Stanley depth and the lcm-lattice

# Stanley depth and the lcm-lattice

Bogdan Ichim Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit 5, C.P. 1-764, 014700 Bucharest, Romania Lukas Katthän Goethe-Universität Frankfurt, FB Informatik und Mathematik, 60054 Frankfurt am Main, Germany  and  Julio José Moyano-Fernández Universitat Jaume I, Campus de Riu Sec, Departamento de Matemáticas & Institut Universitari de Matemàtiques i Aplicacions de Castelló, 12071 Castellón de la Plana, Spain
###### Abstract.

In this paper we show that the Stanley depth, as well as the usual depth, are essentially determined by the lcm-lattice. More precisely, we show that for quotients of monomial ideals , both invariants behave monotonic with respect to certain maps defined on their lcm-lattice. This allows simple and uniform proofs of many new and known results on the Stanley depth. In particular, we obtain a generalization of our result on polarization presented in [IKMF15]. We also obtain a useful description of the class of all monomial ideals with a given lcm-lattice, which is independent from our applications to the Stanley depth.

###### Key words and phrases:
Monomial ideal; lcm-lattice; Stanley depth; Stanley decomposition.
###### 2010 Mathematics Subject Classification:
Primary: 05E40
The first author was partially supported by project PN-II-RU-TE-2012-3-0161, granted by the Romanian National Authority for Scientific Research, CNCS - UEFISCDI. The second author was partially supported by the German Research Council DFG-GRK 1916. The third author was partially supported by the Spanish Government Ministerio de Economía y Competitividad (MINECO), grants MTM2012-36917-C03-03 and MTM2015-65764-C3-2-P, as well as by Universitat Jaume I, grant P1-1B2015-02.

## 1. Introduction

Let be a field, a -graded -algebra and a finitely generated -graded -module. The Stanley depth of , denoted , is a combinatorial invariant of which was introduced by Apel in [Ape03a] and has attracted the attention of many researchers [HP06, HVZ09, BHK10, OY11, DGKM16]. We refer the reader to the survey of Herzog [Her13] for an introduction to this subject.

One line of research on the Stanley depth was motivated by a conjecture of Stanley from 1982 [Sta82, Conjecture 5.1], which states that , see also [Gar80, Remark 5.2] and [Sta79, p. 149]. In the following, we refer to this inequality as Stanley’s inequality. Thus, a mayor aim of this study is to establish relations between the depth and the Stanley depth. At a first glance, one might not expect any deep connection between them, at these invariants seem to be very different in nature. On the one hand, we have the depth, which is an algebraic invariant (homological in nature), and on the other hand we have the Stanley depth, which is a purely combinatorial invariant. Nevertheless, several parallel results for the depth and the Stanley depth have been found, see for example [Rau07, Cim09, HJZ10, Ish12, SF17]. A counterexample to the original Stanley conjecture was recently given by Duval, Goeckner, Klivans and Martin in [DGKM16]. However, there still seems to be a deep and interesting connection between these two invariants, which is yet to be fully understood.

The Stanley depth is defined in terms of certain combinatorial decompositions of the module , which are called Stanley decompositions. It is also worth mentioning that these Stanley decompositions have a separate life in applied mathematics. Sturmfels and White [SW91] have shown that Stanley decompositions may be used to describe finitely generated graded algebras, for example rings of invariants under some group action. Recently this has found applications in the normal form theory for systems of differential equations (see Murdock [Mur02], Murdock and Sanders [MS07], Sanders [San07]).

Most of the research on the Stanley depth concentrates on the particular case of a module of the form for two monomial ideals in the polynomial ring . In this paper we will also work in this setting. For the sake of simplicity, we restrict this introduction to the case of modules of the form (most of the results are later proven in a more general setup).

The lcm-lattice of an ideal is the lattice of all least common multiples of subsets of the (minimal) generators of , ordered by divisibility, see Gasharov et al. [GPW99]. It is a finite atomistic lattice that is known to encode a lot of information about . In particular, it encodes the structure of the minimal free resolution of over and thus determines the Betti numbers and the projective dimension of [GPW99, Theorem 3.3]. More precisely, what is shown in [GPW99] is the following: Let and be two monomial ideals. Then, given a free resolution of and a surjective join-preserving map which is bijective on the atoms, one can construct a free resolution of by a certain relabeling procedure. In particular, the projective dimension of is bounded above by the projective dimension of . In this paper we obtain the analogous statement for the Stanley depth.

