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Affine quadrics and the Picard group of the motivic category

  • Alexander Vishik (a1)

Abstract

In this paper we study the subgroup of the Picard group of Voevodsky’s category of geometric motives $\operatorname{DM}_{\text{gm}}(k;\mathbb{Z}/2)$ generated by the reduced motives of affine quadrics. Our main tools here are the functors of Bachmann [On the invertibility of motives of affine quadrics, Doc. Math. 22 (2017), 363–395], but we also provide an alternative method. We show that the group in question can be described in terms of indecomposable direct summands in the motives of projective quadrics over $k$ . In particular, we describe all the relations among the reduced motives of affine quadrics. We also extend the criterion of motivic equivalence of projective quadrics.

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[Bac17] Bachmann, T., On the invertibility of motives of affine quadrics , Doc. Math. 22 (2017), 363395.
[Bac18] Bachmann, T., Motivic and real étale stable homotopy theory , Compos. Math. 154 (2018), 883917.
[BV18] Bachmann, T. and Vishik, A., Motivic equivalence of affine quadrics , Math. Ann. 371 (2018), 741751.
[CM06] Chernousov, V. and Merkurjev, A., Motivic decomposition of projective homogeneous varieties and the Krull-Schmidt theorem , Transform. Groups 11 (2006), 371386.
[EKM08] Elman, R., Karpenko, N. and Merkurjev, A., The algebraic and geometric theory of quadratic forms , Amer. Math. Soc. Colloq. Publ. 56 (2008), 435 pp.
[Har77] Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977).
[Hu05] Hu, P., On the Picard group of the stable A1 -homotopy theory , Topology 44 (2005), 609640.
[Kar00] Karpenko, N., Criteria of motivic equivalence for quadratic forms and central simple algebras , Math. Ann. 317 (2000), 585611.
[Lam05] Lam, T. Y., Introduction to quadratic forms over fields, Graduate Studies in Mathematics, vol. 67 (American Mathematical Society, Providence, RI, 2005).
[Mor04] Morel, F., On the motivic 𝜋0 of the sphere spectrum , in Axiomatic, enriched and motivic homotopy theory (Kluwer, Dordrecht, 2004), 219260.
[Ros90] Rost, M., Some new results on the Chow groups of quadrics, Preprint (1990), www.math.uni-bielefeld.de/∼rost/chowqudr.html.
[Ros98] Rost, M., The motive of a Pfister form, Preprint (1998), www.math.uni-bielefeld.de/∼rost/motive.html.
[Vis98] Vishik, A., Integral motives of quadrics, MPIM Preprint 1998-13 (1998), www.mpim-bonn.mpg.de/node/263.
[Vis04] Vishik, A., Motives of quadrics with applications to the theory of quadratic forms , in Geometric methods in the algebraic theory of quadratic forms: Summer School, Lens, 2000, Lecture Notes in Mathematics, vol. 1835, ed. Tignol, J.-P. (Springer, Berlin, 2004), 25101.
[Vis11] Vishik, A., Excellent connections in the motives of quadrics , Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), 183195.
[Voe95] Voevodsky, V., Bloch-Kato conjecture for $\mathbb{Z}/2$ -coefficients and algebraic Morava K-theories, Preprint (1995), www.math.uiuc.edu/K-theory/0076.
[Voe00] 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), 188238.
[Voe03] Voevodsky, V., Motivic cohomology with ℤ/2-coefficients , Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59104.
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Affine quadrics and the Picard group of the motivic category

  • Alexander Vishik (a1)

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