Arithmetical rank of squarefree monomial ideals of small arithmetic degree

DOI 10.1007/s10801-008-0142-3

Arithmetical rank of squarefree monomial ideals of small arithmetic degree

Kyouko Kimura·Naoki Terai·Ken-ichi Yoshida

Received: 30 May 2007 / Accepted: 9 May 2008 / Published online: 29 May 2008

© Springer Science+Business Media, LLC 2008

Abstract In this paper, we prove that the arithmetical rank of a squarefree monomial idealIis equal to the projective dimension ofR/I in the following cases: (a)I is an almost complete intersection; (b) arithdegI =regI; (c) arithdegI=indegI+1.

We also classify all almost complete intersection squarefree monomial ideals in terms of hypergraphs, and use this classification in the proof in case (c).

Keywords Arithmetical rank·Almost complete intersection·Alexander duality· Regularity·Arithmetic degree·Initial degree

1 Introduction

Throughout this paper, letR=k[x₁, . . . , x_n]be a polynomial ring over a fieldkwith the unique homogeneous maximal idealm=(x1, . . . , xn)R, and letI be a homoge- neous ideal ofR, unless otherwise specified. Then the arithmetical rank ofI, denoted by araI, is defined as follows:

araI:=min

r∈N:there exista1, . . . , ar∈I such that

(a1, . . . , ar)=√ I

.

K. Kimura·K. Yoshida (

Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan e-mail:[email protected]

K. Kimura

e-mail:[email protected]

N. Terai

Department of Mathematics, Faculty of Culture and Education, Saga University, Saga 840-8502, Japan

e-mail:[email protected]

This paper deals with the problem of computing the arithmetical rank of a monomial ideal (that is, the minimal number of equations needed to define the variety associated to a monomial ideal).

A trivial upper bound on araI is the minimal number of generators ofI, denoted byμ(I ). On the other hand, it is well known that heightI gives a lower bound for araI. An ideal I satisfying araI =heightI is said to be a set-theoretic complete intersection. LetH_Iⁱ(R)denote theith local cohomology module ofRwith support atV (I ). Then the cohomological dimension ofI is defined by cd(I )=max{i∈Z : H_Iⁱ(R)=0}. From the expression of the local cohomology modules in terms of ˇCech complex, one can easily see that cd(I )≤araI.

Now assume that I is a squarefree monomial ideal of R. Then Lyubeznik [9]

showed that cd(I )=pd_RR/I, the projective dimension ofR/I. We also note that heightI ≤pd_RR/I always holds, and that equality holds if and only if R/I is Cohen–Macaulay. Combining all inequalities stated above, we have

heightI≤pd_RR/I=cd(I )≤araI≤μ(I ). (1.1) In particular, ifI is a set-theoretic complete intersection, then R/I is Cohen–

Macaulay. So, we consider the following fundamental question:

Question LetI be a squarefree monomial ideal ofR. When does araI=pd_RR/I hold? In particular, suppose thatR/I is Cohen–Macaulay. When isIa set-theoretic complete intersection?

Barile proved the equality for certain classes of squarefree monomial ideals in [1–6]. We remark that it does not always hold as was shown by Yan [15]. He showed that araI=4 for the squarefree monomial idealI generated by monomials

x1x2x3, x1x2x4, x1x3x5, x1x4x6, x1x5x6, x2x3x6, x2x4x5, x2x5x6, x3x4x5, x3x4x6, which is the Stanley–Reisner ideal of the triangulation ofP²(R)with six vertices.

However, when chark=2,R/I is Cohen–Macaulay and pd_RR/I =heightI=3<

4=araI. In particular,I is not a set-theoretic complete intersection. In this example, the deviationd(I )=μ(I )−heightI=7 is rather big. So, in this paper, we focus our attention on ideals with “small deviation” (e.g., almost complete intersection ideals) and on the Alexander dual of such ideals.

Before stating our results, we recall several definitions. Let M be an arbitrary noetherian gradedR-module, and let

0→

j∈Z

R(−j )^βtj(M)→ · · · →

j∈Z

R(−j )^β1j(M)→

j∈Z

R(−j )^β0,j(M)→M→0 be a graded minimal free resolution ofM overR, whereR(−j ) is a graded free R-module whosenth graded piece is given byRn−j, andt=pd_RM, the projective dimension ofMover R. The regularity and the initial degree ofM are defined as follows:

regM=max{j−i∈Z : β_ij(M)=0};

indegM=min{j∈Z : β_0j(M)=0},

i.e., indegM is equal to the minimal degree of the generators of M. Note that indegM≤regM.