###### Theorem (Corollary of Theorem 4.5).

Let and be two proper monomial ideals in two polynomial rings in resp. variables. If there exists a surjective join-preserving map , such that , then

 n−sdepthSS/I≥n′−sdepthS′S′/I′.

By we denote the minimal elements of the lattices. In view of this result, we define the Stanley projective dimension, , analogously to the Stanley depth (cf. Definition 2.3). In particular, it easily follows that .

###### Example 1.1.

For and consider the two ideals

 I :=⟨x3,x2y,xy2,y3⟩⊂S  and I′ :=⟨yzv,xzv,xy2v,x2y2z⟩⊂S′.

Their lcm-lattices are depicted in Figure 1. We define a map by setting , , and every other monomial in is mapped to the monomial in which is at the same place in Figure 1.

This map is join-preserving and surjective, so Theorem 4.5 applies. It is clear that and so . It follows that , or equivalently that .

Theorem 4.5 has a number of important consequences. First of all, it shows that two ideals with isomorphic lcm-lattices have the same Stanley projective dimension. Thus, this invariant is determined by the isomorphism type of the lcm-lattice. In particular, the lcm-lattice of an ideal is invariant under polarization. Hence Theorem 4.5 generalizes the main result of [IKMF15], where we showed that the Stanley projective dimension is invariant under polarization.

Next, we present a simple and uniform proof for upper bounds on the Stanley projective dimension (i.e. lower bounds on the Stanley depth) in terms of the number of generators in Proposition 5.2. We also characterize the extremal case and prove that Stanley’s inequality holds for ideals with , where is the number of generators of . As another application we study generic deformations in Proposition 5.5 and the forming of colon ideals in Proposition 5.8. The latter allows us to give in Corollary 5.9 a uniform proof that both depth and Stanley depth are bounded by the dimensions of the associated prime ideals. Moreover, in Proposition 5.12 we show that for studying the Stanley depth one may always assume that the ideal under consideration is generated in a single degree.

We further identify some operations on ideals, e.g. passing to the radical, that yield surjective join-preserving maps on the lcm-lattice, so we obtain inequalities for the Stanley projective dimension in these cases. As all our proofs rest on Theorem 4.5 and we showcase an analogous result for the usual projective dimension (Theorem 4.9), we obtain the same bounds as for the usual projective dimension. While these results are well-known, it is relevant that we obtain uniform proofs for both depth and Stanley depth, thus explaining the observed parallel behavior.

In the way of studying the relation of ideals to their lcm-lattices, we also get a result of independent interest. In Theorem 3.4 we give a complete description of the class of all monomial ideals with a given lcm-lattice. This result also allows the easy construction of monomial ideals with a prescribed lcm-lattice, which we consider very useful for the study of examples. Theorem 3.4 extends results obtained in [Map13] to the (more general) case of not necessarily atomistic lattices, which is needed for our applications to both depth and Stanley depth.

Finally, the fact that both the projective dimension and the Stanley projective dimension are determined by the lcm-lattice allows us further to formulate open questions about the depth and the Stanley depth completely in terms of finite lattices. So one can try to apply notions and techniques from this field to approach these questions. In the last section we indicate some of these ideas. In particular, this enables us to reduce the study of infinitely many monomial ideals to finitely many finite lattices. This paves the way to computations. In several computational experiments, we have classified all lcm-lattices of ideals with up to five generators and found that the projective dimension and the Stanley projective dimension of coincide for these lattices; this is presented in [IKMF16].

In another follow-up paper [Kat15], the second author applied Theorem 4.5 to show that many questions about the Stanley depth can be reduced to a very special class of ideals. In particular, Stanley’s inequality holds for ideals with up to seven generators and for quotients of the polynomial ring by ideals with up to six generators.

## 2. Preliminaries

### 2.1. Finite lattices and semilattices

Let us recall some definitions and facts about finite lattices and semilattices. We refer the reader to [DP02] for more background information. A join-semilattice is a partially ordered set such that, for any , there is a unique least upper bound called the join of and . A lattice is a join-semilattice with the additional property that for any , there is a unique greatest lower bound called the meet of and .

Every finite join-semilattice has a unique maximal element . Moreover, a finite join-semilattice is a lattice if and only if it has a minimal element. So we can associate to every finite join-semilattice a canonical lattice by adjoining a minimal element . All lattices and semilattices in the sequel will be assumed to be finite.

