(1)MIRCEA CIMPOEAS¸ Communicated by Vasile Brˆınz˘anescu This is a survey of the results obtained by the author regarding the Stanley depth of monomial ideals and the Stanley’s conjecture

MIRCEA CIMPOEAS¸ Communicated by Vasile Brˆınz˘anescu

This is a survey of the results obtained by the author regarding the Stanley depth of monomial ideals and the Stanley’s conjecture.

AMS 2010 Subject Classification: Primary: 13H10, Secondary: 13P10.

Key words: Stanley depth, Stanley’s conjecture, monomial ideal.

1. INTRODUCTION

Let K be a field and S=K[x₁, . . . , x_n] the polynomial ring overK. Let M be a Zⁿ-gradedS-module. A Stanley decomposition of M is a direct sum D:M =L_r

i=1m_iK[Z_i] asK-vector space, where m_i ∈M,Z_i ⊂ {x₁, . . . , x_n} such that m_iK[Z_i] is a free K[Z_i]-module. We define sdepth(D) = min^r_i=1|Z_i| and sdepth_S(M) = max{sdepth(D)| D is a Stanley decomposition of M}, see [1]. The number sdepth(M) is called theStanley depth ofM. Herzog, Vladoiu and Zheng show in [11] 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(S/I) and sdepth_S(I) and to find some relation between them.

Csaba Biro, David M. Howard, Mitchel T. Keller, William T. Trotter and Stephen J. Young proved that ifm= (x₁, . . . , x_n)⊂S, then sdepth(m) =_n

2

, see [2, Theorem 2.2]. We extended this result to monomial ideals generated by powers of variables, see [7, Theorem 1.3]. More precisely, we proved in [5]

that sdepth(m^k)≤l

n k+1

m

, for any positive integer k and we conjectured that the equality holds. Shen proved that if I is a complete intersection monomial ideal minimally generated by m monomials, then sdepth(I) = n−_m

2

, see [17, Theorem 2.4]. Okazaki proved that if I is an arbitrary monomial ideal minimally generated by m monomials, then sdepth(I) ≥ n−_m

2

, see [13, Theorem 2.1].

In [6], we compute the Stanley depth for the quotient ring of a square free Veronese ideal and we give some bounds for the Stanley depth of a square

REV. ROUMAINE MATH. PURES APPL.,58(2013),2,205–212

free Veronese ideal. In particular, it follows that both satisfy the Stanley’s conjecture. In [9] we proved that if I is a monomial ideal, almost complete intersection, then the Stanley conjecture holds for I and S/I. In [10], we give several bounds for sdepth_S(I+J), sdepth_S(I ∩J), sdepth_S(S/(I+J)), sdepth_S(S/(I∩J)), sdepth_S(I :J) and sdepth_S(S/(I :J)), where I, J ⊂S= K[x1, . . . , xn] are monomial ideals. Also, we give some equivalent forms of Stanley conjecture for I and S/I, whereI ⊂S is a monomial ideal.

2. STANLEY DEPTH

FOR MONOMIAL COMPLETE INTERSECTION IDEALS

We proved in [7] the following key result.

Lemma 2.1 ([8, Lemma 1.1]). Let v₁, . . . , v_m ∈ K[x₂, . . . , x_m] be some monomials and let a be a positive integer. Let I = (x^a₁v1, v2,· · · , vm) and I⁰= (x^a+1₁ v1, v2,· · ·, vm). Then sdepth(I) = sdepth(I⁰).

Csaba Biro, David M. Howard, Mitchel T. Keller, William T. Trotter and Stephen J. Young proved the following theorem.

Theorem 2.2 ([2, Theorem 2.2]). If m = (x₁, . . . , x_n) ⊂ S, then sdepth(m) =_n

2

.

As a direct consequence of Lemma 2.1 and Theorem 2.2, we get.

Theorem2.3 ([8, Theorem 1.3]). Leta1, . . . , anbe some positive integers.

