Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Arakelov motivic cohomology I


Authors: Andreas Holmstrom and Jakob Scholbach
Journal: J. Algebraic Geom. 24 (2015), 719-754
DOI: https://doi.org/10.1090/jag/648
Published electronically: April 23, 2015
MathSciNet review: 3383602
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Abstract | References | Additional Information

Abstract: This paper introduces a new cohomology theory for schemes of finite type over an arithmetic ring. The main motivation for this Arakelov-theoretic version of motivic cohomology is the conjecture on special values of $ L$-functions and zeta functions formulated by the second author. Taking advantage of the six functors formalism in motivic stable homotopy theory, we establish a number of formal properties, including pullbacks for arbitrary morphisms, pushforwards for projective morphisms between regular schemes, localization sequences, $ h$-descent. We round off the picture with a purity result and a higher arithmetic Riemann-Roch theorem.

In a sequel to this paper, we relate Arakelov motivic cohomology to classical constructions such as arithmetic $ K$ and Chow groups and the height pairing.


References [Enhancements On Off] (What's this?)

  • [Ayo07] Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I, Astérisque 314 (2007), x+466 pp. (2008) (French, with English and French summaries). MR 2423375 (2009h:14032)
  • [Beĭ87] A. A. Beĭlinson,
    Height pairing between algebraic cycles, $ K$-theory, arithmetic and geometry (Moscow, 1984-1986), volume 1289 of Lecture Notes in Math., Springer, Berlin, 1987, pp. 1-25. MR 0923131 (89h:11027)
  • [BGF12] José Ignacio Burgos Gil and Elisenda Feliu, Higher arithmetic Chow groups, Comment. Math. Helv. 87 (2012), no. 3, 521-587. MR 2980520, https://doi.org/10.4171/CMH/262
  • [BGKK07] J. I. Burgos Gil, J. Kramer, and U. Kühn, Cohomological arithmetic Chow rings, J. Inst. Math. Jussieu 6 (2007), no. 1, 1-172. MR 2285241 (2008f:14039)
  • [BS09] Ulrich Bunke and Thomas Schick, Smooth $ K$-theory, Astérisque 328 (2009), 45-135 (2010) (English, with English and French summaries). MR 2664467 (2012a:19015)
  • [Bur94] José Ignacio Burgos, A $ C^\infty $ logarithmic Dolbeault complex, Compositio Math. 92 (1994), no. 1, 61-86. MR 1275721 (95g:32056)
  • [Bur97] Jose Ignacio Burgos, Arithmetic Chow rings and Deligne-Beilinson cohomology, J. Algebraic Geom. 6 (1997), no. 2, 335-377. MR 1489119 (99d:14015)
  • [BW98] Jose Ignacio Burgos and Steve Wang, Higher Bott-Chern forms and Beilinson's regulator, Invent. Math. 132 (1998), no. 2, 261-305. MR 1621424 (99j:14008), https://doi.org/10.1007/s002220050224
  • [CD09] Denis-Charles Cisinski and Frédéric Déglise,
    Triangulated categories of mixed motives, 2009.
  • [CD12] Denis-Charles Cisinski and Frédéric Déglise, Mixed Weil cohomologies, Adv. Math. 230 (2012), no. 1, 55-130. MR 2900540, https://doi.org/10.1016/j.aim.2011.10.021
  • [Dég08] Frédéric Déglise, Around the Gysin triangle. II, Doc. Math. 13 (2008), 613-675. MR 2466188 (2009m:14025)
  • [Dég11] Frédéric Déglise,
    Orientation theory in the arithmetic case (in preparation),
    2011.
  • [Del71] Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5-57 (French). MR 0498551 (58 #16653a)
  • [Den94] Christopher Deninger, Motivic $ L$-functions and regularized determinants, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 707-743. MR 1265547 (94m:11077)
  • [EV88] Hélène Esnault and Eckart Viehweg,
    Deligne-Beĭlinson cohomology, Beĭlinson's conjectures on special values of $ L$-functions, volume 4 of Perspect. Math., Academic Press, Boston, MA, 1988, pp. 43-91.MR 0944991 (89k:14008)
  • [Fel10] Elisenda Feliu, Adams operations on higher arithmetic $ K$-theory, Publ. Res. Inst. Math. Sci. 46 (2010), no. 1, 115-169. MR 2662616 (2011h:14030), https://doi.org/10.2977/PRIMS/3
  • [FL85] William Fulton and Serge Lang, Riemann-Roch algebra, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 277, Springer-Verlag, New York, 1985. MR 801033 (88h:14011)
  • [FM12] M. Flach and B. Morin, On the Weil-étale topos of regular arithmetic schemes, Doc. Math. 17 (2012), 313-399. MR 2946826