We say that an element covers another element , if and there exists no other element , such that . An element is called an atom if it covers the minimal element in . Equivalently, the atoms are the minimal elements of (in the sense that there are no smaller elements). We call atomistic, if every element can be written as a join of atoms.

A meet-irreducible element is an element which is covered by exactly one other element. This terminology is justified by noting that is meet-irreducible if and only if implies or for where the meet is taken in . A join-preserving map is a map with for all . Note that every join-preserving map preserves the order.

### 2.2. The lcm-lattice and lcm-closed subsets

Let be a polynomial ring. A monomial is a product of powers of variables of . In particular, is a monomial, but is not. We write for the set ot monomials of . Note that forms a -basis of .

Recall from [GPW99] that the lcm-lattice of a monomial ideal is the set of all least common multiples of subsets of the minimal set of generators of , together with a minimal element which is usually identified with and regarded as the lcm of the empty set. For our scope, we need to modify this notion in several ways. First, we need to consider non-minimal generating sets, second we need a reasonable replacement of the lcm-lattice for a pair of ideals, and finally we want that our modified definition yields isomorphic lattices for all principal ideals, including the unit ideal. To this end we give the following definition.

###### Definition 2.1.

We call a finite set of monomials lcm-closed, if the least common multiple (lcm) of every non-empty subset of is also contained in .

The lcm-closure of a finite set , denoted by , is defined as the set of all monomials that can be obtained as the least common multiple (lcm) of some non-empty subset of

Note that and that if and only if is lcm-closed. Every lcm-closed set can be regarded as a join-semilattice, where the order is given by divisibility and the join is the lcm. We will often consider the associated lattice of an lcm-closed set , where is an additional minimal element. The element could be regarded as the lcm of the empty set, but we do not identify with .

Note that if is a proper monomial ideal, then , where is a minimal generating set of .

###### Remark 2.2.
1. Following [GPW99], for we get , but . While this difference is minor, we think that the latter is in fact a more convenient definition of the lcm-lattice of . For example, the lcm-lattice of should be isomorphic to the lcm-lattice of a principal ideal.

2. The associated lattice of an lcm-closed set is atomistic if and only if the minimal elements of form the minimal set of generators of some monomial ideal (in fact, ). So in general, could be regarded as an lcm-lattice associated to a not necessarily minimal set of generators of . The reason why we consider non-minimal generating sets is that we are going to consider maps of lcm-lattices. Even if we start with a minimal set of generators of some monomial ideal, its image might not be minimal anymore.

### 2.3. Stanley depth and maps changing it

Consider the polynomial ring endowed with the multigraded structure. Let be a finitely generated graded -module, and let be a homogeneous element in . Let be a subset of the set of indeterminates of . The -submodule of is called a Stanley space of if is free (as -submodule). A Stanley decomposition of is a finite family

 D=(K[Zi],λi)i∈I

in which and is a Stanley space of for each with

 M=⨁i∈IλiK[Zi]

as a multigraded -vector space. This direct sum carries the structure of an -module and has therefore a well-defined depth. The Stanley depth of is defined to be the maximal depth of a Stanley decomposition of .

In the same fashion we introduce the following definition.

###### Definition 2.3.

The Stanley projective dimension of is the minimal projective dimension of a Stanley decomposition of .

Note that by the Auslander-Buchsbaum formula. While this definition might seem redundant, it turns out that our results (for example, see Theorem 4.5) are more naturally stated in terms of the Stanley projective dimension. Further, Stanley’s inequality is equivalent to the following:

 pdim M≥spdim M. (⋆)

The original Stanley conjecture [Sta82] stated that (2.3) holds for all finitely generated modules. As mentioned above, this was recently disproved by Duval et al. [DGKM16].

In the proof of our main result we use a certain type of poset maps which was first introduced in [IKMF15]. Before we recall the definition, let us introduce a notation. For with , we define

 ρg(a):=#{j∈[n]:aj=gj}.
###### Definition 2.4.

[IKMF15, Definition 3.1] Let and . A monotonic map is said to change the Stanley depth by with respect to and , if it satisfies the following two conditions:

1. .

2. For each interval , the (restricted) preimage can be written as a finite disjoint union of intervals, such that

 ρg(bi)≥ρg′(b′)+ℓ  ~{}for all i.

Those maps were profusely studied in [IKMF15]. For the reader’s convenience we recall a key result, which motivates the above definition and is used in the sequel.

