WHAT IS … STANLEY DEPTH?

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

1. History and background

Richard P. Stanley is well known for his fundamental and important contributions to combinatorics and its relationship to algebra and geometry, in particular in the theory of simplicial complexes. Two kinds of simplicial complexes play central roles in combinatorics: partitionable complexes and Cohen–Macaulay complexes. Stanley posed a central conjecture relating these two notions: Are all Cohen–Macaulay simplicial complexes partitionable? In a 1982 Inventiones Mathematicae paper [4], Stanley defined what is now called the Stanley depth of a graded module over a graded commutative ring. Stanley depth is a geometric invariant of a module that, by a conjecture of Stanley, relates to an algebraic invariant of the module, called simply the depth. It is shown in [2] that this conjecture implies his conjecture about partitionable Cohen–Macaulay simplicial complexes. Our aim here is to introduce the notion of the Stanley depth.

Let 𝕂 be a field and S = 𝕂[x_{1}, …, x_{n}] the 𝕂-algebra of polynomials over 𝕂 in n indeterminates x_{1}, …, x_{n}. We may write x = {x_{1}, …, x_{n}} and denote S by 𝕂[x] for convenience. A monomial in S is a product x^{a} = x_{1}^{a_{1}} … x_{n}^{a_{n}} for a vector a = (a_{1}, …, a_{n}) ∈ ω^{n} of nonnegative integers. The set of all monomials in S, denoted by Mon(S), forms a 𝕂-basis of S.

A monomial ideal I of S is an ideal generated by monomials in S, so that there exists a subset A ⊆ Mon(S) with I = ⟨x^{a} | x^{a} ∈ A⟩. The Hilbert basis theorem now implies that there exists a finite subset B ⊆ A such that I = ⟨x^{b} | x^{b} ∈ B⟩. Therefore, every monomial ideal I of S may be written in the form I = ⟨x^{a_{1}}, …, x^{a_{ℓ}}⟩, where x^{a_{1}}, …, x^{a_{ℓ}} ∈ Mon(S). Monomial ideals are the most manageable among all types of ideals due to their simple structure.

Clearly, every monomial ideal I in S is a 𝕂-subspace of S. The monomials in I form a 𝕂-basis of I. Let us characterize monomials that lie in a given monomial ideal. In order to do this, fix a monomial ideal I = ⟨x^{a_{1}}, …, x^{a_{ℓ}}⟩ in S. Then a monomial x^{b} lies in I if and only if there exists t with 1 ≤ t ≤ ℓ such that x^{a_{t}} | x^{b}. This is equivalent to x^{b} = x^{a_{t}}x^{c} or b = a_{t} + c, for some c ∈ ω^{n}. Thus, the set ⋃_{t=1}^{ℓ}(a_{t} + ω^{n}), which is a set of lattice points in ω^{n}, consists of the exponents of all monomials in I. For example, the set of all monomials in the ideal I = ⟨x_{1}x_{2}^{2}, x_{1}^{2}x_{2}⟩ of S = 𝕂[x_{1}, x_{2}] is {x^{b} | b ∈ (1, 2) + ω^{2}} ∪ {x^{b} | b ∈ (2, 1) + ω^{2}}. The exponents of these monomials have been shown in Figure 1 by the blue points.

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

Key words and phrases. Monomial ideal, Stanley space, Stanley decomposition, Stanley depth, Depth, Stanley conjecture.

[figure]

Figure 1. The blue points mark the exponents of all monomials belonging to the ideal I = ⟨x_{1}x_{2}^{2}, x_{1}^{2}x_{2}⟩ of S = 𝕂[x_{1}, x_{2}].

2. Stanley depth

Let x^{a} be a monomial in S = 𝕂[x] and fix it. The 𝕂-subspace of S whose basis consists of all monomials x^{a}u, where u is a monomial in 𝕂[z], z ⊆ x, is called a Stanley space of dimension |z| and is denoted by x^{a}𝕂[z]. Here |z| denotes the number of elements of z. For example, if S = 𝕂[x_{1}, x_{2}], then x_{1}^{2}x_{2}𝕂, x_{1}^{3}x_{2}𝕂[x_{1}], x_{1}x_{2}^{2}𝕂[x_{2}] and x_{1}^{2}x_{2}^{2}𝕂[x_{1}, x_{2}] are all Stanley spaces with dimensions 0, 1, 1 and 2, respectively. In Figure 2, the red, orange and blue points consist of the exponents of all monomials in x_{1}^{3}x_{2}𝕂[x_{1}], x_{1}x_{2}^{2}𝕂[x_{2}] and x_{1}^{2}x_{2}^{2}𝕂[x_{1}, x_{2}], respectively. Also x_{1}^{2}x_{2}𝕂 consists of only one monomial, namely x_{1}^{2}x_{2}. We draw the exponent of this monomial by a violet point in Figure 2.

[figure]

Figure 2. The violet, red, orange and blue points consist of the exponents of all monomials which belong to x_{1}^{2}x_{2}𝕂, x_{1}^{3}x_{2}𝕂[x_{1}], x_{1}x_{2}^{2}𝕂[x_{2}] and x_{1}^{2}x_{2}^{2}𝕂[x_{1}, x_{2}], respectively.

