Abstract Hilbert schemes

In analogy with classical projective algebraic geometry, Hilbert functors c an be defined for objects in any Abelian category. We study the moduli prob lem for such objects. Using Grothendieck's general framework. We show that with suitable hypotheses the Hilbert functor is representable by an algebra ic space locally of finite type over the base field. For the category of th e graded modules over a strongly Noetherian graded ring, the Hilbert functo r of graded modules with a fixed Hilbert series is represented by a commuta tive projective scheme. For the projective scheme corresponding to a suitab le noncommutative graded algebra, the Hilbert functor is represented by a c ountable union of commutative projective schemes.