The Ohm-Rush content function III: Completion, globalization, and power-content algebras

TL;DR

This paper investigates the Ohm-Rush property in ring homomorphisms, especially in completions and flat extensions, revealing dimension-dependent behaviors and introducing a new intermediate property.

Contribution

It characterizes when the completion map is Ohm-Rush in Noetherian local rings and explores globalization conditions, introducing a new property between Ohm-Rush and weak content algebras.

Findings

Completion map is Ohm-Rush in dimension one

Completion map is not Ohm-Rush in higher dimensions

New intermediate property between Ohm-Rush and weak content algebras

Abstract

One says that a ring homomorphism $R \rightarrow S$ is Ohm-Rush if extension commutes with arbitrary intersection of ideals, or equivalently if for any element $f\in S$, there is a unique smallest ideal of $R$ whose extension to $S$ contains $f$, called the content of $f$. For Noetherian local rings, we analyze whether the completion map is Ohm-Rush. We show that the answer is typically `yes' in dimension one, but `no' in higher dimension, and in any case it coincides with the content map having good algebraic properties. We then analyze the question of when the Ohm-Rush property globalizes in faithfully flat modules and algebras over a 1-dimensional Noetherian domain, culminating both in a positive result and a counterexample. Finally, we introduce a notion that we show is strictly between the Ohm-Rush property and the weak content algebra property.