ON THE STANLEY DEPTH OF WEAKLY POLYMATROIDAL IDEALS

M. R. POURNAKI, S. A. SEYED FAKHARI, AND S. YASSEMI

Abstract. Let 𝕂 be a field and S = 𝕂[x_{1}, …, x_{n}] be the polynomial ring in n variables over the field 𝕂. In this paper, it is shown that Stanley’s conjecture holds for I and S/I if I is a product of monomial prime ideals or I is a high enough power of a polymatroidal or a stable ideal generated in a single degree.

1. Introduction

Let 𝕂 be a field and S = 𝕂[x_{1}, …, x_{n}] be the polynomial ring in n variables over the field 𝕂. Let M be a finitely generated ℤ^{n}-graded S-module. Let u ∈ M be a homogeneous element and Z ⊆ {x_{1}, …, x_{n}}. The 𝕂-subspace u𝕂[Z] generated by all elements uv with v ∈ 𝕂[Z] is called a Stanley space of dimension |Z| if it is a free 𝕂[ℤ]- module. Here, as usual, |Z| denotes the number of elements of Z. A decomposition D of M as a finite direct sum of Stanley spaces is called a Stanley decomposition of M. The minimum dimension of a Stanley space in D is called Stanley depth of D and is denoted by sdepth(D). The quantity

sdepth(M) := max {sdepth(D) | D is a Stanley decomposition of M}

is called Stanley depth of M. Stanley [10] conjectured that

depth(M) ≤ sdepth(M)

for all ℤ^{n}-graded S-modules M. For a reader friendly introduction to the Stanley depth, we refer the reader to [8].

Let I be a monomial ideal of S whose Rees algebra is R(I) and let 𝔪 = (x_{1}, …, x_{n}) be the graded maximal ideal of S. Then the 𝕂-algebra R(I)/𝔪R(I) is called the fibre ring, and its Krull dimension is called the analytic spread of I and is denoted by ℓ(I). This invariant is a measure for the growth of the number of generators of the powers of I. Indeed, for k ≫ 0, the Hilbert function H(R(I)/𝔪R(I), 𝕂, k) = dim_{𝕂}(I^{k}/𝔪I^{k}), which counts the number of generators of the powers of I, is a polynomial function of degree ℓ(I) − 1.

Let I be a weakly polymatroidal ideal of S which is generated in a single degree and ℓ(I) its analytic spread. In this paper, we show that sdepth(I) ≥ n − ℓ(I) + 1 and sdepth(S/I) ≥ n − ℓ(I) (see Theorem 2.5) and we conclude that if I is a product of monomial prime ideals of S, then I and S/I satisfy Stanley’s conjecture. We also show that if either I is a polymatroidal ideal or it is a stable ideal of S which is generated in a single degree, then I^{k} and S/I^{k} satisfy Stanley’s conjecture for k ≫ 0 (see Corollaries 2.7 and 2.11).

2000 Mathematics Subject Classification. Primary: 13C15, 05E99; Secondary: 13C13.

Key words and phrases. Monomial ideal, Stanley depth, Stanley conjecture, Polymatroidal ideal.

The research of M. R. Pournaki and S. Yassemi was in part supported by a grant from IPM (No.

91130115 and No. 91130214).

2. The results

In this paper, we deal with polymatroidal ideals. They were introduced in [4] and represent a natural generalization of matroidal ideals. In the following, we define polymatroidal ideals, and for more detailed information, we refer the reader to [4, 5, 6].

Definition 2.1. Let I be a monomial ideal of S = 𝕂[x_{1}, …, x_{n}] which is generated in a single degree, and assume that G(I) is the set of minimal monomial generators of I. The ideal I is called polymatroidal if the following exchange condition is satisfied: For monomials u = x_{1}^{a_{1}} … x_{n}^{a_{n}} and v = x_{1}^{b_{1}} … x_{n}^{b_{n}} belonging to G(I) and for every i with a_{i} > b_{i}, one has j with a_{j} < b_{j} such that x_{j}(u/x_{i}) ∈ G(I).

Weakly polymatroidal ideals are generalizations of polymatroidal ideals, and they are defined as follows.

Definition 2.2 ([5], Definition 12.7.1). A monomial ideal I of S = 𝕂[x_{1}, …, x_{n}] is called weakly polymatroidal if for every two monomials u = x_{1}^{a_{1}} … x_{n}^{a_{n}} and v = x_{1}^{b_{1}} … x_{n}^{b_{n}} in G(I) such that a_{1} = b_{1}, …, a_{t−1} = b_{t−1} and a_{t} > b_{t} for some t, there exists j > t such that x_{t}(v/x_{j}) ∈ I.