Let I be a fixed monomial ideal of S = 𝕂[x]. A decomposition D of I as a finite direct sum of Stanley spaces is called a Stanley decomposition of I. It is known that there exists at least one Stanley decomposition of I. For example,

D_{1} : I = x_{1}^{2}x_{2}𝕂 ⊕ x_{1}^{3}x_{2}𝕂[x_{1}] ⊕ x_{1}x_{2}^{2}𝕂[x_{2}] ⊕ x_{1}^{2}x_{2}^{2}𝕂[x_{1}, x_{2}]

is a Stanley decomposition for I = ⟨x_{1}x_{2}^{2}, x_{1}^{2}x_{2}⟩ in S = 𝕂[x_{1}, x_{2}]. The two other Stanley decompositions of I may be written as follows:

D_{2} : I = x_{1}^{2}x_{2}𝕂 ⊕ x_{1}^{3}x_{2}𝕂 ⊕ x_{1}^{4}x_{2}𝕂[x_{1}] ⊕ x_{1}x_{2}^{2}𝕂[x_{2}] ⊕ x_{1}^{2}x_{2}^{2}𝕂[x_{1}, x_{2}],
D_{3} : I = x_{1}x_{2}^{2}𝕂[x_{2}] ⊕ x_{1}^{2}x_{2}𝕂[x_{1}, x_{2}].

[figure]

Figure 3. The left-hand side figure demonstrates the Stanley decomposition D_{2}, while the one on the right corresponds to D_{3}.

For a given Stanley decomposition D of I, the minimum dimension of a Stanley space in D is called the Stanley depth of D and is denoted by sdepth(D). Therefore, in the above example we have sdepth(D_{1}) = 0, sdepth(D_{2}) = 0 and sdepth(D_{3}) = 1.

Finally, the Stanley depth of I is defined to be

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

Thus, in the above example, sdepth(I) is at least equal to one. One may easily show that, indeed, sdepth(I) = 1.

If I^{c} denotes the 𝕂-subspace of S = 𝕂[x] generated by all monomials of S which do not belong to I, then we may define a Stanley decomposition as well as the Stanley depth of I^{c}. The Stanley depth of I^{c} is denoted by sdepth(S/I), instead of sdepth(I^{c}), since, as 𝕂-subspaces, S = I ⊕ I^{c} and, hence, S/I ≅ I^{c}. For example,

D : I^{c} = x_{1}x_{2}𝕂 ⊕ 𝕂[x_{1}] ⊕ x_{2}𝕂[x_{2}]

is a Stanley decomposition of I^{c} for I = ⟨x_{1}x_{2}^{2}, x_{1}^{2}x_{2}⟩ in S = 𝕂[x_{1}, x_{2}]. Here we have sdepth(D) = 0 and one may easily show that sdepth(S/I) = 0.

Stanley [4] defined what is now called the Stanley depth of a ℤ^{n}-graded module M over the ring S. Note that if M is of the form I or S/I, where I is a monomial ideal, then this latter notion coincides with the Stanley depth defined above. In general, no algorithm for the computation of the Stanley depth is known. However, in the particular case when M is of the form I/J, where J ⊆ I are monomial ideals of S, such an algorithm was given by Herzog, Vladoiu, and Zheng (see [3]). They describe how to compute the Stanley depth in this case. Undoubtedly, this is one of the most important contributions to the theory so far. It provides the only method known to compute the Stanley depth and, for example, it was used to compute the Stanley depth for a complete intersection ideal. It is known that the Stanley depth of the monomial ideal I = ⟨x_{1}^{a_{1}}, …, x_{m}^{a_{m}}⟩ in the ring S = 𝕂[x_{1}, …, x_{n}] is equal to n − m + ⌈m/2⌉.

The depth of a module is one of its important algebraic invariants. For a ℤ^{n}- graded module M over the ring S, the depth of M, denoted by depth(M), is the supremum of the lengths ℓ of all sequences a_{1}, …, a_{ℓ} of ℤ-homogenous elements of the ideal ⟨x_{1}, …, x_{n}⟩, for which M ≠ ⟨a_{1}, …, a_{ℓ}⟩M and a_{i} is not a zero divisor of M/⟨a_{1}, …, a_{i−1}⟩M, for i = 1, …, ℓ. Stanley conjectured that depth(M) ≤ sdepth(M) holds for every nonzero finitely generated ℤ^{n}-graded module M over S. Certain cases of the Stanley conjecture have been proven, for example, for monomial ideals generated by square free monomials of S = 𝕂[x_{1}, …, x_{n}] where n ≤ 5. The question is still largely open with a huge amount of active research behind it. For further reading we refer to [1], [2] and [3].

Acknowledgments

The authors would like to thank Professor Jürgen Herzog and Professor Rahim Zaare-Nahandi for reading this note and encouraging us to publish it.

References

[1] J. Apel, On a conjecture of R. P. Stanley, I. Monomial ideals, and II. Quotients modulo monomial ideals, J. Algebraic Combin. 17 (2003), no. 1, 39–56, 57–74.

[2] J. Herzog, A. Soleyman Jahan, S. Yassemi, Stanley decompositions and partitionable simplicial complexes, J. Algebraic Combin. 27 (2008), no. 1, 113–125.

[3] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra, to appear.

[4] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982), no. 2, 175–193.

M. R. Pournaki, Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran.

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

S. A. Seyed Fakhari, Department of Mathematics, Shahid Beheshti University G. C., Evin, Tehran 19834, Iran.

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

M. Tousi, Department of Mathematics, Shahid Beheshti University G. C., Evin, Tehran 19834, Iran.

E-mail address: mtousi@ipm.ir

URL: http://math.ipm.ac.ir/mtousi/

S. Yassemi, School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran.

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