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Keywords:
Grothendieck ring of varieties; assembler; motivic measure
Grothendieck ring of varieties; assembler; motivic measure
Summary:
The Grothendieck ring of varieties admits well-known realization maps to, say, mixed Hodge structures or compactly supported $\ell$-adic cohomology. These are often called motivic measures. Zakharevich and Campbell have developed a spectral refinement of the Grothendieck ring of varieties. We develop a realization map from their refinement to the $K$-theory of mixed motives. Composing the latter with the standard realizations of motives, this yields spectral refinements of all the usual motivic measures.
References:
[1] Beilinson, A., Vologodsky, V.: A DG guide to Voevodsky's motives. Geom. Funct. Anal. 17, no. 6, 1709–1787. MR 2399083 DOI 10.1007/s00039-007-0644-5
[2] Blumberg, A., Mandell, M.: Algebraic K-theory and Abstract Homotopy Theory. Adv. Math. 226, no. 4, 3760–3812. MR 2764905 DOI 10.1016/j.aim.2010.11.002
[3] Bondarko, M.: Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general. J. of K-theory 6, i. 03, p. 387–504 DOI 10.1017/is010012005jkt083
[4] Campbell, J.: The K-theory spectrum of varieties. Trans. Amer. Math. Soc. 371, no. 11, 7845–7884. MR 3955537 DOI 10.1090/tran/7648
[5] Cisinski, D.-C., Déglise, F.: Local and stable homological algebra in Grothendieck abelian categories. Homology Homotopy Appl. 11, no. 1, 219–260. MR 2529161 DOI 10.4310/HHA.2009.v11.n1.a11
[6] Cisinski, D.-C., Déglise, F.: Triangulated categories of mixed motives. Springer Monographs in Mathematics. Springer, Cham, 2019 MR 3971240
[7] Campbell, J., Wolfson, J., Zakharevich, I.: Derived $\ell$-adic zeta functions. Adv. Math. 354, 106760, 53. MR 3993931 DOI 10.1016/j.aim.2019.106760
[8] Campbell, J., Zakharevich, I.: Devissage and Localization for the Grothendieck Spectrum of Varieties. Adv. Math., 411, Paper No. 108710, 80. MR 4512395
[9] Déglise, F., Mazzari, N.: The rigid syntomic ring spectrum. J. Inst. Math. Jussieu 14, no. 4, 753–799. MR 3394127 DOI 10.1017/S1474748014000152
[10] Friedlander, E., Voevodsky, V.: Bivariant cycle cohomology. Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, NJ, pp. 138–187
[11] Gillet, H., Soulé, C.: Descent, motives and K-theory. Journal für die reine und angewandte Mathematik 478, pp. 127–176
[12] Huber, A.: Mixed motives and their realization in derived categories. Lecture Notes in Mathematics, vol. 1604, Springer-Verlag, Berlin, 1995. MR 1439046
[13] Huber, A.: Realization of Voevodsky's motives. J. Algebraic Geom. 9, no. 4, 755–799. MR 1775312
[14] Ivorra, F.: Réalisation $l$-adique des motifs triangulés géométriques. I. Doc. Math. 12, 607–671. MR 2377240
[15] Ivorra, F.: Perverse, Hodge and motivic realizations of étale motives. Compos. Math. 152, no. 6, 1237–1285. MR 3518311 DOI 10.1112/S0010437X15007812
[16] Kelly, S.: Voevodsky motives and $ldh$-descent. Astérisque, no. 391, 125. MR 3673293
[17] Lurie, J.: Higher Algebra. https://people.math.harvard.edu/~lurie/papers/HA.pdf
[18] Mazza, C., Voevodsky, V., Weibel, C.: Lecture notes on motivic cohomology. Clay Mathematics Monographs, vol. 2, American Mathematical Society, Providence, Clay Mathematics Institute, Cambridge, 2006. MR 2242284
[19] Nekovář, J., Nizioł, W.: Syntomic cohomology and $p$-adic regulators for varieties over $p$-adic fields. Algebra Number Theory 10, no. 8, 1695–1790, With appendices by Laurent Berger and Frédéric Déglise. MR 3556797
[20] Robalo, M.: K-theory and the bridge from motives to noncommutative motives. Advances in Mathematics 269, 399–550 DOI 10.1016/j.aim.2014.10.011
[21] Röndigs, O.: The Grothendieck ring of varieties and algebraic K-theory of spaces. Motivic Geometry, The Open Book Series, Vol. 6, 165–196, Mathematical Sciences Publishers
[22] Sosnilo, V.: Theorem of the heart in negative K-theory for weight structures. Doc. Math., 24, 2137–2158, MR 4033821 DOI 10.4171/dm/722
[23] Suslin, A., Voevodsky, V.: Relative cycles and Chow sheaves. Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, 2000, pp. 10–86. MR 1764199
[24] Toën, B.: Lectures on DG-categories. Topics in algebraic and topological K-theory, Lecture Notes in Math., vol. 2008, Springer, Berlin, pp. 243–302
[25] Toën, B.: The homotopy theory of $dg$-categories and derived Morita theory. Invent. Math. 167, 615–667. MR 2276263 DOI 10.1007/s00222-006-0025-y
[26] Voevodsky, V.: Triangulated categories of motives over a field. Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, pp. 138–187
[27] Vologodsky, V.: Hodge realizations of 1-motives and the derived Albanese. J. K-Theory 10, no. 2, 371–412. MR 3004172 DOI 10.1017/is011012008jkt178
[28] Weibel, C.: The K-book. An introduction to algebraic K-theory, Graduate Studies in Mathematics, vol. 145, American Mathematical Society, Providence, 2013, MR 3076731
[29] Zakharevich, I.: The K-theory of assemblers. Adv. Math., 304, 1176–1218, MR 3558230 DOI 10.1016/j.aim.2016.08.045
[30] Zakharevich, I.: The annihilator of the Lefschetz motive. Duke Math. J. 166, 1989-2022, MR 3694563
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