Comparing motives of smooth algebraic varieties

Grigory Garkusha

doi:10.1016/j.crma.2018.11.006

Homological algebra/Algebraic geometry

Comparing motives of smooth algebraic varieties
[Comparaison des motifs de variétés algébriques lisses]

Présenté par : Claire Voisin

Grigory Garkusha ¹

¹ Department of Mathematics, Swansea University, Fabian Way, Swansea SA1 8EN, UK

Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1100-1105.

Résumé
Abstract

Étant donné un corps parfait de caractéristique exponentielle e, nous montrons que les Cor-, $K_{0}^{\oplus}$ -, $K_{0}$ - et $K_{0}$ -motifs des variétés algébriques lisses à coefficients dans $Z [1 / e]$ sont localement quasi isomorphes deux à deux. De plus, nous démontrons que leurs catégories triangulées de motifs à coefficients dans $Z [1 / e]$ sont équivalentes. Une application est donnée pour la suite spectrale motivique bivariante.

Given a perfect field of exponential characteristic e, the Cor-, $K_{0}^{\oplus}$ -, $K_{0}$ - and $K_{0}$ -motives of smooth algebraic varieties with $Z [1 / e]$ -coefficients are shown to be locally quasi-isomorphic to each other. Moreover, it is proved that their triangulated categories of motives with $Z [1 / e]$ -coefficients are equivalent. An application is given for the bivariant motivic spectral sequence.

Reçu le : 2017-10-10
Accepté le : 2018-11-08
Publié le : 2018-11-01

DOI : 10.1016/j.crma.2018.11.006

Affiliations des auteurs :

Grigory Garkusha ¹

¹ Department of Mathematics, Swansea University, Fabian Way, Swansea SA1 8EN, UK

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     title = {Comparing motives of smooth algebraic varieties},
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     pages = {1100--1105},
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     volume = {356},
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     doi = {10.1016/j.crma.2018.11.006},
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Grigory Garkusha. Comparing motives of smooth algebraic varieties. Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1100-1105. doi : 10.1016/j.crma.2018.11.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.11.006/

Bibliographie
Cité par

[1] G. Garkusha Reconstructing rational stable motivic homotopy theory (preprint) | arXiv

[2] G. Garkusha; I. Panin K-motives of algebraic varieties, Homol. Homotopy Appl., Volume 14 (2012) no. 2, pp. 211-264

[3] G. Garkusha; I. Panin The triangulated category of K-motives $D K_{-}^{eff} (k)$ , J. K-Theory, Volume 14 (2014) no. 1, pp. 103-137

[4] G. Garkusha; I. Panin On the motivic spectral sequence, J. Inst. Math. Jussieu, Volume 17 (2018) no. 1, pp. 137-170

[5] D. Grayson Weight filtrations via commuting automorphisms, K-Theory, Volume 9 (1995), pp. 139-172

[6] P. Hu On the Pickard group of the $A^{1}$ -stable homotopy category, Topology, Volume 44 (2005), pp. 609-640

[7] J.F. Jardine Fields Lectures: Presheaves of Spectra, 2007 www-home.math.uwo.ca/~jardine/papers/Fields-02.pdf (Also available online at)

[8] A. Suslin On the Grayson spectral sequence, Tr. Mat. Inst. Steklova, Volume 241 (2003) no. 2, pp. 218-253 (Russian). English transl. in Proc. Steklov Inst. Math., 241, 2003, pp. 202-237

[9] A. Suslin; V. Voevodsky Bloch–Kato conjecture and motivic cohomology with finite coefficients, Banff, Alberta, Canada, June 7–19, 1998 (B.B. Gordon; J.D. Lewis; S. Müller-Stach; S. Saito; N. Yui, eds.) (Nato Science Series C Math. Phys. Sci.), Volume vol. 548, Kluwer Academic Publishers, Dordrecht, The Netherlands (2000), pp. 117-189

[10] V. Voevodsky Triangulated category of motives over a field (V. Voevodsky; A. Suslin; E. Friedlander, eds.), Cycles, Transfers and Motivic Homology Theories, Annals of Mathematics Studies, Princeton University Press, 2000

[11] M.E. Walker Motivic Cohomology and the K-Theory of Automorphisms, University of Illinois at Urbana-Champaign, IL, USA, 1996 (PhD Thesis)

[12] M.E. Walker Thomason's theorem for varieties over algebraically closed fields, Trans. Amer. Math. Soc., Volume 356 (2003) no. 7, pp. 2569-2648

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