  Let $R$ be an algebra over a ring $\Bbbk$, $T$ an $R$-algebra, $M$ a finitely
generated projective $R$-module, and $N$ a $T$-module. Let $G$ be a linearly
reductive group scheme over $\Bbbk$ equipped with a representation
$\rho:\underline{G}_{R}\rightarrow
\underline{\textrm{Aut}}_{\textrm{Mod}(R)}(M)$. For the graded $T$-algebra $A$,
defined as $ A := \left( S_{T}^{\bullet} (M^{\vee} \otimes_{R}N )\right)^{G}, $
we determine the conditions under which the graded $T$-algebra $A$ is finitely
generated, finitely presented, or flat. Furthermore, we establish the
conditions under which a closed embedding of $\textrm{Proj} \ A$ into a
projective space exists. Since we do not impose any Noetherian hypotheses, our
results generalize those in the literature, providing new powerful tools
regarding moduli problems.
