On the étale fundamental groups of arithmetic schemes, revised

Feng-Wen An

Published 2009-10-26, updated 2010-09-09Version 4

In this paper we will give a computation of the \'{e}tale fundamental group of an integral arithmetic scheme. For such a scheme, we will prove that the \'{e}tale fundamental group is naturally isomorphic to the Galois group of the maximal formally unramified extension over the function field. It consists of the main theorem of the paper. Here, formally unramified will be proved to be arithmetically unramified which is defined in an evident manner and coincides with that in algebraic number theory. Hence, formally unramified has an arithmetic sense. At the same time, such a computation coincides with the known result for a normal noetherian scheme.