The arithmetic local Nori fundamental group

TL;DR

This paper introduces the local Nori fundamental group scheme for algebraic structures over perfect fields, explores its properties, analogies with Galois groups, and proposes conjectures related to classical theorems.

Contribution

It defines the local Nori fundamental group scheme, analyzes its properties, and formulates conjectures analogous to Neukirch-Uchida and Abhyankar.

Findings

The local Nori fundamental group scheme has many similarities to the absolute Galois group.

The local fundamental group of a normal variety is a quotient of that of an open subset.

Proposed conjectures extend classical theorems to the local Nori fundamental group context.

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.