NEW CLASSES OF SET-THEORETIC COMPLETE INTERSECTION MONOMIAL IDEALS

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

Abstract. Let Δ be a simplicial complex and χ be an s-coloring of Δ. Biermann and Van Tuyl have introduced the simplicial complex Δ_{χ}. As a corollary of Theorems 5 and 7 in their 2013 article, we obtain that the Stanley–Reisner ring of Δ_{χ} over a field is Cohen–Macaulay. In this note, we generalize this corollary by proving that the Stanley–Reisner ideal of Δ_{χ} over a field is set-theoretic complete intersection. This also generalizes a result of Macchia.

1. Statement of the main theorem

Let us start this note with some notions of combinatorics. A simplicial complex Δ on the set of vertices V = {v_{1}, …, v_{n}} is a collection of subsets of V which is closed under taking subsets; that is, if F ∈ Δ and F′ ⊆ F, then also F′ ∈ Δ. Every element F ∈ Δ is called a face of Δ, and a facet of Δ is a maximal face of Δ with respect to inclusion. It is clear that all facets of Δ determines it. When F_{1}, …, F_{t} are all facets of Δ, we write Δ = ⟨F_{1}, …, F_{t}⟩. In this case, we say χ is an s-coloring of Δ when χ is a partition of V, say V = W_{1} ∪ ⋯ ∪ W_{s} (where the sets W_{j} are allowed to be empty), such that for every 1 ≤ i ≤ t and every 1 ≤ j ≤ s the inequality |F_{i} ∩W_{j}| ≤ 1 holds true. Biermann and Van Tuyl [3] have defined a new simplicial complex Δ_{χ} on the set of vertices {v_{1}, …, v_{n}, w_{1}, …, w_{s}} with faces σ ∪ τ, where σ is a face of Δ and τ is any subset of {w_{1}, …, w_{s}} such that for every w_{j} ∈ τ, we have σ ∩ W_{j} = ∅. It is shown in [3] that the facets of Δ_{χ} are in the form F ∪ F′, where F is any face of Δ and F′ = {w_{i} | F ∩ W_{i} = ∅}. The simplicial complex Δ_{χ} is generally larger than Δ. For example, if Δ = ⟨{v_{1}, v_{2}, v_{3}}, {v_{3}, v_{4}}, {v_{4}, v_{5}, v_{6}}⟩ is the simplicial complex on the set of vertices V = {v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6}} and χ is the 3-coloring of Δ given by V = W_{1} ∪ W_{2} ∪ W_{3}, where W_{1} = {v_{1}, v_{4}}, W_{2} = {v_{2}, v_{5}}, and W_{3} = {v_{3}, v_{6}}, then we have

2000 Mathematics Subject Classification. 13F55, 13A15.

Key words and phrases. Simplicial complex, s-Coloring, Arithmetical rank, Set-theoretic complete intersection ideal, Lyubeznik resolution.

The research of M. R. Pournaki, S. A. Seyed Fakhari, and S. Yassemi was in part supported by a grant from IPM (No. 93130115, No. 93130422, and No. 93130214).

Δ_{χ} = ⟨{w_{1}, w_{2}, w_{3}}, {v_{1}, w_{2}, w_{3}}, {v_{2}, w_{1}, w_{3}}, {v_{3}, w_{1}, w_{2}},
{v_{4}, w_{2}, w_{3}}, {v_{5}, w_{1}, w_{3}}, {v_{6}, w_{1}, w_{2}}, {v_{1}, v_{2}, w_{3}},
{v_{1}, v_{3}, w_{2}}, {v_{2}, v_{3}, w_{1}}, {v_{3}, v_{4}, w_{2}}, {v_{4}, v_{5}, w_{3}},
{v_{4}, v_{6}, w_{2}}, {v_{5}, v_{6}, w_{1}}, {v_{1}, v_{2}, v_{3}}, {v_{4}, v_{5}, v_{6}}⟩.

