Infinite intersections of open subschemes and the Hilbert scheme of points

We study infinite intersections of open subschemes and the corresponding infinite intersection of Hilbert schemes. If {Uα} is the collection of open subschemes of a variety X containing the fixed point P, then we show that the Hilbert functor of flat and finite families of is given by the infinite intersection , where is the Hilbert functor of flat and finite families on Uα. In particular, we show that the Hilbert functor of flat and finite families on is representable by a scheme.