It is clear from the above definition that every polymatroidal ideal is weakly polymatroidal.

Lemma 2.3. Let I be a monomial ideal of S = 𝕂[x_{1}, …, x_{n}] which is generated in a single degree. Then for every 1 ≤ i ≤ n, we have ℓ((I : x_{i})) ≤ ℓ(I).

Proof. It is enough to show that for every integer k ≥ 1, μ(I^{k}) ≥ μ((I : x_{i})^{k}), where μ(I) denotes the number of minimal generators of I. Now assume that I is generated in degree p and G(I) = {u_{1}, …, u_{s}} is the set of minimal monomial generators of I. Without loss of generality, we may assume that there exists 0 ≤ t ≤ s such that u_{1}, …, u_{t} are divisible by x_{i} and u_{t+1}, …, u_{s} are not divisible by x_{i}. Let u′_{j} = u_{j}/x_{i} (1 ≤ j ≤ t).

For every integer k ≥ 1, we define an injective map f from G((I : x_{i})^{k}) to G(I^{k}), and this completes the proof. In order to do this, let u ∈ G((I : x_{i})^{k}). Then we may write u as below, where 0 ≤ q ≤ k:

u = u′_{i_{1}} … u′_{i_{q}}u_{i_{q+1}} … u_{i_{k}}.

Note that q = kp − deg(u), and therefore q is independent from the above representation. Therefore, we may define

f(u) := x_{i}^{q}u = u_{i_{1}} … u_{i_{q}}u_{i_{q+1}} … u_{i_{k}} ∈ I^{k}.

Since I^{k} is generated in degree pk, f(u) ∈ G(I^{k}). We now prove that f is injective. Assume that there exist u, v ∈ G((I : x_{i})^{k}) such that f(u) = f(v). Then by definition of f, for every j ≠ i, we have deg_{x_{j}}(u) = deg_{x_{j}}(v). Hence, if deg_{x_{i}}(u) > deg_{x_{i}}(v), then v|u and if deg_{x_{i}}(v) > deg_{x_{i}}(u), then u|v and in the both cases we derive a contradiction because u, v ∈ G((I : x_{i})^{k}). Therefore, deg_{x_{i}}(u) = deg_{x_{i}}(v) and so u = v, which implies that f is injective. □

For proving our main result, we need the following lemma.

Lemma 2.4. Let I be a weakly polymatroidal ideal of S = 𝕂[x_{1}, …, x_{n}] which is generated in a single degree. Then (I : x_{1}) satisfies the same property.

Proof. It is clear from the definition that (I : x_{1}) is a weakly polymatroidal ideal. Therefore, we prove that it is generated in a single degree. Suppose that G(I) = {u_{1}, …, u_{s}} is the set of minimal monomial generators of I, and let deg(u_{i}) = k. Without loss of generality, we may assume that u_{1}, …, u_{t} are divisible by x_{1} and u_{t+1}, …, u_{s} are not divisible by x_{1}, where 1 ≤ t ≤ s. Let v_{i} = u_{i}/x_{1} (1 ≤ i ≤ t). We claim that (I : x_{1}) is generated by v_{1}, …, v_{t}. In order to prove the claim, let v ∈ (I : x_{1}) be a monomial. Then x_{1}v ∈ I and so there exists 1 ≤ i ≤ s in such a way that u_{i} divides x_{1}v. If 1 ≤ i ≤ t, then v is divisible by v_{i} and therefore v ∈ (v_{1}, …, v_{t}). Therefore, we may assume that i ≥ t+1. Now u_{i} is not divisible by x_{1} and so u_{i}|v. By Definition 2.2, there exists j ≥ 2 such that x_{1}u_{i}/x_{j} ∈ I. Since deg(x_{1}u_{i}/x_{j}) = k, there exists 1 ≤ p ≤ t such that u_{p} = x_{1}u_{i}/x_{j} and hence v_{p} = u_{i}/x_{j} divides v and therefore v ∈ (v_{1}, … v_{t}). This proves the claim and completes the proof of the lemma. □

We are now in the position to state and prove our main result.

Theorem 2.5. Let I be a weakly polymatroidal ideal of S = 𝕂[x_{1}, …, x_{n}] which is generated in a single degree. Then we have the following assertions:

(i) sdepth(I) ≥ n − ℓ(I) + 1 and sdepth(S/I) ≥ n − ℓ(I).

(ii) depth(S/I) ≥ n − ℓ(I).

