ON A CONJECTURE OF R. P. STANLEY PART II – QUOTIENTS MODULO MONOMIAL IDEALS

JOACHIM APEL

Abstract. In 1982 Richard P. Stanley conjectured that any finitely generated ℤ^{n}-graded module M over a finitely generated ℕ^{n}-graded 𝕂-algebra R can be decomposed in a direct sum M = ⨁_{i=1}^{t} ν_{i}S_{i} of finitely many free modules ν_{i}S_{i} which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the S_{i} have to be subalgebras of R of dimension at least depth M.

We will study this conjecture for modules M = R/I, where R is a polynomial ring and I a monomial ideal. We will present a necessary and sufficient condition for squarefree monomial Cohen-Macaulay rings to fulfill Stanley’s Conjecture. Moreover, we prove that Stanley’s Conjecture holds for the quotient modulo any generic monomial ideal, the quotient modulo any monomial ideal in at most three variables, and for any cogeneric Cohen-Macaulay ring. The presented theorems are constructive and allow to extract explicit algorithms for the computation of Stanley decompositions. Further, we will discuss some relationships between our algebraic results and combinatorial properties of related complexes. Finally, we will give an outlook to Stanley decompositions of arbitrary graded polynomial modules. In particular, we obtain a more general result in the 3-variate case.

1. Introduction

This is the second of two articles studying some aspects of a conjecture formulated by Richard P. Stanley in 1982. For the reader’s convenience we repeat some parts of the introduction of Part I (see [A]).

Conjecture 1 ([St82], 5.1). Let R be a finitely-generated ℕ^{n}-graded 𝕂-algebra (where R_{0} = 𝕂 as usual), and let M be a finitely-generated ℤ^{n}-graded R-module. Then there exist finitely many subalgebras S_{1}, …, S_{t} of R, each generated by algebraically independent ℕ^{n}-homogeneous elements of R, and there exist ℤ^{n}-homogeneous elements ν_{1}, …, ν_{t} of M, such that

M = ⨁_{i=1}^{t} ν_{i}S_{i},

where dim S_{i} ≥ depth M for all i, and where ν_{i}S_{i} is a free S_{i}-module (of rank one). Moreover, if 𝕂 is infinite and under a given specialization to an ℕ-grading of R is generated by R_{1}, then we can choose the (ℕ^{n}-homogeneous) generators of each S_{i} to lie in R_{1}.

Research at MSRI is supported in part by NSF grant DMS-9701755. Further, the author was supported by a research fellowship of the Deutsche Forschungsgemeinschaft (DFG).

In [A] this conjecture could be proved to hold for a series of cases where M is an ideal of a polynomial ring. In particular, the conjecture was shown for ideals in rings with at most three variables and for monomial ideals, which are generic with respect to one variable. In this paper we will study the case M = 𝕂[X]/I with an emphasize on Cohen-Macaulay rings. The verification of Conjecture 1 for Cohen-Macaulay modules would contribute to answers of further combinatorial problems, in particular, it would prove that Cohen-Macaulay complexes are partitionable (see [St79]).

Definition 1. Let R be a finitely-generated ℕ^{n}-graded 𝕂-algebra and let M be a finitely-generated ℤ^{n}-graded R-module. For each nonnegative integer d let Δ_{d} denote the set of all direct sum decompositions M = ⨁_{i=1}^{t} ν_{i}S_{i} of M in finitely many free S_{i}-modules ν_{i}S_{i} (i ∈ {1, …, t}), where the S_{1}, …, S_{t} are subalgebras of R each of them generated by at least d algebraically independent ℕ^{n}-homogeneous elements of R and the ν_{1}, …, ν_{t} are ℤ^{n}-homogeneous elements of M.

If not all sets Δ_{d} are empty then we call the largest d such that Δ_{d} ≠ ∅ the Stanley dimension of M and denote it by Sdim M.

For ℕ^{n}-graded polynomial rings R = 𝕂[x_{1}, …, x_{m}] the existence of Sdim M follows e.g. from results of Riquier related to the solution space of systems of partial differential equations ([Ri], see also [T]). In the context of the special case studied in this paper Stanley’s Conjecture reads as follows.

Conjecture 2. Let R = 𝕂[X] be a polynomial ring in the variables X = {x_{1}, …, x_{n}} over a field 𝕂. Then for any monomial ideal I ⊂ R holds

Sdim R/I ≥ depth R/I . (1)

That is, there exists a finite decomposition (called Stanley decomposition throughout this paper) of R/I of the following type

R/I = ⨁_{i=1}^{t} u_{i}𝕂[Z_{i}], (2)

where the u_{i} are residue classes of (monic) monomials modulo I, Z_{i} ⊆ X and |Z_{i}| ≥ depth(XR, R/I) = depth R/I for all i = 1, …, t.

In this paper we will ask for classes of modules M = R/I for which there exist decompositions (2) satisfying the condition |Z_{i}| ≥ min_{𝔭∈ Ass M} dim R/𝔭. In the most interesting case of Cohen-Macaulay rings M this is exactly Condition (1). However, in general, Condition (1) is weaker and it is not surprising that we will meet ideals, see e.g. Example 1, where our condition fails. After declaration of some notions and notations in Section 2 and deriving a trivial upper bound on the Stanley dimension in Section 3 we will start our considerations with quotients modulo squarefree ideals in Section 4. We will give a sufficient condition for squarefree monomial ideals I such that M = R/I possesses a decompositions (2) satisfying |Z_{i}| ≥ min_{𝔭∈ Ass M} dim R/𝔭. In the case of a pure dimensional I this condition is also necessary (Proposition 2). That reduces (or better reformulates) Stanley’s conjecture for squarefree Cohen-Macaulay rings to our condition. Section 5 deals with nonsquarefree ideals. We will prove the inequality Sdim R/I ≤ Sdim R/\sqrt{I} for arbitrary monomial ideals (Theorem 1). Moreover, we will generalize Proposition 2 to the nonsquarefree case (Proposition 3 and Theorem 2). However, here we will obtain only sufficient conditions.

In particular, we will show that Stanley’s Conjecture holds for the quotient R/I for any generic monomial ideal (Section 6). Moreover, we will prove Stanley’s Conjecture for the case of at most three variables (Section 7) and for all cogeneric Cohen-Macaulay rings (Section 8). All presented propositions and theorems contain (at least indirect) also an algorithm for the construction of Stanley decompositions for ideals satisfying the assumptions. However often, the derived algorithms will be impractical by the dependence on a suitable enumeration of the primary components. But in the cases of generic, 3-variate, and cogeneric Cohen-Macaulay ideals the corresponding theorems provide explicite information about a suitable arrangement of the primary components. This leads to efficient algorithms for the construction of Stanley decompositions in these cases.

In Section 9 we will relate our results to combinatorial properties of the Stanley-Reisner complex and the Scarf complex corresonding to a monomial ideal. Finally, we will give an outlook to the general case of arbitrary graded polynomial modules in Section 10 and generalize our results in the 3-variate case.

2. Notation

First we will introduce some notions and notations used throughout this paper. By writing A ⊂ B we mean that A is a (not necessarily nonempty) subset of B but A ≠ B. A ⊆ B stands for A = B or A ⊂ B. |A| denotes the number of elements of a finite set A.

A term x_{1}^{i_{1}} … x_{n}^{i_{n}}, i.e. a monomial with coefficient 1, is called a monic monomial. For a monic monomial t = x_{1}^{i_{1}} … x_{n}^{i_{n}} we introduce the notation supp t = {x_{j} : 1 ≤ j ≤ n , i_{j} > 0} for the Support of t. Since we will need to work only with monic monomials throughout this paper, for the sake of simplicity, we will use the short cut monomial for monic monomial.

By a minimal monomial t of a set A ⊂ 𝕂[x_{1}, …, x_{n}] we will always mean that t is a monic monomial contained in A and that \frac{t}{x} ∉ A for all x ∈ supp t. So a minimal generator t of a monomial ideal I is a monic monomial such that t but no proper divisor of t belongs to I.

Consider a nonzero monomial ideal I ⊂ R = 𝕂[x_{1}, …, x_{n}] with irredundant primary decomposition I = 𝔮_{1} ∩ 𝔮_{2} ∩ ⋯ ∩ 𝔮_{k}, that is, the intersection of any all but one of the primary ideals is strictly larger than I. Note, frequently the additional assumption that all primary ideals have to be maximal, i.e. the associated prime ideals 𝔭_{i} of the primary components 𝔮_{i} are pairwise distinct, is subsumed by the notion irredundant primary decomposition. We emphasize that we do not make this additional assumption, unless in places where it is explicitly stated, since for monomial ideals it is often much more convenient to work with the (upto the order of components uniquely determined) irredundant decomposition in irreducible ideals, i.e. primary ideals generated only by powers of variables.

Let Ass M = {𝔭_{1}, …, 𝔭_{k}}, where 𝔭_{i} belongs to the primary ideal 𝔮_{i}, denote the set of associated primes of M = R/I. 𝔮_{i} is called an embedded component if there exists j ≠ i such that 𝔭_{j} ⊂ 𝔭_{i}. By a minimal associated prime 𝔭_{i} we mean that 𝔭_{j} ⊄ 𝔭_{i} for all 𝔭_{j} ∈ Ass M.

A set of variables Y ⊆ X is called an algebraically independent set of I iff I∩𝕂[Y] = {0}. If, in addition, no set of variables properly containing Y is an independent set of I then Y is called a maximal algebraically independent set of I. A monomial prime ideal^{1} 𝔭 possesses exactly one maximal algebraically independent set, namely the set Y = {x ∈ X | x ∉ 𝔭}. Any 𝔭-primary ideal 𝔮 has the same uniquely determined maximal algebraically independent set Y as 𝔭, |Y | is called the (Krull-)dimension of R/𝔭 and denoted by dim R/𝔭. Throughout this paper the notion Y_{i} will always refer to the maximal algebraically independent set of the associated prime 𝔭_{i} corresponding to the primary component 𝔮_{i} of I, 1 ≤ i ≤ k. A set Y is algebraically independent for I iff it is algebraically independent for some associated prime ideal of M and it is maximal algebraically independent for I iff it is maximal algebraically independent for some minimal associated prime ideal of M. Accordingly, the dimension of M is defined by dim M = max_{𝔭_{i}∈ Ass M} dim R/𝔭_{i}. In this paper we will need to refer frequently to min_{𝔭_{i}∈ Ass M} dim R/𝔭_{i} which is an upper bound for the depth of M (c.f. [E, SV]). An ideal I satisfying dim M = min_{𝔭_{i}∈ Ass M} dim R/𝔭_{i} is called pure dimensional or simply pure. Cohen-Macaulay rings M are defined by the condition dim M = depth M in which case we call I a Cohen-Macaulay ideal. From the previous observations it follows immediately that any Cohen-Macaulay ideal is pure.

Finally, we will need the notions of (strongly) generic and cogeneric monomial ideals introduced by Bayer, Peeva and Sturmfels and generalized by Miller, Sturmfels and Yanagawa. A monomial ideal I is called strongly generic if no two distinct minimal generators m and m′ of I have the same positive degree in any variable x ∈ X (see [BPS]). If the monomial ideal I satisfies only the weaker assumption that for any two distinct minimal generators m and m′ which have the same degree in some variable x ∈ X, there is a third minimal generator m′′ which strictly divides lcm(m, m′), i.e. supp lcm(m, m′) = supp \frac{lcm(m,m′)}{m′′}, then I is called generic (see [MSY]). Generic monomial ideals I allow a simple characterization for R/I being a Cohen-Macaulay ring, which is the case iff I is pure dimensional iff I has no embedded primary components (c.f. [MSY, Theorem 2.5]). If I is (strongly) generic and 𝔭 is an arbitrary monomial prime ideal then the ideal I_{(𝔭)} obtained by localization of I at 𝔭 is (strongly) generic, too (c.f. [MSY, Remark 2.1]). If the Alexander dual I^{a} of the monomial ideal I is (strongly) generic then I is called (strongly) cogeneric. That is, I is cogeneric if whenever two distinct irreducible components 𝔮_{i} and 𝔮_{j} of I have a minimal generator in common then there exists an irreducible component 𝔮_{m} of I which is contained in the ideal sum 𝔮_{i} + 𝔮_{j} and has no minimal generator in common with this sum. Various characterizations of cogeneric Cohen-Macaulay monomial ideals were given by Miller, Sturmfels and Yanagawa (see [MSY, Theorem 4.9]). In this paper we will make use of one of them which in view of Alexander duality can be formulated as:

1Actually this implies that all minimal generators of 𝔭 are variables.

