From the Geometric Invariant Theory book [Mumford - Fogarty - Kirwan], we have the following theorem
([MFK,Theorem 1.1) Let $X$ be an affine scheme over a field $k$, let $G$ be a reductive algebraic group over $k$, let $\sigma:G\times X \to X$ be an action of $G$ on X. Then, a universal categorical quotient $(Y,\Phi)$ of $X$ by $G$ exists.
To fix ideas let's suppose $X:=\operatorname{Spec} A$ for a finitely generated $k$-algebra $A$.
Since $G$ is reductive, we have that $A^G$ is a finitely generated $k$-algebra.
I don't see where the finite generation of $A^G$ is used in the proof of the theorem? For instance we can drop the assumption that $G$ is reductive but then we still need to require that : $A^G$ is finitely generated $k$-algebra in order to conclude that $\operatorname{Spec} A^G$ is the categorical quotient. Where this assumption is used in the proof of the theorem above?
Thank you for your help.
As noted in the comments, if one is working with (locally) finite type schemes over $k$, then one needs $A^G$ to be finitely generated in order for the quotient $\operatorname{Spec} A^G$ to belong to the category of (locally) finite type schemes over $k$ and therefore be the categorical quotient.