Skip to main content Accessibility help
×
Home

Étale motives

  • Denis-Charles Cisinski (a1) and Frédéric Déglise (a2)

Abstract

We define a theory of étale motives over a noetherian scheme. This provides a system of categories of complexes of motivic sheaves with integral coefficients which is closed under the six operations of Grothendieck. The rational part of these categories coincides with the triangulated categories of Beilinson motives (and is thus strongly related to algebraic $K$ -theory). We extend the rigidity theorem of Suslin and Voevodsky over a general base scheme. This can be reformulated by saying that torsion étale motives essentially coincide with the usual complexes of torsion étale sheaves (at least if we restrict ourselves to torsion prime to the residue characteristics). As a consequence, we obtain the expected results of absolute purity, of finiteness, and of Grothendieck duality for étale motives with integral coefficients, by putting together their counterparts for Beilinson motives and for torsion étale sheaves. Following Thomason’s insights, this also provides a conceptual and convenient construction of the $\ell$ -adic realization of motives, as the homotopy $\ell$ -completion functor.

Copyright

References

Hide All
[Ayo07]Ayoub, J., Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (I, II), Astérisque 314, 315 (2007).
[Ayo14]Ayoub, J., La réalisation étale et les opérations de Grothendieck, Ann. Sci. École Norm. Sup. (4) 47 (2014), 1145.
[BBD82]Beilinson, A. A., Bernstein, J. and Deligne, P., Faisceaux pervers, Astérisque 100 (1982), 5171.
[Bei87]Beilinson, A. A., Height pairing between algebraic cycles, in K-theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Mathematics, vol. 1289 (Springer, Berlin, 1987), 125.
[Bon13]Bondarko, M. V., On weights for relative motives with integral coefficients, Preprint (2013),arXiv:1304.2335 [math.AG].
[Bon14]Bondarko, M. V., Weights for relative motives: relation with mixed complexes of sheaves, Int. Math. Res. Not. IMRN 2014 (2014), 47154767.
[BS01]Balmer, P. and Schlichting, M., Idempotent completion of triangulated categories, J. Algebra 236 (2001), 819834.
[Cis13]Cisinski, D.-C., Descente par éclatements en K-théorie invariante par homotopie, Ann. of Math. (2) 177 (2013), 425448.
[CD09]Cisinski, D.-C. and Déglise, F., Local and stable homological algebra in Grothendieck abelian categories, Homology, Homotopy Appl. 11 (2009), 219260.
[CD12]Cisinski, D.-C. and Déglise, F., Triangulated categories of mixed motives, Preprint (2012),arXiv:0912.2110v3 [math.AG].
[CD15]Cisinski, D.-C. and Déglise, F., Integral mixed motives in equal characteristic, Doc. Math. (2015), to appear, arXiv:1410.6359 [math.AG].
[Con07]Conrad, B., Deligne’s notes on Nagata compactifications, J. Ramanujan Math. Soc. 22 (2007), 205257.
[Dég07]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 (Cambridge University Press, Cambridge, 2007), 138205.
[Dég08a]Déglise, F., Around the Gysin triangle II, Doc. Math. 13 (2008), 613675.
[Dég08b]Déglise, F., Motifs génériques, Rend. Semin. Mat. Univ. Padova 119 (2008), 173244.
[Dég12]Déglise, F., Coniveau filtration and motives, in Regulators, Contemporary Mathematics, vol. 571 (American Mathematical Society, Providence, RI, 2012), 5176.
[Dég14]Déglise, F., Orientation theory in arithmetic geometry, Preprint (2014), http://perso.ens-lyon.fr/frederic.deglise/docs/2014/RR.pdf.
[Del01]Deligne, P., Voevodsky lectures on cross functors, Notes, http://www.math.ias.edu/∼vladimir/seminar.html (2001).
[DG02]Dwyer, W. G. and Greenlees, J. P. C., Complete modules and torsion modules, Amer. J. Math. 124 (2002), 199220.
[EGA2]Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Publ. Math. Inst. Hautes Études Sci. 8 (1961).
[EGA4]Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Publ. Math. Inst. Hautes Études Sci. 20, 24, 28, 32 (1964–1967).
[Eke90]Ekedahl, T., On the adic formalism, in The Grothendieck Festschrift, Vol. II, Progress in Mathematics, vol. 87 (Birkhäuser, Boston, MA, 1990), 197218.
[Fuj02]Fujiwara, K., A proof of the absolute purity conjecture (after Gabber), in Algebraic geometry 2000, Azumino (Hotaka), Advanced Studies in Pure Mathematics, vol. 36 (Mathematical Society of Japan, Tokyo, 2002), 153183.
