On the subalgebra of invariant elements: Finiteness and immersions

Abstract

[EN]Given the n-dimensional affine space over an arbitrary commutative ring k, G a group scheme flat and finitely presented over k and X ⊂ Ank a G-invariant affine and closed subscheme, we prove that the GIT quotient X/G is of finite type over k, even if k is not noetherian, provided G is linearly reductive. This is well-known if X → X/G is faithfully flat, which does not hold in general. We also explore the infinite- dimensional case. Concretely, we consider a G − k finitely presented projective module M and an arbitrary k-module N. We prove, under certain conditions on k and G, that the degrees of the generators of (S• (M ∨ ⊗ N ))G and the degrees of the generators of the ideal of relations are bounded. We encode this property into the notion of partially generated graded (pgg) algebra and we give their main properties. In particular, we prove the existence of canonical equivariant immersions of spectra of pgg algebras in certain projective spaces.