Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-24T13:04:29.123Z Has data issue: false hasContentIssue false
  • English
  • Français

Chow motives without projectivity

Part of: Abelian categories (Co)homology theory Arithmetic problems. Diophantine geometry $K$-theory in geometry

Published online by Cambridge University Press:  04 September 2009

Jörg Wildeshaus
Jörg Wildeshaus*
Affiliation:
LAGA, UMR 7539, Institut Galilée, Université Paris 13, Avenue Jean-Baptiste Clément, F-93430 Villetaneuse, France (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In a recent paper, Bondarko [Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), Preprint (2007), 0704.4003] defined the notion of weight structure, and proved that the category DMgm(k) of geometrical motives over a perfect field k, as defined and studied by Voevodsky, Suslin and Friedlander [Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000)], is canonically equipped with such a structure. Building on this result, and under a condition on the weights avoided by the boundary motive [J. Wildeshaus, The boundary motive: definition and basic properties, Compositio Math. 142 (2006), 631–656], we describe a method to construct intrinsically in DMgm(k) a motivic version of interior cohomology of smooth, but possibly non-projective schemes. In a sequel to this work [J. Wildeshaus, On the interior motive of certain Shimura varieties: the case of Hilbert–Blumenthal varieties, Preprint (2009), 0906.4239], this method will be applied to Shimura varieties.

Keywords

weight structuresweight filtrationsChow motivesgeometrical motivesmotives for modular formsboundary motiveinterior motive

MSC classification

Secondary: 14F42: Motivic cohomology; motivic homotopy theory 14F20: Étale and other Grothendieck topologies and (co)homologies 14F25: Classical real and complex (co)homology 14F30: $p$-adic cohomology, crystalline cohomology 14G35: Modular and Shimura varieties 18E30: Derived categories, triangulated categories 19E15: Algebraic cycles and motivic cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 5 , September 2009 , pp. 1196 - 1226
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]André, Y., Une introduction aux motifs, Panoramas et Synthèses, vol. 17 (Société Mathématique de France, Paris, 2004).Google Scholar
[2]Balmer, P. and Schlichting, M., Idempotent completion of triangulated categories, J. Algebra 236 (2001), 819–834.CrossRefGoogle Scholar
[3]Beilinson, A. A., Higher regulators of modular curves, in Applications of algebraic K-theory to algebraic geometry and number theory. Proceedings of the AMS–IMS–SIAM joint summer research conference held June 12–18, 1983, Contemporary Mathematics, vol. 55, eds S. J. Bloch, R. Keith Dennis, E. M. Friedlander and M. R. Stein (American Mathematical Society, Providence, RI, 1986), 1–34.Google Scholar
[4]Beilinson, A. A., Bernstein, J. and Deligne, P., Faisceaux pervers, in Analyse et topologie sur les espaces singuliers (I), Astérisque, vol. 100, eds B. Teissier and J. L. Verdier (Société Mathématique de France, Paris, 1982).Google Scholar
[5]Bondarko, M. V., Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), Preprint (2007), http://arXiv.org/abs/0704.4003.Google Scholar
[6]Bondarko, M. V., Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura, J. Inst. Math. Jussieu 8 (2009), 39–97.CrossRefGoogle Scholar
[7]Deligne, P., Formes modulaires et représentations ℓ-adiques, in Sém. Bourbaki, Exposé 355, Lecture Notes in Mathematics, vol. 179 (Springer, Berlin, 1969), 139–172.Google Scholar
[8]Deligne, P., Résumé des premiers exposés de A. Grothendieck, in Groupes de Monodromie en Géométrie Algébrique (SGA 7I), Lecture Notes in Mathematics, vol. 288 (Springer, Berlin, 1972), 1–24.CrossRefGoogle Scholar
[9]Deligne, P. and Goncharov, A. B., Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. École Norm. Sup. (4) 38 (2005), 1–56.CrossRefGoogle Scholar
[10]Friedlander, E. M. and Voevodsky, V., Bivariant cycle cohomology, in Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000).Google Scholar
[11]Ghate, E., Adjoint L-values and primes of congruence for Hilbert modular forms, Compositio Math. 132 (2002), 243–281.CrossRefGoogle Scholar
[12]Gillet, H. and Soulé, C., Descent, motives and K-theory, J. Reine Angew. Math. 478 (1996), 127–176.Google Scholar
[13]Guillén, F. and Navarro Aznar, V., Un critère d’extension des foncteurs définis sur les schémas lisses, Publ. Math. Inst. Hautes Études Sci. 95 (2002), 1–91.CrossRefGoogle Scholar
[14]Huber, A., Realization of Voevodsky’s motives, J. Algebraic Geom. 9 (2000), 755–799, Corrigendum, 13 (2004), 195–207.Google Scholar
[15]Kimura, S.-I., Chow groups are finite dimensional, in some sense, Math. Ann. 331 (2005), 173–201.CrossRefGoogle Scholar
[16]Lecomte, F., Réalisation de Betti des motifs de Voevodsky, C. R. Acad. Sci. Paris Ser. I Math. 346 (2008), 1083–1086.CrossRefGoogle Scholar
[17]Nizioł, W., Semistable conjecture via K-theory, Duke Math. J. 141 (2008), 151–178.CrossRefGoogle Scholar
[18]Rapoport, M. and Zink, Th., Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math. 68 (1982), 21–101.CrossRefGoogle Scholar
[19]Schneider, P., Introduction to the Beilinson conjectures, in Beilinson’s conjectures on special values of L-functions, Perspectives in Mathematics, vol. 4, eds M. Rapoport, N. Schappacher and P. Schneider (Academic Press, New York, 1988), 1–35.Google Scholar
[20]Scholl, A. J., Motives for modular forms, Invent. Math. 100 (1990), 419–430.CrossRefGoogle Scholar
[21]Scholl, A. J., L-functions of modular forms and higher regulators, in preparation (the version dated May 15, 1997 is available at http://www.dpmms.cam.ac.uk/∼ajs1005/mono/index.html).Google Scholar
[22]Serre, J.-P., Motifs, in Journées Arithmétiques (Luminy, 1989), Astérisque, vol. 198–200 (Société Mathématique de France, Paris, 1991), 333–349.Google Scholar
[23]Tsuji, T., p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999), 233–411.CrossRefGoogle Scholar
[24]Voevodsky, V., Triangulated categories of motives, in Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000).Google Scholar
[25]Voevodsky, V., Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not. 2002 (2002), 351–355.CrossRefGoogle Scholar
[26]Voevodsky, V., Suslin, A. and Friedlander, E. M., Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000).Google Scholar
[27]Wildeshaus, J., The boundary motive: definition and basic properties, Compositio. Math. 142 (2006), 631–656.CrossRefGoogle Scholar
[28]Wildeshaus, J., On the interior motive of certain Shimura varieties: the case of Hilbert–Blumenthal varieties, Preprint (2009), http://arXiv.org/abs/0906.4239, submitted.Google Scholar