On the subalgebra of invariant elements: finiteness and immersions

Abstract

Let $R$ be an algebra over a ring $\mathbb{k}$, $T$ an $R$-algebra, $M$ a finitely generated projective $R$-module, and $N$ a $T$-module. Let $G$ be a linearly reductive group scheme over $\mathbb{k}$ equipped with a representation $\rho :{\underline{G}}_R\to {\underline{\text{Aut}}}_{\text{Mod}(R)}(M)$. 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 $\text{Proj}A$ 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.

arXiv commentMost of the results have been generalized. For instance, $\mathbb{k}$ is now a commutative ring instead of a field of characteristic zero. The title and the abstract have changed as well