  • [Gon05] Alexander B. Goncharov, Regulators, Handbook of $ K$-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 295-349. MR 2181826 (2006j:11092), https://doi.org/10.1007/3-540-27855-9_8
  • [GRS08] Henri Gillet, Damian Rössler, and Christophe Soulé, An arithmetic Riemann-Roch theorem in higher degrees, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 6, 2169-2189 (English, with English and French summaries). MR 2473633 (2010b:14048)
  • [GS90a] Henri Gillet and Christophe Soulé, Arithmetic intersection theory, Inst. Hautes Études Sci. Publ. Math. 72 (1990), 93-174 (1991). MR 1087394 (92d:14016)
  • [GS90b] Henri Gillet and Christophe Soulé, Characteristic classes for algebraic vector bundles with Hermitian metric. I, Ann. of Math. (2) 131 (1990), no. 1, 163-203. MR 1038362 (91m:14032a), https://doi.org/10.2307/1971512
  • [GS90c] Henri Gillet and Christophe Soulé, Characteristic classes for algebraic vector bundles with Hermitian metric. II, Ann. of Math. (2) 131 (1990), no. 2, 205-238. MR 1043268 (91m:14032b), https://doi.org/10.2307/1971493
  • [Hir03] Philip S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003. MR 1944041 (2003j:18018)
  • [Hub00] Annette Huber, Realization of Voevodsky's motives, J. Algebraic Geom. 9 (2000), no. 4, 755-799. MR 1775312 (2002d:14029)
  • [Jar00] J. F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445-553 (electronic). MR 1787949 (2002b:55014)
  • [Jos06] Jürgen Jost, Compact Riemann surfaces, An introduction to contemporary mathematics, 3rd ed., Universitext, Springer-Verlag, Berlin, 2006. MR 2247485 (2007b:32024)
  • [Mor11] Baptiste Morin, Zeta functions of regular arithmetic schemes at $ s=0$, Duke Math. J. 163 (2014), no. 7, 1263-1336. MR 3205726, https://doi.org/10.1215/00127094-2681387
  • [MV99] Fabien Morel and Vladimir Voevodsky, $ {\bf A}^1$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45-143 (2001). MR 1813224 (2002f:14029)
  • [Nek94] Jan Nekovář, Beĭlinson's conjectures, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 537-570. MR 1265544 (95f:11040)
  • [Rio07] Joël Riou, Opérations sur la $ K$-théorie algébrique et régulateurs via la théorie homotopique des schémas, C. R. Math. Acad. Sci. Paris 344 (2007), no. 1, 27-32 (French, with English and French summaries). MR 2286583 (2007k:19006), https://doi.org/10.1016/j.crma.2006.11.011
  • [Rio10] Joël Riou, Algebraic $ K$-theory, $ {\bf A}^1$-homotopy and Riemann-Roch theorems, J. Topol. 3 (2010), no. 2, 229-264. MR 2651359 (2011f:19001), https://doi.org/10.1112/jtopol/jtq005
  • [Roe99] Damian Roessler, An Adams-Riemann-Roch theorem in Arakelov geometry, Duke Math. J. 96 (1999), no. 1, 61-126. MR 1663919 (2000a:14029), https://doi.org/10.1215/S0012-7094-99-09603-5
  • [RSØ10] Oliver Röndigs, Markus Spitzweck, and Paul Arne Østvær, Motivic strict ring models for $ K$-theory, Proc. Amer. Math. Soc. 138 (2010), no. 10, 3509-3520. MR 2661551 (2011h:14024), https://doi.org/10.1090/S0002-9939-10-10394-3
  • [Sch12] Jakob Scholbach, Arakelov motivic cohomology II, J. Algebraic Geom. (to appear).
  • [Sch13] Jakob Scholbach,
    Special $ L$-values of geometric motives.
    Preprint (Feb. 2013), available at http://arxiv.org/abs/1003.1215.
  • [Sou92] C. Soulé, Lectures on Arakelov geometry, With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer, Cambridge Studies in Advanced Mathematics, vol. 33, Cambridge University Press, Cambridge, 1992. MR 1208731 (94e:14031)
  • [Tak05] Yuichiro Takeda, Higher arithmetic $ K$-theory, Publ. Res. Inst. Math. Sci. 41 (2005), no. 3, 599-681. MR 2153537 (2006i:14022)


Additional Information

Andreas Holmstrom
Affiliation: Institut des Hautes Études Scientifiques Le Bois-Marie, 35 Route de Chartres, F-91440 Bures-sur-Yvette, France
Email: andreas.holmstrom@gmail.com

Jakob Scholbach
Affiliation: Universität Münster, Mathematisches Institut, Einsteinstrasse 62, D-48149 Münster, Germany
Email: jakob.scholbach@uni-muenster.de

DOI: https://doi.org/10.1090/jag/648
Received by editor(s): October 10, 2012
Received by editor(s) in revised form: June 26, 2013
Published electronically: April 23, 2015
Article copyright: © Copyright 2015 University Press, Inc.

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