The cleanness of (symbolic) powers of Stanley-Reisner ideals

Let Δ be a pure simplicial complex on the vertex set [n] = {1,..., n} and IΔ its Stanley-Reisner ideal in the polynomial ring S = K[x1,..., xn]. We show that Δ is a matroid (complete intersection) if and only if S/IΔ(m) (S/IΔ(m)) is clean for all m ∈ N and this is equivalent to saying that S/IΔ(m) (S/IΔ(m), respectively) is Cohen-Macaulay for all m ∈ N. By this result, we show that there exists a monomial ideal I with (pretty) cleanness property while S/Im or S/Im is not (pretty) clean for all integer m ≥ 3. If dim(Δ) = 1, we also prove that S/IΔ(2) Δ (S/IΔ2) is clean if and only if S/IΔ(2) (S/IΔ2, respectively) is Cohen-Macaulay.