###### Proposition 2.5 ([Ikmf15, Proposition 3.3]).

Let , and be two polynomial rings and let be monomial ideals. Consider a monotonic map and let be the map defined by . Set , . Choose and , such that every minimal generator of and divides , and every minimal generator of and divides . If changes the Stanley depth by with respect to and , then

• and are monomial ideals, and

• .

## 3. Labelings and lcm-closed sets

In this section we present several results on lcm-closed sets that are later needed for the proof of the main results of this paper. Throughout the section, let be a fixed polynomial ring.

A labeling of a finite lattice is a map , i.e. an assignment of a monomial to each element of . Now, for a finite lcm-closed set we define a labeling as follows: For the minimal and maximal elements , we define and . For every other we set

 wG(m):=1mgcd{p∈G:p>m}.

This labeling was introduced in [Map13, Eq. (3.3)]. It satisfies the following inversion formula:

###### Proposition 3.1.

For it holds that

 m=∏q∈¯¯¯¯Gq≱mwG(q).

Note that the formula for evaluates to , but .

###### Proof.

In [Map13, Proposition 3.6], Mapes proves that this formula holds for the minimal elements of , under the additional assumption that is atomistic. However, the argument given there actually shows the formula in the generality claimed here. ∎

The next corollary gives a characterization of a lcm-closed sets of squarefree monomials in terms of .

###### Corollary 3.2.

Let be a finite lcm-closed set. Then contains only squarefree monomials if and only if is squarefree for every and for all .

###### Proof.

If the given conditions are satisfied, then all elements of are squarefree, since by Proposition 3.1 every element of is a product of different monomials for some . On the other hand, if every element of is squarefree, then in particular the lcm of all elements is a squarefree monomial. But this is the product of all the , so the claimed properties follow. ∎

We now come to our first key result, which in particular gives a complete description of those pairs that come from a monomial ideal and extends Theorem 3.2 and Proposition 3.6 in [Map13] to not necessarily atomistic lattices.

###### Definition 3.3.

A labeling (on a finite lattice ) is admissible if it satisfies the following two conditions:

1. for incomparable .

2. if is meet-irreducible and .

We consider two pairs and of finite lattices with labelings to be isomorphic if and the isomorphism maps to .

###### Theorem 3.4.

The map is a bijection between the set of finite lcm-closed sets , and the set of isomorphism classes of pairs where is a finite lattice and is a admissible labeling. The inverse map is given by mapping a pair to the set

 G(L,w):=⎧⎪ ⎪⎨⎪ ⎪⎩∏b∈Lb≱aw(b):a∈L⎫⎪ ⎪⎬⎪ ⎪⎭ (3.1)

This theorem allows the very simple construction of ideals with a given lcm-lattice. Indeed, for a given lattice one only needs to choose an admissible labeling. Moreover, considering the possible admissible labelings one gets an overview over the class of all monomial ideals with a fixed lcm-lattice.

###### Proof of Theorem 3.4..

Let denote the set of all finite lcm-closed subsets of and let the set of isomorphism classes of pairs where is a finite lattice and is a labeling. Moreover, let denote the set where we assume the labeling to be admissible. We denote the two maps of the claim by and .

Proposition 3.1 shows that is the identity map, so in particular is injective. Therefore, to prove the claim it is sufficient to show that the image of is , in other words we need to show that

1. is admissible for each , and

2. for each , there exists a with .

The first item is proven in Lemma 3.4 and Lemma 3.5 of [Map13]. Again, in [Map13] the result is only claimed for atomistic lattices, but the argument given there holds in our general setup.

For the second item, our candidate for is as given by Equation 3.1. So we need to show that is lcm-closed, and (under this isomorphism), . In [Map13, Theorem 2.3] it is shown that in the case that is atomistic. However, the argument given there does not directly apply to the general situation, so we need a new proof.

Consider defined by and

 ψ(a):=∏b∈Lb≱aw(b)

for . The labeling being admissible implies that for each variable such that , the set of such that forms a chain . Set for , where if . For let be the minimal index , such that . As a notation, for a monomial we define to be the exponent of in . We extend this definition to by setting for all . Then

 ordi(ψ(a))=ordi(s(i,a)−1∏j=1w(aij)). (3.2)

In particular, if , then for , because of the inequality .

Let . We claim that if and only if . It is clear from the definition that implies , so assume that . Every non-maximal element in a finite lattice is the meet of the set of meet-irreducible elements greater than or equal to it. So, in order to show , we may prove the following: Each meet-irreducible element which is greater than or equal to is also greater than or equal to . So consider such an element . As , there exists an index such that .

