We first consider the sheaf of holomorphic functions $\mathcal{O}(\mathbb{C}^n)$ on $\mathbb{C}^n$. By [Oka coherence theorem](https://en.wikipedia.org/wiki/Oka_coherence_theorem), $\mathcal{O}(\mathbb{C}^n)$ is coherent over itself.


Now we consider a finite group $G$ acting on $\mathbb{C}^n$ and let $\pi: \mathcal{C}^n\to \mathbb{C}^n/G$ be the projection. We define a sheaf $\bar{\mathcal{O}}$ on $\mathbb{C}^n/G$ as follows: for any open subset $U\subset \mathbb{C}^n/G$, we define
$$
\bar{\mathcal{O}}(U):=\{f\in \mathcal{O}(\pi^{-1}(U))|f \text{ is }G-\text{invariant.}\}
$$



> 
> 
> > 
> > My question is: is this sheaf $\bar{\mathcal{O}}$ coherent over itself too?
> > 
> > 
> > 
> 
> 
> 



Do we have the Oka coherence theorem for finite group actions?
This would be true. You need two facts:


1. Grauert's theorem that coherent sheaves are preserved by proper direct images. This implies
$\pi\_\*\mathcal{O}\_{\mathbb{C}^n}$ is coherent.
2. Sub modules of coherent sheaves are coherent. Therefore
$$\tilde{\mathcal{O}} = \pi\_\*\mathcal{O}\_{\mathbb{C}^n}^G\subset \pi\_\*\mathcal{O}\_{\mathbb{C}^n}$$
is coherent