Lemma 1 ([MSY] Theorem 4.9 c). A cogeneric monomial ideal I with irredundant decomposition I = 𝔮_{1} ∩ 𝔮_{2} ∩ ⋯ ∩ 𝔮_{k} in irreducible components is Cohen-Macaulay iff I is pure dimensional and for any irreducible components 𝔮_{i} and 𝔮_{j} such that codim(𝔭_{i} + 𝔭_{j}) > codimI + 1 there exists an irreducible component 𝔮_{l} of I, where l ∉ {i, j}, satisfying 𝔮_{l} ⊂ 𝔮_{i} + 𝔮_{j}.

It is easy to observe that we have the following analogue to the generic case. If I is (strongly) cogeneric and 𝔭 is an arbitrary monomial prime ideal then the ideal I_{(𝔭)} obtained by localization of I at 𝔭 is (strongly) cogeneric, too.

3. An Upper Bound for Sdim M

Let m_{1}, …, m_{l} denote the minimal generators of I. We introduce the notion S = {u ∈ T | ∀i ∈ {1, …, l} : m_{i} ∤ u} for the set of all standard monomials modulo I. Recall that the set of minimal generators is the reduced Gröbner basis of I with respect to any admissible term order ≺. Furthermore, the residue classes of S modulo I form a 𝕂-vector space basis of M. In the following we will freely identify M and spann_{𝕂}S. Decomposition (2) has to satisfy u_{i}v ∈ S for all monic monomials v ∈ 𝕂[Z_{i}], i = 1, …, t. In particular, for arbitrary i ∈ {1, …, t} and u ∈ u_{i}𝕂[Z_{i}] the set Z_{i} has to be algebraically independent for the ideal quotient I : (u) and, consequently,

(3) Sdim M ≤ min_{u∈S} dim R/(I : (u)) = min_{𝔭_{i}∈ Ass M} dim R/𝔭_{i}

which shows that depth M and Sdim M share the upper bound at the right hand side. Let us start with a simple example illustrating that the above inequality can be strict.

Example 1. Consider the ideal I = (xz, yz, xu, yu) = (x, y) ∩ (z, u) ⊂ 𝕂[x, y, z, u]. There are two possibilities for two-dimensional subalgebras of 𝕂[x, y, z, u] which can appear in a direct summand containing the monomial 1, namely either 𝕂[z, u] or 𝕂[x, y]. Choosing 𝕂[z, u] there will be no way to find two-dimensional subalgebras for the direct summands containing the standard monomials x and y. Avoiding this problem by choosing 𝕂[x, y] will only shift the problem to the standard monomials z and u. A best decomposition (2), i.e. one with |Z_{i}| = Sdim R/I, is R/I = 𝕂[x, y] ⊕ z𝕂[z, u] ⊕ u𝕂[u]. But, this is already a Stanley decomposition since depth R/I = 1.

From I : (u) = ⋂_{i=1}^{k} (𝔮_{i} : (u)) and 𝔮_{i} : (u) = \{ \begin{matrix}𝔮_{i} : & if u ∉ 𝔭_{i} \\ 𝔮′_{i} : & if u ∈ 𝔭_{i} \ 𝔮_{i} \\ R : & if u ∈ 𝔮_{i} \end{matrix} , where 𝔮′_{i} is

a 𝔭_{i}-primary ideal properly containing 𝔮_{i}, we deduce that the maximal algebraically independent sets of I : (u) are exactly the maximal algebraically independent sets Y_{i} of the associated primes 𝔭_{i} ∈ Ass R/I which satisfy u ∈ (⋂_{𝔭_{j}⊂𝔭_{i}} 𝔮_{j}) \ 𝔮_{i}.

Now, we can describe the problem we met in Example 1 as follows. Let I be a pure dimensional monomial ideal with irredundant decomposition I = 𝔮_{1} ∩ 𝔮_{2} in maximal primary ideals. Assume, there are two elements s, t ∈ 𝔮_{1} \ 𝔮_{2} and two elements s′, t′ ∈ 𝔮_{2} \ 𝔮_{1} such that gcd(s, t) = gcd(s′, t′) =: m. Then it follows m ∉ 𝔮_{1} ∪ 𝔮_{2}. We will show that under certain conditions we have Sdim R/I < dim R/I. In the case \frac{lcm(s,t)}{gcd(s,t)} ∈ ⟨Y_{2}⟩ it will follow s𝕂[Y_{2}]∩t𝕂[Y_{2}] ≠ \{0\}. Hence, in case Sdim R/I = dim R/I the Stanley decomposition must contain a direct summand u𝕂[Y_{2}] such that s, t ∈ u𝕂[Y_{2}]. In particular, also m ∈ u𝕂[Y_{2}]. Now suppose, we have also \frac{lcm(s′,t′)}{gcd(s′,t′)} ∈ ⟨Y_{1}⟩. Then application of the same arguments shows that the Stanley decomposition must contain a second direct summand v𝕂[Y_{1}] such that s′, t′, m ∈ v𝕂[Y_{1}]. Obviously, this is impossible simultaneously.

In some sense this is the typical scenario for monomial ideals I with the property Sdim R/I < min_{𝔭_{i}∈ Ass M} dim R/𝔭_{i}. This behavior is always caused by contradicting requirements posed by a set of standard monomials modulo I to the direct summand which should contain their greatest common divisor.

4. The Squarefree Case

Let us first consider the case that the monomial ideal I is squarefree. Then all primary components 𝔮_{i}, i = 1, …, k, of I are even prime, i.e. 𝔮_{i} = 𝔭_{i}, and I has no embedded components. In the pure dimensional case we have the following nice relationship.

Lemma 2. Let I ⊂ R = 𝕂[X] be a pure squarefree monomial ideal such that Sdim M = dim M, where M = R/I. Then the direct summands of an arbitrary Stanley decomposition are in a natural bijection with the associated primes 𝔭_{1}, …, 𝔭_{k} of M, more precisely, each Stanley decomposition of M has the form

M = ⨁_{i=1}^{k} t_{i} 𝕂[Y_{i}] . (4)

Proof: Consider an arbitrary Stanley decomposition

M = ⨁_{i=1}^{m} t_{i} 𝕂[Z_{i}].

From Sdim M = dim M it follows |Z_{i}| ≥ dim M and in view of purity we obtain Z_{i} ∈ {Y_{1}, …, Y_{k}} for all i = 1, …, m. Let 1 ≤ j ≤ k and consider a monomial t ∈ M which belongs to all but the j-th associated prime of M. Furthermore, let i be such that t ∈ t_{i} 𝕂[Z_{i}]. Then we must have Z_{i} = Y_{j} since, otherwise, t𝕂[Z_{i}]∩I ≠ {0}. Hence, for each associated prime 𝔭_{j} the Stanley decomposition contains at least one direct summand such that Z_{i} = Y_{j}.

Now, consider two direct summands t_{i}𝕂[Z_{i}] and t_{l}𝕂[Z_{l}] satisfying Z_{i} = Z_{l} = Y_{j} for some j ∈ {1, …, k}. It follows supp t_{i} ⊆ Y_{j} and supp t_{l} ⊆ Y_{j} since, otherwise, (t_{i}𝕂[Z_{i}] + t_{l}𝕂[Z_{l}])∩I ≠ {0}. Hence, lcm(t_{i}, t_{l}) ∈ t_{i}𝕂[Z_{i}]∩t_{l}𝕂[Z_{l}] which implies i = l since the sum is direct. □

Proposition 1. Let I ⊂ R be a squarefree monomial ideal. If the associated primes of the module M = R/I can be ordered in such a way that each set 𝔭_{1} ∩ ⋯ ∩ 𝔭_{i−1} \ 𝔭_{i} contains exactly one minimal monomial t_{i}, i = 2, …, k, then it holds Sdim M = min_{p_{i}∈ Ass M} dim R/𝔭_{i}. Moreover, setting t_{1} = 1 we obtain the Stanley decomposition

M = ⨁_{i=1}^{k} t_{i} 𝕂[Y_{i}] . (5)

Proof: First, we want to show t_{i}𝕂[Y_{i}] ∩ t_{j}𝕂[Y_{j}] = {0} for all 1 ≤ i < j ≤ k. From t_{i} ∉ 𝔭_{i} it follows t_{i}𝕂[Y_{i}] ∩ 𝔭_{i} = {0}. Furthermore, t_{j}𝕂[Y_{j}] ⊆ 𝔭_{i} since t_{j} ∈ 𝔭_{1} ∩ ⋯ ∩ 𝔭_{i} ∩ ⋯ ∩ 𝔭_{j−1} and the claim follows. So far, we observed that the sum at the right hand side of (5) is direct. It remains to show the equality. Consider an arbitrary monomial t ∈ M. There is a uniquely determined index i such that t ∉ 𝔭_{i} but t ∈ 𝔭_{l} for all 1 ≤ l < i. Then t_{i} | t. Moreover, t ∈ 𝕂[Y_{i}] since 𝔭_{i} ∩ supp t = ∅. This proves the inclusion “⊆” and the other inclusion is obvious.

In conclusion, Sdim M ≥ min_{p_{i}∈ Ass M} dim R/𝔭_{i} which has to be equality in view of Inequality (3). □

Example 2 ([Rei] Remark 3). Consider Reisner’s example I = (x_{1}x_{2}x_{3}, x_{1}x_{2}x_{4}, x_{1}x_{3}x_{5}, x_{1}x_{4}x_{6}, x_{1}x_{5}x_{6}, x_{2}x_{3}x_{6}, x_{2}x_{4}x_{5}, x_{2}x_{5}x_{6}, x_{3}x_{4}x_{5}, x_{3}x_{4}x_{6}) ⊂ 𝕂[x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}] of a monomial ideal which is Cohen-Macaulay iff the characteristic of 𝕂 is not 2. Localization at the prime ideal 𝔭 = (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) yields I_{(𝔭)} = (x_{1}x_{4}, x_{1}x_{5}, x_{2}x_{3}, x_{2}x_{5}, x_{3}x_{4}). The associated primes of the module M_{(𝔭)} = R_{(𝔭)}/I_{(𝔭)} are 𝔭_{1} = (x_{1}, x_{2}, x_{3}), 𝔭_{2} = (x_{1}, x_{2}, x_{4}), 𝔭_{3} = (x_{1}, x_{3}, x_{5}), 𝔭_{4} = (x_{2}, x_{4}, x_{5}), 𝔭_{5} = (x_{3}, x_{4}, x_{5}). Application of Proposition 1 yields M_{(𝔭)} = 𝕂[x_{4}, x_{5}] ⊕ x_{3}𝕂[x_{3}, x_{5}] ⊕ x_{2}𝕂[x_{2}, x_{4}] ⊕ x_{1}𝕂[x_{1}, x_{3}] ⊕ x_{1}x_{2}𝕂[x_{1}, x_{2}].

Now, we ask for a Stanley decomposition of the quotient M = R/I. Since no confusion is possible we will denote the associated prime ideal 𝔭_{i} ∩ R, i = 1, …, 5, by the same symbol 𝔭_{i} for the sake of better readability. The missing associated primes of I containing x_{6} are 𝔭_{6} = (x_{1}, x_{4}, x_{6}), 𝔭_{7} = (x_{1}, x_{5}, x_{6}), 𝔭_{8} = (x_{2}, x_{3}, x_{6}), 𝔭_{9} = (x_{2}, x_{5}, x_{6}), and 𝔭_{10} = (x_{3}, x_{4}, x_{6}). One easily observes that the sequence 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5} cannot be extended by appending one of the primes 𝔭_{6}, …, 𝔭_{10} towards a sequence allowing the application of the above proposition. Moreover, we checked using a computer program that our proposition is not applicable to the intersection of any more than 5 associated prime ideals of M.

