Splittings and Symbolic Powers of square-free monomial Ideals

Splittings and Symbolic powers of square-free monomial Ideals

Jonathan Montaño Jonathan Montaño
Department of Mathematical Sciences
New Mexico State University
PO Box 30001
Las Cruces, NM 88003-8001
jmon@nmsu.edu
 and  Luis Núñez-Betancourt Luis Núñez-Betancourt
Centro de Investigación en Matemaáticas
Guanajuato, Gto., México
luisnub@cimat.mx
Abstract.

We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with respect to a morphism which resembles the Frobenius map and that exists in all characteristics. Using these methods, we recover a result by Hoa and Trung which states that the normalized -invariants and the Castelnuovo-Mumford regularity of the symbolic powers converge. In addition, we give a sufficient condition for the equality of the ordinary and symbolic powers of this family of ideals, and relate it to Conforti-Cornuéjols conjecture. Finally, we interpret this condition in the context of linear optimization.

Key words and phrases:
monomial ideals, ordinary powers, symbolic powers, Castelnuovo-Mumford regularity
2010 Mathematics Subject Classification:
Primary ; Secondary .
2010 Mathematics Subject Classification:
05C65, 90C27, 13F20, 13F55, 13A35
The first author was supported byt the Simons Travel Grant from the Simons Foundation.
The second author was partially supported by NSF Grant 1502282 and CONACyT Grant 284598.

1. Introduction

Symbolic powers of ideals have been studied intensely over the last two decades (see [DDSG18] for a recent survey). Particular attention has been given to square-free monomial ideals, as in this setting methods from combinatorics, convex geometry, and linear optimization can be utilized to investigate properties of symbolic powers. For instance, the Cohen-Macaulay property of all symbolic powers of a square-free monomial ideal is characterized in terms of the combinatoric structure of its underlying simplicial complex [TT11, TT12, Var11]. In addition, the symbolic Rees algebra of monomial ideals is Noetherian [HHT07]; we note that this phenomenon does not hold for an arbitrary ideal in a polynomial ring [Rob85].

In this article, we propose a technique to deal with questions about symbolic powers of square-free monomial ideals. Specifically, we study the symbolic Rees algebras of an square-free monomial ideal via methods in prime characteristic. This combination, to the best of our knowledge, has not been previously used in combinatorial commutative algebra. In order for our results to hold over fields of arbitrary characteristic, we consider the map that raises every monomial to a power and resembles the Frobenius map. Considering this map, we obtain that the symbolic Rees algebra and the symbolic associated graded algebra are split in this general context (see Theorem 4.3). In particular, these symbolic algebras are -pure in prime characteristic (see Corollary 4.4).

Motivated by the behavior of the Castelnuovo-Mumford regularity for powers of ideals [CHT99, Kod00], Herzog, Hoa, and Trung [HT02] asked whether the limit

exists for every homogeneous ideal in a polynomial ring . It is known that the function is bounded by a linear function on if is a monomial ideal. This follows because the regularity of a monomial ideal is bounded by the degree of the least common multiple of the generators [BH95, HT98].

Herzog and Hoa showed that the limit above exists for square-free monomial ideals. In fact, they showed a stronger version for the -invariants [HT10, Theorems 4.7 and 4.9]. As a first consequence of our methods, we recover this result, providing an alternative proof.

Theorem A (see Theorem 3.5 and Corollary 3.6).

Let be a square-free monomial ideal. Then,

exists for every . In particular,

exists.

The function is in fact a linear quasi-polynomial if is a monomial ideal [HT02]. In addition, as a consequence of our previous result, we recover properties of this quasi-polynomial in Corollary 3.8. Along the way of proving Theorem A, we showed that

for every . In addition, we showed that

for every (see Theorem 3.4). Similar results were previoulsy obtained for the Stanley depth of symbolic powers [SF17].

Conforti and Cornuéjols [CC90] made a conjecture in the context of linear optimization. This conjecture was translated as a characterization of the set of square-free monomials ideals whose symbolic and ordinary powers are equal [GRV09, GVV07]. The following definition is needed to state this conjecture.

Definition B.

A square-free monomial ideal of height is König if there exists a regular sequence of monomials in of length . The ideal is said to have the packing property if every ideal obtained from by setting any number of variables equal to or is König.

The Conforti-Cornuéjols conjecture can be stated as follows.

Conjecture C ([Cc90]).

A square-free monomial ideal is packed if and only if for every

We point out that one direction of the conjecture is already know. Explicitly, if for every , then is packed.

