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Abstract
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Let \Delta be a pure simplicial complex on the vertex set [n] = {1,...,n} and I_\Delta its Stanley-Reisner ideal in the polynomial ring S = K[x_1,..., x_n]. We show that \Delta is a matroid (complete intersection) if and only if S/I_\Delta ^(m) (S/I_\Delta^m) is clean for all m \in N and this is equivalent to saying that S/I_\Delta ^(m) (S/I_\Delta^m, respectively) is Cohen-Macaulay for all m \in 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(\Delta) = 1, we also prove that S/I_\Delta ^(2) (S/I_\Delta^2) is clean if and only if S/I_\Delta ^(2) (S/I_\Delta^2, respectively) is Cohen-Macaulay.
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