Ideal containments under flat extensions

Let φ:S=k[y0,...,yn]→R=k[y0,...,yn] be given by yi→fi where f0,...,fn is an R-regular sequence of homogeneous elements of the same degree. A recent paper shows for ideals, IΔ⊆S, of matroids, Δ, that IΔ(m)⊆Ir if and only if φ⁎(IΔ)(m)⊆φ⁎(IΔ)r where φ⁎(IΔ) is the ideal generated in R by φ(IΔ). We prove this result for saturated homogeneous ideals I of configurations of points in Pn and use it to obtain many new counterexamples to I(rn−n+1)⊆Ir from previously known counterexamples.