ON THE STANLEY DEPTH OF MONOMIAL IDEALS
MIRCEA CIMPOEAŞ
Communicated by Vasile Brı̂nzănescu
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 , . . . , xn ] the polynomial ring over K. Let
n -graded S-module. A Stanley decomposition of M is a direct sum
M be a ZL
D : M = ri=1 mi K[Zi ] as K-vector space, where mi ∈ M , Zi ⊂ {x1 , . . . , xn }
such that mi K[Zi ] is a free K[Zi ]-module. We define sdepth(D) = minri=1 |Zi |
and sdepthS (M ) = max{sdepth(D)| D is a Stanley decomposition of M }, see
[1]. The number sdepth(M ) is called the Stanley depth of M . 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 sdepthS (S/I) and sdepthS (I) and to find some relation between
them.
Csaba Biro, David M. Howard, Mitchel T. Keller, William T. Trotterand
Stephen J. Young proved that if m = (x1 , . . . , xn ) ⊂ S, then sdepth(m) = n2 ,
see [2, Theorem 2.2]. We extended this result to monomial ideals generated
by powers of variables,
l see
m [7, Theorem 1.3]. More precisely, we proved in [5]
n
k
that sdepth(m ) ≤ k+1 , 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
,
see
[13,
2
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
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Mircea Cimpoeaş
2
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 sdepthS (I + J), sdepthS (I ∩ J), sdepthS (S/(I + J)),
sdepthS (S/(I ∩ J)), sdepthS (I : J) and sdepthS (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, where I ⊂ 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 , . . . , vm ∈ K[x2 , . . . , xm ] be some
monomials and let a be a positive integer. Let I = (xa1 v1 , v2 , · · · , vm ) and
0
I 0 = (xa+1
1 v1 , v2 , · · · , vm ). Then sdepth(I) = sdepth(I ).
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 , . . . , xn ) ⊂ S, then
sdepth(m) = n2 .
As a direct consequence of Lemma 2.1 and Theorem 2.2, we get.
Theorem 2.3 ([8, Theorem 1.3]). Let a1 , . . . , an be some positive integers.
Then:
lnm
.
sdepth((xa11 , . . . , xann )) = sdepth((x1 , . . . , xn )) =
2
m
a1
a
m
In particular, sdepth((x1 , . . . , xm )) = n − 2 for any 1 ≤ 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 , . . . , vm }. Let S 0 = S[xn+1 ] and I 0 = (v1 , . . . , vm−1 , xn+1 vm ). Then
sdepth(I 0 ) ≤ sdepth(I) + 1.
In the condition of the above Corollary, Shen proved in [17] the other
inequality and, as a direct consequence he obtained.
Theorem 2.6 ([17, Theorem 2.4]). Let I 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.
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On the Stanley depth of monomial ideals
207
Theorem 2.7 ([13, Theorem 2.1]). Let I ⊂ S be a monomial ideal (minimally) generated by m monomials. Then
jmk
sdepth(I) ≥ max{1, n −
}.
2
We recall also the following result of Asia Rauf, see [16, Theorem 1.1].
Theorem 2.8. Let I ⊂ S be 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 minimally 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 , . . . , xn ] be a
monomial ideal, minimally generated by m monomials, k = max{|P | : P ∈
Ass(S/I)}, and s ≥ k be an integer. Then
l
m
s
1. If m ≤ s − 1 + k−1
, 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 for I.
Corollary 2.13 ([9, Corollary 1.9]). Let I be a monomial almost complete intersection ideal. Then Stanley’s conjecture holds for S/I and I.
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Mircea Cimpoeaş
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3. STANLEY DEPTH OF SQUAREFREE VERONESE IDEALS
In this section we will follow [6].
Theorem 3.1 ([6, Theorem 1.1]).
(1) sdepth(S/In,d ) = d − 1.
(2) d ≤ sdepth(In,d ) ≤ n−d
d+1 + d.
Since depth(S/In,d ) = d − 1, we get the following corollary.
Corollary 3.2 ([6, Corollary 1.2]). In,d and S/In,d satisfy the Stanley
conjecture. Also,
sdepth(In,d ) ≥ sdepth(S/In,d ) + 1.
Let k ≤ n be two positive integers. We denote An,k = {F ⊂ [n]| |F | = k}.
With this notations, we have the following well known result from combinatorics.