Then:

sdepth((x^a₁¹, . . . , x^a_nⁿ)) =sdepth((x₁, . . . , x_n)) =ln 2 m

. In particular, sdepth((x^a₁¹, . . . , x^a_m^m)) =n−_m

2

for any1≤m≤n.

We also proved in [8] the following results.

Theorem 2.4 ([8, Theorem 2.1]). Let I ⊂S be a complete intersection monomial ideal. Then sdepth(I) = sdepth(√

I).

Corollary 2.5 ([8, Corrolary 1.3]). Let I ⊂S be a monomial ideal with G(I) ={v₁, . . . , v_m}. Let S⁰ =S[x_n+1]and I⁰ = (v₁, . . . , vm−1, x_n+1v_m).Then sdepth(I⁰)≤sdepth(I) + 1.

In the condition of the above Corollary, Shen proved in [17] the other inequality and, as a direct consequence he obtained.

Theorem2.6 ([17, Theorem 2.4]). LetI be a complete intersection mono- mial ideal minimally generated by m monomials. Then, sdepth(I) =n−_m

2

. Okazaki gived in [13] an interesting result related with the previous.

Theorem 2.7 ([13, Theorem 2.1]). Let I ⊂S be a monomial ideal (min- imally) generated by m monomials. Then

sdepth(I)≥max{1, n−jm 2

k }.

We recall also the following result of Asia Rauf, see [16, Theorem 1.1].

Theorem2.8. LetI ⊂Sbe a monomial complete intersection, minimally generated by m monomials. Then sdepth(S/I) =n−m.

Two main results from [7] are the following.

Proposition 2.9 ([7, Proposition 1.2]). Let I ⊂S be a monomial ideal, minimally generated by m monomials. Then sdepth(S/I)≥n−m.

Note that the above Proposition is a counterpart for Theorem 2.6, since the equality sdepth(S/I) =n−m holds for a monomial complete intersection ideal I ⊂ S, minimally generated by m monomials. Therefore, monomial complete intersection ideals have the minimal Stanley depth, w.r.t. number of generators.

Theorem 2.10 ([7, Theorem 1.4]). Let I ⊂S be a monomial ideal such that I = v(I :v), for a monomial v ∈ S. Then sdepth(S/I) = sdepth(S/(I : v))and sdepth(I) = sdepth(I :v).

Theorem 2.11 ([7, Theorem 2.6]). Let I ⊂S be a monomial ideal min- imally generated by at most 3 generators. Then I and S/I satisfy the Stanley conjecture.

Our main results from [9] are the following.

Theorem 2.12 ([9, Theorem 1.8]). Let I ⊂ S = K[x₁, . . . , x_n] be a monomial ideal, minimally generated by m monomials, k = max{|P| : P ∈ Ass(S/I)}, and s≥k be an integer. Then

1. If m≤s−1 +l

s k−1

m

, then sdepth(S/I)≥n−s.

2. If m≤2s−3 +l

2s−2 k−1

m

, then sdepth(I)≥n−s+ 1.

If depth(S/I) =n−s then(1) and (2) imply the Stanley conjecture for S/I, respectively forI.

Corollary 2.13 ([9, Corollary 1.9]). Let I be a monomial almost com- plete intersection ideal. Then Stanley’s conjecture holds for S/I and I.

3. STANLEY DEPTH OF SQUAREFREE VERONESE IDEALS In this section we will follow [6].

Theorem 3.1 ([6, Theorem 1.1]). (1) sdepth(S/I_n,d) =d−1.

(2) d≤sdepth(I_n,d)≤ ^n−d_d+1 +d.

Since depth(S/I_n,d) =d−1, we get the following corollary.

Corollary 3.2 ([6, Corollary 1.2]). I_n,d and S/I_n,d satisfy the Stanley conjecture. Also,

sdepth(I_n,d)≥sdepth(S/I_n,d) + 1.

