Notes on the quasi-galois closed schemes

TL;DR

This paper explores the concept of quasi-galois closed schemes, providing a new criterion for their identification using affine coverings without relying on affine structures, with applications to étale fundamental groups.

Contribution

It introduces a simplified approach to characterize quasi-galois closed schemes using affine coverings in a fixed field, avoiding complex affine structure definitions.

Findings

Provides a sufficient and necessary condition for quasi-galois closed schemes.

Develops a method using affine coverings to analyze quasi-galois closed schemes.

Avoids complex affine structure definitions in scheme analysis.

Abstract

Let $f:X\to Y$ be a surjective morphism of integral schemes. Then $X$ is said to be quasi-galois closed over $Y$ by $f$ if $X$ has a unique conjugate over $Y$ in an algebraically closed field. Such a notion has been applied to the computation of \'etale fundamental groups. In this paper we will use affine coverings with values in a fixed field to discuss quasi-galois closed and then give a sufficient and essential condition for quasi-galois closed. Here, we will avoid using affine structures on a scheme since their definition looks copious and fussy.