## 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

### MSC:

 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14C25 Algebraic cycles

### Keywords:

Milnor K-theory; Chow group of zero cycles

Zbl 0588.00014