Module scheme/Invariant module/Algebra/Fact

Let ${}G$ be a finite group acting on a ${}K$-algebra ${}S$ by ${}K$-algebra automorphisms with invariant ring ${}R$. Let ${}\rho$ be a linear representation on the vector space ${}V$, and let

${}M={\left(S\otimes _{K}V\right)}^{\rho }\,$

be the invariant ${}R$-module.

Then there exists a natural ${}\mathbb {N}$-graded ${}R$-algebra homomorphism

$\operatorname {Sym} {\left(M\right)}\longrightarrow {\left(S\otimes _{K}V\right)}^{\beta \times \rho },$

which is an isomorphism in zeroth and first degree.

If ${}V\subseteq {{\mathbb {A} }_{K}^{d}}$ is the locus where ${}\beta$ is free, and ${}U$ its image, then the corresponding morphism of spectra

$Z_{\rho }\longrightarrow \operatorname {Spec} {\left(\operatorname {Sym} {\left(M\right)}\right)}$

is over ${}U$ an isomorphism of vector bundles.

Write a proof