Square-free Grobner degenerations

Let I be a homogeneous ideal of S = K[x(1), ..., x(n)] and let J be an initial ideal of I with respect to a term order. We prove that if J is radical then the Hilbert functions of the local cohomology modules supported at the homogeneous maximal ideal of S/I and S/J coincide. In particular, depth(S/I) = depth(S/J) and reg(S/I) = reg(S/J).