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


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