Motivated by this conjecture Hà asked111 BIRS-CMO workshop on Ordinary and Symbolic Powers of Ideals Summer of 2017, Casa Matemática Oaxaca, Mexico. if there exists a number , in terms of , such that if for , then the equality holds for every . If the answer to this question is , then Conforti-Cornuéjols . This is expected as a similar property is known for integral closure of ordinary powers. Specifically, it suffices to verify the equality up to the analytic spread of minus one [Sin07] (see also [RRV03]). In our next main result, we answer Hà’s question.

Theorem D (see Theorem 4.8).

Let be a square-free monomial ideal and its minimal number of generators. Then, for every if and only if for every .

The previous result gives a finite algorithm to verify Conjecture C for a specific monomial ideal. We refer to the work of Gitler, Valencia and Villarreal [GVV07, Remark 3.5] for a different algorithm to verify Conjecture C. In Example 4.9 we show that the bound given by in Theorem D is sharp.

In Section 5, we recall the ideas from linear optimization that originally gave rise to Conjecture C. In particular, we translate Theorem D to this context in Theorem 5.2, showing that the Max-Flow-Min-Cut property of clutters can be verified with a finite process (see [BT82, Corollary 2.3] for a related result).

2. Notation

In this section we set up the notation used throughout the entire manuscript. We assume is a standard graded polynomial ring over the field and . The ideal is assumed to be a monomial ideal.

For a fixed we set . We denote by the ideal of generated by .

We note that and are isomorphic as rings. Then, the category of -modules is naturally equivalent to the category of -modules. For an -module, , we denote by the corresponding -module. Given that , obtains a structure of -module via restriction of scalars. In addition, corresponds to under this isomorphism. We consider the analogous notation for . We often refer to the containment and .

For a vector we denote by the monomial .

Definition 2.1.

Given , we denote by the th-symbolic power of :

We now consider algebras associated to ordinary and symbolic powers of ideals.

Definition 2.2.

We consider the following graded algebras.

  1. The Rees algebra of : .

  2. The associated graded algebra of : .

  3. The the symbolic Rees algebra of : .

  4. The symbolic associated graded algebra of : .

3. Castelnuovo-Mumford regularity and -invaratians

In this section we study the graded structure of the symbolic powers. In particular, we prove Theorem A. The techniques here are inspired by methods in prime characteristic used to bound -invariants of -pure graded rings [HR76, DSNnB18]. In particular, the results proved in this section are motivated by the fact that the symbolic Rees and associated graded algebra are -pure for every prime (see Corollary 4.4).

If is a graded -module, we denote by

the -th -invariant of for . The Castelnuovo-Mumford regularity of is defined as

We now introduce the notation necessary to formalize the idea of -roots.

Remark 3.1.

Let be a square-free monomial and its minimal primes. We have

We now define a splitting from to which is inspired by the trace map in prime characteristic. In fact, these maps are the same if and .

Definition 3.2.

For , we define the -homomorphism induced by

Where . We note that restricted to is the identity. Then, is isomorphic to a direct summand of . We also consider the analogous map . We now show that our splitting is compatible with symbolic powers.

Lemma 3.3.

Let be a square-free monomial ideal. Then,

for every and .

Proof.

We first prove our claim when is a prime monomial ideal for some . In this case, we have that is generated as a -vector space by

Let such that , then and . Set , since , we have and hence as desired. Now, let be an arbitrary square-free monomial ideal and let be its minimal primes, then Therefore,

hence the result follows. ∎

As a consequence of the previous result we obtain the following relations on depths and -invariants of symbolic powers; these relations are key ingredients in the proof of Theorem A.

Theorem 3.4.

Let be a square-free monomial ideal and . Then

  1. .

  2. for every .

Proof.

By Lemma 3.3 the natural map splits for every and with the splitting as in Lemma 3.3. Therefore, the module is a direct summand of for every . We note that

(3.1)

We conclude that implies that , and hence . Therefore,

which proves the first part.

From Equation (3.1), we have

and the second part follows. ∎

As a consequence of Theorem 3.4 we recover the following limits for -invariants of symbolic powers.

Theorem 3.5 ([Ht10, Theorem 4.7]).

Let be a square-free monomial ideal. Then,

exists for every .

Proof.

Fix . The sequence has an upper bound [HHT07, 3.3], then so does . Set for every and .

If , we have that for every and the claim follows. We now assume that and show that We note that Theorem 3.4(2) implies

(3.2)

Fix and let such that It suffices to show for every as this implies that . Let for some and , then . Since , we obtain and then, and so Applying Inequality (3.2) with we have , which finishes the proof. ∎

As a corollary we obtain that the related limit for the Castelnuovo-Mumford regularity of symbolic powers exists.

