K-theory, local cohomology and tangent spaces to Hilbert schemes

Abstract: By using K-theory, we construct a map from the tangent space to the Hilbert scheme at a point Y to the local cohomology group. And we use this map to answer affirmatively(after slight modification) a question by Mark Green and Phillip Griffiths in [8] on constructing a map from the tangent space to the Hilbert scheme to the tangent space to the cycle group.