Letk≤nbe two positive integers. We denoteA_n,k ={F ⊂[n]| |F|=k}.

With this notations, we have the following well known result from combina- torics.

Theorem 3.3. For any positive integers d≤n such that d≤n/2, there exists a bijective map Φ_n,d :A_n,d →A_n,d, such that Φ_n,d(F)∩F =∅, for any F ∈A_n,d.

As a consequence, we get the following corollary.

Corollary 3.4 ([6, Corollary 1.5]). Let n, dbe two positive integers such that 2d+ 1≤n≤3d. Then sdepth(In,d) =d+ 1.

We also, give the following conjecture.

Conjecture 3.5. For any positive integers d ≤ n such that d ≤ n/2, sdepth(I_n,d) =j

n−d d+1

k +d.

In [12], Mitchel T. Keller, Yi-Huang Shen, Noah Streib and Stephen J.

Young give a patial positive answer to the above conjecture. More precisely, they proved the following result.

Theorem 3.6 ([12, Theorem 1.1]). Let I_n,d be the squarefree Veronese ideal in S, generated by all squarefree monomials of degree d.

(1)If1≤d≤n <5d+4, then the Stanley depthsdepth(In,d) = jn−d

d+1

k +d.

(2) If d≥1 and n≥5d+ 4 thend+ 3≤sdepth(In,d)≤j

n−d d+1

k +d.

In [5] we proved the following results, related with those above.

Theorem 3.7 ([5, Theorem 2.2]). Let k be a positive integer. Then sdepth(m^k)≤

n k+ 1

. In particular, if k≥n−1, then sdepth(m^k) = 1.

Proposition 3.8 ([5, Proposition 2.3]). Let I ⊂S be a monomial ideal.

Then

sdepth(m^kI) = 1 f or k0.

We also conjectured that:

Conjecture 3.9. sdepth(m^k) =l

n k+1

m .

Regarding this conjecture, Bruns, Krattenthaler and Uliczka proved in [4] that the Hilbert depth of the k-th power of m is exactly l

n k+1

m

. Since the Hilbert depth is always ≥than the Stanley depth we cannot conclude the equality for the Stanley depth, although the conjecture seems to be true. For the definition and basic properties of the Hilbert depth of a graded, respectively multigraded, module, see [3].

4.SEVERAL INEQUALITIES REGARDING SDEPTH In this section, we present the main result from [10]. We denote S = K[x1, . . . , xn] the ring of polynomials in n variables, where n ≥ 2. For a monomial u ∈ S, we denote supp(u) = {x_i : x_i|u}. We begin this section by recalling the following results. Let I ⊂ S⁰ = K[x₁, . . . , x_r], J ⊂ S⁰⁰ = K[xr+1, . . . , xn] be monomial ideals, where 1 ≤ r < n. Then, we have the following inequalities:

Proposition 4.1 ([10, Proposition 1.1]).

(1) sdepth_S(IS∩J S)≥sdepth_S⁰(I) + sdepth_S⁰⁰(J). ([14, Lemma 1.1]).

(2) sdepth_S(S/(IS +J S)) ≥ sdepth_S⁰(S⁰/I) + sdepth_S⁰⁰(S⁰⁰/J). ([15, Theorem 3.1]).

(3) depth_S(S/(IS∩J S))−1 = depth_S(S/(IS+J S)) = depth_S⁰(S⁰/I) + depth_S⁰⁰(S⁰⁰/J). ([14, Lemma 1.1]).

We give similar results, in the following theorem.

Theorem 4.2 ([10, Theorem 1.3]). Let I ⊂ S⁰ = K[x1, . . . , xr], J ⊂ S⁰⁰=K[x_r+1, . . . , x_n]be monomial ideals, where1≤r < n. Then, we have the following inequalities:

(1) sdepth_S(IS)≥sdepth_S(IS+J S)≥min{sdepth_S(IS),sdepth_S⁰⁰(J) + sdepth_S⁰(S⁰/I)}.

