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arXiv · 1711.06898

The arithmetic local Nori fundamental group

M. Romagny·F. Tonini·L. Zhang
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Abstract

In this paper we introduce the local Nori fundamental group scheme of a reduced scheme or algebraic stack over a perfect field $k$. We give particular attention to the case of fields: to any field extension $K/k$ we attach a pro-local group scheme over $k$. We show how this group has many analogies, but also some crucial differences, with the absolute Galois group. We propose two conjectures, analogous to the classical Neukirch-Uchida Theorem and Abhyankar Conjecture, providing some evidence in their favor. Finally we show that the local fundamental group of a normal variety is a quotient of the local fundamental group of an open, of its generic point (as it happens for the \'etale fundamental group) and even of any smooth neighborhood.

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BibTeXRIS

M. Romagny, F. Tonini, L. Zhang. 2017-11-18. The arithmetic local Nori fundamental group. https://arxiv.org/abs/1711.06898

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