Proof. We prove (i) and (ii) simultaneously by induction on n and k, where k is the degree of generators of I. Let G(I) = {u_{1}, …, u_{s}} be the set of minimal monomial generators of I, and let deg(u_{i}) = k. If n = 1, then I is a principal ideal, and so we have ℓ(I) = 1, sdepth(I) = 1, and depth(S/I) = sdepth(S/I) = 0. Therefore, in this case, the inequalities in (i) and (ii) are trivial. If k = 1, then I is a complete intersection and so ℓ(I) = s. In this case, the inequality in (ii) is trivial, and the inequalities in (i) follow from [9, Theorem 1.1] and [7, Proposition 3.4]. We now consider n ≥ 2 and k ≥ 2. Assume that there exists a variable x_{j} such that

x_{j} ∉ ⋃_{i=1}^{s} Supp(u_{i}),

where for a monomial u ∈ S, Supp(u) is the set of variables which divide u. Hence, x_{j} is regular over S/I and so depth(S/I) = depth(S_{j}/IS_{j}) + 1, where S_{j} is the polynomial ring obtained from S by deleting the variable x_{j}. Therefore, the induction hypothesis on n implies that depth(S/I) ≥ n − ℓ(I). On the other hand, by [9, Theorem 1.1] and [7, Lemma 3.6], we conclude that sdepth(S/I) = sdepth(S_{j}/IS_{j})+1 and sdepth(I) = sdepth(IS_{j}) + 1. Therefore, using the induction hypothesis on n, we conclude that sdepth(I) ≥ n − ℓ(I) + 1 and sdepth(S/I) ≥ n − ℓ(I). Therefore, we may assume that

⋃_{i=1}^{s} Supp(u_{i}) = {x_{1}, …, x_{n}}.

Let S′ = 𝕂[x_{2}, …, x_{n}], and consider I′ = I ∩ S′ and I′′ = (I : x_{1}). Now I = I′S′ ⊕ x_{1}I′′S and S/I = (S′/I′S′) ⊕ x_{1}(S/I′′S) and therefore by the definition of Stanley depth we have

(1)  sdepth(I) ≥ min{sdepth_{S′}(I′S′), sdepth_{S}(I′′)}

and

(2)  sdepth(S/I) ≥ min{sdepth_{S′}(S′/I′S′), sdepth_{S}(S/I′′)}.

On the other hand, by applying the depth lemma on the exact sequence

0 ⟶ S/(I : x_{1}) ⟶ S/I ⟶ S/(I, x_{1}) ⟶ 0,

we conclude that

(3)  depth(S/I) ≥ min{depth_{S′}(S′/I′S′), depth_{S}(S/I′′)}.

Using Lemmas 2.3 and 2.4 and the induction hypothesis on k, we now conclude that depth_{S}(S/I′′) ≥ n − ℓ(I), sdepth_{S}(I′′) ≥ n − ℓ(I) + 1, and sdepth_{S}(S/I′′) ≥ n − ℓ(I).

Note that I′S′ is a weakly polymatroidal ideal of S′ which is generated in a single degree. Since

x_{1} ∈ ⋃_{i=1}^{s} Supp(u_{i})

and the generators of I′S are not divisible by x_{1}, using [5, Lemma 10.3.19], we conclude that ℓ(I′S′) ≤ ℓ(I) − 1, and therefore, by our induction hypothesis on n, we conclude that

sdepth_{S′}(I′S′) ≥ (n − 1) − ℓ(I′S′) + 1 ≥ (n − 1) − (ℓ(I) − 1) + 1 = n − ℓ(I) + 1

and similarly sdepth_{S′}(S′/I′S′) ≥ n − ℓ(I) and depth_{S′}(S′/I′S′) ≥ n − ℓ(I). Now the inequalities (1), (2), and (3) complete the proof of the theorem. □

It is known and easy to prove that ht(I) ≤ ℓ(I) for every monomial ideal I. In the following corollary, we give a stronger lower bound for the analytic spread of a weakly polymatroidal ideal which is generated in a single degree.

Corollary 2.6. Let I be a weakly polymatroidal ideal of S = 𝕂[x_{1}, …, x_{n}] which is generated in a single degree. Then

max{ht(𝔭) | 𝔭 ∈ Ass(S/I)} ≤ ℓ(I).

