Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T08:06:15.282Z Has data issue: false hasContentIssue false
  • English
  • Français

Reciprocity sheaves

Part of: $K$-theory in geometry $K$-theory in number theory (Co)homology theory Higher algebraic $K$-theory

Published online by Cambridge University Press:  14 July 2016

Bruno Kahn ,
Shuji Saito ,
Takao Yamazaki  and
Kay Rülling
Bruno Kahn
Affiliation:
IMJ-PRG, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, France email [email protected]
Shuji Saito
Affiliation:
Interactive Research Center of Science, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1 Okayama, Meguro, Tokyo 152-8551, Japan email [email protected]
Takao Yamazaki
Affiliation:
Institute of Mathematics, Tohoku University, Aoba, Sendai 980-8578, Japan email [email protected]
Kay Rülling
Affiliation:
Freie Universität Berlin, Arnimallee 7, 14195 Berlin, Germany 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.

We start developing a notion of reciprocity sheaves, generalizing Voevodsky’s homotopy invariant presheaves with transfers which were used in the construction of his triangulated categories of motives. We hope that reciprocity sheaves will eventually lead to the definition of larger triangulated categories of motivic nature, encompassing non-homotopy invariant phenomena.

Keywords

presheaves with transfershomotopy invarianceWeil reciprocity

MSC classification

Primary: 19E15: Algebraic cycles and motivic cohomology
Secondary: 14F42: Motivic cohomology; motivic homotopy theory 19D45: Higher symbols, Milnor $K$-theory 19F15: Symbols and arithmetic
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 9 , September 2016 , pp. 1851 - 1898
Copyright
© The Authors 2016 

References

Barbieri-Viale, L. and Kahn, B., On the derived category of 1 motives , Astérisque 381 (2016), 1–254.Google Scholar
Binda, F. and Saito, S., Relative cycles with moduli and regulator maps, Preprint (2014),arXiv:1412.0385.Google Scholar
Bourbaki, N., Algèbre commutative (Masson, Paris, 1985).Google Scholar
Chatzistamatiou, A. and Rülling, K., Higher direct images of the structure sheaf in positive characteristic , Algebra Number Theory 5 (2011), 693–775.Google Scholar
Chatzistamatiou, A. and Rülling, K., Hodge–Witt cohomology and Witt-rational singularities , Doc. Math. 17 (2012), 663–781.Google Scholar
Colliot-Thélène, J.-L., Hoobler, R. and Kahn, B., The Bloch–Ogus–Gabber theorem, Fields Institute Communications, vol. 16 (American Mathematical Society, Providence, RI, 1997), 31–94.Google Scholar
Colliot-Thélène, J. L., Sansuc, J.-J. and Soulé, C., Torsion dans le groupe de Chow de codimension deux , Duke Math. J. 50 (1983), 763–801.Google Scholar
Déglise, F., Finite correspondences and transfers over a regular base , in Algebraic cycles and motives, vol. 1, London Mathematical Society Lecture Note Series, vol. 343, eds Nagel, J. and Peters, C. (Cambridge University Press, Cambridge, 2007), 138–205.Google Scholar
Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique: étude globale élémentaire de quelques classes de morphismes , Publ. Math. Inst. Hautes Études Sci. 8 (1961), 5–222.Google Scholar
Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique: étude locale des schémas et des morphismes de schémas (quatrième partie) , Publ. Math. Inst. Hautes Études Sci. 32 (1967), 5–361.Google Scholar
Ekedahl, T., On the multiplicative properties of the de Rham–Witt complex I , Ark. Mat. 22 (1984), 185–239.Google Scholar
Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, vol. 2 (Springer, Berlin, 1998).Google Scholar
Grayson, D., Universal exactness in algebraic K-theory , J. Pure Appl. Algebra 36 (1985), 139–141.Google Scholar
Gros, M., Classes de Chern et classes de cycles en cohomologie de Hodge–Witt logarithmique , Mém. Soc. Math. Fr. (N.S.) 21 (1985), 1–87.Google Scholar
Illusie, L., Complexe de de Rham–Witt et cohomologie cristalline , Ann. Sci. Éc. Norm. Supér. (4) 12 (1979), 501–661.Google Scholar
Ivorra, F. and Rülling, K., $K$ -groups of reciprocity functors, J. Algebraic Geom., to appear, Preprint (2012), arXiv:1209.1217.Google Scholar
Kahn, B., Foncteurs de Mackey à réciprocité, Preprint (1991), arXiv:1210.7577.Google Scholar
Kerz, M. and Saito, S., Chow group of 0-cycles with modulus and higher dimensional class field theory, Duke Math. J., to appear, Preprint (2011), arXiv:1304.4400.Google Scholar
Mazza, C., Voevodsky, V. and Weibel, C., Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2 (American Mathematical Society, Providence, RI, 2006); Clay Mathematics Institute, Cambridge, MA.Google Scholar
Rülling, K., The generalized de Rham–Witt complex over a field is a complex of zero-cycles , J. Algebraic Geom. 16 (2007), 109–169.Google Scholar
Russell, H., Albanese varieties with modulus over a perfect field , Algebra Number Theory 7 (2013), 853–892.Google Scholar
Serre, J.-P., Groupes algébriques et corps de classes (Hermann, Paris, 1959).Google Scholar
Spiess, M. and Szamuely, T., On the Albanese map for smooth quasiprojective varieties , Math. Ann. 235 (2003), 1–17.Google Scholar
Suslin, A. and Voevodsky, V., Singular homology of abstract algebraic varieties , Invent. Math. 123 (1996), 61–94.Google Scholar
Suslin, A. and Voevodsky, V., Relative cycles and Chow sheaves , in Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000), 10–85.Google Scholar
Voevodsky, V., Cohomological theory of presheaves with transfers , in Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000), 87–137.Google Scholar
Voevodsky, V., Triangulated categories of motives over a field , in Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000), 188–238.Google Scholar