The Galois closure for rings and some related constructions

TL;DR

This paper explores the Galois closure of finite projective algebras over rings, addressing open questions by constructing intermediate closures, and generalizing and analyzing their properties.

Contribution

It proves the existence of intermediate $S_n$-closures and extends previous results on Galois closures for rings.

Findings

Existence of intermediate $S_n$-closures established.

Generalizations of properties of Galois closures.

New results on the structure of $G(A/R)$ and related constructions.

Abstract

Let $R$ be a ring and let $A$ be a finite projective $R$-algebra of rank $n$. Manjul Bhargava and Matthew Satriano have recently constructed an $R$-algebra $G(A/R)$, the Galois closure of $A/R$. Many natural questions were asked at the end of their paper. Here we address one of these questions, proving the existence of the natural constructions they call intermediate $S_n$-closures. We will also study properties of these constructions, generalizing some of their results, and proving new results both on the intermediate $S_n$-closures and on $G(A/R)$.