Theorem 3.3. For any positive integers d ≤ n such that d ≤ n/2, there
exists a bijective map Φn,d : An,d → An,d , such that Φn,d (F ) ∩ F = ∅, for any
F ∈ An,d .
As a consequence, we get the following corollary.
Corollary 3.4 ([6, Corollary 1.5]). Let n, d be two positive integers such
that 2d + 1 ≤ n ≤ 3d. Then sdepth(In,d ) = d + 1.
We also, give the following conjecture.
Conjecture
3.5.
j
k For any positive integers d ≤ n such that d ≤ n/2,
n−d
sdepth(In,d ) = d+1 + 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 In,d be the squarefree Veronese
ideal in S, generated by all squarefree monomials of degree d.
j
k
+d.
(1) If 1 ≤ d ≤ n < 5d+4, then the Stanley depth sdepth(In,d ) = n−d
j
k d+1
(2) If d ≥ 1 and n ≥ 5d + 4 then d + 3 ≤ sdepth(In,d ) ≤ n−d
d+1 + 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
n
k
sdepth(m ) ≤
.
k+1
In particular, if k ≥ n − 1, then sdepth(mk ) = 1.
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On the Stanley depth of monomial ideals
209
Proposition 3.8 ([5, Proposition 2.3]). Let I ⊂ S be a monomial ideal.
Then
sdepth(mk I) = 1 f or k 0.
We also conjectured that:
Conjecture 3.9. sdepth(mk ) =
l
n
k+1
m
.
Regarding this conjecture, Bruns, Krattenthaler and Uliczka
in
l proved
m
n
[4] that the Hilbert depth of the k-th power of m is exactly k+1 . 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) = {xi : xi |u}. We begin this section
by recalling the following results. Let I ⊂ S 0 = K[x1 , . . . , xr ], J ⊂ S 00 =
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) sdepthS (IS ∩ JS) ≥ sdepthS 0 (I) + sdepthS 00 (J). ([14, Lemma 1.1]).
(2) sdepthS (S/(IS + JS)) ≥ sdepthS 0 (S 0 /I) + sdepthS 00 (S 00 /J). ([15,
Theorem 3.1]).
(3) depthS (S/(IS ∩ JS)) − 1 = depthS (S/(IS + JS)) = depthS 0 (S 0 /I) +
depthS 00 (S 00 /J). ([14, Lemma 1.1]).
We give similar results, in the following theorem.
Theorem 4.2 ([10, Theorem 1.3]). Let I ⊂ S 0 = K[x1 , . . . , xr ], J ⊂
= K[xr+1 , . . . , xn ] be monomial ideals, where 1 ≤ r < n. Then, we have the
following inequalities:
(1) sdepthS (IS) ≥ sdepthS (IS + JS) ≥ min{sdepthS (IS), sdepthS 00 (J) +
sdepthS 0 (S 0 /I)}.
(2) sdepthS (S/IS) ≥ sdepthS (S/(IS ∩ JS)) ≥ min{sdepthS (S/IS),
sdepthS 00 (S 00 /J) + sdepthS 0 (I)}.
S 00
Corollary 4.3 ([5, Corrolary 1.8]). With the notations above, we have
the followings:
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Mircea Cimpoeaş
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(1) If the Stanley conjecture hold for I and J, then the Stanley conjecture
holds for IS ∩ JS.
(2) If the Stanley conjecture hold for S 0 /I and S 00 /J, then the Stanley
conjecture holds for S/(IS + JS).
(3) If the Stanley conjecture hold for I, J and S 0 /I or for I, J and S 00 /J,
then the Stanley conjecture holds for (IS + JS).
(4) If the Stanley conjecture hold for S 0 /I, S 00 /J and I or S 0 /I, S 00 /J and
I and J, then the Stanley conjecture holds for S/(IS ∩ JS).
In the following, we consider 1 ≤ s ≤ r + 1 ≤ n three integers, with n ≥ 2.
We denote S 0 := K[x1 , . . . , xr ], S 00 := K[xs , . . . , xn ] and S := K[x1 , . . . , xn ].
Let p := r − s + 1. With these notations, we generalize some results of Proposition 4.1 and Theorem 4.2.
Theorem 4.4 ([10, Theorem 2.2]). Let I ⊂ S 0 and J ⊂ S 00 be two monomial ideals. Then:
(1) sdepthS (IS ∩ JS) ≥ sdepthS 0 (I) + sdepthS 00 (J) − p = sdepthS (IS) +
sdepthS (JS) − n.
