cohomology I, arXiv:1012.2523v2 (2010).Google Scholar
Hinich, Vladimir and Schechtman, Vadim, On homotopy limit of homotopy algebras, in K-theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Math, Volume 1289, pp. 240–264 (Springer, Berlin, 1987) MR 923138 (89d:55052).Google Scholar
Hovey, Mark, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165(1) (2001), 63–127.Google Scholar
Hbl, Reinhold and Yekutieli, Amnon, Adelic Chern forms and application, Amer. J. Math. 121(4) (1999), 797–839.Google Scholar
Jannsen, Uwe, Mixed motives and algebraic K-theory, Lecture Notes in Mathematics, Volume 1400 (Springer-Verlag, Berlin, 1990).CrossRefGoogle Scholar
Langer, Andreas, On the syntomic regulator for products of elliptic curves, J. Lond. Math. Soc. (2) 84(2) (2011), 495–513MR 2835341 (2012k:19005).Google Scholar
Levine, Marc,$K$-theory and motivic cohomology of schemes, I, http://wwwuni-duede/%7Ebm0032/publ/KthyMotI1201pdf, 2004.Google Scholar
Le Stum, Bernard, Cohomologie rigide et variétés abéliennes, C. R. Acad. Sci. Paris Sér. I Math. 303(20) (1986), 989–992.Google Scholar
Neeman, Amnon, Triangulated categories, Annals of Mathematics Studies, Volume 148 (Princeton University Press, Princeton, NJ, 2001).Google Scholar
Panin, Ivan, Oriented cohomology theories of algebraic varieties II (After I Panin and A Smirnov), Homology, Homotopy Appl. 11(1) (2009), 349–405MR 2529164 (2011c:14063).Google Scholar
Soulé, Christophe, Régulateurs, Astérisque(133–134) (1986), 237–253 Seminar Bourbaki, Vol. 1984/85 MR 837223 (87g:11158).Google Scholar
Tamme, Georg, Karoubi’s relative Chern character, the rigid syntomic regulator, and the Bloch-Kato exponential map, http://arxivorg/abs/11114109v1, 2011.Google Scholar
Voevodsky, Vladimir, Homology of schemes, Selecta Math. (N.S.) 2(1) (1996), 111–153MR 1403354 (98c:14016).Google Scholar