Levine, Marc Motivic cohomology. (English) Zbl 07823054 Beliaev, Dmitry (ed.) et al., International congress of mathematicians 2022, ICM 2022, Helsinki, Finland, virtual, July 6–14, 2022. Volume 3. Sections 1–4. Berlin: European Mathematical Society (EMS). 2048-2106 (2023). Summary: We give a survey of the development of motivic cohomology, motivic categories, and some of their recent descendants.For the entire collection see [Zbl 1532.00037]. MSC: 14F42 Motivic cohomology; motivic homotopy theory 14C15 (Equivariant) Chow groups and rings; motives 19D55 \(K\)-theory and homology; cyclic homology and cohomology 19D45 Higher symbols, Milnor \(K\)-theory 55P42 Stable homotopy theory, spectra Keywords:motivic cohomology; motives; \(K\)-theory; algebraic cycles; motivic homotopy theory × Cite Format Result Cite Review PDF Full Text: DOI References: [1] A. Ananyevskiy, SL-oriented cohomology theories. In Motivic homotopy theory and refined enumerative geometry, pp. 1-19, Contemp. Math. 745, Amer. Math. Soc., Providence, RI, 2020. · Zbl 1442.14078 [2] A. Ananyevskiy, G. Garkusha, and I. Panin, Cancellation theorem for framed motives of algebraic varieties. Adv. Math. 383 (2021), 107681, 38 pp. · Zbl 1471.14050 [3] A. Ananyevskiy, M. Levine, and I. Panin, Witt sheaves and the Á-inverted sphere spectrum. J. Topol. 10 (2017), no. 2, 370-385. · Zbl 1378.14021 [4] A. Ananyevskiy and A. Neshitov, Framed and MW-transfers for homotopy mod-ules. Selecta Math. (N.S.) 25 (2019), 26. · Zbl 1436.14042 [5] J. Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanes-cents dans le monde motivique. I. Astérisque 314 (2007), x+466 pp. · Zbl 1146.14001 [6] J. Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanes-cents dans le monde motivique. II. Astérisque 315 (2007), vi+364 pp. · Zbl 1153.14001 [7] T. Bachmann, The generalized slices of Hermitian K-theory. J. Topol. 10 (2017), no. 4, 1124-1144. · Zbl 1453.14065 [8] T. Bachmann, B. Calmès, F. Déglise, J. Fasel, and P. A. Østvaer, Milnor-Witt Motives. (2020), arXiv:2004.06634. [9] T. Bachmann and E. Elmanto, Voevodsky’s slice conjectures via Hilbert schemes. Algebr. Geom. 8 (2021), no. 5, 626-636. · Zbl 1471.19003 [10] T. Bachmann and K. Wickelgren, A 1 -Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections. 2021, arXiv:2002.01848. [11] T. Bachmann and K. Wickelgren, On quadratically enriched excess and residual intersections. 2021, arXiv:2112.05960. [12] J. Barge and F. Morel, Groupe de Chow des cycles orientés et classe d’Euler des fibrés vectoriels. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 4, 287-290. · Zbl 1017.14001 [13] A. A. Beilinson, Higher regulators and values of L-functions. Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Nov. Dostizh. 24 (1984), no. 2, 181-238; J. Sov. Math. 30, (1985), no. 2, 2036-2070. · Zbl 0588.14013 [14] A. A. Beilinson, Height pairing between algebraic cycles. In K-theory, arithmetic and geometry (Moscow, 1984-1986), pp. 1-25, Lecture Notes in Math. 1289, Springer, Berlin, 1987. · Zbl 0651.14002 [15] P. Berthelot and A. Ogus, Notes on crystalline cohomology. