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Voevodsky’s motives and Weil reciprocity. (English) Zbl 1309.19009

Duke Math. J. 162, No. 14, 2751-2796 (2013); corrigendum ibid. 164, No. 10, 2093-2098 (2015).
M. Somekawa [K-Theory 4, No. 2, 105–119 (1990; Zbl 0721.14003)] defined \(K\)-groups \(K(k;G_1,\dots,G_n)\), where \(G_1,\dots, G_n\) are semi-abelian varieties over the field \(k\), and stated that, in the conjectured abelian category \({\mathcal M}{\mathcal M}\) of mixed motives over \(k\), there should be an isomorphism of the form \[ K(k; G_1,\dots, G_n)\simeq \text{Ext}^n_{{\mathcal M}{\mathcal M}}(\mathbb{Z},G_1[-1]\otimes\cdots\otimes G_n[-1]), \] where \(G_1[-1],\dots, G_n[-1]\) are the corresponding 1-motives.
In this paper the authors prove a “real” isomorphism from a group, which is a generalization of \(K(k;G_1,\dots, G_n)\), and an Hom in the existing category \({\mathbf D}{\mathbf M}^{\text{eff}}_-\), defined by Voevodsky, of effective motivic complexes. More precisely \[ k(k,{\mathcal F}_1,\dots,{\mathcal F}_n)\simeq\text{Hom}_{{\mathbf D}{\mathbf M}^{\text{eff}_-}}(\mathbb{Z},{\mathcal F}_1[0],\dots,{\mathcal F}-n[0]), \] where \({\mathcal F}_1,\dots,{\mathcal F}_n\) are homotopy invariant sheaves with transfers. In the case \({\mathcal F}_1={\mathcal F}_2=\cdots={\mathcal F}_n= {\mathbf G}_m\) the left-hand side is isomorphic to the usual Milnor \(K\)-group \(K^M_n(k)\), while the right-hand side is motivic cohomology. Therefore, for a perfect field \(k\), the above formula yields the Suslin-Voevodsky isomorphism for motivic cohomology \(K^M_n(k)\simeq H^n_M(k,\mathbb{Z}(n))\).
The above result has the following application to algebraic cycles
Theorem 1. Let \(X_1,\dots, X_n\) be quasi-projective schemes over a field \(k\) of characteristic 0 and let \(X= X_1\times X_2\times\cdots\times X_n\). For every \(r\geq 0\) there is an isomorphism \[ K(k;\underline{\mathrm{CH}}_0(X_1,\dots, \underline{\mathrm{CH}}_0(X_n),\,{\mathbf G}_M,\dots,{\mathbf G}_M)\simeq \mathrm{CH}_{-r}(X,r)), \] where \(\underline{\mathrm{CH}}_0(X_i)\), for \(1\leq i\leq n\), are the homotopy invariant Nisnevich sheaves with transvers \(U\to \mathrm{CH}_0(X_i\times_k k(U))\) and \(\mathrm{CH}_i(X,j)\), for \(i,j\in\mathbb{Z}\), are Bloch’s higher Chow groups.
Special cases of Theorem 1 were previously known, when the \(X_i\)’s are smooth projective and \(r= 0\) or \(n=1\).

MSC:

19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
14F42 Motivic cohomology; motivic homotopy theory
19D45 Higher symbols, Milnor \(K\)-theory
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)

Citations:

Zbl 0721.14003
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References:

