Pretty cleanness and filter-regular sequences

Abstract

Let K be a field and S = K[x 1, …, x n ]. Let I be a monomial ideal of S and u 1, …, u r be monomials in S. We prove that if u 1, …, u r form a filter-regular sequence on S/I, then S/I is pretty clean if and only if S/(I, u 1, …, u r ) is pretty clean. Also, we show that if u 1, …, u r form a filter-regular sequence on S/I, then Stanley’s conjecture is true for S/I if and only if it is true for S/(I, u 1, …, u r ). Finally, we prove that if u 1, …, u r is a minimal set of generators for I which form either a d-sequence, proper sequence or strong s-sequence (with respect to the reverse lexicographic order), then S/I is pretty clean.