Do we have the Oka coherence theorem for finite group actions?

Question

We first consider the sheaf of holomorphic functions $\mathcal{O}(\mathbb{C}^n)$ on $\mathbb{C}^n$. By 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?

Answer 1

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