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1978. · Zbl 0383.14010 [16] B. Bhatt and A. Mathew, p-adic étale twists and algebraic K-theory. Work in progress. [17] B. Bhatt, M. Morrow, and P. Scholze, Integral p-adic Hodge theory. Publ. Math. Inst. Hautes Études Sci. 128 (2018), 219-397. · Zbl 1446.14011 [18] B. Bhatt, M. Morrow, and P. Scholze, Topological Hochschild homology and integral p-adic Hodge theory. Publ. Math. Inst. Hautes Études Sci. 129 (2019), 199-310. · Zbl 1478.14039 [19] F. Binda, D. Park, and R. A. Østvaer, Triangulated categories of logarithmic motives over a field. 2020, arXiv:2004.12298. [20] F. Binda and S. Saito, Relative cycles with moduli and regulator maps. J. Inst. Math. Jussieu 18 (2019), no. 6, 1233-1293. · Zbl 1439.14038 [21] F. Binda et al., In Motivic homotopy theory and refined enumerative geometry. Workshop on Motivic Homotopy Theory and Refined Enumerative Geometry, May 14-18, 2018, Universität Duisburg-Essen, Essen, Germany, edited by F. Binda, M. Levine, M. T. Nguyen and O. Röndigs, Contemporary Mathematics, 745, American Mathematical Society, Providence, RI, 2020. [22] S. Bloch, K 2 and algebraic cycles. Ann. of Math. (2) 99 (1974), 349-379. · Zbl 0298.14005 [23] S. Bloch, K 2 of Artinian Q-algebras, with application to algebraic cycles. Comm. Algebra 3 (1975), 405-428. · Zbl 0327.14002 [24] S. Bloch, Algebraic cycles and higher K-theory. Adv. Math. 61 (1986), no. 3, 267-304. · Zbl 0608.14004 [25] S. Bloch, The moving lemma for higher Chow groups. J. Algebraic Geom. 3 (1994), no. 3, 537-568. · Zbl 0830.14003 [26] S. Bloch and H. Esnault, An additive version of higher Chow groups. Ann. Sci. Éc. Norm. Supér. (4) 36 (2003), no. 3, 463-477. · Zbl 1100.14014 [27] S. Bloch and H. Esnault, The additive dilogarithm. Kazuya Kato’s fiftieth birthday. Doc. Math. Extra Vol. (2003), 131-155. · Zbl 1052.11048 [28] S. Bloch and K. Kato, p-adic étale cohomology. Publ. Math. Inst. Hautes Études Sci. 63 (1986), 107-152. · Zbl 0613.14017 [29] S. Bloch and S. Lichtenbaum, A Spectral Sequence for Motivic Cohomology. Preprint, 1995, http://www.math.uiuc.edu/K-theory/0062/. [30] S. Bloch and A. Ogus, Gersten’s conjecture and the homology of schemes. Ann. Sci. Éc. Norm. Supér. (4) 7 (1974), 181-201. · Zbl 0307.14008 [31] B. Calmès and J. Fasel, The category of finite Chow-Witt correspondences. Preprint, 2014, arXiv:1412.2989. [32] J-L. Cathelineau, Remarques sur les différentielles des polylogarithmes uniformes. Ann. Inst. Fourier (Grenoble) 46 (1996), no. 5, 1327-1347. · Zbl 0861.19003 [33] D-C. Cisinski and F. Déglise, Triangulated categories of mixed motives. Springer Monogr. Math., Springer, Cham, 2019. · Zbl 07138952 [34] A. Dold and R. Thom, Quasifaserungen und unendliche symmetrische Produkte. Ann. of Math. (2) 67 (1958), 239-281. · Zbl 0091.37102 [35] W. G. Dwyer and E. M. Friedlander, Algebraic and etale K-theory. Trans. Amer. Math. Soc. 292 (1985), no. 1, 247-280. · Zbl 0581.14012 [36] E. Elmanto, M. Hoyois, A. A. Khan, V. Sosnilo, and M. Yakerson, Framed trans-fers and motivic fundamental classes. J. Topol. 13 (2020), no. 2, 460-500. · Zbl 1444.14050 [37] E. Elmanto, M. Hoyois, A. A. Khan, V. Sosnilo, and M. Yakerson, Modules over algebraic cobordism. Forum Math. Pi 8 (2020), e14, 44 pp. · Zbl 1458.14027 [38] E. Elmanto, M. Hoyois, A. A. Khan, V. Sosnilo, and M. Yakerson, Motivic infinite loop spaces. Camb. J. Math. 9 (2021), no. 2, 431-549. · Zbl 1535.14057 [39] J. Fasel, Groupes de Chow-Witt. Mém. Soc. Math. Fr. (N.S.) 113, 2008. · Zbl 1190.14001 [40] J-M. Fontaine and W. Messing, p-adic periods and p-adic étale cohomology. In Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), pp. 