On the conditions ensuring the freedom of topological modules over algebras of holomorphic functions

Let $\mathcal{O}(U)$ be the Fréchet algebra of holomorphic functions on a domain $U\subset\mathbb{C}$, endowed with the compact-open topology. In this paper we study certain properties ensuring the freedom of finitely generated Fréchet $\mathcal{O}(U)$-modules. In particular, it is proved that each flat finitely generated Fréchet $\mathcal{O}(U)$-module is free.