Pedro Sancho

Characterization of Quasi-Coherent Modules that are Module Schemes

Let R be a commutative ring with unit, and let E be an R-module. We say the functor of R-modules E, defined by E(B) = E ⊗ R B, is a quasi-coherent R-module, and its dual E* is an R-module scheme. Both types of R-module functors are essential for the development of the theory of the linear representations of an affine R-group. We prove that a quasi-coherent R-module E is an R-module scheme if and only if E is a projective R-module of finite type, and, as a consequence, we also characterize finitely generated projective R-modules.

Homogeneous Hilbert scheme

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 Formula math. 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.