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Milnor \(K\)-theory is the simplest part of algebraic \(K\)-theory. (English) Zbl 0776.19003
It is well known that \(K_ 0(X)\otimes_{\mathbb{Z}}\mathbb{Q}\cong\oplus_ p\text{CH}^ p(X)\otimes_{\mathbb{Z}}\mathbb{Q}\), where \(X\) is a smooth algebraic variety, \(K_ 0(X)\) is the Grothendieck group of vector bundles on \(X\), and \(\text{CH}^ p(X)\) is the Chow group of codimension \(p\) cycles mod rational equivalence. S. Bloch [Adv. Math. 61, 267- 304 (1986; Zbl 0608.14004)] has defined groups \(\text{CH}^ p(X,n)\) (\(n\) an integer \(\geq 0\)) and proved that \(K_ n(X)\otimes_{\mathbb{Z}}\mathbb{Q}\cong\oplus_ p\text{CH}^ p(X,n)\otimes_{\mathbb{Z}}\mathbb{Q}\) for \(n\geq 0\). Motivated in part by these considerations, Yu. P. Nesterenko and A. A. Suslin [Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 1, 121-146 (1989; Zbl 0668.18011)] have proved that if \(F\) is a field (writing \(\text{CH}^ p(F,n)\) for \(\text{CH}^ p(\text{Spec}(F),n))\), then \(\text{CH}^ p(F,n)=0\) for \(p>n\), and \(\text{CH}^ n(F,n)\cong K^ M_ n(F)\) (where \(K^ M\) denotes Milnor \(K\)-Theory).
In the present paper the author reproves the result of Nesterenko and Suslin. His proof differs from that of Nesterenko and Suslin in its use of explicit rational curves in \(\mathbb{A}^{n+1}_ F\) to verify relations in \(\text{CH}^ n(F,n)\).

MSC:
19D45 Higher symbols, Milnor \(K\)-theory
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
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[1] Bass, H. and Tate, J.: The Milnor ring of a global field, in H. Bass (ed),Algebraic K-Theory II Lecture Notes in Math. 342, Springer, New York (1973), pp. 349–446. · Zbl 0299.12013
[2] Bloch, S.: Algebraic cycles and higherK-theory,Adv. Math. 61 (1986), 267–304. · Zbl 0608.14004 · doi:10.1016/0001-8708(86)90081-2
[3] Eilenberg, S. and MacLane S.: Acyclic models,Amer. J. Math. 75 (1953), 189–199. · Zbl 0050.17205 · doi:10.2307/2372628
[4] Fulton, W.:Intersection Theory, Ergebnisse ser. 3: v. 2, Springer-Verlag, Berlin (1984). · Zbl 0541.14005
[5] Kato, K.: A generalization of local class field theory by usingK-groups, II,J. Fac. Sci. Univ. Tokyo 27(3) (1980), 603–683. · Zbl 0463.12006
[6] Milnor, J.: AlgebraicK-theory and quadratic forms,Invent. Math. 9 (1970), 318–344. · Zbl 0199.55501 · doi:10.1007/BF01425486
[7] Serre, J.-P.: Homologie singulière des espaces fibrés,Ann. of Math. 54 (1951), 425–505. · Zbl 0045.26003 · doi:10.2307/1969485
[8] Suslin, A. A.: Reciprocity laws and the stable rank of polynomial rings,Math. USSR-Izv. 15(3) (1980), 589–623. · Zbl 0452.13007 · doi:10.1070/IM1980v015n03ABEH001270
[9] Nesterenko, Yu. P. and Suslin, A. A.: Homology of the full linear group over a local ring, and Milnor’sK-theory.Math. USSR-Izv. 34 (1990), 121–145. · Zbl 0684.18001 · doi:10.1070/IM1990v034n01ABEH000610
[10] Suslin, A. A.: Mennicke symbols and and their applications in theK-theory of fields, in R. K. Dennis (ed),Algebraic K-Theory I Lecture Notes in Math., 966, Springer, New York (1982), pp. 334–351.
[11] Bloch, S.: Algebraic cycles and the Lie algebra of mixed Tate motives,J. Amer. Math. Soc. 4 (1991), 771–791. · Zbl 0762.14002 · doi:10.1090/S0894-0347-1991-1102577-2
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