Corollary 3.6 ([Ht10, Theorem 4.9]).

Let be a square-free monomial ideal. Then,

Proof.

The result follows from Theorem 3.5 and the following equalities

Remark 3.7.

If is a square-free monomial ideal, then

Let denote the smallest degree of a nonzero element of . The Waldschmidt constant is defined by . Then,

We recall that is a Noetherian algebra [HHT07, 3.2]; therefore, agrees with a linear quasi-polynomial for . As a consequence of Theorem 3.4 we obtain that the leading coefficient of this quasi-polynomial is constant for square-free monomial ideals and an bound for .

Corollary 3.8.

[HT10, Theorem 4.9] Let be a square-free monomial ideal. Then, is equal to a constant for and

Proof.

From Corollary 3.6 it follows that must be equal to for . For the second claim, we observe that

for . We conclude that for every . ∎

The previous results, together with results for matroids [MT17] and low dimension [HT16], motivated Minh and Trung [MT17] to ask the following question.

Question 3.9 ([Mt17]).

Let be a square-free monomial ideal. Is a linear polynomial for ?

It is a classical result that for any homogeneous ideal , is a linear function for (see [CHT99, Kod00]). In general, not much is known about the invariant besides the fact that it is non-negative [TW05, 3.3]. Corollary 3.8 provides an upper bound for for a wide family of square-free monomial ideals.

Corollary 3.10.

Let be a square-free monomial ideal such that for every , then . In particular, this holds for bipartite edge ideals.

Proof.

The result follows by the assumption and Corollary 3.8. The case of edge ideals follows because they satisfy for every [SVV94, Theorem 5.9]. ∎

4. Associated Graded Algebras and Equality of Symbolic and Ordinary Powers

In this section, we study the graded algebras defined in Definition 2.2. This is in order to prove Theorem D. Our strategy is the following. We first show that the symbolic Rees and associated graded algebras split from its rings of -roots in Theorem 4.3. Then, in Theorem 4.5, we characterize the equality of symbolic and ordinary powers in terms of this splitting. Finally, we use this characterization to prove Theorem D. We start with introducing rings of -roots for the Rees and associated graded algebras.

Notation 4.1.

We set

and

We consider the ideals

and

A classical result states that is reduced if and only if for every [HHTZ08, Corollary 1.6]. As a consequence of this result, and its proof, one has that is a radical ideal. We set,

Remark 4.2.
  1. By Remark 3.1 we have the inclusion

  2. Since is monomial, we have that . Then,

Theorem 4.3.

Let be a square-free monomial ideal. Then, the maps induced by the observation in Remark 3.1

split for every .

Proof.

Fix and let be the splitting in Definition 3.2. We define

to be the homogeneous morphism of -modules induced by if divides , and otherwise. The map is well-defined because

for every by Lemma 3.3, and it is -linear since is -linear. If , then because is a splitting. We conclude that is also a splitting.

Consider the ideal as in Notation 4.1. By Lemma 3.3 we obtain

for every . Therefore, , i.e., is compatible with . This induces a splitting . The conclusion follows. ∎

Theorem 4.3 has the following consequence if the field has positive characteristic.

Corollary 4.4.

Let be a square-free monomial ideal. If is a perfect field of prime characteristic , then and are -pure.

Proof.

We note that the rings involved are -finite. Then, they are -pure if and only if they are -split [HR74, Corollary 5.3]. Since is perfect, we have that and correspond to the rings of -roots of and respectively. In addition, is the ideal of that corresponds to . Since is radical, is a reduced ring. Then, corresponds to the ring of -roots of . Then, the result follows from Theorem 4.3 with . ∎

The following Theorem provides necessary and sufficient conditions for the equality of ordinary and symbolic powers of square-free monomial ideals.

Theorem 4.5.

Let be a square-free monomial ideal. Then, the following are equivalent.

  1. for every .

  2. for some and every .

  3. for every

Proof.

Clearly implies . We now assume and prove . We first prove that is a radical ideal. Let be a monomial such that then there exists such that . We observe that . We note that the assumption in implies . Therefore,

A decreasing induction on shows . Then, is reduced. As a consequence, for every [HHTZ08, Corollary 1.6].

Now, we assume that . By Lemma 3.3, we have for every , therefore follows. ∎

We now state an open problem given by Hà at the BIRS-CMO workshop on Ordinary and Symbolic Powers of Ideals during the summer of 2017 at Casa Matemática Oaxaca.

Problem 4.6 (Hà).