Although many properties of the simplicial complex Δ_{χ} are known, it might be complicated in some sense comparing with Δ.

We now switch to the connection of the above-mentioned notions with commutative algebra. Let Δ be a simplicial complex on the set of vertices V = {v_{1}, …, v_{n}}. The Stanley–Reisner ideal of Δ over a field 𝕂 is the ideal I_{Δ} of R = 𝕂[x_{1}, …, x_{n}], the polynomial ring in n variables over 𝕂, which is defined as follows:

I_{Δ} = ⟨ ∏_{v_{i}∈F} x_{i} | F ∉ Δ⟩.

Also, the Stanley–Reisner ring of Δ over 𝕂 is defined as R/I_{Δ}. One may ask about the arithmetical rank of the Stanley–Reisner ideal of a simplicial complex and whether it being set-theoretic complete intersection. We recall that for a given unitary commutative Noetherian ring R and for an ideal I of R, the arithmetical rank of I is denoted by ara(I) and is defined as the smallest integer t for which there exist a_{1}, …, a_{t} ∈ R such that \sqrt{⟨a_{1}, …, a_{t}⟩} = \sqrt{I}. The inequality ht(I) ≤ ara(I) holds true generally and in the case of equality, I is called set-theoretic complete intersection. The arithmetical rank of monomial ideals of R = 𝕂[x_{1}, …, x_{n}] has been studied by several authors (see, for example, [1, 2, 6, 8, 9]).

We are now in the position to state the main theorem of this note. In the next section, we give a proof of the main theorem.

Main Theorem. Let Δ be a simplicial complex and χ be an s-coloring of Δ. Then the Stanley–Reisner ideal of Δ_{χ} over a field is set-theoretic complete intersection.

Before closing this section, we would like to point out that our main theorem generalizes two results of [3] and [9]. Let us explain some more things in detail to see this.

The main theorem as a generalization of a result of [3]. Let R = 𝕂[x_{1}, …, x_{n}] and I be a squarefree monomial ideal of R. Lyubeznik [8] has proven that pd_{R}(R/I) ≤ ara(I), where pd_{R}(R/I) denotes the projective dimension of R/I. Thus

ht(I) ≤ pd_{R}(R/I) ≤ ara(I).

This implies that, if I is set-theoretic complete intersection, then R/I is Cohen– Macaulay. Combining the above observation with our main theorem gives the following corollary, which is also a corollary of Theorems 5 and 7 in the article [3] of Biermann and Van Tuyl.

Corollary I ([3], Theorems 5 and 7). Let Δ be a simplicial complex and χ be an scoloring of Δ. Then the Stanley–Reisner ring of Δ_{χ} over a field is Cohen–Macaulay.

The main theorem as a generalization of a result of [9]. Let G be a finite simple graph with the set of vertices V (G) = {v_{1}, …, v_{n}}. To the graph G, one can associate an ideal I(G) ⊆ 𝕂[x_{1}, …, x_{n}] generated by all monomials x_{i}x_{j} such that v_{i} and v_{j} are adjacent. The ideal I(G) is called the edge ideal of G over 𝕂. We recall that A ⊆ V (G) is an independent set in G if none of its elements are adjacent. Based on this notion, the independence simplicial complex of G is defined by

Δ_{G} = {A ⊆ V (G) | A is an independent set in G}.

Note that Δ_{G} is precisely the simplicial complex with the Stanley–Reisner ideal I(G). Cook and Nagel [4] have defined the fully clique-whiskered graphs in the following way. For a given graph G, C ⊆ V (G) is called a clique of G if every pair of vertices of C are adjacent in G. Let χ be a partition of V (G), say V (G) = W_{1} ∪ ⋯ ∪ W_{s}, such that W_{i} is a clique of G for every 1 ≤ i ≤ s. Add new vertices y_{1}, …, y_{s} and new edges vy_{i} for every v ∈ W_{i} and every 1 ≤ i ≤ s. The resulting graph is called a fully cliquewhiskered graph of G. One can easily see that the independence simplicial complex of this latter graph is Δ_{χ} and so our main theorem gives the following corollary, which is Proposition 4.1 in the article [9] of Macchia.