Then there exists a such that (where ). Now implies that . But as remarked above, the fact that implies that , hence .

It follows that is injective, as implies and thus .

Further, we claim that equals the lcm of and . For this, first note that if and only if and . This implies that for all . Therefore

 ordi(ψ(a∨b))=max(ordi(ψ(a)),ordi(ψ(b)))=ordi(lcm(ψ(a),ψ(b)))

for all , hence .

Summarizing, we have shown that is an injective map which preserves the join. The latter implies that lcm-closed, i.e. . Hence induces an isomorphism .

It remains to show that for all . By definition of , we have to show that if and for . We handle both cases together by proving that

 ordi(gcd(ψ(b):b>a))=ordi(ψ(a))+ordi(w(a))

for each . We compute

 ordi(gcd(ψ(b):b>a)) =min{ordi(ψ(b)):b>a} =min{ordi(s(i,b)−1∏j=1w(aij)):b>a} =ordi(k−1∏j=1w(aij))

where . We compute further:

 k=min(s(i,b):b>a) =min{min{j:aij≥b}:b>a} =min{j:aij>a} ={s(i,a)+1if a=ais(i,a),s(i,a)otherwise.

Note that in the second case it holds that (otherwise because is admissible).

Recall that . So we conclude that

 ordi(gcd( ψ(b):b>a))= =⎧⎨⎩ordi(∏s(i,a)−1j=1w(aij))+ordi(w(ais(i,a)))if a=ais(i,a),ordi(∏s(i,a)−1j=1w(aij))otherwise =ordi(ψ(a))+ordi(w(a)).

## 4. Invariants and surjective join-preserving maps

This section contains the main results of this paper. They are presented in Subsection 4.2 and Subsection 4.3. Here we show that the Stanley depth, as well as the usual depth, are determined by the lcm-lattice. Subsection 4.1 contains several related technical results. We end with an example which shows that the -graded Hilbert depth is not determined by the lcm-lattice.

### 4.1. The structure of surjective join-preserving maps

In this Subsection we prove some structural results on surjective join-preserving maps; these will be needed in the sequel. The first two structural lemmata will be useful in Subsection 4.2.

Let be a surjective join-preserving map of finite lattices. We define as .

###### Lemma 4.1.

The map has the following properties:

1. is the identity and is (thus) injective.

2. is monotonic, i.e. implies for .

3. For and , it holds that if and only if .

###### Proof.

The first claim is immediate from the fact that preserves joins. For the second, note that for any . Thus is contained in the preimage of and hence . Now assume that . Then

 δ†(a)≤δ†(a)∨δ†(b)≤δ†(a∨b)=δ†(b).

For the last claim, note that implies , and implies (since is monotonic). ∎

###### Lemma 4.2.

Let and be two finite lcm-closed sets of squarefree monomials in two polynomial rings. Assume that there exists a surjective join-preserving map with such that for all . Then there exists a ring homomorphism sending a subset of the variables injectively to the variables of and the other variables to . This map satisfies for .

###### Proof.

As and consist only of squarefree monomials, it holds that all values of and are squarefree and pairwise coprime by Corollary 3.2.

We define as follows: For every , choose many variables dividing and let them map bijectively to the variables dividing . The remaining variables of are mapped to one. By construction, for it holds that

 Ψ(wG(m))={wG′(m′)~{}if~{}m∈δ†(¯¯¯¯¯¯G′)~{}and~{}m=δ†(m′);1~{}if~{}m∉δ†(¯¯¯¯¯¯G′).

Using Proposition 3.1 we conclude that

 Ψ(m)=Ψ(∏q∈¯¯¯¯Gq≱mwG(q))=∏q∈¯¯¯¯Gq≱mΨ(wG(q))=∏q′∈¯¯¯¯¯G′q′≱δ(m)wG′(q′)=δ(m).

where . For the third equality, we used part (3) of Lemma 4.1. Note that the last equality holds because . ∎

The next two structural lemmata will be used in Subsection 5.1. Fix a meet-irreducible element and let denote the unique element covering it. We consider the equivalence relation on defined by setting and any other element is equivalent only to itself.

###### Lemma 4.3.

There is a natural lattice structure on , such that the canonical surjection preserves the join. Moreover, if is atomistic and is not an atom, then is atomistic.

###### Proof.

Let denote the equivalence class of an element . We define . To show that this is well-defined we have to prove that and implies . For this, we distinguish the cases that either or and similarly for . One easily sees that each case is either trivial or follows from the observation that for all .

