On the subalgebra of invariant elements: finiteness and immersions

Abstract: 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.