On the étale fundamental groups of arithmetic schemes, revised

Details On the étale fundamental groups of arithmetic schemes, revised On the étale fundamental groups of arithmetic schemes, revised

2009-10-26

arxiv.org/abs/0910.4646v4

Mathematics

Algebraic Geometry

Revised. 23 pages

Scientific paper

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.

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