Abelianized fundamental group of the affine space over a finite field and big Witt vectors in several variables

TL;DR

This paper characterizes abelian coverings of algebraic varieties over finite fields using Cartier divisors and describes the abelianized fundamental group of affine space via big Witt vectors in multiple variables, extending classical concepts.

Contribution

It introduces a new description of abelian coverings through principal Cartier divisors and generalizes big Witt vectors to multiple variables for affine space over finite fields.

Findings

The geometric Galois group is given by the k-valued points of a Cartier dual.

The abelianized fundamental group of affine n-space is described by big Witt vectors in n variables.

Provides a new framework connecting algebraic geometry and Witt vectors in several variables.

Abstract

Let $X$ be a normal proper variety over a perfect field $k$. We describe abelian coverings of X in terms of the functor $\underline{\rm HDiv}_X$ of principal relative Cartier divisors on $X$. If the base field $k$ is finite, the geometric Galois group of the maximal abelian extension of the function field of $X$ is given by the $k$-valued points of the Cartier dual of the completion of $\underline{\rm HDiv}_X$. As another application, we present the geometric abelianized fundamental group of the affine $n$-space over a finite field by the group of big Witt vectors in $n$ variables, a generalization of the (usual) big Witt vectors.