The -operation on inherits associativity, commutativity and idempotency from the join of , cf. [DP02, Thm. 2.10]. Moreover, inherits a minimal element from , so it is in fact a lattice. It is clear that preserves this join. The last statement is also clear as is a bijection on the atoms. ∎

###### Lemma 4.4.

Let be finite lattices and a join-preserving map.

1. If is not injective, then there exists a meet-irreducible element such that .

2. If for some meet-irreducible element , then factors through .

###### Proof.

(1) There exists a maximal element such that the pre-image of has at least two elements, that is . Choose another element , . Then by maximality, as . It is easy to see that the interval is mapped to , so we may choose such that and is covered by . We claim that this is meet-irreducible. Assume to the contrary that there exists another element covering . Then

 δ(c)=δ(c∨a)=δ(c)∨δ(a)=δ(c)∨δ(b)=δ(c∨b)

As , it follows from our choice of that and thus , a contradiction.

(2) It is clear that factors though set-theoretically, i.e. there exists a map such that . So we only need to show that preserves the join. This is an easy computation:

 ¯δ(¯¯b∨¯¯c)=¯δ(¯¯¯¯¯¯¯¯¯¯¯b∨c)=δ(b∨c)=δ(b)∨δ(c)=¯δ(¯¯b)∨¯δ(¯¯c)

for . ∎

### 4.2. Stanley projective dimension and surjective join-preserving maps

In this Subsection we prove the following theorem, which is the main result of this paper.

###### Theorem 4.5.

Let and be four lcm-closed sets of monomials in two (possibly) different polynomial rings, such that . Assume further that there exists a surjective join-preserving map with such that . Then

 spdimS⟨G⟩/⟨H⟩≥spdimS′⟨G′⟩/⟨H′⟩.

For monomial ideals and , the theorem applies in particular to and . However, in general we do not assume that the sets come from minimal sets of generators.

A particular case is if and are the polarizations of and , respectively. Therefore, the theorem is a generalization of the authors’ result on polarization [IKMF15, Theorem 4.4]. This does not diminish the importance of [IKMF15, Theorem 4.4], since it is required in the proof of Theorem 4.5. We give a small example to demonstrate that the assumption is necessary.

###### Example 4.6.

Let , and . There is a surjective join-preserving map defined by mapping to and every other element to itself. It holds that , and indeed the conclusion of Theorem 4.5 does not hold:

 spdimS⟨G⟩/⟨H⟩=spdimSS=0≱1=spdimS⟨x,y⟩=spdimS⟨G′⟩/⟨H′⟩

Before we give the proof of Theorem 4.5 we prepare two lemmata.

###### Lemma 4.7.

Let be two squarefree monomial ideals. Let be the images of and under the map sending to . Then we have

 spdimS[Y]I/J≥spdimSI′/J′.

This extends [Cim08, Lemma 2.2], which shows only the case .

###### Proof.

Let and let be the -submodule of those elements having positive -degree. Every Stanley decomposition of restricts to a Stanley decomposition of , hence .

On the other hand, we have

 M>0=(I∩⟨Y⟩)/(J∩⟨Y⟩)≅(I:Y)/(J:Y)=(I:Y∞)/(J:Y∞),

where for the last equality we use that and are squarefree. But and the same holds for , hence . By [IMF14, Proposition 5.1] we conclude that and the claim follows. ∎

The second lemma comprises the main part of the proof of the theorem.

###### Lemma 4.8.

Let be two finite lcm-closed sets of squarefree monomials, such that . Let be a fixed element. Then there exist two other finite lcm-closed sets of squarefree monomials in one additional variable , such that the following holds:

1. There is an isomorphism of lattices, such that and for every it holds that

 degwG′(δ(c)):={degwG(c) if c≠m,degwG(c)+1 if c=m.
2. .

3. .

###### Proof.

Consider the map of monomials given by

 ~δ(c)={c if c∣m,Y⋅c if c∤m.

We define and as the images of resp.  under this map. It is easy to see that is injective and preserves the lcm of monomials. Thus , both sets are lcm-closed, and induces an isomorphism . Moreover, it follows from the definitions that

 wG′(m)=1mgcd{p∈G′:p>m}=Y⋅wG(m)

and for every other . Part (1) of the lemma is then proven. Part (2) follows straight from the fact that is monotonic.

(3) Let and . The inequality “” follows from Lemma 4.7, as and are the images of and