# The Ohm-Rush content function II. Noetherian rings, valuation domains, and base change

## TL;DR

This paper advances the understanding of Ohm-Rush algebras by establishing new results on their properties over Noetherian rings, valuation domains, and under base change, with implications for algebraic dimension and prime spectrum topology.

## Contribution

It proves that faithfully flat weak content algebras over Noetherian rings are semicontent and characterizes content algebras over valuation domains, extending previous work.

## Key findings

- Faithfully flat weak content algebra over Noetherian rings is semicontent.
- In valuation domains, content algebra maps correspond to isomorphisms of value groups.
- Inclusion of valuation domains is a content algebra iff the induced map on value groups is an isomorphism.

## Abstract

The notion of an Ohm-Rush algebra, and its associated content map, has connections with prime characteristic algebra, polynomial extensions, and the Ananyan-Hochster proof of Stillman's conjecture. As further restrictions are placed (creating the increasingly more specialized notions of weak content, semicontent, content, and Gaussian algebras), the construction becomes more powerful. Here we settle the question in the affirmative over a Noetherian ring from our previous article of whether a faithfully flat weak content algebra is semicontent (and over an Artinian ring of whether such an algebra is content), though both questions remain open in general. We show that in content algebra maps over Pr\"ufer domains, heights are preserved and a dimension formula is satisfied. We show that an inclusion of nontrivial valuation domains is a content algebra if and only if the induced map on value groups is an isomorphism, and that such a map induces a homeomorphism on prime spectra. Examples are given throughout, including results that show the subtle role played by properties of transcendental field extensions.