On the subalgebra of invariant elements: finiteness and immersions

TL;DR

This paper investigates the algebraic properties of invariant subalgebras under group actions, establishing conditions for finiteness, presentation, flatness, and embeddings without Noetherian assumptions, thus broadening the scope of moduli theory tools.

Contribution

It generalizes existing results by removing Noetherian constraints, providing new criteria for finiteness, presentation, flatness, and embeddings of invariant subalgebras in a broad algebraic setting.

Findings

Conditions for finite generation of invariant algebras

Criteria for finite presentation and flatness

Existence of closed embeddings into projective space

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.