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 $k$. 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 étale fundamental group) and even of any smooth neighborhood.

17 pages

References in corpus (1)

• On a purely inseparable analogue of the Abhyankar Conjecture for affine curves

Cited by in corpus (1)

• A generalized Abhyankar's conjecture for simple Lie algebras in characteristic $p>5$