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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 \(\text{CH}^ n(F,n)\).

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)
Full Text: DOI
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