179-207, Contemp. Math. 67, Amer. Math. Soc., Providence, RI, 1987. · Zbl 0632.14016 [41] E. M. Friedlander, Étale K-theory. I. Connections with etale cohomology and algebraic vector bundles. Invent. Math. 60 (1980), no. 2, 105-134. · Zbl 0519.14010 [42] E. M. Friedlander, Étale K-theory. II. Connections with algebraic K-theory. Ann. Sci. Éc. Norm. Supér. (4) 15 (1982), no. 2, 231-256. · Zbl 0537.14011 [43] E. M. Friedlander and A. Suslin, The spectral sequence relating algebraic K-theory to motivic cohomology. Ann. Sci. Éc. Norm. Supér. (4) 35 (2002), no. 6, 773-875. · Zbl 1047.14011 [44] G. Garkusha and I. Panin, On the motivic spectral sequence. J. Inst. Math. Jussieu 17 (2018), no. 1, 137-170. · Zbl 1390.19006 [45] G. Garkusha and A. Neshitov, Fibrant resolutions for motivic Thom spectra. 2018, arXiv:1804.07621. [46] G. Garkusha, A. Neshitov, and I. Panin, Framed motives of relative motivic spheres. Trans. Amer. Math. Soc. 374 (2021), no. 7, 5131-5161. · Zbl 1484.14048 [47] G. Garkusha and I. Panin, Homotopy invariant presheaves with framed transfers. Cambridge J. Math. 8 (2020), no. 1, 1-94. · Zbl 1453.14066 [48] G. Garkusha and I. Panin, Framed motives of algebraic varieties (after V. Voevod-sky). J. Amer. Math. Soc. 34 (2021), no. 1, 261-313. · Zbl 1491.14034 [49] T. Geisser, Motivic cohomology over Dedekind rings. Math. Z. 248 (2004), no. 4, 773-794. · Zbl 1062.14025 [50] T. Geisser and L. Hesselholt, Topological cyclic homology of schemes. In Alge-braic K-theory (Seattle, WA, 1997), pp. 41-87, Proc. Sympos. Pure Math. 67, Amer. Math. Soc., Providence, RI, 1999. · Zbl 0953.19001 [51] T. Geisser and L. Hesselholt, On the K-theory and topological cyclic homology of smooth schemes over a discrete valuation ring. Trans. Amer. Math. Soc. 358 (2006), no. 1, 131-145. · Zbl 1087.19003 [52] T. Geisser and M. Levine, The K-theory of fields in characteristic p. Invent. Math. 139 (2000), no. 3, 459-493. · Zbl 0957.19003 [53] H. Gillet, Riemann-Roch theorems for higher algebraic K-theory. Adv. Math. 40 (1981), no. 3, 203-289. · Zbl 0478.14010 [54] D. R. Grayson, Weight filtrations via commuting automorphisms. K-Theory 9 (1995), no. 2, 139-172. · Zbl 0826.19003 [55] J. J. Gutiérrez, C. Röndigs, M. Spitzweck, and P. A. Østvaer, Motivic slices and coloured operads. J. Topol. 5 (2012), no. 3, 727-755. · Zbl 1258.18012 [56] C. Haesemeyer and C. A. Weibel, The norm residue theorem in motivic coho-mology. Ann. of Math. Stud. 200, Princeton University Press, Princeton, NJ, 2019. · Zbl 1433.14001 [57] M. Hanamura, Mixed motives and algebraic cycles. I. Math. Res. Lett. 2 (1995), no. 6, 811-821. · Zbl 0867.14003 [58] M. Hanamura, Mixed motives and algebraic cycles. III. Math. Res. Lett. 6 (1999), no. 1, 61-82. · Zbl 0968.14004 [59] M. Hanamura, Mixed motives and algebraic cycles. II. Invent. Math. 158 (2004), no. 1, 105-179. · Zbl 1068.14022 [60] L. Hesselholt, On the p-typical curves in Quillen’s K-theory. Acta Math. 177 (1996), no. 1, 1-53. · Zbl 0892.19003 [61] J. Hornbostel and M. Wendt, Chow-Witt rings of classifying spaces for symplectic and special linear groups. J. Topol. 