Infinite Intersections of open subschemes and the Hilbert scheme of points

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