Depth and Stanley Depth of Multigraded Modules

Abstract

Abstract. We study the behavior of depth and Stanley depth along short exact sequences of multigraded modules and under reduction modulo an element.

Stanley depth of monomial ideals with small number of generators

“…In the third section, we prove that if I ⊂ K [ 1 2 3 ] is saturated, then sdepth(I) ≥ 2, see Proposition 3.1. As a consequence, sdepth(I) ≥ sdepth(K [ 1 2 3 ]/I) + 1 for any monomial ideal in I ⊂ K [ 1 2 3 ], see Corollary 3.1, thereby giving in this special case an affirmative answer to a question raised by Rauf in [13].…”

Depth and Stanley depth of the edge ideals of square paths and square cycles

“…In this work we focus on the relationship between sdepth I and sdepth S/I for an arbitrary monomial ideal I. In [19], Rauf showed that if I is a complete intersection monomial ideal, then sdepth(I) > sdepth(S/I). Subsequently, Herzog's survey article [5] stated the weaker inequality sdepth(I) ≥ sdepth(S/I) (for arbitrary monomial ideals) as Conjecture 64, noting that the inequality is strict in all known cases.…”

Stanley depth of powers of the edge ideal of a forest

“…Finally, we state and prove a generalization of our main theorem. We remark that the proofs of the main theorem and its extension are minor modifications of the original proofs for depth, replacing depth by sdepth and the depth lemma by [7,Lemma 2.2]. Nevertheless, here we give complete proofs in order to make the treatment clear and self-contained.…”