But it holds Sdim M = dim M since M = 𝕂[x_{4}, x_{5}, x_{6}]⊕x_{3}𝕂[x_{3}, x_{5}, x_{6}]⊕x_{2}𝕂[x_{2}, x_{3}, x_{5}]⊕x_{1}𝕂[x_{1}, x_{3}, x_{6}]⊕x_{1}x_{2}𝕂[x_{1}, x_{2}, x_{5}]⊕x_{1}x_{4}𝕂[x_{1}, x_{3}, x_{4}]⊕x_{1}x_{5}𝕂[x_{1}, x_{4}, x_{5}]⊕x_{2}x_{4}𝕂[x_{2}, x_{4}, x_{6}] ⊕ x_{2}x_{6}𝕂[x_{1}, x_{2}, x_{6}] ⊕ x_{3}x_{4}𝕂[x_{2}, x_{3}, x_{4}] is a Stanley decomposition of M.

There is a strong relationship between Proposition 1 and shellable (nonpure) simplicial complexes. In view of this connection it is absolutely not surprising that we could not apply Proposition 1 in the previous example. We will discuss this relationship in more detail in Section 9.

The next remark concerns monomial ideals satisfying the assumption of Proposition 1 which are not pure dimensional. In this case the proposition proves the existence of a Stanley decomposition whose summands are in bijection with the associated primes but in contrast to Lemma 2 it does not imply that each Stanley decomposition has this nice property.

Example 3. Examples for Stanley decompositions of M = R/I, where I = (x, y) ∩ (y, z, u) ⊂ 𝕂[x, y, z, u], are M = 𝕂[z, u] ⊕ x𝕂[x] and M = 𝕂[z] ⊕ u𝕂[u] ⊕ zu𝕂[z, u] ⊕ x𝕂[x]. While the first decomposition is of the type described in Proposition 1, the second is not.

The generalization of Proposition 1 which could be applied successfully in Example 2 can be expressed as follows:

Proposition 2. Let I be a squarefree monomial ideal with associated primes 𝔭_{1}, …, 𝔭_{k}. By setting

J_{1} := R,
J_{i+1} := J_{i} ∩ 𝔭_{i} + (a ∈ J_{i} : t_{i} ∤ a)R, (i = 1, …, k),

a sequence t_{1}, t_{2}, …, t_{k} of standard monomials modulo I defines a descending sequence J_{1} ⊇ J_{2} ⊇ ⋯ ⊇ J_{k+1} of overideals of I. If the associated primes of I can be ordered in such a way that for suitable monomials t_{1}, t_{2}, …, t_{k} we have

1. t_{1} = 1 and t_{i} is a minimal monomial of the set J_{i} \⋃_{j=1}^{i−1} t_{j}𝕂[Y_{j}] for i = 2, …, k,

2. J_{k+1} = I + UR, where U = ⋃_{j=1}^{k−1} (t_{j}𝕂[Y_{j}] ∩ J_{j+1}), and

3. for each monomial t ∈ J_{k+1} \ I there exists a uniquely determined index i ∈ {1, …, k} such that t ∈ t_{i} 𝕂[Y_{i}].

Then it holds Sdim M = min_{p_{i}∈ Ass M} dim R/𝔭_{i} and M = ⨁_{i=1}^{k} t_{i} 𝕂[Y_{i}] is a Stanley decomposition of M.

Moreover, if I is pure and the above conditions are not satisfiable for any permutation of associated primes then we have Sdim M < dim M.

Before we come to the proof let us explain the idea behind the rather technical conditions formulated in the proposition. If I satisfies even the stronger assumptions of Proposition 1 then defining t_{i}, i ∈ {2, …, k}, to be the unique minimal monomial contained in 𝔭_{1} ∩ ⋯ ∩ 𝔭_{i−1} \ 𝔭_{i} leads to the inclusion (a ∈ J_{i} : t_{i} ∤ a)R ⊆ J_{i} ∩ 𝔭_{i} and, hence, J_{i+1} = 𝔭_{1} ∩ ⋯ ∩ 𝔭_{i} for all i ∈ {1, …, k}. If I does not satisfy the assumptions of Proposition 1 then possible additional minimal elements of J_{i} \ 𝔭_{i} have to be kept in J_{i+1} for later use. Note that the difference between the ideal (a ∈ J_{i} : t_{i} ∤ a)R and the quotient J_{i}/(t_{i}) is larger than only the fact that both ideals live in different rings since the ideal (a ∈ J_{i} : t_{i} ∤ a)R typically contains nonzero multiples of t_{i}. If there are such multiples which do not belong to 𝔭_{i} they cause an unpleasant behavior, namely, t_{i} 𝕂[Y_{i}] will intersect with J_{i+1} nontrivially. The importance of Condition 3 becomes obvious already in the simple Example 1. Condition 2 ensures that no element of the intersection t_{i} 𝕂[Y_{i}] ∩ J_{i+1} will belong to a second module t_{j} 𝕂[Y_{j}]. Note, that this can only happen for “high” dimensional M, more precisely, only if dim M ≥ |supp t_{i}| + 2. Similarly, while in “small” dimensions all monomials t_{i} are minimal generators of J_{i} this need not to be the case in higher dimensions. The problems are demonstrated in the following example of a four-dimensional strongly generic monomial Cohen-Macaulay ideal.

Example 4. Consider the ideal I = (x_{1}x_{3}x_{5}, x_{2}x_{4}x_{6}) with irredundant prime decomposition I = (x_{1}, x_{2}) ∩ (x_{2}, x_{3}) ∩ (x_{3}, x_{4}) ∩ (x_{4}, x_{5}) ∩ (x_{5}, x_{6}) ∩ (x_{1}, x_{4}) ∩ (x_{2}, x_{5}) ∩ (x_{3}, x_{6}) ∩ (x_{1}, x_{6}).

The following two ideal sequences J_{1}, …, J_{10} are induced by different sequences of standard terms.

i  t_{i}  Y_{i}  J_{i}
1  1  {x_{3}, x_{4}, x_{5}, x_{6}} (1)
2  x_{1}  {x_{1}, x_{4}, x_{5}, x_{6}} (x_{1}, x_{2})
3  x_{2}  {x_{1}, x_{2}, x_{5}, x_{6}} (x_{2}, x_{1}x_{3})
4  x_{1}x_{3} {x_{1}, x_{2}, x_{3}, x_{6}} (x_{1}x_{3}, x_{2}x_{3}, x_{2}x_{4})
5  x_{2}x_{4} {x_{1}, x_{2}, x_{3}, x_{4}} (x_{2}x_{3}, x_{2}x_{4}, x_{1}x_{3}x_{4}, x_{1}x_{3}x_{5})
6  x_{2}x_{3} {x_{2}, x_{3}, x_{5}, x_{6}} (x_{2}x_{3}, x_{1}x_{3}x_{4}, x_{1}x_{3}x_{5}, x_{2}x_{4}x_{5}, x_{2}x_{4}x_{6})
7 x_{1}x_{3}x_{4} {x_{1}, x_{3}, x_{4}, x_{6}} (x_{2}x_{3}x_{4}, x_{1}x_{2}x_{3}, x_{1}x_{3}x_{4}, x_{1}x_{3}x_{5}, x_{2}x_{4}x_{5}, x_{2}x_{4}x_{6})
8 x_{2}x_{4}x_{5} {x_{1}, x_{2}, x_{4}, x_{5}} (x_{1}x_{2}x_{3}, x_{1}x_{3}x_{5}, x_{2}x_{3}x_{4}, x_{2}x_{4}x_{5}, x_{2}x_{4}x_{6})
9 x_{2}x_{3}x_{4}x_{5} {x_{2}, x_{3}, x_{4}, x_{5}} (x_{1}x_{2}x_{3}, x_{1}x_{3}x_{5}, x_{2}x_{3}x_{4}, x_{2}x_{4}x_{6})
10  (x_{1}x_{2}x_{3}, x_{1}x_{3}x_{5}, x_{2}x_{3}x_{4}, x_{2}x_{4}x_{6})
1  1  {x_{3}, x_{4}, x_{5}, x_{6}} (1)
2  x_{1}  {x_{1}, x_{4}, x_{5}, x_{6}} (x_{1}, x_{2})
3  x_{2}  {x_{1}, x_{2}, x_{5}, x_{6}} (x_{2}, x_{1}x_{3})
4  x_{1}x_{3} {x_{1}, x_{2}, x_{3}, x_{6}} (x_{1}x_{3}, x_{2}x_{3}, x_{2}x_{4})
5  x_{2}x_{3} {x_{1}, x_{2}, x_{3}, x_{4}} (x_{2}x_{3}, x_{2}x_{4}, x_{1}x_{3}x_{4}, x_{1}x_{3}x_{5})
6 x_{2}x_{3}x_{6} {x_{2}, x_{3}, x_{5}, x_{6}} (x_{2}x_{4}, x_{1}x_{3}x_{4}, x_{1}x_{3}x_{5}, x_{2}x_{3}x_{5}, x_{2}x_{3}x_{6})
7 x_{1}x_{3}x_{4} {x_{1}, x_{3}, x_{4}, x_{6}} (x_{2}x_{4}, x_{1}x_{3}x_{4}, x_{1}x_{3}x_{5}, x_{2}x_{3}x_{5}, x_{1}x_{2}x_{3}x_{6})
8  x_{2}x_{4} {x_{1}, x_{2}, x_{4}, x_{5}} (x_{2}x_{4}, x_{1}x_{3}x_{5}, x_{2}x_{3}x_{5}, x_{1}x_{2}x_{3}x_{6})
9 x_{2}x_{3}x_{5} {x_{2}, x_{3}, x_{4}, x_{5}} (x_{1}x_{3}x_{5}, x_{2}x_{3}x_{4}, x_{2}x_{4}x_{6}, x_{2}x_{3}x_{5}, x_{1}x_{2}x_{3}x_{6})
10  (x_{1}x_{3}x_{5}, x_{2}x_{3}x_{4}, x_{2}x_{4}x_{6}, x_{2}x_{3}x_{5}x_{6}, x_{1}x_{2}x_{3}x_{6})

Only the first sequence satisfies the assumptions of our proposition and, hence, M = t_{1}𝕂[Y_{1}] ⊕ ⋯ ⊕ t_{9}𝕂[Y_{9}] is a Stanley decomposition of M. Note, the only minimal generator x_{2}x_{3}x_{4} of J_{9} which is not contained in (x_{1}, x_{6}) cannot be used as t_{9} since it is already contained in the summand x_{2}x_{4}𝕂[x_{1}, x_{2}, x_{3}, x_{4}]. The second sequence violates Condition 2 since x_{1}x_{2}x_{3} is contained in the intersection x_{1}x_{3} 𝕂[x_{1}, x_{2}, x_{3}, x_{6}] ∩ x_{2}x_{3} 𝕂[x_{1}, x_{2}, x_{3}, x_{4}]. Nevertheless, Condition 3 holds.

It is clear that any violation of Condition 2 prevents the result from being a Stanley decomposition. However, if one could detect any “bad” sequence of prime ideals and/or choice of standard monomials by checking only the final Condition 3 one could use a simpler formulation of the proposition. Our example refutes such a possibility.

Nevertheless, it is an interesting question if the application of Condition 2 and the necessity of choosing monomials t_{i} which are not minimal generators of J_{i} can be avoided by application of “good” strategies for ordering the primes and choosing the monomials. At least in the above example, and in some more general situations discussed in Sections 6 - 8, the answer is positive.