12 (2019), no. 3, 916-966. · Zbl 1444.14018 [62] M. Hoyois, The six operations in equivariant motivic homotopy theory. Adv. Math. 305 (2017), 197-279. · Zbl 1400.14065 [63] M. Hoyois, The localization theorem for framed motivic spaces. Compos. Math. 157 (2021), no. 1, 1-11. · Zbl 1455.14042 [64] A. Huber, Mixed motives and their realization in derived categories. Lecture Notes in Math. 1604, Springer, Berlin, 1995. · Zbl 0938.14008 [65] A. Huber and S. Müller-Stach, Periods and Nori motives. With contributions by Benjamin Friedrich and Jonas von Wangenheim. Ergeb. Math. Grenzgeb. (3) 65, Springer, Cham, 2017. · Zbl 1369.14001 [66] T. Ito, K. Kato, C. Nakayama, and S. Usui, On log motives. Tunis. J. Math. 2 (2020), no. 4, 733-789. · Zbl 1444.14015 [67] R. Iwasa and W. Kai, Chern classes with modulus. Nagoya Math. J. 236 (2019), 84-133. · Zbl 1436.19005 [68] U. Jannsen, Mixed motives, motivic cohomology, and Ext-groups. In Proceed-ings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pp. 667-679, Birkhäuser, Basel, 1995. · Zbl 0856.14006 [69] B. Kahn, H. Miyazaki, S. Saito, and T. Yamazaki, Motives with modulus, I: Modulus sheaves with transfers for non-proper modulus pairs. Épijournal Géom. Algébrique 5 (2021), 1, 46 pp. · Zbl 1506.19002 [70] B. Kahn, H. Miyazaki, S. Saito, and T. Yamazaki, Motives with modulus, II: modulus sheaves with transfers for proper modulus pairs. Épijournal Géom. Algébrique 5 (2021), 2, 31 pp. · Zbl 1468.19005 [71] B. Kahn, H. Miyazaki, S. Saito, and T. Yamazaki, Motives with modulus, III: the categories of motives. 2021, arXiv:2011.11859. [72] B. Kahn, S. Saito, and T. Yamazaki, Reciprocity sheaves. With two appendices by Kay Rülling. Compos. Math. 152 (2016), no. 9, 1851-1898. · Zbl 1419.19001 [73] B. Kahn, S. Saito, and T. Yamazaki, Reciprocity sheaves, II. 2021, arXiv:1707.07398. [74] W. Kai, A moving lemma for algebraic cycles with modulus and contravariance. Int. Math. Res. Not. IMRN 1 (2021), 475-522. · Zbl 1480.14004 [75] W. Kai and H. Miyazaki, Suslin’s moving lemma with modulus. Ann. K-Theory 3 (2018), no. 1, 55-70. · Zbl 1386.14046 [76] J. L. Kass and K. Wickelgren, An arithmetic count of the lines on a smooth cubic surface. Compos. Math. 157 (2021), no. 4, 677-709. · Zbl 1477.14085 [77] J. L. Kass and K. Wickelgren, The class of Eisenbud-Khimshiashvili-Levine is the local A 1 -Brouwer degree. Duke Math. J. 168 (2019), no. 3, 429-469. · Zbl 1412.14014 [78] K. Kato, Milnor K-theory and the Chow group of zero cycles. In Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, CO, 1983), pp. 241-253, Contemp. Math. 55, Amer. Math. Soc., Providence, RI, 1986. · Zbl 0603.14009 [79] K. Kato, On p-adic vanishing cycles (application of ideas of Fontaine-Messing). Adv. Stud. Pure Math. 10 (1987), 207-251. · Zbl 0645.14009 [80] K. Kato and S. Saito, Global class field theory of arithmetic schemes. In Appli-cations of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, CO, 1983), pp. 255-331, Contemp. Math. 55, Amer. Math. Soc., Provi-dence, RI, 1986. · Zbl 0614.14001 [81] M. Kerz and S. Saito, Chow group of 0-cycles with modulus and higher-dimen-sional class field theory. Duke Math. J. 165 (2016), no. 15, 2811-2897. · Zbl 1401.14148 [82] M. Kurihara, A note on p-adic étale cohomology. Proc. Japan Acad. Ser. A Math. Sci. 63 (1987), no. 7, 275-278. · Zbl 0647.14006 [83] M. Levine, Mixed motives. Math. Surveys Monogr. 57, American Mathematical Society, Providence, RI, 1998. · Zbl 0902.14003 [84] M. Levine, Techniques of localization in the theory of algebraic cycles. J. Alge-braic Geom. 10 (2001), no. 2, 299-363. · Zbl 1077.14509 [85] M. Levine, The homotopy coniveau tower. J. Topol. 