Homogeneous Hilbert scheme

Let $S$ be a locally noetherian scheme and $R$ an $\mathbb{N}$-graded $\mathcal O_S$-algebra of finite type. We say that $X=\operatorname{Spec}R$ is a homogeneous variety over $S$. In this paper we prove that the functor

is representable by an $S$-scheme that is a disjoint union of locally projective schemes over $S$. The proof is very simple, and it only makes use of the theory of graded modules and standard flatness criteria. From this, one obtains an elementary construction (which does not make use of cohomology) of the ordinary Hilbert scheme of a locally projective $S$-scheme.