Example 4 (continued). After reordering the primes according to I = (x_{1}, x_{2}) ∩ (x_{2}, x_{3}) ∩ (x_{3}, x_{4}) ∩ (x_{1}, x_{4}) ∩ (x_{4}, x_{5}) ∩ (x_{2}, x_{5}) ∩ (x_{5}, x_{6}) ∩ (x_{3}, x_{6}) ∩ (x_{1}, x_{6}) even Proposition 1 becomes applicable and yields

i  t_{i}  Y_{i}  J_{i}
1  1  {x_{3}, x_{4}, x_{5}, x_{6}} (1)
2  x_{1}  {x_{1}, x_{4}, x_{5}, x_{6}} (x_{1}, x_{2})
3  x_{2}  {x_{1}, x_{2}, x_{5}, x_{6}} (x_{2}, x_{1}x_{3})
4  x_{2}x_{3} {x_{2}, x_{3}, x_{5}, x_{6}} (x_{1}x_{3}, x_{2}x_{3}, x_{2}x_{4})
5  x_{1}x_{3} {x_{1}, x_{2}, x_{3}, x_{6}} (x_{1}x_{3}, x_{2}x_{4})
6 x_{1}x_{3}x_{4} {x_{1}, x_{3}, x_{4}, x_{6}} (x_{2}x_{4}, x_{1}x_{3}x_{4}, x_{1}x_{3}x_{5})
7  x_{2}x_{4} {x_{1}, x_{2}, x_{3}, x_{4}} (x_{2}x_{4}, x_{1}x_{3}x_{5})
8 x_{2}x_{4}x_{5} {x_{1}, x_{2}, x_{4}, x_{5}} (x_{1}x_{3}x_{5}, x_{2}x_{4}x_{5}, x_{2}x_{4}x_{6})
9 x_{2}x_{3}x_{4}x_{5} {x_{2}, x_{3}, x_{4}, x_{5}} (x_{1}x_{3}x_{5}, x_{2}x_{4}x_{6}, x_{2}x_{3}x_{4}x_{5})
10  (x_{1}x_{3}x_{5}, x_{2}x_{4}x_{6})

Proof of Proposition 2: First, we want to show that the sum ⨁_{i=1}^{k} t_{i} 𝕂[Y_{i}] is direct. So, in contrary, let us assume the existence of a monomial t ∈ t_{i} 𝕂[Y_{i}]∩t_{j} 𝕂[Y_{j}] for some integers i and j such that 1 ≤ i < j ≤ k. At the one hand side t ∉ J_{k+1} according to condition 3. But at the other hand side from condition 2 and t_{j} 𝕂[Y_{j}] ⊆ J_{j} ⊆ J_{i+1} we can deduce t ∈ J_{k+1}. This is a contradiction and, hence, t_{i} 𝕂[Y_{i}] ∩ t_{j} 𝕂[Y_{j}] = {0} for all i ≠ j.

The inclusion M ⊇ ⨁_{i=1}^{k} t_{i} 𝕂[Y_{i}] is an immediate consequence of t_{i} ∉ 𝔭_{i} for all i = 1, …, k. It remains to show the inclusion M ⊆ ⨁_{i=1}^{k} t_{i} 𝕂[Y_{i}]. Let t ∈ M be an arbitrary standard monomial modulo I, i.e. t ∉ I. If t ∈ J_{k+1} then t ∈ ⨁_{i=1}^{k} t_{i} 𝕂[Y_{i}] by condition 3. Otherwise, there exists a uniquely determined index i, 1 ≤ i ≤ k, such that t ∈ J_{i} \ J_{i+1} and one easily observes t ∈ t_{i} 𝕂[Y_{i}]. This completes the proof of the first part of the statement.

The second part of the statement is a corollary to Lemma 2. Assume Sdim M = dim M. Then by the lemma each Stanley decomposition of M has the form M = ⨁_{i=1}^{k} t_{i} 𝕂[Y_{i}]. Now, let j_{1}, …, j_{k} be a permutation of the integers {1, …, k} such that t_{j_{l}} ∤ t_{j_{m}} for all 1 ≤ m < l ≤ k. Then the permutation 𝔭_{j_{1}}, …, 𝔭_{j_{k}} of associated primes and the monomial sequence t_{j_{1}}, …, t_{j_{k}} fulfill the assumptions of the proposition. □

Example 2 (continued). Based on the permutation 𝔭_{1}, 𝔭_{2}, 𝔭_{6}, 𝔭_{4}, 𝔭_{10}, 𝔭_{9}, 𝔭_{8}, 𝔭_{3}, 𝔭_{5}, 𝔭_{7} of Ass M the sequence 1, x_{3}, x_{2}, x_{1}, x_{1}x_{2}, x_{1}x_{4}, x_{1}x_{5}, x_{2}x_{4}, x_{2}x_{6}, x_{3}x_{4} of monomials fulfills the assumptions of Proposition 2 and confirms the Stanley decomposition presented earlier.

Proposition 2 reduces the question if at least all Cohen-Macaulay rings R/I defined by a squarefree monomial ideal I satisfy Stanley’s conjecture 2 to the question if each squarefree monomial Cohen-Macaulay ideal possesses a permutation of associated primes such that conditions 1-3 are satisfied for a suitable choice of standard monomials t_{1}, …, t_{k}.

Besides purity another important property of Cohen-Macaulay rings consists in the connectedness in codimension 1, i.e. for any two associated primes 𝔭 and 𝔭′ of R/I, there is a sequence 𝔭 = 𝔭_{j_{1}}, 𝔭_{j_{2}}, …, 𝔭_{j_{r}} = 𝔭′ of associated primes of R/I such that codim (𝔭_{j_{i}} + 𝔭_{j_{i+1}}) = 1 + codimI for all i ∈ {1, …, r − 1}. Unfortunately, even the condition that I is a pure dimensional squarefree monomial ideal which is connected in codimension 1 is not strong enough to ensure Sdim R/I = dim R/I. For a counter example consider I′ = 𝔭_{1} ∩ 𝔭_{2} ∩ 𝔭_{3} ∩ 𝔭_{4} ∩ 𝔭_{5} ∩ 𝔭_{6}, where the prime ideals are defined as in Example 2. Here we have Sdim R/I′ = 2 < dim R/I′ = 3. At the other hand side Example 11 will show that connectedness in codimension 1 is also not necessary for the validity of Sdim R/I = dim R/I.

5. The Nonsquarefree Case

In this section we study quotients modulo monomial ideals which are not necessarily squarefree. Typically, we loose the bijection between associated primes of M = R/I and the direct summands of a Stanley decomposition of M even in the case that I is pure and Sdim M = dim M. But since an element contained in exactly one primary component of I can only belong to a direct summand which makes use of the maximal algebraically independent set of this primary ideal, at least we can conclude that the correspondence remains surjective.

Theorem 1. Let I ⊂ R be a monomial ideal and M = R/I. Furthermore, let \sqrt{I} = ⋂_{𝔭∈ Ass M} 𝔭 be the radical of I and M̃ = R/\sqrt{I}. Then it holds Sdim M ≤ Sdim M̃.

Proof: Let S denote the set of standard monomials modulo I and S̃ the set of standard monomials modulo \sqrt{I}. Recall the identifications M = spann_{𝕂}S and M̃ = spann_{𝕂}S̃. Consider an arbitrary Stanley decomposition M = ⨁_{i=1}^{m} t_{i} 𝕂[W_{i}] of M. For each monomial u ∈ M̃ we define Z_{u} := W_{i}, where i ∈ {1, …, m} is the uniquely determined index such that u^{ν} ∈ t_{i} 𝕂[W_{i}] for all sufficiently high powers of u. Note, supp u ⊆ Z_{u} for the so-defined sets Z_{u} and, therefore, M̃ ⊇ ∑_{u∈M̃} u 𝕂[Z_{u}]. The other inclusion holds by construction.

Finally, we have to show that any two summands u 𝕂[Z_{u}] and v 𝕂[Z_{v}] either intersect trivially or are contained in the summand gcd(u, v)𝕂[Z_{gcd(u,v)}]. By construction there exist uniquely determined i, j ∈ {1, …, m} such that u^{r} ∈ t_{i} 𝕂[W_{i}] and v^{s} ∈ t_{j} 𝕂[W_{j}] for large enough exponents r and s. Moreover, Z_{u} = W_{i} and Z_{v} = W_{j}. Since the summands originate from a Stanley decomposition either we have i = j or t_{j} 𝕂[Z_{v}] ∩ t_{i} 𝕂[Z_{u}] = {0}. In the latter case it follows v^{s} 𝕂[Z_{v}] ∩ u^{r} 𝕂[Z_{u}] = {0} and, hence, v 𝕂[Z_{v}]∩u 𝕂[Z_{u}] = {0}. In the remaining case i = j we have u^{r}, v^{s} ∈ t_{i} 𝕂[W_{i}] for all sufficiently large integers r and s. Hence, also gcd(u, v)^{r} = gcd(u^{r}, v^{r}) ∈ t_{i} 𝕂[W_{i}] for all sufficiently large exponents r. By construction Z_{gcd(u,v)} = W_{i} and, consequently, u 𝕂[Z_{u}] + v 𝕂[Z_{v}] ⊆ gcd(u, v) 𝕂[Z_{gcd(u,v)}]. In conclusion, removing all redundant summands leads to a decomposition M̃ = ⨁_{i=1}^{l} v_{i} 𝕂[Z_{v_{i}}] which proves Sdim M̃ ≥ min_{i=1}^{l} |Z_{v_{i}}| ≥ Sdim M. □

Of course, there are ideals I for which the inequality is proper. For instance, equality can never hold in the case that I has embedded components and Sdim M̃ = dim M̃. The following example illustrates that we need not to have equality even in the pure case.

Example 5. Consider the monomial ideal I = (x_{1}^{2}, x_{2}^{2}, x_{3})∩(x_{2}, x_{3}, x_{4})∩(x_{3}, x_{4}^{2}, x_{5}) = (x_{3}, x_{2}^{2}x_{5}, x_{2}^{2}x_{4}^{2}, x_{1}^{2}x_{4}x_{5}, x_{1}^{2}x_{4}^{2}, x_{1}^{2}x_{2}x_{5}) and its radical \sqrt{I} = (x_{1}, x_{2}, x_{3}) ∩ (x_{2}, x_{3}, x_{4}) ∩ (x_{3}, x_{4}, x_{5}). Application of Proposition 1 gives M̃ = 𝕂[x_{4}, x_{5}] ⊕ x_{1} 𝕂[x_{1}, x_{5}] ⊕ x_{2} 𝕂[x_{1}, x_{2}]. However, Sdim M < 2 since there is an unresolvable conflict between the elements x_{1}^{2}x_{4} and x_{2}^{2}x_{4} at the one hand side and the elements x_{4}^{2} and x_{4}x_{5} at the other hand side. While the first two elements require a direct summand containing x_{4} 𝕂[x_{1}, x_{2}] the other require x_{4} 𝕂[x_{4}, x_{5}], a contradiction. A Stanley decomposition of M is M = 𝕂[x_{4}, x_{5}] ⊕ x_{1} 𝕂[x_{1}, x_{5}] ⊕ x_{2} 𝕂[x_{1}, x_{2}] ⊕ x_{1}x_{4} 𝕂[x_{4}, x_{5}] ⊕ x_{2}x_{4} 𝕂[x_{4}, x_{5}] ⊕ x_{2}x_{5} 𝕂[x_{5}] ⊕ x_{1}x_{2}x_{4} 𝕂[x_{4}, x_{5}] ⊕ x_{1}x_{2}x_{5} 𝕂[x_{5}] ⊕ x_{1}^{2}x_{4} 𝕂[x_{1}, x_{2}] ⊕ x_{2}^{2}x_{4} 𝕂[x_{2}] ⊕ x_{1}x_{2}^{2}x_{4} 𝕂[x_{2}].

The next example demonstrates a situation where the Stanley dimensions of M and M̃ coincide.