Corollary II ([9], Proposition 4.1). Let G be a finite simple graph. Then the edge ideals of fully clique-whiskered graphs of G over a field are all set-theoretic complete intersections.

2. Proof of the main theorem

We are now ready to prove the main theorem. In order to do this, suppose that the set of vertices of Δ is V = {v_{1}, …, v_{n}} and V = W_{1} ∪ ⋯ ∪ W_{s} is the s-coloring of Δ given by χ. Also, let R = 𝕂[x_{1}, …, x_{n}] be the polynomial ring in n variables over a field 𝕂. Suppose that {v_{1}, …, v_{n}, w_{1}, …, w_{s}} is the set of vertices of Δ_{χ}, and consider the Stanley–Reisner ideal I_{Δ_{χ}} of Δ_{χ} over 𝕂 as a monomial ideal in the polynomial ring R′ = 𝕂[x_{1}, …, x_{n}, y_{1}, …, y_{s}].

Lemma A. We have ht(I_{Δ_{χ}}) = n.

Proof. One can easily see that

I_{Δ_{χ}} = I_{Δ} + ⟨x_{i}y_{j} | v_{i} ∈ W_{j}, 1 ≤ j ≤ s⟩.

Note that the ideal ⟨x_{1}, …, x_{n}⟩ is a minimal prime ideal of I_{Δ_{χ}}. Now [3, Theorem 5] implies that ht(I_{Δ_{χ}}) = n, as required. □

Let us introduce the Lyubeznik resolution, which is defined in [7]. For every monomial ideal I of a polynomial ring R, there is an explicit free resolution of R/I, which is defined as follows. Assume that G(I) = {m_{1}, …, m_{p}} is the minimal set of monomial generators of I. For every integer t ≥ 1 and every sequence 1 ≤ ℓ_{1} < ⋯ < ℓ_{t} ≤ p, consider the symbol e_{ℓ_{1}···ℓ_{t}}. Let T_{t} be the free R-module generated by all symbols e_{ℓ_{1}···ℓ_{t}}, 1 ≤ ℓ_{1} < ⋯ < ℓ_{t} ≤ p. The differential map d_{t} is also defined by

d_{t}(e_{ℓ_{1}···ℓ_{t}}) = ∑_{k=1}^{t}(−1)^{k−1} \frac{lcm(m_{ℓ_{1}}, …, m_{ℓ_{t}})}{lcm(m_{ℓ_{1}}, …, m̂_{ℓ_{k}}, …, m_{ℓ_{t}})}e_{ℓ_{1}...ℓ̂_{k}...ℓ_{t}}.

Then the complex (T_{•}, d_{•}) is a free resolution of R/I and is called the Taylor resolution of I (see, for example, [5, Theorem 7.1.1]). Now fix an order m_{1} > ⋯ > m_{p} on the set of minimal monomial generators of I. The symbol e_{ℓ_{1}···ℓ_{t}} is called Ladmissible if m_{q} does not divide lcm(m_{ℓ_{k}}, …, m_{ℓ_{t}}) for every 1 ≤ k < t and every 1 ≤ q < ℓ_{k}. Then the Lyubeznik resolution of I with respect to the above order of the minimal monomial generators is a subcomplex of the Taylor resolution generated by all L-admissible symbols. Since being L-admissible depends on an order of minimal monomial generators of I, a Lyubeznik resolution also depends on it. Finally, we say that an L-admissible symbol e_{ℓ_{1}···ℓ_{t}} is maximal if e_{k_{1}···k_{t′}} is not L-admissible whenever {ℓ_{1}, …, ℓ_{t}} ⊊ {k_{1}, …, k_{t′}}.

Lemma B. There is a Lyubeznik resolution of I_{Δ_{χ}} with length ≤ n.

