Milnor K-theory and the Chow group of zero cycles. (English) Zbl 0603.14009

Applications of algebraic K-theory to algebraic geometry and number theory, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Boulder/Colo. 1983, Part I, Contemp. Math. 55, 241-253 (1986).
[For the entire collection see Zbl 0588.00014.]
Main result: for any smooth variety X of dimension \(d\) over a field k, the Chow group of zero cycles \(CH^ d(X)\) is isomorphic with \(H^ d(X,{\mathcal K}^ M_ d)\). Here \({\mathcal K}^ M_ d\) is the Milnor K-sheaf \({\mathcal O}^*_ X\otimes...\otimes {\mathcal O}^*_ X/{\mathcal J}\), where \({}^*\) denotes as usual the invertible elements, the tensor product is taken \(d\quad times,\) and \({\mathcal J}\) is the subsheaf generated locally by sections of the form \(f_ 1\otimes...\otimes f_ d\) such that \(f_ i+f_ j=1\) for some \(i\neq j\). As noted by the author, C. Soulé obtained independently a general result for cycles of any dimension, but involving also torsion (and k being infinite).
Reviewer: M.Stoia


14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14C25 Algebraic cycles


Zbl 0588.00014