Example 6. Let us consider the strongly generic monomial Cohen-Macaulay ideal I = (x, y^{3}z^{2}u) ⊂ 𝕂[x, y, z, u]. I has the irredundant decomposition in irreducible components I = (x, y^{3}) ∩ (x, z^{2}) ∩ (x, u). M = 𝕂[z, u] ⊕ y 𝕂[z, u] ⊕ y^{2} 𝕂[z, u] ⊕ y^{3} 𝕂[y, u] ⊕ y^{3}z 𝕂[y, u] ⊕ y^{3}z^{2} 𝕂[y, z] is a Stanley decomposition of M = R/I (see our forthcoming Proposition 3). Application of Theorem 1 yields the decomposition M̃ = 𝕂[z, u]⊕y 𝕂[y, u]⊕yz 𝕂[y, z] of the quotient modulo the radical \sqrt{I} = (x, yzu) = (x, y)∩(x, z)∩(x, u). Since the Stanley dimension of M̃ cannot exceed the dimension of M̃ this is already a Stanley decomposition.

In this example we can go also in the opposite direction, i.e. we can complete the above Stanley decomposition of M̃ to a Stanley decomposition M = 𝕂[z, u] ⊕ y 𝕂[y, u] ⊕ yz 𝕂[y, z] ⊕ yzu 𝕂[z, u] ⊕ y^{2}zu 𝕂[z, u] of M.

Let us discuss the abilities and the limits of the methods applied in the proof of Theorem 1. Given a Stanley decomposition of M we can decompose M̃ and we have some guaranteed quality of the resulting decomposition, i.e. it is not worse than this of M. However if Sdim M < Sdim M̃ then we will not obtain a Stanley decomposition of M̃, in general. For instance, replacing the first direct summand in the Stanley decomposition of M in Example 5 according to 𝕂[x_{4}, x_{5}] = 𝕂[x_{4}] ⊕ x_{5} 𝕂[x_{5}] ⊕ x_{4}x_{5} 𝕂[x_{4}, x_{5}] will maintain the Stanley decomposition property for M but lifting the new decomposition to a decomposition of M̃ will not provide a Stanley decomposition anymore. This is of course not surprising but what one really would like to have is the opposite construction which could be applied successfully in both Examples 5 and 6, namely to complete a given Stanley decomposition of M̃ to a Stanley decomposition of M by adding some direct summands. There seems to be a good chance that this is always possible. However, the answer to this question remains open. It follows a generalization of Proposition 1 to the nonsquarefree case.

Proposition 3. Let I ⊂ R be a monomial ideal which possesses an irredundant primary decomposition I = 𝔮_{1} ∩ ⋯ ∩ 𝔮_{k} such that 𝔮_{1} ∩ ⋯ ∩ 𝔮_{i−1} \ 𝔮_{i}, i = 2, …, k, contains exactly one minimal monomial t_{i}. Then Sdim M = min_{𝔭_{i}∈ Ass M} dim R/𝔭_{i}, where M = R/I. Moreover, a Stanley decomposition of M can be obtained by removing all redundant summands from the inner sums of

M = ⨁_{i=1}^{k} ∑_{t∈𝔮_{1}∩···∩𝔮_{i−1}\𝔮_{i}} t𝕂[Y_{i}] , (6)

where 𝔮_{1} ∩ ⋯ ∩ 𝔮_{i−1} := R for i = 1.

Proof: The equality in (6) is ensured by construction. That the first big sum is direct follows in a similar way as in the proof of Proposition 1. Formally, set t_{1} = 1. It remains to show that for any i ∈ {1, …, k} and s, t ∈ 𝔮_{1} ∩ ⋯ ∩ 𝔮_{i−1} \ 𝔮_{i} it holds gcd(s, t) ∈ 𝔮_{1} ∩ ⋯ ∩ 𝔮_{i−1} \ 𝔮_{i} and s 𝕂[Y_{i}] + t 𝕂[Y_{i}] ⊆ gcd(s, t) 𝕂[Y_{i}] or the modules s 𝕂[Y_{i}] and t 𝕂[Y_{i}] intersect trivially. First observe t_{i} | gcd(s, t) and, hence, gcd(s, t) ∈ 𝔮_{1} ∩ ⋯ ∩ 𝔮_{i−1} \ 𝔮_{i}. Suppose lcm(s, t) ∈ s 𝕂[Y_{i}] ∩ t 𝕂[Y_{i}]. Then it follows deg_{x} s = deg_{x} t = deg_{x} gcd(s, t) for all x ∉ Y_{i} and, hence, s 𝕂[Y_{i}] + t 𝕂[Y_{i}] ⊆ gcd(s, t) 𝕂[Y_{i}]. □

Example 7. Consider the strongly generic monomial Cohen-Macaulay ideal I = (x^{2}, y^{2}z^{2}, yzw, w^{2}) = (x^{2}, y, w^{2})∩(x^{2}, z, w^{2})∩(x^{2}, y^{2}, w)∩(x^{2}, z^{2}, w) ⊂ R = 𝕂[x, y, z, w].

With the notation of Proposition 3 the sequence t_{1} = 1, t_{2} = y, t_{3} = yz, t_{4} = y^{2}z provides the Stanley decomposition M = 𝕂[z]⊕y 𝕂[y]⊕yz 𝕂[z]⊕y^{2}z 𝕂[y]⊕x 𝕂[z]⊕ w 𝕂[z] ⊕ xw 𝕂[z] ⊕ yw 𝕂[y] ⊕ xy 𝕂[y] ⊕ xyw 𝕂[y] ⊕ xyz 𝕂[z] ⊕ xy^{2}z 𝕂[y].

However, the proposition will not be applicable to the irredundant decomposition I = (x^{2}, y^{2}, w^{2}, yw) ∩ (x^{2}, z^{2}, w^{2}, zw) in maximal primary ideals.

The last statement shows that it is sometimes advisable to work with irreducible rather than with maximal primary ideals and this will become even more apparent in the next sections. However, it should be mentioned that there are also situations were the use of maximal primary ideals is preferable. Consider, for instance, the simple case that I is a primary ideal. While application of Proposition 3 to the primary decomposition I = 𝔮 immediately yield Sdim R/I = dim R/I and a Stanley decomposition of R/I it may happen that no permutation of the irreducible components of I allows the application of the proposition, a simple example showing this behavior is I = (x^{2}, y^{2}, z, u) ∩ (x, y, z^{2}, u^{2}).

Next, we generalize Proposition 2 to the case of nonsquarefree ideals I = 𝔮_{1} ∩ ⋯ ∩ 𝔮_{k}. Instead of a sequence t_{1}, …, t_{k} of standard monomials modulo I we use a sequence T_{1}, …, T_{k} of finite sets of standard monomials modulo I for the definition of a descending sequence J_{1} ⊇ J_{2} ⊇ ⋯ ⊇ J_{k+1} of overideals of I by setting

(7) J_{1} := R , J_{i+1} := J_{i} ∩ 𝔮_{i} + (a ∈ J_{i} : ∀t ∈ T_{i} : t ∤ a)R, (i = 1, …, k).

Theorem 2. Let I be a monomial ideal with irredundant primary decomposition I = 𝔮_{1} ∩ ⋯ ∩ 𝔮_{k}. Furthermore, assume that there exists a sequence T_{1}, …, T_{k} of finite sets of standard monomials modulo I, where for all i = 1, …, k we have

1. T_{i} ⊂ J_{i} \ 𝔮_{i}, T_{i} ∩ (T_{j}𝕂[Y_{j}]) = ∅ for all j ∈ {1, …, i − 1},

2. for any two distinct elements t, s ∈ T_{i} there exists a variable x ∈ 𝔭_{i} such that deg_{x} t ≠ deg_{x} s, 3. all t ∈ T_{i} are minimal in the sense that for all x ∈ Y_{i} dividing t it holds \frac{t}{x} ∉ J_{i}

or \frac{t}{x} ∈ ⋃_{j=1}^{i−1} T_{j}𝕂[Y_{j}], such that the ideal sequence J_{1} ⊇ J_{2} ⊇ ⋯ ⊇ J_{k+1} ⊇ I induced by (7) satisfies

4. J_{k+1} = I + UR, where U = ⋃_{j=1}^{k−1} (T_{j}𝕂[Y_{j}] ∩ J_{j+1}), and

5. for each monomial s ∈ J_{k+1} \ I there exist a uniquely determined index i ∈ {1, …, k} and a uniquely determined monomial t ∈ T_{i} such that s ∈ t 𝕂[Y_{i}].

Then it holds Sdim M = min_{p_{i}∈ Ass M} dim R/𝔭_{i} and M = ⨁_{i=1}^{k} ⨁_{t∈T_{i}} t 𝕂[Y_{i}] is a Stanley decomposition of M.

Proof: The inclusion M ⊇ ∑_{i=1}^{k} ∑_{t∈T_{i}} t 𝕂[Y_{i}] is an immediate consequence of condition 1. Let s be a standard monomial modulo I. If there exists i ∈ {1, …, k} such that s ∈ J_{i} \ J_{i+1} then s ∈ t𝕂[Y_{i}] for some t ∈ T_{i}. Otherwise, s ∈ J_{k+1} \ I and the existence of i and t ∈ T_{i} such that s ∈ t𝕂[Y_{i}] follows by condition 5. This proves also the other inclusion M ⊆ ∑_{i=1}^{k} ∑_{t∈T_{i}} t 𝕂[Y_{i}].

It remains to show that the sums are even direct. Each inner sum is direct according to conditions 2 and 3. That the outer sum is direct follows in full analogy to the proof of Proposition 2 from conditions 4 and 5. □

We have to leave open the question whether there is an analogy to the opposite direction of Proposition 2. Let us just remark why application of the same proof idea would not work here. No permutation of the sets T_{1}, …, T_{k} resulting from a given Stanley decomposition of M need to satisfy the assumptions of the theorem. Since the primary components were not assumed to be maximal, typically, we can form different set sequences T_{1}, …, T_{k} corresponding to a given Stanley decomposition of M, even the number of sets may vary. But also this additional freedom is still insufficient for saving our proof idea from the squarefree case.

Theorem 2 generalizes the previous Propositions 1, 2 (without its opposite direction) and 3. Later, in Section 9 we will see that each of the special cases considered in the propositions has a particular combinatorial meaning. There is one more particular case which deserves a separate presentation since all large classes of ideals for which we will prove Stanley’s Conjecture in the subsequent sections will satisfy even its stronger assumptions.

Corollary 1. Let I be a monomial ideal with irredundant primary decomposition I = 𝔮_{1} ∩ ⋯ ∩ 𝔮_{k}. Further, let J_{1} := R and J_{i+1} := 𝔮_{1} ∩ ⋯ ∩ 𝔮_{i} for i ∈ {1, …, k}. Finally, for each i ∈ {1, …, k} define T_{i} to be the set of all monomials t ∈ J_{i} \ 𝔮_{i} such that \frac{t}{x} ∉ J_{i} for all variables x ∈ (supp t) ∩ Y_{i}.

If any two distinct monomials belonging to the same set T_{i} differ in the degree of at least one variable x ∈ 𝔭_{i}, i ∈ {1, …, k}, then it holds Sdim M = min_{p_{i}∈ Ass M} dim R/𝔭_{i} and M = ⨁_{i=1}^{k} ⨁_{t∈T_{i}} t 𝕂[Y_{i}] is a Stanley decomposition of M.

Proof: It is an easy excersise to show that the introduced sets T_{i} of standard monomials are finite and satisfy conditions 1-3 of Theorem 2. The ideals J_{i} introduced in the corollary are the same as induced from T_{1}, …, T_{k} by (7). Hence, also the conditions 4 and 5 hold. □

Corollary 2. Let I ⊂ R be a monomial ideal and M = R/I. If dim M ≤ 1 then it holds Sdim M = min_{𝔭∈ Ass M} dim R/𝔭.

