Almost complete intersections and Stanley's conjecture

Abstract

Let K be a field and I a monomial ideal of the polynomial ring S = K[x 1 , . . . , xn]. We show that if either: 1) I is almost complete intersection, 2) I can be generated by less than four monomials; or 3) I is the Stanley-Reisner ideal of a locally complete intersection simplicial complex on [n], then Stanley's conjecture holds for S/I.

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

“…The similar characterizations of being Buchsbaum and generalized Cohen-Macaulay were also studied by them. Minh and Trung in [12] proved that for simplicial complex ∆ with dim(∆) = 1, I (2) ∆ is Cohen-Macaulay if and only if diam(∆) ≤ 2, where diam(∆) denotes the diameter of ∆. We pursue this line of research further.…”

“…Let ∆ be a 1-dimensional simplicial complex and I = I ∆ ⊂ S. Minh and Trung in [12] studied under which conditions S/I (2) and S/I 2 are Cohen-Macaulay. In this section we will give a characterization for the Cohen-Macaulayness of S/I (2) and S/I 2 in terms of the cleanness property.…”

“…This length is infinite if there is no path connecting them. The diameter of G, diam(G), is defined by (c) ⇒ (a) By [14,Theorem 3.10] it is enough to show that S/(I (2) ) p is clean. Let I = I ∆ = ∩ t i=1 P F i be a primary decomposition of I.…”

“…, r n−2 } with r 1 < r 2 < · · · < r n−2 , then we set (F, 1) := {(r i , 1) | r i ∈ F } and (F, 2 j ) := {(r j , 2)}∪ {(r i , 1) | 1 ≤ i ≤ n − 2, i = j}. Note that (I (2) ) p is a monomial ideal in a polynomial ring T = K[x (1,1) , . .…”

Monomial Ideals, Computations and Applications

“…In a very recent paper [14], Bandari, Divaani-Aazar and Soleyman Jahan showed that if I S D KOEx 1 ; : : : ; x n is a monomial ideal generated by monomials u 1 ; u 2 ; : : : ; u t , then S=I is pretty clean (and hence satisfies Stanley's conjecture) if either: u 1 ; u 2 ; : : : ; u t is a filter-regular sequence, or d -sequence, or I is an almost complete intersection. In a very recent paper [14], Bandari, Divaani-Aazar and Soleyman Jahan showed that if I S D KOEx 1 ; : : : ; x n is a monomial ideal generated by monomials u 1 ; u 2 ; : : : ; u t , then S=I is pretty clean (and hence satisfies Stanley's conjecture) if either: u 1 ; u 2 ; : : : ; u t is a filter-regular sequence, or d -sequence, or I is an almost complete intersection.…”