For a squarefree monomial ideal I of R, the arithmetic degree, denoted by arithdegI, is coincident with the number of prime components of I. It is known that regI≤arithdegI. See [7,8].

Schenzel–Vogel [11] and Schmitt–Vogel [12] showed that araI=pd_RR/I for the squarefree monomial idealI with indegI=arithdegI. One of the motivation for our study is to generalize this result.

Theorem 1.1 (See also Theorems2.1,5.1and6.1) LetRbe a polynomial ring over a fieldk, and letI be a squarefree monomial ideal ofR which satisfies one of the following conditions:

(1) μ(I )≤pd_RR/I+1(e.g.,I is an almost complete intersection).

(2) arithdegI=regI. (3) arithdegI=indegI+1.

Then we have that araI=pd_RR/I.

Let us explain the organization of this paper. In Section2, we consider the question in the case of almost complete intersection ideals (i.e.,μ(I )=heightI +1); see Theorem2.1.

In Section 3, we introduce the notion of hypergraphs associated to squarefree monomial ideals. In the next section, we classify almost complete intersection square- free monomial ideals in terms of hypergraphs; see Theorem4.4. As an application, we compute some invariants (the regularity, analytic spread etc.) for such ideals.

In Section5, we consider the question in the case of arithdegI =regI (those ideals satisfying this condition are obtained as the Alexander dual ideals of square- free monomial ideals withμ(I )=pd_RR/I); see Theorem5.1. The main tool in our argument is the Schmitt–Vogel method in [12].

Finally in Section6, we consider the question in the Alexander dual case of al- most complete intersection squarefree monomial ideals; see Theorem6.1. We use Theorems4.4,5.1in the proof of Theorem6.1.

2 Arithmetical rank of almost complete intersection squarefree monomial ideals

A homogeneous idealIof a polynomial ringRis said to be an almost complete inter- section (resp. a complete intersection) ifμ(I )=heightI+1 (resp.μ(I )=heightI).

LetI be a squarefree monomial ideal ofR. Then

heightI≤pd_RR/I ≤araI≤μ(I ) (2.1) holds as stated in the introduction. In particular, ifI is a complete intersection, then araI=heightI, and so there is nothing to do any more. On the other hand, ifI is an

almost complete intersection, then we have

0≤μ(I )−pd_RR/I≤1.

The purpose of this section is to determine the arithmetical rank in this situation.

Before stating our result, let us recall the definition of a Taylor resolution. Let I =(m₁, . . . , m_μ)be a monomial ideal with the minimal set of monomial genera- torsG(I )= {m₁, . . . , m_μ}. Then the Taylor resolutionF_•ofI is a finite graded free complex of the following shape:

F_•: 0−→F_μ−→^dμ F_μ−1−→ · · · −→F₁−→^d1 F₀−→R/I −→0, where

F_p=

1≤1<···<p≤μ

R e_1···p,

d_p(e_1···p)= p

i=1

(−1)ⁱ lcm(m1, . . . , m_p) lcm(m1, . . . ,m_i, . . . , m_p)e

1···i···p.

That is, the free basis ofF_p is{e_1···p}with dege_1···p=deg lcm(m1, . . . , m_p). It is known thatF_•is not necessarily the minimal graded free resolution ofR/I. This implies that pd_RR/I≤μ(I ), and that ifF_•is minimal, the equality holds. Note that the converse is also true.

Theorem 2.1 (See also [2, Corollary 1]) IfI is a squarefree monomial ideal of R withμ(I )≤pd_RR/I+1, then we have araI =pd_RR/I. In particular, ifI is an almost complete intersection, then the same formula holds.

In order to prove the theorem, we need the following lemma.

Lemma 2.2 LetI be a squarefree monomial ideal ofR. Then pd_RR/I≤μ(I )−1 if and only if araI≤μ(I )−1. In other words, pd_RR/I=μ(I )if and only if araI= μ(I ).

Proof It is enough to show that araI ≤μ(I ) −1 holds whenever pd_RR/I ≤ μ(I )−1. To do that, letG(I )= {m₁, . . . , m_μ}be the minimal set of monomial gen- erators ofI, whereμ=μ(I ). Now suppose that pd_RR/I≤μ−1. Then since the Taylor resolution ofIis not minimal, we may assume thatm1m2· · ·m_μ−1is divisible bym_μ. For eachi=1, . . . , μ−1, lets_i be theith elementary symmetric polynomial inm1, m2, . . . , mμ−1. Since everymj (j=1, . . . , μ−1)is a root of the polynomial

(X−m₁)(X−m₂)· · ·(X−m_μ−1)=X^μ−1−s₁X^μ−2+ · · · +(−1)^μ−1s_μ−1, we get

m^μ_j⁻¹=s₁m^μ_j⁻²− · · · +(−1)^μs_μ−1∈(s₁, . . . , s_μ−2, s_μ−1).