Proof: Let I = 𝔮_{1} ∩⋯∩𝔮_{k} be an irredundant decomposition in irreducible components and consider an arbitrary fixed l ∈ {2, …, k}. Since dim R/𝔮_{l} ≤ dim R/I ≤ 1 any two monomials of (𝔮_{1} ∩ ⋯ ∩ 𝔮_{l−1})\𝔮_{l} which coincide in the degree of all variables belonging to 𝔭_{l} can differ only in the degree of at most one variable. Therefore, one is multiple of the other. Now, the assertion follows from Corollary 1. □

Example 8. Let I be an arbitrary Borel-fixed ideal and consider M = R/I. If the irreducible components 𝔮_{1}, …, 𝔮_{k} are enumerated according to decreasing dimension of R/𝔮_{i} then for any l ∈ {2, …, k} we have the inclusion 𝔭_{1} ∪ ⋯ ∪ 𝔭_{l−1} ⊆ 𝔭_{l} for the associated prime ideals according to [E, Corollary 15.25]. Hence, we can apply Corollary 1.

This example reflects the well-known fact that Conjecture 1 holds for n = 1 and, hence, also for higher n when I is in generic coordinates. In this situation such a decomposition was given first by Rees, whose decomposition satisfies both parts of Conjecture 1 in case n = 1 and 𝕂 infinite (see [Ree]). Another decomposition satisfying the first part of Conjecture 1 for arbitrary 𝕂 and n = 1 is due to Baclawski and Garsia (see [BG]).

Example 9 ([MSY] Example 4.18b). Consider the monomial ideal I = (a^{4}, b, c) ∩ (a^{2}, b^{4}, d) ∩ (a, b^{3}, e) ∩ (a^{3}, b^{2}, e^{2}) ⊂ 𝕂[a, b, c, d, e], which is cogeneric and Cohen-Macaulay.

By Proposition 1 we obtain the Stanley decomposition M̃ = 𝕂[d, e] ⊕ c 𝕂[c, e] ⊕ cd 𝕂[c, d] of the quotient M̃ = R/\sqrt{I}. Proposition 3 is not applicable but Corollary 1 yields the Stanley decomposition M = 𝕂[d, e] ⊕ a 𝕂[d, e] ⊕ a^{2} 𝕂[d, e] ⊕ a^{3} 𝕂[d, e] ⊕ c 𝕂[c, e] ⊕ ac 𝕂[c, e] ⊕ b 𝕂[c, e] ⊕ b^{2} 𝕂[c, e] ⊕ b^{3} 𝕂[c, e] ⊕ ab 𝕂[c, e] ⊕ ab^{2} 𝕂[c, e] ⊕ ab^{3} 𝕂[c, e]⊕cd 𝕂[c, d]⊕cde 𝕂[c, d]⊕bde 𝕂[c, d]⊕bd 𝕂[c, d]⊕b^{2}d 𝕂[c, d]⊕abd 𝕂[c, d]⊕ abde 𝕂[c, d] ⊕ a^{2}b 𝕂[c, d] ⊕ a^{2}c 𝕂[c, d] ⊕ acd 𝕂[c, d] ⊕ a^{2}be 𝕂[c, d] ⊕ a^{2}ce 𝕂[c, d] ⊕ acde 𝕂[c, d].

6. Generic Monomial Ideals

The central result of this section consists in proving the following theorem.

Theorem 3. Let I ⊂ R be a generic monomial ideal and M = R/I. Then Sdim M = min_{𝔭∈ Ass M} dim R/𝔭. Furthermore, let I = 𝔮_{1} ∩ ⋯ ∩ 𝔮_{k} be an irredundant decomposition in irreducible components, where the irreducible components are ordered in such a way that x_{1}^{a} ∈ 𝔮_{i} implies x_{1}^{a} ∈ 𝔮_{j} for all positive integers a and all integers 1 ≤ i < j ≤ k. Moreover, components with common minimal generator^{2} in x_{1} are ordered by recursive application of analogous conditions to x_{2}, x_{3}, … until the tie is broken. Then for each l ∈ {2, …, k} the set (𝔮_{1} ∩ ⋯ ∩ 𝔮_{l−1})\𝔮_{l} contains exactly one minimal monomial.

2All components containing no pure power of x_{1} are also considered to have the minimal generator in x_{1} in common.

For the proof we will need the following simple fact.

Remark 1. Let I ⊂ 𝕂[x_{1}, …, x_{n}] be a monomial ideal and m_{1}, …, m_{l} its minimal generators ordered in such a way that deg_{x_{1}} m_{i} ≤ deg_{x_{1}} m_{i+1} for all i ∈ {1, …, l−1}. Furthermore, let r ∈ {1, …, l} be such that deg_{x_{1}} m_{r−1} < deg_{x_{1}} m_{r} and set d := deg_{x_{1}} m_{r}. Then the overideal I_{d} ⊇ I generated by {x_{1}^{d}, m_{1}|_{x_{1}=1}, …, m_{r−1}|_{x_{1}=1}}, where m_{i}|_{x_{1}=1} is obtained from m_{i} by setting x_{1} = 1 (1 ≤ i < r), has the following properties:

1. if I is (strongly) generic, then so is I_{d},

2. each irreducible component of I which has x_{1}^{d} as a minimal generator is also an irreducible component of I_{d},

3. x_{1}^{d} is a minimal generator of each irreducible component of I_{d}.

The given generating set of I_{d} needs not to be minimal, nevertheless, the first and the third property are obvious. The second property follows immediately by repeated application of the well-known formula

(ts)R + J = (tR + J) ∩ (sR + J), (8)

which is valid for arbitrary monomial ideals J and arbitrary monomials t and s such that gcd(t, s) = 1. The statement of the remark extends to the ideal I_{∞} := (m_{1}|_{x_{1}=1}, …, m_{l}|_{x_{1}=1}). In this case the second condition has to be understood as: each irreducible component of I whose associated prime ideal does not contain x_{1} is an irreducible component of I_{∞}. The third condition reads as x_{1} ∉ 𝔭 for all associated primes of R/I_{∞}. Obviously, I_{∞} can be identified with I_{(x_{2},... ,x_{n})}, the localization of I at the prime ideal (x_{2}, …, x_{n}), in a natural way. In this sense Remark 1 generalizes Remark 2.1 from [MSY].

Lemma 3. Let the monomial ideal I ⊂ 𝕂[x_{1}, …, x_{n}] be generic with irredundant decomposition I = 𝔮_{1} ∩ ⋯ ∩ 𝔮_{k} in irreducible components, where the components are ordered as in Theorem 3. Furthermore, let d be a positive integer such x_{1}^{d} ∉ I but x_{1}^{d} is a minimal generator for some irreducible component of I and let l ∈ {2, …, k} be minimal such that x_{1}^{d} ∈ 𝔮_{l}. Define Ĩ:= ⋂_{i=1}^{l−1} 𝔮_{i} as the intersection of all irreducible components which do not contain x_{1}^{d}. Then the set Ĩ \ 𝔮, where 𝔮 is an arbitrary irreducible component of I having x_{1}^{d} as a minimal generator, contains exactly one minimal monomial t. Moreover, x_{1}^{d}t is a minimal generator of I.

Proof: Recall by construction 𝔮 = 𝔮_{l} is one possible choice, but there may be others.

Consider two minimal monomials t and t′ of Ĩ \ 𝔮. By construction it follows immediately τ, τ′ ∈ I, where τ := lcm(x_{1}^{d}, t) = x_{1}^{d}t and τ′ := lcm(x_{1}^{d}, t′) = x_{1}^{d}t′. Let m and m′ be minimal generators of I such that m | τ and m′ | τ′. Since \frac{τ}{x_{1}} , \frac{τ′}{x_{1}} ∉ 𝔮 we have deg_{x_{1}} m = deg_{x_{1}} m′ = d. By minimality of t and t′ it follows \frac{τ}{x_{i}} ∉ I for all x_{i} ∈ supp t and \frac{τ′}{x_{i}} ∉ I for all x_{i} ∈ supp t′. Hence, τ = m and τ′ = m′.

If m = m′ then it will follow also t = t′ since two minimal monomials of Ĩ\𝔮 cannot differ only in the degree of x_{1}. So let us assume m ≠ m′. Then by genericity of I there exists a minimal generator m′′ of I which satisfies deg_{x_{i}} m′′ < max(deg_{x_{i}} m, deg_{x_{i}} m′) for all x_{i} ∈ supp m′′. Hence, deg_{x_{1}} m′′ < d and, consequently, m′′ |_{x_{1}=1}∈ 𝔮. But since m′′ divides lcm(τ, τ′) we have also that m′′ |_{x_{1}=1} divides lcm(t, t′). This implies lcm(t, t′) ∈ 𝔮 and since 𝔮 is irreducible we must have t ∈ 𝔮 or t′ ∈ 𝔮, a contradiction.

In summary, we observed that Ĩ\ 𝔮 contains exactly one minimal monomial t and that x_{1}^{d}t is a minimal generator of I. □

Proof of Theorem 3: We proceed by induction on the number of variables n. The initial case n = 1 is trivial. So let us assume that the statement holds for (n − 1) variables. In particular, the statement holds for the localization I_{∞} and for each ideal I_{d} introduced in Remark 1, in the latter case just consider the quotient I_{d}/(x_{1}) and lift the result.

Fix an arbitrary l ∈ {2, …, k} and consider the set D = (𝔮_{1} ∩ ⋯ ∩ 𝔮_{l−1}) \ 𝔮_{l}. If x_{1} ∉ 𝔭_{l} then the assertion follows immediately by the corresponding property of I_{∞}. So, consider the case that x_{1}^{d} is a minimal generator of 𝔮_{l} for some positive integer d. If x_{1}^{d} ∉ 𝔮_{l−1} the assertion follows from Lemma 3. So, finally, it remains to consider the case x_{1}^{d} ∈ 𝔮_{l−1}. Let r ∈ {1, …, l−1} be the minimal index such that x_{1}^{d} ∈ 𝔮_{r}. If r = 1 then the assertion follows immediately by the corresponding property of I_{d}. Suppose, 1 < r < l. Then it follows the existence of a uniquely determined minimal monomial t ∈ (𝔮_{1} ∩ ⋯ ∩ 𝔮_{r−1}) \ 𝔮_{l} from Lemma 3. According to Remark 1 I_{d} has the form I_{d} = Ĩ∩ 𝔮_{r} ∩ ⋯ ∩ 𝔮_{l′}, where l′ ∈ {l, …, k} is the maximal index such that x_{1}^{d−1} ∉ 𝔮_{l′} and Ĩ is a certain monomial overideal of 𝔮_{1} ∩ ⋯ ∩ 𝔮_{r−1} with the property that x_{1}^{d} is a minimal generator for all its irreducible components. Applying the induction assumption to I_{d} it follows the existence of a uniquely determined minimal monomial s ∈ (Ĩ∩ 𝔮_{r} ∩ ⋯ ∩ 𝔮_{l−1}) \ 𝔮_{l}. Since 𝔮_{1} ∩ ⋯ ∩ 𝔮_{r−1} ∩ I_{d} = 𝔮_{1} ∩ ⋯ ∩ 𝔮_{l−1} any element of D must be a multiple of lcm(t, s). The simple observation lcm(t, s) ∈ D finishes the proof of the second part of Theorem 3 and the first part follows immediately in view of Proposition 3. □

7. The 3-variate Case

Another class of modules M = R/I for which the validity of Stanley’s Conjecture can be proved using the results presented in Section 5 are the quotients modulo monomials ideals in at most three variables.

Theorem 4. Let I ⊂ R = 𝕂[x_{1}, x_{2}, x_{3}] be a monomial ideal and M = R/I. Then it holds Sdim M = min_{𝔭∈ Ass M} dim R/𝔭. Moreover, let I = 𝔮_{1}∩⋯∩𝔮_{k} be an irredundant decomposition in irreducible components, where dim R/𝔮_{i} ≥ dim R/𝔮_{j} for all integers i and j such that 1 ≤ i < j ≤ k. For all l ∈ {2, …, k} set D_{l} := (𝔮_{1} ∩ ⋯ ∩ 𝔮_{l−1}) \ 𝔮_{l} and define T_{l} := \{t ∈ D_{l} : t is a monomial and ∀x_{i} ∈ supp t \ 𝔭_{l} : \frac{t}{x_{i}} ∉ D_{l}\}. Then for any l ∈ {2, …, k} and any t, t′ ∈ T_{l} either it holds t = t′ or there exists x_{i} ∈ 𝔭_{l} such that deg_{x_{i}} t ≠ deg_{x_{i}} t′.