(2) sdepthS (S/(IS + JS)) ≥ sdepthS 0 (S 0 /I) + sdepthS 00 (S 00 /J) − p =
sdepthS (S/IS) + sdepthS (S/JS) − n.
(3) sdepthS (IS+JS) ≥ min{sdepthS (IS), sdepthS 00 (J)+sdepthS 0 (S 0 /I)−
p} = min{sdepthS (IS), sdepthS (JS) + sdepthS (S/IS) − n}.
(4) sdepthS (S/(IS ∩ JS)) ≥ min{sdepthS (S/IS), sdepthS 00 (S 00 /J) +
sdepthS 0 (I) − p} = min{sdepthS (S/IS), sdepthS (S/JS) + sdepthS (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 ∩ · · · ∩
pCk , be
the irredundant minimal decomposition of I. If we denote Pj =
Cj for
1 ≤ j ≤ k, we have Ass(S/I) = {P1 , . . . , Pk }. In particular, if I is squarefree,
Cj = Pj for all j. Denote dj = ht(Pj ), where 1 ≤ i ≤ k. We may assume
that d1 ≥ d2 ≥ · · · ≥ dk . We obtain the following bounds for sdepthS (I) and
sdepthS (S/I).
Corollary 4.5 ([10, Corollary 2.13]).
(1) n − bd1 /2c ≥ sdepthS (I) ≥ n − bd1 /2c − · · · − bdk /2c.
(2) n − d1 ≥ sdepthS (S/I) ≥ n − bd1 /2c − · · · − bdk−1 /2c − dk .
In a more general case,
√ let I = Q1 ∩ · · · ∩ Qk be the primary irredundant
decomposition of I, Pi = Qi and denote qj = sdepthS (Qj ) and dj = ht(Pj ).
We may assume that d1 ≥ d2 ≥ · · · ≥ dk . Note that qj ≤ n − dj /2, since
Pj = (Qj : uj ), where uj ∈ S is a monomial, and therefore sdepthS (Qj ) ≤
sdepthS (Pj ), by Proposition 2.7(1). On the other hand, we obviously have
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On the Stanley depth of monomial ideals
211
sdepthS (S/Qj ) = sdepthS (S/Pj ). With these notations, we obtain the following bounds for sdepthS (I) and sdepthS (S/I).
Corollary 4.6 ([10, Corollary 2.14]).
(1) n − bd1 /2c ≥ sdepthS (I) ≥ q1 + · · · + qk − n(k − 1).
(2) n − d1 ≥ sdepthS (S/I) ≥ min{n − d1 , q1 − d2 , q1 + q2 − d3 − n, . . . ,
q1 + · · · + qk−1 − dk − n(k − 2)}.
Example 4.7 ([10, Example 2.15]). Let I = Q1 ∩Q2 ∩Q3 ⊂ S := K[x1 , . . . ,
2
2
3 3 3
3
2
x7 ], where
p Q1 = (x1 , . . . , x5 ), Q2 = (x4 , x5 , x6 ) and Q3 = (x6 , x6 x7 , x7 ). Denote
Pj = Qj . Note that q3 = sdepthS (Q3 ) = sdepthK[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 ≥ sdepthS (I) ≥ q1 + q2 + q3 − 14 = 3
and 2 = 7−d1 ≥ sdepthS (S/I) ≥ min{7−d1 , q1 −d2 , q1 +q2 −d3 −7} = min{7−
5, 5 − 3, 5 + 6 − 2 − 7} = 2. Thus sdepthS (I) ∈ {3, 4, 5} and sdepthS (S/I) = 2.
On the other hand, depthS (S/I) ≤ min{n−depthS (S/Pj ) : j = 1, 2, 3} = 2.
In particular, we have sdepthS (I) ≥ depthS (I) and sdepthS (S/I) ≥ depthS (S/I).
Thus, both I and S/I satisfy the Stanley conjecture. In fact, using CoCoA,
we get depthS (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 integer n ≥ 1 and any monomial ideal I ⊂ S = K[x1 , . . . , xn ],
Stanley conjecture holds for I, i.e. sdepthS (I) ≥ depthS (I).
(2) For any integer n ≥ 1 and any monomial ideals I, J ⊂ S, if sdepthS (I+
J) ≥ depthS (I + J), then sdepthS (I) ≥ depthS (I).