Hencem^μ_j⁻¹∈(s1,· · ·, sμ−2, mμ)becausemμ|sμ−1=m1· · ·mμ−1. It follows that I=

(s₁, . . . , s_μ−2, m_μ), and thus araI≤μ−1, as required.

Proof of Theorem 2.1 If pd_RR/I =μ(I ), then Lemma 2.2 implies that araI = μ(I )=pd_RR/I. Otherwise, pd_RR/I =μ(I )−1 by assumption. Then Lemma2.2 implies that araI ≤μ(I )−1=pd_RR/I. But the converse is always true. Hence

araI=pd_RR/I.

Example 2.3 Let I =(x1x2x3, x2x4x6, x3x5x6, x2x3x4x5), then μ(I ) =4, height I = 2 and pd_RR/I = 3. In particular, μ(I ) − pd_RR/I =1 holds, but I is not an almost complete intersection. The proof of the lemma above shows that I=√

(s1, s2, x2x3x4x5), where

s1=x1x2x3+x2x4x6+x3x5x6,

s2=x1x₂²x3x4x6+x1x2x₃²x5x6+x2x3x4x5x₆².

3 Hypergraphs

In this section, we introduce the construction of a particular hypergraph for any given squarefree monomial ideal. In the next section, we will classify all almost complete intersection squarefree monomial ideals using this notion. Furthermore, in Section6, we will use this classification in order to determine the arithmetical rank for the Alexander dual ideals of those ideals.

Let us begin with the definition of hypergraphs associated to squarefree monomial ideals. Let[μ]denote the subset{1, . . . , μ}ofN.

Definition 3.1 LetV= [μ]. We callH⊂2^V a hypergraph with the vertex setV if

F∈H

F =V .

LetI be a squarefree monomial ideal, and letG(I )= {m1, m2, . . . , m_μ}denote the minimal set of monomial generators ofI. For such an idealI, we construct a hypergraphH(I )with the vertex setV= {1,2, . . . , μ}as follows:

F ∈H(I )⇐⇒there existsi (1≤i≤n)such that for allj∈V ,

m_j is divisible byx_i ifj∈F

andmj is not divisible byxi ifj∈V\F . That is,

H(I )=

{j∈V :mj is divisible byxi} :1≤i≤n

.

We callH(I )the hypergraph associated to a squarefree monomial idealI.

Notice that the hypergraphH=H(I )satisfies for allj₁, j₂∈V (j₁=j₂),

there existF₁, F₂∈Hsuch thatj₁∈F₁∩(V \F₂), j₂∈F₂∩(V \F₁). (3.1) Conversely, for any hypergraphHonV= [μ]satisfying condition(3.1), there ex- ists a squarefree monomial idealI in a polynomial ring with enough variables such thatH=H(I ). For example, if we putI_H:=(

FjxF :1≤j ≤μ)in a polyno- mial ringk[xF :F ∈H], thenH=H(I_H)holds. Note that such a choice ofI is not unique. In fact, one can obtain the same hypergraph as the original one if one replaces each variable by a squarefree monomial no two of which have common factors. For example, let I1=(x1x5, x2x5, x3x6, x4x6) and I2=(x1x5, x2x5, x3x6x7, x4x6x7).

Then we have

H(I₁)=H(I₂)= {{1},{2},{3},{4},{1,2},{3,4}}. Definition 3.2 A subsetCofHis said to be a cover ofHif

F∈C

F=V .

In particular,Cis called a minimal cover ofHif no proper subset ofCis a cover ofH.

Hitself is a cover ofH. Assume thatHsatisfies condition (3.1). ThenHis a minimal cover ofHif and only ifHconsists of isolated points.

Note that the cardinality of the minimal cover is not constant in general: for in- stance, for a hypergraph H= {{1,2},{2,4},{1,4},{1,3},{3}} on V = {1,2,3,4}, C1= {{1,2},{2,4},{3}} andC2= {{2,4},{1,3}} are both minimal covers of H. In general, we have

Proposition 3.3 LetI be a squarefree monomial ideal. Then the following two con- ditions are equivalent:

(1) Ihas a prime component of heighth.

(2) H=H(I )has a minimal cover of cardinalityh.

In particular,

heightI=min{C :Cis a(minimal)cover ofH}.

