Gaussian elements of a semicontent algebra

TL;DR

This paper extends the Gaussian property of content functions from univariate polynomials to multivariate polynomials, power series, and base changes over Noetherian rings using the Ohm-Rush framework.

Contribution

It generalizes known results about Gaussian content to broader algebraic contexts, including multivariate and power series cases, and base change scenarios.

Findings

Content function acts as a homomorphism in new contexts

Strong correspondence when base ring is approximately Gorenstein

Results hold under separable algebraically closed field extensions

Abstract

The connection between a univariate polynomial having locally principal content and the content function acting like a homomorphism (the so-called Gaussian property) has been explored by many authors. In this work, we extend several such results to the contexts of multivariate polynomials, power series over a Noetherian ring, and base change of affine $K$-algebras by separable algebraically closed field extensions. We do so by using the framework of the Ohm-Rush content function. The correspondence is particularly strong in cases where the base ring is approximately Gorenstein or the element of the target ring is regular.