The arithmetic local Nori fundamental group

Abstract

In this paper we introduce the local Nori fundamental group scheme of a reduced scheme or algebraic stack over a perfect field k k . We give particular attention to the case of fields: to any field extension K / k K/k we attach a pro-local group scheme over k 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 étale fundamental group) and even of any smooth neighborhood.