Krull dimensions of rings of holomorphic functions

TL;DR

This paper proves that the Krull dimension of the ring of holomorphic functions on a connected complex manifold is at least continuum whenever it is positive, revealing a lower bound on its algebraic complexity.

Contribution

It establishes a lower bound on the Krull dimension of rings of holomorphic functions, linking complex analysis with algebraic dimension theory.

Findings

Krull dimension is at least continuum if positive

Connects algebraic properties of holomorphic function rings with complex geometry

Provides new insights into the structure of function rings on complex manifolds

Abstract

We prove that the Krull dimension of the ring of holomorphic functions of a connected complex manifold is at least continuum if it is positive.