Noncoherence of some rings of holomorphic functions in several variables as an easy consequence of the one-variable case

TL;DR

This paper demonstrates that certain rings of holomorphic functions in multiple variables, such as the polydisc and ball algebras, are not coherent, extending known one-variable results to higher dimensions.

Contribution

It shows that the noncoherence of the disk and Wiener algebras in one variable implies the noncoherence of their multivariable counterparts.

Findings

Polydisc algebra is not coherent.

Ball algebra is not coherent.

Wiener algebra of the polydisc is not coherent.

Abstract

Using the facts that the disk algebra and the Wiener algebra are not coherent, we prove that the polydisc algebra, the ball algebra and the Wiener algebra of the polydisc are not coherent.