Tame coverings of arithmetic schemes

Abstract

The objective of this paper is to investigate tame fundamental groups of schemes of finite type over Spec(Z). More precisely, let X be a connected scheme of finite type over Spec(Z) and let X be a compactification of X, i.e. a scheme which is proper and of finite type over Spec(Z) and which contains X as a dense open subscheme. Then the tame fundamental group of X classifies finite etale coverings of X which are tamely ramified along the boundary X − X, in particular, the tame fundamental group π 1(X, X − X) is a quotient of the etale fundamental group π1(X). Our interest in the tame fundamental group arises from the observation that it seems to be the maximal quotient of π1(X) which is ‘visible’ via class field theory by algebraic cycle theories (see [S-S1], [S] and [S-S2] for more precise statements on ‘tame class field theory’). Coverings of a regular scheme which are tamely ramified along a normal crossing divisor have been studied in [SGA1], [G-M]. In this paper we consider tame ramification along an arbitrary Zariski-closed subset. The main reason for this is the lack of good desingularization theorems in positive and mixed characteristics. A simple imitation of the definition of tame ramification in the normal crossing case proves to be not useful in the more general situation. We give a definition of tameness in the general situation in section 1 and we show that it coincides with the previous one in the normal crossing case. For Galois coverings of normal schemes our definition of tameness coincides with that proposed by Abbes [Ab] and with the notion of “numerical tameness” defined by Chinburg/Erez [C-E]. Naturally, the question ariseswhether the tame fundamental groupπ 1(X, X− X) is independent of the choice of the compactification X.At themoment, we can answer this question only for the maximal pro-nilpotent quotient. In section 2, we consider discrete valuations of higher rank associated to Parshin chains. We investigate the connection between the tameness of a covering and the behaviour of the associated higher dimensional henselian fields and we show that a finite covering of a regular arithmetic scheme with nilpotent Galois group is tamely