Let $X$ be $ \operatorname{Proj}(A)$ for some graded ring A, and let $G$ be a finite group acting on $A$ with morphisms of graded rings; consequently $G$ acts on $X$.
I know I can write $X = \bigcup_{f\in A^+}X_f$ where $X_f=\{p \in X \mid f \notin p\}$.
Is it true that the quotient $X/G$, as a topological space, can be written as $X/G = \bigcup_{f\in A^G}p(X_f)$ ? With $p:X \to X/G$ I intend the projection, while $A^G $ is the subring of $G$ invariants element of $A$.
I tried to prove it, but no idea on how to proceed. Any help/hint would be appreciated!
