Ramification filtration by moduli in higher-dimensional global class field theory

Abstract

In their approach to higher-dimensional global class field theory, Kato and Saito define the class group of a proper arithmetic scheme \bar{X} as an inverse limit C_{KS}(\bar{X}) = \varprojlim_{\mathcal{I}} C_{\mathcal{I}}(\bar{X}) of certain Nisnevich cohomology groups C_{\mathcal{I}}(\bar{X}) taken over all non-zero coherent ideal sheaves \mathcal{I} of \mathcal{O}_{\bar{X}}. The ideal sheaves \mathcal{I} should be regarded as higher-dimensional analogues of the classical moduli \mathfrak{m} on a global field K, which induce a filtration of the idele class group C_K by the ray class groups C_K/C_K^{\mathfrak{m}}. In higher dimensions however, it is not clear how the induced filtration of the abelian fundamental group can be interpreted in terms of ramification. In view of Wiesend's class field theory, we define an easier notion of moduli in higher dimensions only involving curves on the scheme. We then show that both notions agree for moduli that correspond to tame ramification.