(3) For any integers n, m ≥ 1, any monomial ideal I ⊂ S = K[x1 , . . . , xn ],
if u1 , . . . , um ∈ S is a regular sequence on S/I and J = (u1 , . . . , um ), then if:
sdepthS (I + J) ≥ depthS (I + J) ⇒ sdepthS (I) ≥ depthS (I).
(4) For any integers n, m ≥ 1, any monomial ideal I ⊂ S = K[x1 , . . . , xn ],
if u1 , . . . , um ∈ S is a regular sequence on S/I and J = (u1 , . . . , um ), then if:
sdepthS (I + J) = depthS (I + J) ⇒ sdepthS (I) = depthS (I).
(5) For any integer n ≥ 1, any monomial ideal I ⊂ S = K[x1 , . . . , xn ], if
S̄ = S[y], then: sdepthS̄ (I, y) = depthS (I) ⇒ sdepthS (I) = depthS (I).
Acknowledgments. The support from grant ID-PCE-2011-1023 of Romanian Ministry of Education, Research and Innovation is gratefully acknowledged.
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REFERENCES
[1] J. Apel, On a conjecture of R.P. Stanley; Part II – Quotients Modulo Monomial Ideals.
J. Algebraic Combin. 17 (2003), 57–74.
[2] C. Biro, D.M. Howard, M.T. Keller, W.T. Trotter and S.J. Young, Interval partitions
and Stanley depth. J. Combin. Theory Ser. A 117, (2010), 4, 475–482.
[3] W. Bruns, C. Krattenthaler and J. Uliczka, Stanley decompositions and Hilbert depth in
the Koszul complex. J. Commut. Algebra 2 (2010), 3, 327–357.
[4] W. Bruns, C. Krattenthaler and J. Uliczka, Hilbert depth of powers of the maximal ideal.
Commutative Algebra and its Connections to Geometry (PASI 2009), Contemporary
Mathematics 555, Amer. Math. Soc., R.I., 2011, 1–12.
[5] Mircea Cimpoeaş, Some remarks on the Stanley depth for multigraded modules. Le
Mathematiche LXIII (2008), II, 165–171.
[6] Mircea Cimpoeaş, Stanley depth of square free Veronese ideals. Preprint (2009).
http://arxiv.org/pdf/0907.1232.
[7] M. Cimpoeas, Stanley depth of monomial ideals with small number of generators. Cent.
Eur. J. Math. 7 (2009), 4, 629–634.
[8] M. Cimpoeas, Stanley depth for monomial complete intersection. Bull. Math. Soc. Sci.
Math. Roumanie (N.S.) 51(99) (2008), 3, 205–211.
[9] M. Cimpoeas, The Stanley conjecture on monomial almost complete intersection ideals.
Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 55(103) (2012), 1, 35–39.
[10] M. Cimpoeas, Several results regarding Stanley depth. Romanian J. Math. Comp. Sci.
2 (2012), 2, 28–40.
[11] J. Herzog, M. Vladoiu and X. Zheng, How to compute the Stanley depth of a monomial
ideal. J. Algebra 322 (2009), 6, 3151–3169.
[12] Mitchel T. Keller, Yi-Huang Shen, Noah Streib and Stephen J. Young, On the Stanley
depth of squarfree Veronese ideals. J. Algebraic Combin. 33 (2011), 2, 313–324.
[13] R. Okazaki, A lower bound of Stanley depth of monomial ideals., J. Commut. Algebra
3 (2011), 1, 83–88.
[14] A. Popescu, Special Stanley decompositions. Bull. Math. Soc. Sci. Math. Roumanie
(N.S.) 53(101) (2010), 4, 363–372.
[15] A. Rauf, Depth and sdepth of multigraded module. Comm. Algebra 38 (2010), 2, 773–
784.
[16] A. Rauf, Stanley Decompositions, Pretty Clean Filtrations and Reductions Modulo Regular Elements. Bull. Math. Soc. Sc. Math. Roumanie (N.S.) 50(98) (2007) 347–354.
[17] Y. Shen, Stanley depth of complete intersection monomial ideals and upper-discrete partitions. J. Algebra 321 (2009), 1285–1292.
Received 19 March 2013
“Simion Stoilow” Institute of Mathematics,
Research unit 5, P.O. Box 1-764,
Bucharest 014700, Romania
[email protected]