How to compute the Stanley depth of a monomial ideal

Abstract

Let J ⊂ I be monomial ideals. We show that the Stanley depth of I/J can be computed in a finite number of steps. We also introduce the fdepth of a monomial ideal which is defined in terms of prime filtrations and show that it can also be computed in a finite number of steps. In both cases it is shown that these invariants can be determined by considering partitions of suitable finite posets into intervals.

Stanley depth of monomial ideals with small number of generators

“…Let K be a field and S = K [8] that this invariant can be computed in a finite number of steps if M = I/J, where J ⊂ I ⊂ S are monomial ideals. There are two important particular cases.…”

“…If I ⊂ S is a monomial ideal, we are interested in computing sdepth(S/I) and sdepth(I). There are some papers regarding this problem, like [5,8,10,12,14]. Stanley's conjecture says that sdepth(S/I) ≥ depth(S/I), or in the general case, sdepth(M) ≥ depth(M), where M is a finitely generated multigraded S-module.…”

Graph and depth of a monomial squarefree ideal

“…Using the induction hypothesis on s we get sdepth S J ≥ depth S J, and so sdepth S I ≥ min{sdepth S I , sdepth S J}≥min{depth S I 1 , depth S I 2 } =depth S I. The proof follows from Theorem 1.8 and the above proposition, the reduction to the case J = m being given by [1,Lemma 3.6].…”

“…Let I ⊂ S be a monomial ideal of S, u ∈ I a monomial and uK [Z], Z ⊂ {x 1 The Stanley Conjecture [11] says that sdepth I ≥ depth I. This conjecture holds for arbitrary monomial squarefree ideals if n ≤ 5 by [7] (see especially the arXiv version), or for intersections of four monomial prime ideals by [5], [8].…”

Stanley depth of squarefree Veronese ideals

“…Herzog, Vladoiu and Zheng show in [5] that sdepth(M ) can be computed in a finite number of steps if M = I/J, where J ⊂ I ⊂ S are monomial ideals. There are two important particular cases, I and S/I.…”