Proof. Let > be the pure lexicographic order induced by x_{1} > ⋯ > x_{n} > y_{1} > ⋯ > y_{s} (see [5, Example 2.1.2(c)]), and consider the Lyubeznik resolution of I_{Δ_{χ}} induced by >. We show that this specific Lyubeznik resolution of I_{Δ_{χ}} has length ≤ n. In order to do this, let m_{1} > ⋯ > m_{p} be the minimal monomial generators of I_{Δ_{χ}} and e_{ℓ_{1}···ℓ_{k}} be a maximal L-admissible symbol. We use induction on n to show that k ≤ n. This completes the proof of the lemma.

One can easily check that k ≤ n is true for n = 1, 2. Thus we may assume that n ≥ 3. Note that there exists a unique integer, say i_{0}, with 1 ≤ i_{0} ≤ s, such that v_{1} ∈ W_{i_{0}}. We claim that there exists an integer 1 ≤ i ≤ n such that x_{i} divides m_{ℓ_{1}} but does not divide lcm(m_{ℓ_{2}}, …, m_{ℓ_{k}}). In order to prove the claim, since e_{ℓ_{1}...ℓ_{k}} is a maximal L-admissible symbol, by using [6, Lemma 3], it follows that ℓ_{1} = 1. This implies that x_{1} divides m_{ℓ_{1}}. Now, by the definition of L-admissible symbol, we conclude that there is a variable which divides m_{ℓ_{1}} but does not divide lcm(m_{ℓ_{2}}, …, m_{ℓ_{k}}). If all of the variables which divide m_{ℓ_{1}} belong to the set {x_{1}, …, x_{n}}, then we are done. Otherwise, we have m_{ℓ_{1}} = x_{1}y_{i_{0}} since x_{1} | m_{ℓ_{1}} and v_{1} ∈ W_{i_{0}}. By the choice of >, we conclude that x_{1} does not divide lcm(m_{ℓ_{2}}, …, m_{ℓ_{k}}), and so the claim holds true. By the claim, we may assume that x_{i} divides m_{ℓ_{1}} but does not divide lcm(m_{ℓ_{2}}, …, m_{ℓ_{k}}). Let

Δ′ = Δ \ {v_{i}} = {F ∈ Δ | v_{i} ∉ F }

be a simplicial complex on the set of vertices V′ = V \ {v_{i}} and χ′ be an s-coloring of Δ′ given by V′ = W′_{1} ∪ ⋯ ∪ W′_{s}, where W′_{j} = W_{j} \ {v_{i}} for every 1 ≤ j ≤ s. It is clear that m_{ℓ_{2}}, …, m_{ℓ_{k}} belong to the set of minimal monomial generators of I_{Δ′_{χ′}} and they determine an L-admissible symbol of I_{Δ′_{χ′}}. Since Δ′ has n − 1 vertices, our induction hypothesis implies that k − 1 ≤ n − 1 and thus k ≤ n. □

We also need to use the following known result of Kimura.

Lemma C ([6], Theorem 1). Let I be a monomial ideal of a polynomial ring and λ be the minimum length of the Lyubeznik resolutions of I. Then ara(I) ≤ λ.

Now, by combining Lemmas B and C, we get ara(I_{Δ_{χ}}) ≤ n. Therefore, Lemma A implies that

n = ht(I_{Δ_{χ}}) ≤ pd_{R}(R/I_{Δ_{χ}}) ≤ ara(I_{Δ_{χ}}) ≤ n,

and so ht(I_{Δ_{χ}}) = ara(I_{Δ_{χ}}) = n. This means that I_{Δ_{χ}} is a set-theoretic complete intersection ideal of R′ and completes the proof of the main theorem. □

Acknowledgment

The authors would like to thank the referee for a careful reading of this note and for valuable comments.

References

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M. R. Pournaki, Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.

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

S. A. Seyed Fakhari, School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.

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

S. Yassemi, School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.

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