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

Let Δ be a pure simplicial complex on the vertex set [n] = {1,..., n} and I _Δ its Stanley-Reisner ideal in the polynomial ring S = K[x ₁,..., x _n]. We show that Δ is a matroid (complete intersection) if and only if S/I _Δ ^(m) (S/I _Δ ^(m)) is clean for all m ∈ N and this is equivalent to saying that S/I _Δ ^(m) (S/I _Δ ^(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/I ^m or S/I ^m is not (pretty) clean for all integer m ≥ 3. If dim(Δ) = 1, we also prove that S/I _Δ ⁽²⁾ Δ (S/I _Δ ²) is clean if and only if S/I _Δ ⁽²⁾ (S/I _Δ ², respectively) is Cohen-Macaulay.