Prime filtrations of monomial ideals and polarizations

We show that an arbitrary monomial ideal I is pretty clean if and only if its polarization Ip is clean. This yields a new characterization of pretty clean monomial ideals in terms of the arithmetic degree, and it also implies that a multicomplex is shellable if and only the simplicial complex corresponding to its polarization is (non-pure) shellable. We also discuss Stanley decompositions in relation to prime filtrations.