[1] R. Akhtar, Milnor \(K\)-theory of smooth varieties , \(K\)-Theory 32 (2004), 269-291. · Zbl 1083.19001
[2] M. Artin, A. Grothendieck, and J.-L. Verdier, Théorie des topos et cohomologie étale des schémas , Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), Lecture Notes in Math. 269 , Springer, Berlin, 1972. · Zbl 0234.00007
[3] L. Barbieri-Viale and B. Kahn, On the derived category of \(1\)-motives , preprint, [math.AG]. 1009.1900v1
[4] H. Bass and J. Tate, “The Milnor ring of a global field” in Algebraic K-Theory, II: “Classical” Algebraic K-Theory and Connections with Arithmetic (Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972) , Lecture Notes in Math. 342 , Springer, Berlin, 1973, 349-446.
[5] S. Bloch, Algebraic cycles and higher K-theory , Adv. Math. 61 (1986), 267-304. · Zbl 0608.14004
[6] F. Déglise, Modules homotopiques avec transferts et motifs génériques , Ph.D. dissertation, Université Paris Diderot-Paris 7, 2002.
[7] E. Friedlander and V. Voevodsky, “Bivariant cycle cohomology” in Cycles, Transfers, and Motivic Homology Theories , Ann. of Math. Stud. 143 , Princeton Univ. Press, Princeton, 2000, 138-187. · Zbl 1019.14011
[8] A. Huber and B. Kahn, The slice filtration and mixed Tate motives , Compos. Math. 142 (2006), 907-936. · Zbl 1105.14022
[9] B. Kahn, “The decomposable part of motivic cohomology and bijectivity of the norm residue homomorphism” in Algebraic K-Theory, Commutative Algebra, and Algebraic Geometry (Santa Margherita Ligure, 1989) , Contemp. Math. 126 , Amer. Math. Soc., Providence, 1992, 79-87. · Zbl 0759.19005
[10] B. Kahn, Nullité de certains groupes attachés aux variétés semi-abéliennes sur un corps fini; application , C. R. Math. Acad. Sci. Paris 314 (1992), 1039-1042. · Zbl 0785.14027
[11] B. Kahn, Somekawa’s \(K\)-groups and Voevodsky’s Hom groups , preprint, [math.AG]. 1009.4554v1
[12] B. Kahn and R. Sujatha, Birational motives, I , preprint, . Accessed 23 September 2010. · Zbl 1388.14025
[13] K. Kato and S. Saito, Unramified class field theory of arithmetical surfaces , Ann. of Math. (2) 118 (1983), 241-275. · Zbl 0562.14011
[14] S. Kelly, Triangulated categories of motives in positive characteristic , Ph.D. dissertation, Université Paris 13, Paris, 2012.
[15] M. Levine, Techniques of localization in the theory of algebraic cycles , J. Algebraic Geom. 10 (2001) 299-363. · Zbl 1077.14509
[16] S. MacLane, Categories for the Working Mathematician , Grad. Texts in Math. 5 , Springer, New York, 1971.
[17] C. Mazza, V. Voevodsky, and C. Weibel, Lecture Notes on Motivic Cohomology , Clay Math. Monogr. 2 , Amer. Math. Soc., Providence, 2006. · Zbl 1115.14010
[18] S. Mochizuki, Motivic interpretation of Milnor \(K\)-groups attached to Jacobian varieties , Hokkaido Math. J. 41 (2012), 1-10. · Zbl 1275.19004
[19] D. Quillen, “Higher algebraic \(K\)-theory. I” in Algebraic K-Theory, I: Higher K-Theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) , Lecture Notes in Math. 341 , Springer, Berlin, 85-147. · Zbl 0292.18004
[20] N. Ramachandran, Duality of Albanese and Picard \(1\)-motives , \(K\)-Theory 22 (2001), 271-301. · Zbl 0983.14003
[21] W. Raskind and M. Spieß, Milnor \(K\)-groups and zero-cycles on products of curves over \(p\)-adic fields , Compos. Math. 121 (2000), 1-33. · Zbl 0985.14003
[22] J.-P. Serre, Morphismes universels et variété d’Albanese , Séminaire Chevalley 4 (1958-1959), 1-22.
[23] J.-P. Serre, Groupes algébriques et corps de classes , Hermann, Paris, 1959. · Zbl 0097.35604
[24] M. Somekawa, On Milnor \(K\)-groups attached to semi-abelian varieties , \(K\)-Theory 4 (1990), 105-119. · Zbl 0721.14003
[25] M. Spieß and T. Szamuely, On the Albanese map for smooth quasi-projective varieties , Math. Ann. 325 (2003), 1-17. · Zbl 1077.14026
[26] M. Spieß and T. Yamazaki, A counterexample to generalizations of the Milnor-Bloch-Kato conjecture , J. K-Theory 4 (2009), 77-90. · Zbl 1186.19004
[27] A. 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) , NATO Sci. Ser. C Math. Phys. Sci. 548 , Kluwer, Dordrecht, 2000, 117-189. · Zbl 1005.19001
[28] A. Suslin and V. Voevodsky, Singular homology of abstract algebraic varieties , Invent. Math. 123 (1996), 61-94. · Zbl 0896.55002
[29] J. Thévenaz, A visit to the kingdom of Mackey functors , Bayreuth. Math. Schr. 33 (1990), 215-241. · Zbl 0703.20006
[30] V. Voevodsky, “Cohomological theory of presheaves with transfers” in Cycles, Transfers, and Motivic Homology Theories , Ann. of Math. Stud. 143 , Princeton Univ. Press, Princeton, 2000, 87-137. · Zbl 1019.14010
[31] V. Voevodsky, “Triangulated categories of motives over a field” in Cycles, Transfers, and Motivic Homology Theories , Ann. of Math. Stud. 143 , Princeton Univ. Press, Princeton, 2000, 188-238. · Zbl 1019.14009
[32] V. Voevodsky, Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic , Int. Math. Res. Not. IMRN 2002 , no. 7, 351-355. · Zbl 1057.14026
[33] V. Voevodsky, Cancellation theorem , Doc. Math. 2010 , Extra volume: Andrei A. Suslin sixtieth birthday, 671-685. · Zbl 1202.14022
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