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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)
##### Keywords:
higher Chow groups; Milnor $$K$$-Theory; rational curves
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##### References:
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