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ABSTRACT: Let S be a locally noetherian scheme and R an N-graded O S -algebra of finite type. We say that X = Spec R is a homogeneous variety over S. In this paper we prove that the functor is representable by an S-scheme which is a disjoint union of locally projective schemes over S. The proof is very simple and it only makes use of the the-ory 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.
Proceedings of the American Mathematical Society 01/2007; 136(03). · 0.61 Impact Factor