Let be a square-free monomial ideal. Find a number , in terms of , such that for every implies for every

This problem strongly related to Conforti-Cornuéjols (Conjecture C). In fact, if satisfies the conclusion of Problem 4.6, then Conjecture C follows [DDSG18, Remark 4.19]. Hà also asked for an optimal value for . In Example 4.9, we prove that our bound is sharp..

If one assumes Conjecture C, then would work [HM10, Remark 4.8]. As a consequence of our methods, we solve Problem 4.6 by giving in terms of the number of generators of in Theorem 4.8. For the proof of this result, we need the following well-known lemma. We include its proof for the sake of completeness.

We denote by the minimal number of generators of . If is generated by the monomials , we denote by the ideal generated by .

Lemma 4.7.

Let be a monomial ideal. If , then .

Proof.

Let and a minimal set of generators of Let be natural numbers such that then by assumption there must exist such that Therefore,

This shows that . To obtain the other containment, we observe that . ∎

Theorem 4.8.

Let be a square-free monomial ideal. If for every , then for every .

Proof.

We show that

for every and then the result follows by Theorem 4.5, . By assumption and Lemma 3.3 this inclusion holds for , as for these values . We fix . Then,

and hence by Lemma 4.7. Therefore, by induction

finishing the proof. ∎

We point out that Theorem 4.8 relates to an open problem stated by Francisco, Hà, and Mermin [FHM13, Problem 5.14(a)]. The following example shows that the number given n Theorem 4.8 is sharp.

Example 4.9.

Let for some , and let

Then for every , whereas [LTar, Corollary 4.5].

Corollary 4.10.

Let be a square-free monomial ideal. Then, for every if and only if for .

Proof.

Let . Since and are regular of the same dimension, we have that as -modules [Fed83, Lemma 1.6 (1)]. Furthermore, standard computations show that the map induced by

where , is a generator of as -module (this is a standard computation in prime characteristic when ). We now focus on the case . We stress that we are not making any assumption on the characteristic of the field. We note that if and only if

because is isomorphic to and is monomial. In addition, is equivalent to for every [Fed83, Lemma 1.6 (2)]. Since , the result follows by Theorem 4.5. ∎

We note that Theorem 4.8 and Corollary 4.10 give an algorithm to verify Conjecture C for a specific ideal.

5. Applications to linear optimization

In this brief section we translate our result to the context of linear programming. For more on this topic, we refer to [HT18].

A clutter is a collection of subsets of such that every two elements of are incomparable with respect to inclusion. We denote by the matrix with entries equal to 0 or 1, such that its columns are the incidence vectors of the sets in . Given , by the Strong Duality Theorem we have the following equality of dual linear programs.

(5.1)

where . We say that packs for if Equation (5.1) has optimal solutions and . The clutter satisfies the Max-Flow-Min-Cut (MFMC) property if it packs for every .

We now recall a lemma that allows us to rephrase Theorem 4.8 in this context.

Lemma 5.1 ([Fhm13, Lemma 5.9]).

Set

Then,

  1. if and only if .

  2. if and only if .

We are now ready to present the main theorem of this section which is related to previous results in integer programming [BT82, Corollary 2.3].

Theorem 5.2.

The clutter packs for every if and only if satisfies the MFMC property.

Proof.

Let denote the column vectors of . We denote by the ideal of the polynomial ring generated by the monomials .

From the definitions, the MFMC property implies the packing property. We focus on the other direction. We assume packs for every . Let be a minimal monomial generator of for some , then it is clear that it divides . Moreover, by Lemma 5.1, we have . This is because reducing a component of by 1, reduces the optimal solution by at most 1. Therefore, by assumption we have and hence by Lemma 5.1. We conclude that for . By Theorem 4.8 it follows that for every . Now, let arbitrary. Then . We have by Lemma 5.1. ∎

Acknowledgments

The authors started this project after participating in the BIRS-CMO workshop on Ordinary and Symbolic Powers of Ideals Summer of 2017, Casa Matemática Oaxaca, Mexico, where they learned new open problems on the subject. The authors thank the organizers: Chris Francisco, Tai Hà, and Adam Van Tuyl. They also thank Craig Huneke for suggestions that improved Theorem 4.8 and Rafael H. Villarreal for suggesting Remark 3.7. The authors would like to thank Hailong Dao and Tai Hà for suggestions on an earlier draft. Part of this project was completed in the Mathematisches Forschungsinstitut Oberwolfach (MFO) while the first author was in residence at the institute under the program Oberwolfach Leibniz Fellows. The first author thanks MFO for their hospitality and excellent conditions for conducting research. The second author thanks Jack Jeffries for inspiring conversations.

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