Proof: Again it suffices to prove the second part of the statement. Then the first part is an immediate consequence of Corollary 1.

Assume, there exists an ideal I = 𝔮_{1} ∩ ⋯ ∩ 𝔮_{k} ∈ 𝕂[x_{1}, x_{2}, x_{3}] which violates the assertion. Let l ∈ {2, …, k} be minimal such that T_{l} contains two distinct monomials t and t′ which coincide in the degree in all variables belonging to 𝔭_{l}. Such monomials must differ in the degrees in at least two variables belonging to Y_{l}, hence, it follows dim R/𝔮_{l} ≥ 2. Taking into account the order of irreducible components and the fact that we are in the 3-variate case we can deduce dim R/𝔮_{1} = ⋯ = dim R/𝔮_{l} = 2. Hence, 𝔮_{1} ∩ ⋯ ∩ 𝔮_{l−1} is a principal ideal, obviously this refutes the existence of monomials t and t′ with the assumed properties. □

At the end of this section we present an example which demonstrates that already in four variables we may have Sdim R/I < dim R/I even for pure dimensional monomial ideals which are connected in codimension 1.

Example 10. Consider I = (x, y^{3}) ∩ (u^{2}, y^{2}) ∩ (y, z) ∩ (u, z^{2}) ⊂ 𝕂[x, y, z, u]. The two monomials y^{2}u, y^{2}z^{2} belong to all but the first irreducible component, hence, they require a direct summand containing y^{2}𝕂[z, u]. A similar argument shows that y^{3}, xy^{2} require a direct summand containing y^{2}𝕂[x, y]. Obviously, both direct summands must be equal and since any module t𝕂[Y] satisfying y^{2}𝕂[z, u] + y^{2}𝕂[x, y] ⊆ t𝕂[Y] will have a nontrivial intersection with I, we have an unresolvable conflict proving Sdim R/I < dim R/I.

8. Cogeneric Cohen-Macaulay Monomial Ideals

Example 10 shows that there is no hope to prove a result for cogeneric monomial ideals which is similarly nice as Theorem 3. However, at least in the Cohen-Macaulay case we will be successful.

Theorem 5. For any cogeneric Cohen-Macaulay monomial ideal I ⊂ R it holds Sdim R/I = dim R/I. Further, let I = 𝔮_{1} ∩ ⋯ ∩ 𝔮_{k} be an irredundant decomposition in irreducible components, where the components are ordered in the same was as in Theorem 3. For all l ∈ {2, …, k} set D_{l} := (𝔮_{1} ∩ ⋯ ∩ 𝔮_{l−1}) \ 𝔮_{l} and define T_{l} := \{t ∈ D_{l} : t is a monomial and ∀x_{i} ∈ supp t \ 𝔭_{l} : \frac{t}{x_{i}} ∉ D_{l}\}. Then for any l ∈ {2, …, k} and any t, t′ ∈ T_{l} either it holds t = t′ or there exists x_{i} ∈ 𝔭_{l} such that deg_{x_{i}} t ≠ deg_{x_{i}} t′.

For the proof we will need the following corollary to Lemma 1.

Corollary 3. Let I be a cogeneric Cohen-Macaulay monomial ideal I with irredundant decomposition I = 𝔮_{1} ∩ ⋯ ∩ 𝔮_{k} in irreducible components. Furthermore, let 𝔮_{i} and 𝔮_{j} be irreducible components such that codim(𝔭_{i} + 𝔭_{j}) > codimI + 1. Then there exists an irreducible component 𝔮_{m} of I such that

1. 𝔮_{m} ⊂ 𝔮_{i} + 𝔮_{j},

2. codim(𝔭_{i} + 𝔭_{m}) ≤ codimI + 1,

3. 𝔮_{i} and 𝔮_{m} have no minimal generator in common,

4. x_{r}^{d} ∉ 𝔮_{m} for all minimal generators x_{r}^{d} of 𝔮_{i} such that x_{r}^{d−1} ∉ 𝔮_{j}.

Proof: Lemma 1 yields the existence of 𝔮_{l′} such that 𝔮_{l′} ⊂ 𝔮_{i} +𝔮_{j}. Moreover, taking into account the cogenerity of I we can arrange that 𝔮_{l′} does not contain any of the common minimal generators of 𝔮_{j} and 𝔮_{i} + 𝔮_{j}. If codim(𝔭_{i} + 𝔭_{l′}) > codimI + 1 we repeat this process and after finitely many iterations we obtain an irreducible component 𝔮_{l} such that 𝔮_{l} ⊂ 𝔮_{i} + 𝔮_{j} and codim(𝔭_{i} + 𝔭_{l}) < codim(𝔭_{i} + 𝔭_{j}). By induction on codim(𝔭_{i} + 𝔭_{j}) we can even assume codim(𝔭_{i} + 𝔭_{l}) ≤ codimI + 1. Now, suppose 𝔮_{i} and 𝔮_{l} have a minimal generator in common. Then by definition of cogeneric monomial ideals there exists 𝔮_{m′} ⊂ 𝔮_{i} + 𝔮_{l} ⊆ 𝔮_{i} + 𝔮_{j} having no minimal generator in common with 𝔮_{i} + 𝔮_{l}. If 𝔮_{i} and 𝔮_{m′} still have common minimal generators we repeat the argument with l = m′. Since 𝔮_{i} cannot share any of its minimal generators with more than one of the intermediate components appearing in this process, eventually, we obtain 𝔮_{m} satisfying the first three conditions of the corollary. Finally, the validity of the fourth condition is an immediate consequence of the validity of the first and the third condition. □

Proof of Theorem 5: In analogy to the proof of Theorem 3 we proceed by induction on the number of variables n. Again the initial case n = 1 is trivial. So let us assume that the statement holds for (n − 1) variables.

Suppose, there exists l ∈ {2, …, k} such that T_{l} contains two distinct elements t and t′ which have the same degree in any variable belonging to 𝔭_{l}. Let t̂and t̂′ denote the monomials obtained by setting all variables of 𝔭_{l} equal to 1 in t or t′, respectively. By construction we have lcm(s, t̂) = t and lcm(s, t̂′) = t′, where s := gcd(t, t′) is the greatest common divisor of t and t′. From t, t′ ∈ 𝔮_{1} ∩ ⋯ ∩ 𝔮_{l−1} we conclude that any irreducible component 𝔮_{j}, 1 ≤ j < l, which does not contain s has to contain both t̂ and t̂′ but cannot contain gcd(t̂, t̂′). In conclusion,

(9) ∀j ∈ {1, …, l − 1} : s ∉ 𝔮_{j} ⟹ codim(𝔭_{j} + 𝔭_{l}) ≥ codimI + 2 .

s ∈ 𝔮_{1} ∩ ⋯ ∩ 𝔮_{l−1} is impossible by construction of T_{l}. Hence, there exists j ∈ {1, …, l − 1} such that s ∉ 𝔮_{j}. Application of Corollary 3 yields the existence of an irreducible component 𝔮_{m} with the following three properties: (i) 𝔮_{m} ⊂ 𝔮_{l} + 𝔮_{j}, (ii) codim(𝔭_{m} + 𝔭_{l}) ≤ codimI + 1, and (iii) x_{r}^{a} ∉ 𝔮_{m} for all minimal generators x_{r}^{a} of 𝔮_{l} such that x_{r}^{a−1} ∉ 𝔮_{j}. I_{∞} as a localization of I at a monomial prime ideal is cogeneric. Hence, application of the induction assumption to I_{∞} shows x_{1} ∈ 𝔭_{l}. Therefore, 𝔮_{l} possesses a minimal generator of the form x_{1}^{d} and by the assumption on the order of the irreducible components of I this implies x_{1}^{d−1} ∉ 𝔮_{j} and, therefore, x_{1}^{d} ∉ 𝔮_{m} according to (iii). Consequently, m < l and in view of (9) and (ii) we obtain s ∈ 𝔮_{m}. Application of (i) yields s ∈ 𝔮_{m} ⊂ 𝔮_{j} + 𝔮_{l}, in contradiction to s ∉ 𝔮_{j} and s ∉ 𝔮_{l}. In summary, the assumption that T_{l} contains distinct elements t, t′ which have the same degree in all variables belonging to 𝔭_{l} must be wrong. This proves the second part of the theorem and the validity of the first statement follows by Corollary 1. □

9. Connections to Properties of Related Complexes

Proposition 1 is strongly related to the notion of shellable nonpure simplicial complexes due to Björner and Wachs (see [BW]). In fact, the assumptions of our proposition turn out to be an algebraic translation of Björner’s and Wachs’ notion in terms of the corresponding Stanley-Reisner ring. Our decompositions are directly related to [BW, Formula 2.2] rather than to the decompositions of Stanley-Reisner rings investigated by Björner and Wachs in Section 12 contained in the second part of [BW]. In the pure case the connection can be found also in [St79], where shellable pure complexes were generalized to partitionable pure complexes. In fact, our Proposition 2 is the algebraic counter part to partitionable nonpure complexes. Let us illustrate this fact by application of the opposite direction of Proposition 2 to the example presented at the top of page 149 in [St79].

Example 11 (Stanley). The upper partition [∅, ab], [c, cd], [d, de], [e, ce] of the simplicial complex Δ = ⟨ab, cd, ce, de⟩ corresponds to the direct sum decomposition 𝕂[Δ] = 𝕂[a, b] ⊕ c𝕂[c, d] ⊕ d𝕂[d, e] ⊕ e𝕂[c, e] of the Stanley-Reisner ring of Δ.

In view of Proposition 2 we obtain I = (c, d, e) ∩ (a, b, e) ∩ (a, b, c) ∩ (a, b, d) ⊂ 𝕂[a, b, c, d, e] and the sequence 1, c, d, e of standard monomials modulo I. Further, we compute J_{1} = (1), J_{2} = (c, d, e), J_{3} = (d, e, ac, bc), J_{4} = (e, ac, ad, bc, bd, cd), J_{5} = (ac, ad, ae, bc, bd, be, cd, de). This sequence satisfies the condition of Proposition 2, in particular, the critical monomials cd, de ∈ J_{5} \ I are contained in the direct sum decomposition.

Stanley’s example illustrates that Sdim R/I = dim R/I can hold even for pure monomial ideals which are not connected in codimension 1. Let us remark another immediate consequence of Stanley’s paper. Shellability implies the Cohen-Macaulay property of the Stanley-Reisner ring independent on the field characteristic. Therefore, the attempt to treat Reisner’s Example 2 by Proposition 1 must fail.

The next remark concerns the combinatorial properties of nonsquarefree monomial ideals. Miller, Sturmfels and Yanagawa proved that both the Scarf complex and the Stanley-Reisner complex corresponding to a pure generic monomial ideal are shellable (see [MSY, Theorem 2.5]). The generic monomial ideals are exactly this large class of ideals for which we were able to construct a Stanley decomposition by means of Proposition 3, which is the natural generalization of Proposition 1 to the nonsquarefree case. This motivates to study the relationship between the algebraic properties expressed in the assumptions of Proposition 3 and the shellability of complexes related to a nonsquarefree monomial ideal. There are indications that this connection will be not as close as in the squarefree case. At the one hand side there exist generic monomial ideals which are nonpure and whose Scarf complex is not shellable in the nonpure sense. But in view of Proposition 3 nonpure generic ideals show the same nice behavior as pure generic monomial ideals.

Example 12 ([BPS]Example 4.2). Bayer, Peeva and Sturmfels showed that the Scarf complex of I = (xyz, x^{4}y^{3}, x^{3}y^{5}, y^{4}z^{3}, y^{2}z^{4}, x^{2}z^{2}) = (x, y^{2}) ∩ (x^{2}, y) ∩ (x, y^{4}, z^{4}) ∩ (x, z^{3})∩(y, z^{2})∩(x^{3}, z)∩(x^{4}, y^{5}, z)∩(y^{3}, z) ⊂ 𝕂[x, y, z] is not shellable even in the nonpure sense. But the above order of irreducible components together with the sequence 1, x, y^{2}, y^{4}, x^{2}, xy, x^{3}y, x^{4}y of standard monomials provides an “algebraic shelling” of R/I.