1 (2008), no. 1, 217-267. · Zbl 1154.14005 [86] M. Levine, Aspects of enumerative geometry with quadratic forms. Doc. Math. 25 (2020), 2179-2239. · Zbl 1465.14008 [87] S. Lichtenbaum, Values of zeta-functions at nonnegative integers. In Number theory (Noordwijkerhout, 1983), pp. 127-138, Lecture Notes in Math. 1068, Springer, Berlin, 1984. · Zbl 0591.14014 [88] R. McCarthy, Relative algebraic K-theory and topological cyclic homology. Acta Math. 179 (1997), 197-222. · Zbl 0913.19001 [89] A. S. Merkurjev and A. A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism. Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011-1046, 1135-1136; Math. USSR, Izv. 21 (1983), no. 2, 307-340. · Zbl 0525.18008 [90] J. S. Milne, Motivic cohomology and values of zeta functions. Compos. Math. 68 (1988), no. 1, 59-100, 101-102. · Zbl 0681.14007 [91] H. Miyazaki, Private communication, Sept. 6, 2021. [92] F. Morel, An introduction to A 1 -homotopy theory. In Contemporary developments in algebraic K-theory, pp. 357-441, ICTP Lect. Notes XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004. [93] F. Morel, A 1 -algebraic topology over a field. Lecture Notes in Math. 2052, Springer, Heidelberg, 2012. · Zbl 1263.14003 [94] F. Morel and V. Voevodsky, A 1 -homotopy theory of schemes. Publ. Math. Inst. Hautes Études Sci. 90 (1999), 45-143. · Zbl 0983.14007 [95] D. Mumford, Rational equivalence of 0-cycles on surfaces. J. Math. Kyoto Univ. 9 (1968), 195-204. · Zbl 0184.46603 [96] Yu. P. Nesterenko and A. A. Suslin, Homology of the general linear group over a local ring, and Milnor’s K-theory. Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 121-146; Math. USSR, Izv. 34 (1990), no. 1, 121-145. · Zbl 0668.18011 [97] A. Ogus, Lectures on logarithmic algebraic geometry. Cambridge Stud. Adv. Math. 178, Cambridge University Press, Cambridge, 2018. · Zbl 1437.14003 [98] J. Park, Regulators on additive higher Chow groups. Amer. J. Math. 131 (2009), no. 1, 257-276. · Zbl 1176.14001 [99] P. Pelaez, Multiplicative properties of the slice filtration. Astérisque 335 (2011). · Zbl 1235.14003 [100] P. Pelaez, On the functoriality of the slice filtration. J. K-Theory 11 (2013), no. 1, 55-71. · Zbl 1319.14029 [101] D. Quillen, Higher algebraic K-theory, In Proceedings of the International Congress of Mathematicians (Vancouver, BC, 1974), Vol. 1 pp. 171-176, Canad. Math. Congress, Montreal, QC, 1974. · Zbl 0359.18014 [102] J. Riou, La conjecture de Bloch-Kato (d’aprés M. Rost et V. Voevodsky). Astérisque 361 (2014), Exp. No. 1073, x, 421-463. · Zbl 1366.19001 [103] C. Röndigs and P. A. Østvaer, Modules over motivic cohomology. Adv. Math. 219 (2008), no. 2, 689-727. · Zbl 1180.14015 [104] K. Rülling, The generalized de Rham-Witt complex over a field is a complex of zero-cycles. J. Algebraic Geom. 16 (2007), no. 1, 109-169. · Zbl 1122.14006 [105] S. Saito, Reciprocity sheaves and logarithmic motives. 2021, arXiv:2107.00381. [106] K. Sato, p-adic étale Tate twists and arithmetic duality. With an appendix by Kei Hagihara. Ann. Sci. Éc. Norm. Supér. (4) 40 (2007), no. 4, 519-588. · Zbl 1146.14012 [107] W. Scharlau, Quadratic and Hermitian forms. Grundlehren Math. Wiss. 270, Springer, Berlin, 1985. · Zbl 0584.10010 [108] P. Schneider, p-adic points of motives. In Motives (Seattle, WA, 1991), pp. 225-249, Proc. Sympos. Pure Math. 55, Amer. Math. Soc., Providence, RI, 1994. · Zbl 0814.14023 [109] J.