The ohm-rush content function iii: completion, globalization, and power-content algebras

TLDR: In this article, the authors analyze whether the completion map of a ring homomorphism is Ohm-Rush and 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.

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.