The Milnor-Chow homomorphism revisited. (English) Zbl 1144.19002
The authors give a simplified proof that the natural Milnor-Chow homomorphism \(\rho \colon K^M_n(A) \rightarrow \mathrm{CH}^n(A,n)\) between Milnor \(K\)-theory and higher Chow groups is surjective for essentially smooth (semi-)local \(k\)-algebras \(A\) with infinite residue fields. The proof is based on the properties of the norm map for the Milnor \(K\)-theory of rings and on the theory of Bass and Tate. The obtained result implies some important consequences. In particular, assuming in addition that \(A\) is integral, the Gersten resolution for Milnor \(K\)-theory is exact; assuming the Bloch-Kato conjecture and that \(A\) contains an infinite field of characteristic prime to some integer \(l >0\), the graded ring \(H^*_{\text{é}t}(A, \mu^{\otimes *}_l)\) is generated by elements of degree one. Again under the Bloch-Kato conjecture the main result of the paper implies a result of S. Bloch on the existence of certain divisors on complex varieties. Other corollaries establish some properties of the Witt rings of regular local rings and their extensions.
Reviewer: Vasyl I. Andriychuk (Lviv)
MSC:
| 19D45 | Higher symbols, Milnor \(K\)-theory |
| 14C25 | Algebraic cycles |
| 14F42 | Motivic cohomology; motivic homotopy theory |
| 19E15 | Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) |
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