The root closure of a ring of mixed characteristic

TL;DR

This paper introduces a new closure operation for rings of mixed characteristic, demonstrating that it produces a ring with desirable properties related to Fontaine rings, and provides examples illustrating its significance.

Contribution

The paper defines a novel closure operation for mixed characteristic rings and proves it results in a ring with favorable Fontaine ring properties.

Findings

Closure operation produces rings with good Fontaine ring properties

Non-closed rings lack these properties

Provides explicit example illustrating the importance of the closure

Abstract

We define a closure operation for rings of mixed characteristic and verify that the closure is a ring. We then show that this closure produces a ring with good properties with respect to its Fontaine ring and give an example to show that rings that are not closed in this sense do not satisfy these properties.