Abstract. I J be two squarefree monomial ideals of a polynomial algebra over a field generated in degree ≥ d, resp. ≥ d + 1 . Suppose that I is either generated by three monomials of degrees d and a set of monomials of degrees ≥ d
IntroductionLet K be a field and S = K[x 1 , . . . , x n ] be the polynomial K-algebra in n variables. Let I J be two squarefree monomial ideals of S and suppose that I is generated by squarefree monomials of degrees ≥ d for some positive integer d. After a multigraded isomorphism we may assume either that J = 0, or J is generated in degrees ≥ d + 1. By [5, Proposition 3.1] (see [12, Lemma 1.1]) we have depth S I/J ≥ d. Depth of I/J is a homological invariant and depends on the characteristic of the field K.The purpose of our paper is to study upper bound conditions for depth S I/J. Let B (resp. C) be the set of the squarefree monomials of degrees d + 1 (resp. d + 2) of I \ J. Suppose that I is generated by some squarefree monomials f 1 , . . . , f r of degrees d for some d ∈ N and a set of squarefree monomials E of degree ≥ d + 1. If d = 1 and each monomial of B \ E is the least common multiple of two f i then it is easy to show that depth S I/J = 1 (see Lemma 3). Trying to extend this result for d > 1 we find an obstruction given by Example 2. Our extension given by Lemma 4 is just a special form, but a natural condition seems to be given in terms of the Stanley depth.More precisely, let P I\J be the poset of all squarefree monomials of I \ J with the order given by the divisibility. Let P be a partition of P I\J in intervals [u, v] = {w ∈ P I\J : u|w, w|v}, let us say P I\J = ∪ i [u i , v i ], the union being disjoint. Define sdepth P = min i deg v i and the Stanley depth of I/J given by sdepth S I/J = max P sdepth P , where P runs in the set of all partitions of P I\J (see [5], [20]). Stanley's Conjecture says that sdepth S I/J ≥ depth S I/J. The Stanley depth of I/J is a combinatorial invariant and does not depend on the characteristic of the field K. Stanley's Conjecture holds when J = 0 and I is an intersection of four monomial An equivalent definition for the Stanley depth is:where a Stanley decomposition of a Z−graded (resp. Z n −graded) S − module M is D = (S i , u i ) i∈I , where u i are homogenous elements of M and S i are graded (resp. Z n −graded) K−algebra retracts of S and S i ∩ Ann(u i ) = 0 such that M = ⊕ i S i u i ; and sdepth D is the depth of the S−module ⊕ i S i u i . A more general concept is the one of Hilbert depth of a Z−graded module M, denoted by hdepth 1 (M). Instead of considering equality, we only assume that M ∼ = ⊕S i (−s i ), where s i ∈ Z. One can also construct hdepth n analogously if M is a multigraded (that is Z n ) module. In [9] is presented (and implemented) an algorithm that computes hdepth 1 (M) based on a Theorem of Uliczka [21]; and in [7] was presented an algorithm that computes hdepth n (M). Meanwhile, another algorithm that computes hdepth 1 and more was given in [3]. [9, Proposition 1.9] gives a partial answer to a question of Herzog ...