Prime filtrations of monomial ideals and polarizations

Details Prime filtrations of monomial ideals and polarizations Prime filtrations of monomial ideals and polarizations

2006-05-04

arxiv.org/abs/math/0605119v1

Mathematics

Commutative Algebra

Scientific paper

We show that all monomial ideals in the polynomial ring in at most 3 variables are pretty clean and that an arbitrary monomial ideal $I$ is pretty clean if and only if its polarization $I^p$ 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.

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