(2) sdepth_S(S/IS) ≥ sdepth_S(S/(IS ∩ J S)) ≥ min{sdepth_S(S/IS), sdepth_S⁰⁰(S⁰⁰/J) + sdepth_S⁰(I)}.

Corollary 4.3 ([5, Corrolary 1.8]). With the notations above, we have the followings:

(1)If the Stanley conjecture hold for I andJ, then the Stanley conjecture holds for IS∩J S.

(2) If the Stanley conjecture hold for S⁰/I and S⁰⁰/J, then the Stanley conjecture holds for S/(IS+J S).

(3) If the Stanley conjecture hold for I, J andS⁰/I or forI, J and S⁰⁰/J, then the Stanley conjecture holds for (IS+J S).

(4) If the Stanley conjecture hold forS⁰/I, S⁰⁰/J andI orS⁰/I, S⁰⁰/J and I and J, then the Stanley conjecture holds for S/(IS∩J S).

In the following, we consider 1≤s≤r+ 1≤nthree integers, withn≥2.

We denote S⁰ := K[x₁, . . . , x_r], S⁰⁰ := K[x_s, . . . , x_n] and S := K[x₁, . . . , x_n].

Letp:=r−s+ 1. With these notations, we generalize some results of Propo- sition 4.1 and Theorem 4.2.

Theorem 4.4 ([10, Theorem 2.2]). Let I ⊂S⁰ and J ⊂S⁰⁰ be two mono- mial ideals. Then:

(1) sdepth_S(IS∩J S)≥sdepth_S⁰(I) + sdepth_S⁰⁰(J)−p= sdepth_S(IS) + sdepth_S(J S)−n.

(2) sdepth_S(S/(IS +J S)) ≥ sdepth_S⁰(S⁰/I) + sdepth_S⁰⁰(S⁰⁰/J) −p = sdepth_S(S/IS) + sdepth_S(S/J S)−n.

(3) sdepth_S(IS+J S)≥min{sdepth_S(IS),sdepth_S⁰⁰(J)+sdepth_S⁰(S⁰/I)−

p}= min{sdepth_S(IS),sdepth_S(J S) + sdepth_S(S/IS)−n}.

(4) sdepth_S(S/(IS ∩ J S)) ≥ min{sdepth_S(S/IS),sdepth_S⁰⁰(S⁰⁰/J) + sdepth_S⁰(I)−p}= min{sdepth_S(S/IS),sdepth_S(S/J S) + sdepth_S(IS)−n}.

This Theorem yields different corollaries and new proofs for other results, see [10]. Now, let I ⊂ S be a monomial ideal and let I = C₁ ∩ · · · ∩C_k, be the irredundant minimal decomposition of I. If we denote P_j = p

C_j for 1≤j ≤k, we have Ass(S/I) ={P₁, . . . , Pk}. In particular, if I is squarefree, C_j = P_j for all j. Denote d_j = ht(P_j), where 1 ≤ i ≤ k. We may assume that d₁ ≥d₂ ≥ · · · ≥ d_k. We obtain the following bounds for sdepth_S(I) and sdepth_S(S/I).

Corollary 4.5 ([10, Corollary 2.13]).

(1) n− bd₁/2c ≥sdepth_S(I)≥n− bd₁/2c − · · · − bd_k/2c.

(2) n−d1≥sdepth_S(S/I)≥n− bd₁/2c − · · · − bd_k−1/2c −dk.

In a more general case, letI =Q₁∩ · · · ∩Q_k be the primary irredundant decomposition of I,P_i =√

Q_i and denote q_j = sdepth_S(Q_j) and d_j =ht(P_j).

We may assume that d1 ≥ d2 ≥ · · · ≥ d_k. Note that qj ≤ n−dj/2, since P_j = (Q_j : u_j), where u_j ∈ S is a monomial, and therefore sdepth_S(Q_j) ≤ sdepth_S(P_j), by Proposition 2.7(1). On the other hand, we obviously have

sdepth_S(S/Q_j) = sdepth_S(S/P_j). With these notations, we obtain the follow- ing bounds for sdepth_S(I) and sdepth_S(S/I).