Proof SetG(I )= {m1, . . . , mμ}to be the minimal set of monomial generators ofI. (1)⇒(2): LetP =(x_i1, . . . , x_ih)be a prime component ofI with heighth. Set F= {j∈V :m_j is divisible byx_i}for 1≤≤h. ThenC= {F₁, . . . , F_h}is a mini- mal cover ofH.

(2)⇒(1): LetC= {Fi :1≤i≤h}be a minimal cover ofH. By definition, we may assume that for eachFi (1≤i≤h),

j ∈F_i⇐⇒m_j is divisible byx_i.

Since_h

=1F=V, we haveI =(m₁, . . . , m_μ)⊂(x₁, . . . , x_h)=P. Thus there is a prime componentPofI such thatP=(x_i1, . . . , x_is)⊂P. By the argument as in the proof of(1)⇒(2),C= {Fi₁, . . . , Fi_s} ⊂Cis a minimal cover ofH. The minimality

ofCimplies thatC=CandP =P.

LetHbe a hypergraph. An element ofHis said to be a face inH. The dimension of a faceF inHis defined by dimF =F −1, and the dimension ofH, denoted by dimH, is defined as the maximal dimension of all faces inH. A face F with dimF =1 is said to be an edge. Two edges are said to be disjoint if they do not intersect.

Proposition 3.4 LetI be a squarefree monomial ideal ofR. Then we have dimH(I )≤μ(I )−heightI.

Proof Putd(I )=μ(I )−heightI. SupposeH(I )has a faceF with dimF > d(I ).

For eachj ∈V \F, we chooseG_j∈Hsuch thatj∈G_j. ThenC= {F} ∪ {G_j:j∈ V \F}is a cover of H. SinceC≤V −F +1< μ(I )−d(I ), this contradicts

Proposition3.3.

Example 3.5 The equality dimH(I )=μ(I )−heightI does not necessarily hold.

For example, if we putI=(x₁x₅, x₂x₅, x₃x₆, x₄x₆), thenμ(I )=4, heightI=2 and H(I )= {{1},{2},{3},{4},{1,2},{3,4}}. In particular, dimH(I )=1<2=μ(I )− heightI.

4 Classification of almost complete intersection squarefree monomial ideals In this section, we classify almost complete intersection squarefree monomial ideals in terms of hypergraphs. Let us begin with studying hypergraphs of those ideals.

Lemma 4.1 Assume thatI is an almost complete intersection. Then: (1) dimH(I )=1.

(2) There are no two disjoint edges inH(I ).

Proof (1) Proposition3.4shows that dimH(I )≤1. Moreover, it is easy to see thatI is a complete intersection if and only if dimH(I )=0. Hence dimH(I )=1.

(2) SupposeH=H(I )has two disjoint edgesF1, F2. For eachj ∈V\(F1∪F2), we chooseGj∈Hsuch thatj∈Gj. ThenC= {F1, F2} ∪ {Gj :j∈V \(F1∪F2)} is a cover of H, and C is at most μ(I )−2. This contradicts Proposition 3.3as

heightI=μ(I )−1.

By the lemma above, the hypergraphH=H(I )associated to any almost complete intersection idealIof heighth≥1 can be represented as a simple graphHequipped with some weight functionw:V −→ {0,1}. In other words,His the simple graph H=(V ,H¹), whereH¹= {F ∈H: dimF =1}with the weight functionw:V → {0,1};w(j )=1 if{j} ∈Handw(j )=0 otherwise.

In this paper, we will describe a vertex of the hypergraph (of dimension one) by the following rule: ifw(j )=1; ifw(j )=0.

Proposition 4.2 Assume thatI is an almost complete intersection squarefree mono- mial ideal withh=heightI ≥2. Then the hypergraphH(I )consists of one of the following one-dimensional hypergraphs with finitely many isolated points.

In the picture below,p,pare integers with 2≤p≤hand 1≤p≤h.

(H1)

...

1 2

p

(H2)

...

1 2

p

(H3)

(H4)

(H5)

(H6)

Remark 4.3 An almost complete intersection squarefree monomial ideal of height 1 is of the form (AB1, AB2), where A,B1,B2 are squarefree monomials no two of which have common factors. This ideal corresponds to(H2)withp=1 in the proposition.

Proof of Proposition4.2 PutH=H(I ). Since dimH=1 by Lemma4.1(1),Hcon- sists of vertices and 1-faces (edges). We may assume thatHdoes not contain any isolated points. Then one can easily see thatHis connected by Lemma4.1(2).