-P. Serre, Groupes algébriques et corps de classes. 2nd edn., Publ. Inst. Math. Univ. Nancago 7, Actual. Sci. Ind., 1264, Hermann, Paris, 1984. [110] M. Spitzweck, A commutative P 1 -spectrum representing motivic cohomology over Dedekind domains. Mém. Soc. Math. Fr. (N.S.) 157 (2018), 110 pp. · Zbl 1408.14081 [111] A. Suslin, Lecture, K-theory conference at Luminy, 1988. [112] A. Suslin and V. Voevodsky, Singular homology of abstract algebraic varieties. Invent. Math. 123 (1996), no. 1, 61-94. · Zbl 0896.55002 [113] A. Suslin and V. Voevodsky, Bloch-Kato conjecture and motivic cohomology with finite coefficients. In The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), pp. 117-189, NATO Sci. Ser. C Math. Phys. Sci. 548, Kluwer Acad. Publ., Dordrecht, 2000. · Zbl 1005.19001 [114] J. Tate, Relations between K 2 and Galois cohomology. Invent. Math. 36 (1976), 257-274. · Zbl 0359.12011 [115] B. Totaro, Milnor K-theory is the simplest part of algebraic K-theory. K-Theory 6 (1992), no. 2, 177-189. · Zbl 0776.19003 [116] V. Voevodsky, A 1 -homotopy theory. Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998). Doc. Math. Extra Vol. I (1998), 579-604. · Zbl 0907.19002 [117] V. Voevodsky, Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic. Int. Math. Res. Not. 7 (2002), 351-355. · Zbl 1057.14026 [118] V. Voevodsky, Open problems in the motivic stable homotopy theory. I. Motives. In polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), pp. 3-34, Int. Press Lect. Ser. 3, Int. Press, Somerville, MA, 2002. · Zbl 1047.14012 [119] V. Voevodsky, A possible new approach to the motivic spectral sequence for alge-braic K-theory. In Recent progress in homotopy theory (Baltimore, MD, 2000), pp. 371-379, Contemp. Math. 293, Amer. Math. Soc., Providence, RI, 2002. · Zbl 1009.19003 [120] V. Voevodsky, Motivic cohomology with Z=2-coefficients. Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59-104. · Zbl 1057.14028 [121] V. Voevodsky, Reduced power operations in motivic cohomology. Publ. Math. Inst. Hautes Études Sci. 98 (2003), 1-57. · Zbl 1057.14027 [122] V. Voevodsky, On the zero slice of the sphere spectrum. Tr. Mat. Inst. Steklova 246 (2004); [123] Algebr. Geom. Metody, Svyazi Prilozh. 106-115; reprinted in Proc. Steklov Inst. Math. 3(246) (2004), 93-102. [124] V. Voevodsky, Motivic Eilenberg-Maclane spaces. Publ. Math. Inst. Hautes Études Sci. 112 (2010), 1-99. · Zbl 1227.14025 [125] V. Voevodsky, On motivic cohomology with Z= l-coefficients. Ann. of Math. (2) 174 (2011), no. 1, 401-438. · Zbl 1236.14026 [126] V. Voevodsky, Cancellation theorem. Doc. Math. Extra vol.: Andrei A. Suslin sixtieth birthday (2010), 671-685. · Zbl 1202.14022 [127] V. Voevodsky, Notes on framed correspondences, https://www.uni-due.de/ bm0032/MotSemWS201617/VoevFramedCorr.pdf. [128] V. Voevodsky, A. Suslin, and E. M. Friedlander, Cycles, transfers, and motivic homology theories. Ann. of Math. Stud. 143, Princeton University Press, Princeton, NJ, 2000. · Zbl 1021.14006 [129] C. Zhong, Comparison of dualizing complexes. J. Reine Angew. Math. 695 (2014), 1-39. Marc Levine Universität Duisburg-Essen, Fakultät Mathematik, Campus Essen, 45117 Essen, Germany, marc.levine@uni-due.de · Zbl 1321.19004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.