Corollary 4.6 ([10, Corollary 2.14]).

(1) n− bd₁/2c ≥sdepth_S(I)≥q1+· · ·+qk−n(k−1).

(2) n−d1 ≥sdepth_S(S/I) ≥ min{n−d1, q1−d2, q1+q2−d3 −n, . . . , q₁+· · ·+qk−1−d_k−n(k−2)}.

Example 4.7 ([10, Example 2.15]). LetI =Q1∩Q₂∩Q₃ ⊂S :=K[x1, . . . , x₇], whereQ₁ = (x²₁, . . . , x²₅),Q₂= (x³₄, x³₅, x³₆) andQ₃ = (x³₆, x₆x₇, x²₇). Denote P_j =p

Q_j. Note thatq₃ = sdepth_S(Q₃) = sdepth_K[x6,x7](Q₃∩K[x₆, x₇]) + 5 = 1 + 5 = 6. Also, since Q1 and Q2 are generated by powers of variables, by [8, Theorem 1.3], q₁ = 7− b5/2c = 5 and q₂ = 7− b3/2c = 6. According to Corollary 2.14, we have 5 = 7− bd₁/2c ≥sdepth_S(I)≥q₁+q₂+q₃−14 = 3 and 2 = 7−d₁ ≥sdepth_S(S/I)≥min{7−d₁, q1−d₂, q1+q2−d₃−7}= min{7−

5,5−3,5 + 6−2−7}= 2. Thus sdepth_S(I)∈ {3,4,5}and sdepth_S(S/I) = 2.

On the other hand, depth_S(S/I)≤min{n−depth_S(S/P_j) :j= 1,2,3}= 2.

In particular, we have sdepth_S(I)≥depth_S(I) and sdepth_S(S/I)≥depth_S(S/I).

Thus, both I and S/I satisfy the Stanley conjecture. In fact, using CoCoA, we get depth_S(S/I) = 2.

In the third section of [10], we give several equivalent form of the Stanley conjecture. We exemplify with the following Proposition.

Proposition 4.8 ([10, Proposition 3.1]). The following assertions are equivalent:

(1)For any integern≥1and any monomial idealI ⊂S=K[x₁, . . . , x_n], Stanley conjecture holds for I, i.e. sdepth_S(I)≥depth_S(I).

(2)For any integern≥1and any monomial idealsI, J ⊂S, ifsdepth_S(I+ J)≥depth_S(I+J), then sdepth_S(I)≥depth_S(I).

(3)For any integersn, m≥1, any monomial idealI ⊂S=K[x1, . . . , xn], if u₁, . . . , u_m ∈S is a regular sequence on S/I andJ = (u₁, . . . , u_m), then if:

sdepth_S(I+J)≥depth_S(I+J)⇒sdepth_S(I)≥depth_S(I).

(4)For any integersn, m≥1, any monomial idealI ⊂S=K[x₁, . . . , x_n], if u1, . . . , um ∈S is a regular sequence on S/I andJ = (u1, . . . , um), then if:

sdepth_S(I+J) = depth_S(I+J)⇒sdepth_S(I) = depth_S(I).

(5) For any integer n≥1, any monomial ideal I ⊂S=K[x₁, . . . , x_n], if S¯=S[y], then: sdepthS¯(I, y) = depth_S(I)⇒sdepth_S(I) = depth_S(I).

Acknowledgments. The support from grant ID-PCE-2011-1023 of Romanian Min- istry of Education, Research and Innovation is gratefully acknowledged.

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Received 19 March 2013 “Simion Stoilow” Institute of Mathematics, Research unit 5, P.O. Box 1-764,

Bucharest 014700, Romania mircea.cimpoeas@imar.ro