Let $H$ be the ring of holomorphic functions on $\mathbb{C}$ and $\mathbb{C}[z]$ the ring of polynomial functions on $\mathbb{C}$. Then $H$ is naturally a $\mathbb{C}[z]$-module. Is $H$ flat?
I heard this result from one of my friends but I'm not sure this the real statement. In any case the real statement should be asserting the flatness of some ring of holomorphic functions, and it is proved by Serre. Also it seems like there is an analogous result where $\mathbb{C}$ is replaced by complex manifolds and $\mathbb{C}[z]$ replaced by germs of holomorphic functions.
Is there any result of this kind? Could you please provide references for its proof? Thanks in advance!