Let $G$ be a geometrically reductive algebraic group over an algebraically closed field $k$. Let $X$ be an affine variety over $k$ on which $G$ acts regularly. Then $G$ acts on the coordinate ring $A$ of $X$ by automorphisms, and we denote by $A^G$ the subring consisting of invariant elements.
Problem: Prove that the inclusion $A^G\subset A$ induces a surjective map of the ring spectra.
Background: A reffined statement is that for a reducitve group action on an affine variety, the categorical quotient map is surjective.
In the book "Lectures on Invariant Theory" by Igor Dolgachev, it is proved that such a categorical quotient is "good" in the sense of G.I.T., but it seems that his proof is incomplete and the probleme above is the missing part.