A note on class field theory for two-dimensional local rings (Q2704570)

It is known that central statements of class field theory of higher local fields (even though not easy to formulate and develop) are relatively similar to those in class field theory of one-dimensional fields, whereas class field theory of higher dimensional fields which are not ``entirely'' complete has features quite distant from class field theory of one-dimensional fields.NEWLINENEWLINENEWLINEThe paper illustrates this principle for the fraction field \(K\) of a two-dimensional complete normal local ring \(A\) with finite residue field. The corresponding class field theory, using class field theory of two-dimensional local fields, was established by \textit{Sh. Saito}: the reciprocity map \(\Psi_K\) sends an element of an appropriate idele-class type construction \(C_K\) involving \(K_2\)-groups of two-dimensional local fields \(K_P\) to the abelian part of the absolute Galois group of \(K\). In the previous sentence \(K_P\) stands for the fraction field of the \(P\)-adic completion of the localization of \(A\) with respect to a prime ideal \(P\) of height 1 in \(A\). If \(A\) is not regular, then the field \(K\) has nontrivial abelian extensions in which every prime ideal \(P\) of height 1 is unramified and splits completely. The Galois group of the maximal extension with this property \(K^{cs}\) over \(K\) is contained in the cokernel of the reciprocity map \(\Psi_K\).NEWLINENEWLINENEWLINEThe existence theorem in class field theory of \(K\) is a surjective map \(L\mapsto D_{L/K}= \Psi_K^{-1} (G(L/K))\) from finite abelian extensions \(L\) of \(K\) to open subgroups of finite index in \(C_K\); it is injective iff \(A\) is regular. The norm group \(N_{L/K}C_L\) is contained in \(D_{L/K}\); it is not difficult to show that \(D_{L/K}\) is open of finite index in \(C_K\).NEWLINENEWLINENEWLINEThe paper demonstrates that \(N_{L/K}C_L\neq D_{L/K}\), and that the map \(L\mapsto N_{L/K}C_L\) from finite abelian extension \(L\) of \(K\) to open subgroups of finite index in \(C_K\) is not surjective in general. It further discusses how different the norm group can be from \(D_{L/K}\) by looking at resolution of singularities for Kummer extensions of two-dimensional local rings; here the author uses a method of \textit{H. Tsuchihashi} [Osaka J. Math. 36, No. 3, 615-639 (1999; Zbl 0958.32028)] in the complex analytic case.