## Global class field theory of arithmetic schemes.(English)Zbl 0614.14001

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, 255-331 (1986).
[For the entire collection see Zbl 0588.00014.]
Let K be an algebraic number field, $${\mathcal O}_ K$$ the ring of integers of K, and let $$X=Spec({\mathcal O}_ K)$$. One of the main results of Takagi- Artin class field theory can be written in the form $$\lim_{\leftarrow} H^ 1(X_{Zar},K_ 1({\mathcal O}_ X,I))\cong Gal(K^{ab}/K)$$ where I ranges over all non-zero ideals of $${\mathcal O}_ K$$ and $$K_ 1({\mathcal O}_ X,I)=Ker({\mathcal O}^*_ X\to ({\mathcal O}_ X/I{\mathcal O}_ X)^*)$$. The main aim of the authors is to generalize this to higher dimensional arithmetic schemes.
Let X be a projective integral scheme over $${\mathbb{Z}}$$ of dimension $$d,$$ and let K be the function field of X. Let $$K^ d_ M({\mathcal O}_ X,I)=Ker(K^ M_ d({\mathcal O}_ X)\to K^ M_ d({\mathcal O}_ X/I))$$, where M denotes Milnor K-theory. The authors conjecture that $$\lim_{\leftarrow} H^ d(X_{Zar},K^ M_ d({\mathcal O}_ X,I))\cong Gal(K^{ab}/K)\quad if$$ $$char(K)=0$$ and K has no ordered field structure (with a similar conjecture if $$char(K)>0)$$. They proved this conjecture in an earlier paper if $$d=2$$ [in Galois groups and their representations, Proc. Symp., Nagoya/Jap. 1981, Adv. Stud. Pure Math. 2, 103-152 (1983; Zbl 0544.12011)]. In the present paper they are unable to prove their conjecture, but they do prove a similar result, namely that there is a canonical isomorphism $$\lim_{I,m} C_{I\Sigma}(X)/mC_{I\Sigma}(X)\cong Gal(K^{ab}/K)$$ where I ranges over all non-zero coherent ideals of $${\mathcal O}_ X$$, m ranges over all non-zero integers, and the group $$C_{I\Sigma}(X)$$ (called the Henselian idele class group with modulus $$I\Sigma$$, $$\Sigma$$ the unique archimedean place of $${\mathbb{Q}})$$ is introduced in the present paper.
Reviewer: Leslie G.Roberts

### MSC:

 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 11S31 Class field theory; $$p$$-adic formal groups

### Citations:

Zbl 0588.00014; Zbl 0544.12011