At the other hand side the ideal I considered in Example 1 is pure and satisfies Sdim R/I < dim R/I but its Scarf complex is a square and, hence, shellable.

However, the Stanley-Reisner complex of the ideal from Example 1 is not even partitionable and, hence, there is still a good chance to find a connection for pure ideals which makes use of the shellings of both related complexes.

10. Outlook to Graded Polynomial Modules

Finally, let us generalize our investigations in two directions. As before, let R = 𝕂[X] be a polynomial ring but equipped with a ℕ^{ν}-grading for some ν ∈ {1, …, n}. Furthermore, let F be a free R-module with basis {e_{1}, …, e_{m}} and equipped with a ℤ^{ν}-grading which extends the ℕ^{ν}-grading of R in the sense that deg_{ℤ^{ν}} ue_{j} = deg_{ℕ^{ν}} u+ deg_{ℤ^{ν}} e_{j} for arbitrary homogeneous elements u ∈ R and 1 ≤ j ≤ m. Finally, assume ∑_{i=1}^{ν} d_{i} = 1 for all x ∈ X, where deg_{ℕ^{ν}} x = (d_{1}, …, d_{ν}). Throughout this section we will always assume that the field 𝕂 is infinite.

We consider modules M = F/N, where N is a ℤ^{ν}-graded submodule of F. In the particular case that N is monomial, what here should mean, that N is generated by elements of the form ue_{j}, where u ∈ R is a monomial and 1 ≤ j ≤ m, the previous results extend easily. We subdivide the set B of minimal generators of N in m pairwise disjoint (not necessarily nonempty) sets B_{1}, …, B_{m}, where B_{i} consists of all minimal generators of the form ue_{i}, 1 ≤ i ≤ m. Let I_{i}, i ∈ {1, …, m}, denote the monomial ideal of R generated by the set {u : ue_{i} ∈ B_{i}}. Since, depth(A ⊕ B) = min(depth A, depth B) for arbitrary R-modules A and B we can construct a Stanley decomposition for M by plugging in Stanley decompositions for each module R/I_{i} in the direct sum decomposition M = ⨁_{i=1}^{m} (R/I_{i}) e_{i}.

Now, let N be an arbitrary ℤ^{ν}-graded submodule of F and consider an admissible term order ≺ of F refining the ℤ^{ν}-grading, i.e. for any monic ℤ^{ν}-homogeneous elements a, b ∈ F such that deg_{ℤ^{ν}} a < deg_{ℤ^{ν}} b it holds in_{≺}a ≺ in_{≺}b. As usual, in_{≺}a denotes the initial term of a ∈ F with respect to ≺. The modules F/in_{≺}N and F/N are isomorphic graded 𝕂-spaces via the linear mapping induced by the assignment a + in_{≺}N ↦ a + N for all module monomials a = ue_{i}. In view of this isomorphism any Stanley decomposition of in_{≺}M := F/in_{≺}N is also a Stanley decomposition of M = F/N whenever it holds depth M ≤ depth in_{≺}M. Gräbe proved the inequality depth M ≥ depth in_{≺}M for arbitrary homogeneous M and there are examples where the inequality is proper (see [G, Theorem 4.1]). But we have the additional freedom of applying two preparatory transformations before we fix the order ≺ and the corresponding initial module in_{≺}M. First, we can apply a variable transformation which has to respect the ℕ^{ν}-grading of R. Second, we can transform the basis of F in a ℤ^{ν}-grading respecting way. There are examples where both transformations are necessary, but it is open if they are also sufficient.

Remark 2. Let 𝕂 be infinite. Suppose, that the following two questions have a positive answer:

1. Does Conjecture 2 hold for arbitrary monomial ideals I ⊂ R?

2. Given an arbitrary ℤ^{ν}-graded module M = F/N. Does there exist grading respecting transformations of R and F and an admissible term order ≺ of the transformed module monomials such that depth M = depth in_{≺}M?

Then Stanley’s Conjecture 1 holds for arbitrary ℤ^{ν}-graded polynomial modules M.

The proof is an immediate consequence of our above observations. This article is a step towards the solution of the first question. In the case ν = 1 and F = R a positive answer to the second question was given by Bayer and Stillman who proved that the equality depth M = depth in_{≺}M holds when ≺ is a reverse lexicographical term order and M is in generic coordinates ([BS, Theorem 2.4], see also [E, Section 15.7]). In the extremal case F = R and ν = n the second question has obviously a positive answer since M = in_{≺}M even without application of a preparatory transformation. This gives reason to the hope that question 2 has a general positive answer in the case F = R, when each group of variables of some fixed ℕ^{ν}-degree is transformed generically and an admissible term order ≺ is used which acts as a reverse lexicographical term order on each set of transformed variables of a fixed ℕ^{ν}-degree. Note, if the variables are enumerated in such a way that variables of the same ℕ^{ν}-degree have consecutive numbers then the variable transformations which respect the ℕ^{ν}- grading of R correspond to the subgroup of GL(n, 𝕂) formed by all block-diagonal matrices with blocks corresponding to groups of variables of common ℕ^{ν}-degree. In the 3-variate case it is easy to generalize Theorem 4 in this way.

Theorem 6. Let 𝕂 be an infinite field and R = 𝕂[x_{1}, x_{2}, x_{3}] a ℕ^{ν}-graded polynomial ring over 𝕂, where ν ∈ {1, 2, 3}. Assume that for each of the variables x_{1}, x_{2}, x_{3} the sum of the coordinates of its ℕ^{ν}-degree is 1.

Then Sdim R/I = min_{𝔭∈ Ass R/I} dim R/𝔭 for any ℕ^{ν}-homogeneous ideal I of R.

Proof: The assertion follows from [BS, Theorem 2.4] in case ν = 1 and from Theorem 4 in case ν = 3. So it remains to consider only the case ν = 2, where we can assume without loss of generality that deg_{ℕ^{ν}} x_{1} = (1, 0) and deg_{ℕ^{ν}} x_{2} = deg_{ℕ^{ν}} x_{3} = (0, 1). Using exactly the same arguments as Eisenbud in [E, Chapter 15] one can show that there is a linear variable transformation fixing x_{1} and mapping x_{2} and x_{3} to linearly independent linear combinations of them such that the initial ideal of the transformed ideal with respect to an arbitrary degree compatible admissible term order satisfying x_{3} ≺ x_{2} possesses a minimal generator x_{1}^{α}x_{2}^{β} which has strictly larger x_{2}-degree than any other minimal generator and which has minimal x_{1}-degree among all minimal generators. In fact, concentrating only on the transformation of x_{2} and x_{3} there is a Zariski open subset of GL(2, 𝕂) describing suitable transformations. Further, according to [E, Proposition 15.15] it suffices to check the equality depth R/I = depth R/in_{≺}I for the case that each variable occurs in some minimal generator of in_{≺}I. Let G denote the reduced Gröbner basis of I with respect to ≺. By homogeneity of I each element of G can be written in the form x_{1}^{α′}(x_{2}^{β′} +p(x_{2}, x_{3}))x_{3}^{γ′}, where p(x_{2}, x_{3}) is a homogeneous polynomial in x_{2} and x_{3} of ℕ-degree β′ without x_{2}^{β′}- term.

One easily observes depth R/I ≤ 1. Hence, it remains to prove that whenever in_{≺}I possesses a trivial irreducible component then also I possesses a trivial primary component (i.e. (x_{1}, x_{2}, x_{3}) ∈ Ass R/I). Now, assume that (x_{1}^{a}, x_{2}^{b}, x_{3}^{c}) is an irreducible component of in_{≺}I and consider the ideal J generated by all elements g |_{x_{1}=1} which are obtained by subsituting 1 for x_{1} in the Gröbner basis elements g ∈ G which satisfy deg_{x_{1}} in_{≺}g < a and deg_{x_{3}} in_{≺}g < c. The preparatory transformation ensures that J contains a homogeneous element whose initial term is a power of x_{2}. Moreover, for any homogeneous element h ∈ J we have deg_{x_{2}} in_{≺}h ≥ b or deg_{x_{3}} in_{≺}h ≥ c since, otherwise, there must be a nonnegative integer a′ < a such that x^{a′}h ∈ I which is impossible because of x^{a′}in_{≺}h ∉ in_{≺}I ⊆ (x_{1}^{a}, x_{2}^{b}, x_{3}^{c}). (x_{1}^{a}, x_{3}^{c}) + J is a (x_{1}, x_{2}, x_{3})- primary ideal containing I and by the above observations I is not contained in any higher dimensional primary subideal of (x_{1}^{a}, x_{3}^{c}) + J. Hence, (x_{1}, x_{2}, x_{3}) ∈ Ass R/I and we are through. □

Finally, let us demonstrate the additional problems arising in the rank F > 1 case by two examples.

Example 13. Consider R = 𝕂[x_{1}, x_{2}, x_{3}] with an arbitrary ℕ^{ν}-grading. Further, let rank F = 3 and deg_{ℤ^{ν}} e_{1} = deg_{ℤ^{ν}} e_{2} = deg_{ℤ^{ν}} e_{3} = (0, …, 0). Assume char 𝕂 ≠ 2 and let M = F/N, where N = (x_{1}e_{1} + x_{1}e_{2}, x_{2}e_{1} + x_{2}e_{3}, x_{3}e_{2} + x_{3}e_{3}). Then the elements x_{1}e_{1}, x_{2}e_{1}, x_{3}e_{2}, x_{1}x_{2}e_{2}, x_{1}x_{2}x_{3}e_{3} generate the initial module in_{≺}M = F/in_{≺}N, where ≺ is an arbitrary admissible term order such that e_{3} ≺ e_{2} ≺ e_{1}. Hence, depth in_{≺}M = 1 < depth M and by symmetry reasons the same is true for any admissible term order ≺. However, after an obvious basis transformation we obtain N = (x_{1}ê_{1}, x_{2}ê_{2}, x_{3}ê_{3}) and, hence, depth in_{≺}M = depth M = 2.

Example 14. Consider R = 𝕂[x_{1}, …, x_{n}] with respect to an arbitrary ℕ^{ν}-grading. Furthermore, let rank F = n + 1 and deg_{ℤ^{ν}} e_{1} = (0, …, 0), deg_{ℤ^{ν}} e_{i} = deg_{ℕ^{ν}} x_{i−1} (i = 2, …, n+1) the degrees of the free generators of F. Then the module M = F/N, where N is generated by the elements x_{i}e_{1} + e_{i+1}, i = 1, …, n, is free of rank 1 and has depth n.

If two module monomials ue_{i} and ve_{j} have the same ℤ^{ν}-degree we have to break ties by the order ≺. If ties are broken by ue_{i} ≺ ve_{j} ⇔ i < j then we obtain in_{≺}N = (e_{2}, …, e_{n+1}) and depth in_{≺}M = n = depth M. However, depth in_{≺}M = 0 < depth M if we break ties according to ue_{i} ≺ ve_{j} ⇔ i > j.

Acknowledgment

I thank Bernd Sturmfels for drawing my attention to this subject and Hans-Gert Gräbe and Jürgen Stückrad for helpful comments. I am thankful to the Mathematical Sciences Research Institute and the Mathematics Department of the University of California at Berkeley for their hospitality during the one year research visit in which the work on this project was done. Last but not least I am grateful to the Deutsche Forschungsgemeinschaft for supporting the visit by a research grant.

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Joachim Apel^{3}, Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720, U.S.A.

E-mail address: apel@msri.org

3On leave from Mathematisches Institut, Universität Leipzig, Augustusplatz 10–11, 04109 Leipzig, Germany, E-mail address: apel@mathematik.uni-leipzig.de