[GL00]Geisser, T. and Levine, M., The K-theory of fields in characteristic p, Invent. Math. 139 (2000), 459493.
[GL01a]Geisser, T. and Levine, M., The Bloch–Kato conjecture and a theorem of Suslin–Voevodsky, J. Reine Angew. Math. 530 (2001), 55103.
[GL01b]Goodwillie, T. G. and Lichtenbaum, S., A cohomological bound for the h-topology, Amer. J. Math. 123 (2001), 425443.
[Héb11]Hébert, D., Structure de poids à la Bondarko sur les motifs de Beilinson, Compos. Math. 147 (2011), 14471462.
[Hov01]Hovey, M., Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165 (2001), 63127.
[ILO14]Illusie, L., Lazslo, Y. and Orgogozo, F., Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents, Astérisque 361, 362 (2014).
[Jan88]Jannsen, U., Continuous étale cohomology, Math. Ann. 280 (1988), 207245.
[Jan90]Jannsen, U., Mixed motives and algebraic K-theory, Lecture Notes in Mathematics, vol. 1400 (Springer, Berlin, 1990).
[Lic84]Lichtenbaum, S., Values of zeta-functions at nonnegative integers, in Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), Lecture Notes in Mathematics, vol. 1068 (Springer, Berlin, 1984), 127138.
[McC01]McCleary, J., A user’s guide to spectral sequences, Cambridge Studies in Advanced Mathematics, vol. 58, second edition (Cambridge University Press, Cambridge, 2001).
[MVW06]Mazza, C., Voevodsky, V. and Weibel, C., Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2 (American Mathematical Society, Providence, RI, 2006).
[Mor11]Morel, F., On the Friedlander–Milnor conjecture for groups of small rank, Current developments in mathematics, 2010 (International Press, Somerville, MA, 2011), 4593.
[Nee01]Neeman, A., Triangulated categories, Annals of Mathematics Studies, vol. 148 (Princeton University Press, Princeton, NJ, 2001).
[Qui73]Quillen, D., Higher algebraic K-theory, in Higher K-theories I (Proc. Conf., Battelle Memorial Inst., Seattle, WA, 1972), Lecture Notes in Mathematics, vol. 341 (Springer, 1973), 85147.
[RS14]Rosenschon, A. and Srinivas, V., Étale motivic cohomology and algebraic cycles, J. Inst. Math. Jussieu (2014), doi:10.1017/S1474748014000401.
[Ryd10]Rydh, D., Submersions and effective descent of étale morphisms, Bull. Soc. Math. France 138 (2010), 181230.
[SGA1]Grothendieck, A. and Dieudonné, J., Revêtements étales et groupe fondamental, Documents mathématiques, vol. 3, édition recomposée et annotée (Société mathématique de France, Paris, 2003); Séminaire de Géométrie Algébrique du Bois–Marie 1960–61 (SGA 1), original edition published as Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, vol. 224 (Springer, Berlin, 1971).
[SGA4]Artin, M., Grothendieck, A. and Verdier, J.-L., Théorie des topos et cohomologie étale des schémas, Lecture Notes in Mathematics, vols 269, 270, 305 (Springer, Berlin, 1972–1973); Séminaire de Géométrie Algébrique du Bois–Marie 1963–64 (SGA 4).
[SGA41/2]Deligne, P., Cohomologie étale, Lecture Notes in Mathematics, vol. 569 (Springer, Berlin, 1977); Séminaire de Géométrie Algébrique du Bois–Marie $\text{SGA}4{\textstyle \frac{1}{2}}$.
[SGA5]Grothendieck, A., Cohomologie -adique et fonctions L, Lecture Notes in Mathematics, vol. 589 (Springer, Berlin, 1977); Séminaire de Géométrie Algébrique du Bois–Marie 1965–66 (SGA 5).
[SV96]Suslin, A. and Voevodsky, V., Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996), 6194.
[SV00a]Suslin, A. and Voevodsky, V., Bloch–Kato conjecture and motivic cohomology with finite coefficients, NATO Sciences Series, Series C: Mathematical and Physical Sciences, vol. 548 (Kluwer, Dordrecht, 2000), 117189.
[SV00b]Suslin, A. and Voevodsky, V., Relative cycles and Chow sheaves, Annals of Mathematics Studies, vol. 143 (Princeton University Press, 2000), 1086.
[Voe96]Voevodsky, V., Homology of schemes, Selecta Math. (N.S.) 2 (1996), 111153.
[Voe02]Voevodsky, V., Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not. IMRN 7 (2002), 351355.
[Voe11]Voevodsky, V., On motivic cohomology with Zl-coefficients, Ann. of Math. (2) 174 (2011), 401438.
[VSF00]Voevodsky, V., Suslin, A. and Friedlander, E. M., Cycles, transfers and motivic homology theories, Annals of Mathematics Studies, vol. 143 (Princeton University Press, 2000).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Étale motives

  • Denis-Charles Cisinski (a1) and Frédéric Déglise (a2)

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed