Let $V$ be a vector space over $\mathbb{C}$ and let $G$ be a finite group acting on $V$ via $G \longrightarrow \operatorname{GL}(V)$. We know that $\mathbb{C}[V]^G \subset \mathbb{C}[V]$ is an integral extension.
The question is, whether it is a faithfully flat extension (it's enough to prove that it is flat). I need this to show that the induced map $\operatorname{Spec}(\mathbb{C}[V]) \longrightarrow \operatorname{Spec}\mathbb{C}[V]^G$ equips $\operatorname{Spec}\mathbb{C}[V]^G$ with a quotient topology (I know there is an easier argument).
Thanks in advance.