Proof. Let 𝔭 ∈ Ass(S/I) be given. By [2, Proposition 1.2.13] we have depth(S/I) ≤ n − ht(𝔭), while by Theorem 2.5 we have depth(S/I) ≥ n − ℓ(I). This implies that ht(𝔭) ≤ ℓ(I) for every 𝔭 ∈ Ass(S/I) and completes the proof of the corollary. □

Let I be a monomial ideal of S = 𝕂[x_{1} …, x_{n}]. A classical result by Burch [3] says that

min_{t} depth(S/I^{t}) ≤ n − ℓ(I).

By a theorem of Brodmann [1], the quantity depth(S/I^{t}) is constant for large t. We call this constant value the limit depth of I, and we denote it by lim_{t→∞} depth(S/I^{t}). Brodmann improved the Burch’s inequality by showing that

lim_{t→∞} depth(S/I^{t}) ≤ n − ℓ(I).

Corollary 2.7. Let I be a polymatroidal ideal of S = 𝕂[x_{1}, …, x_{n}]. Then there exists an integer k_{0} ≥ 1 such that for every k ≥ k_{0}, I^{k} and S/I^{k} satisfy Stanley’s conjecture.

Proof. Note that by [5, Theorem 12.6.3], every power of a polymatroidal ideal is again a polymatroidal ideal. Since every polymatroidal ideal is a weakly polymatroidal ideal which is generated in a single degree, Theorem 2.5 implies that for every k ≥ 1, sdepth(I^{k}) ≥ n − ℓ(I^{k}) + 1 = n − ℓ(I) + 1 and sdepth(S/I^{k}) ≥ n − ℓ(I^{k}) = n − ℓ(I). Now applying Burch’s inequality completes the proof. □

Definition 2.8. Let F be a nonempty subset of [n]. We denote by P_{F} the monomial prime ideal (x_{i} | i ∈ F). A transversal polymatroidal ideal is an ideal I of the form

I = P_{F_{1}}P_{F_{2}} … P_{F_{r}},

where F_{1}, …, F_{r} is a collection of nonempty subsets of [n] with r ≥ 1.

It follows from the above definition that the product of transversal polymatroidal ideals is again a transversal polymatroidal ideal and that every transversal polymatroidal ideal is a polymatroidal ideal.

Corollary 2.9. If I is a transversal polymatroidal ideal of S = 𝕂[x_{1}, …, x_{n}], then I and S/I satisfy Stanley’s conjecture.

Proof. Note that by Theorem 2.5, we have sdepth(I) ≥ n−ℓ(I)+1 and sdepth(S/I) ≥ n − ℓ(I). Also, [6, Corollary 3.14] implies that depth(S/I) = n − ℓ(I). Therefore, I and S/I satisfy Stanley’s conjecture. □

One should note that Corollary 2.9 essentially says that if I is a product of some monomial primes, then I and S/I satisfy Stanley’s conjecture.

Definition 2.10. Let u be a monomial in S = 𝕂[x_{1}, …, x_{n}]. We denote by m(u) the maximum number j such that x_{j}|u. Then a monomial ideal I of S is called a stable ideal if for all monomials u ∈ I and all i < m(u) one has x_{i}(u/x_{m(u)}) ∈ I.

It is clear from the above definition that every stable ideal is weakly polymatroidal.

Corollary 2.11. Let I be a stable ideal of S = 𝕂[x_{1}, …, x_{n}] which is generated in a single degree. Then there exists an integer k_{0} ≥ 1 such that for every k ≥ k_{0}, I^{k} and S/I^{k} satisfy Stanley’s conjecture.

Proof. Since every power of a stable ideal is again stable, Theorem 2.5 implies that for every k ≥ 1, sdepth(I^{k}) ≥ n − ℓ(I^{k}) + 1 = n − ℓ(I) + 1 and sdepth(S/I^{k}) ≥ n − ℓ(I^{k}) = n − ℓ(I). Now applying Burch’s inequality completes the proof. □

Acknowledgment

Part of this work was done while S. Yassemi visited the Max Planck Institute for Mathematics (MPIM), Bonn, Germany. He would like to thank the MPIM for sponsoring his visit to Bonn in March–April 2012.

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[9] A. Rauf, Stanley decompositions, pretty clean filtrations and reductions modulo regular elements, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 50(98) (2007), no. 4, 347–354.

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M. R. Pournaki, Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.

 E-mail address: pournaki@ipm.ir
 URL: http://math.ipm.ac.ir/pournaki/

S. A. Seyed Fakhari, Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran.

 E-mail address: fakhari@ipm.ir
 URL: http://math.ipm.ac.ir/fakhari/

S. Yassemi, School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.

 E-mail address: yassemi@ipm.ir
 URL: http://math.ipm.ac.ir/yassemi/