Case 1: The case whereHcontains no cycles.

SinceHis a connected graph without cycles, it is a tree. Moreover,Hdoes not have two disjoint edges, thus it is isomorphic to either(H1)or(H2).

Case 2: The case whereHcontains a cycleC.

If the number of edges ofC(say,m) is bigger than 3, then one can find two disjoint edges. Thus m=3. Since H cannot have edges that do not belong to C, His a triangle as a graph. Then,His isomorphic to one of(H3), . . . , (H6).

Using Proposition4.2, we classify almost complete intersection squarefree mono- mial ideals. We say thatI is isomorphic toJifI is obtained fromJ by renumbering the variables.

Theorem 4.4 LetI be an almost complete intersection squarefree monomial ideal of heightI =h≥2. Then I can be written in one of the following forms, where A₁, A₂, . . . , B₁, B₂, . . . are non-trivial squarefree monomials no two of which have common factors, andp,pare integers with 2≤p≤hand 1≤p≤h.

(1) I₁=(A₁B₁, A₂B₂, . . . , A_pB_p, A_p+1, . . . , A_h, B₁B₂· · ·B_p).

(2) I2=(A1B1, A2B2, . . . , A_pB_p, A_p+1, . . . , A_h, A_h+1B1B2· · ·B_p).

(3) I3=(B1B2, B1B3, B2B3, A4, . . . , Ah+1).

(4) I₄=(A₁B₁B₂, B₁B₃, B₂B₃, A₄, . . . , A_h+1).

(5) I5=(A1B1B2, A2B1B3, B2B3, A4, . . . , A_h+1).

(6) I6=(A1B1B2, A2B1B3, A3B2B3, A4, . . . , Ah+1).

Moreover, R/I is unmixed if and only if I is isomorphic to Ii for some i= 1,3,4,5. When this is the case,R/I is Cohen–Macaulay.

Proof We assign each vertex (resp. edge) inHtoA_i (resp.B_j). We give pictures only for the cases (H2) and (H6).

(H2)

...

A_h+1 A1

A₂

A_p

B₁ B2

B_p

· · · A_p+1 A_h

(H6)

A1

A2 A3

B₁ B₂

B3

· · · A₄ A_h+1

ThenI is isomorphic to one ofIi for 1≤i≤6 by virtue of Proposition4.2.

It is clear thatR/I_i is unmixed if and only ifi=1,3,4 or 5. In I₁, if we put m_h+1=B1· · ·B_pandm_j=A_jB_j for everyj =1, . . . , p, thenm1· · ·m_pis divisible bymh+1. InI3,I4or I5, if we put m1=A1B1B2,m2=A2B2B3andm3=B2B3, thenm₁m₂is divisible bym₃, where we considerA₁=A₂=1 inI₃(resp.A₂=1 in I4). In any case, using the Taylor resolution we obtain that pd_RR/I_i≤μ(I_i)−1=

heightIi, that is,R/Ii is Cohen–Macaulay.

The following corollary gives an answer to the question stated in the introduction in the case of almost complete intersection squarefree monomial ideals. Letr(R/I ) denote the Cohen–Macaulay type ofR/I.

Corollary 4.5 LetI =(m1, . . . , mh+1)be an almost complete intersection square- free monomial ideal withh≥2. Then the following conditions are equivalent:

(1) R/I is Cohen–Macaulay.

(2) R/I is unmixed.

(3) There existsmsuch thatm|m1· · ·m· · ·mh+1. (4) I is a set-theoretic complete intersection.

When this is the case, under the same notation as in Theorem4.4, we have r(R/I₁)=p; r(R/I_i)=2 (i=3,4,5).

Moreover, the regularity is obtained by the following formula:

regI₁= h

i=1

degA_i+ p

j=1

degB_j−min{degA_i : 1≤i≤p} −h+1;

regI_i =

h+1

i=4

degA_i+ 3

j=1

degB_j+max{degA₁,degA₂} −h+1 for eachi=3,4,5, where we considerA1=A2=1 inI3(resp.A2=1 inI4).

Proof We first show that the above four conditions are equivalent. (1)⇒(2)and (2)⇒(3) are clear. By Lemma 2.2, we have (3)⇒(4). (4)⇒(1): from (4), heightI=pd_RR/I. ThusR/I is Cohen–Macaulay.

Secondly, let us determine the Cohen–Macaulay type and the regularity in the case I=I1. We may assumep=h. Setmj=AjBjfor 1≤j ≤handmh+1=B1· · ·Bh. Considering the Taylor resolution ofI1, by [10, Theorem 5.2], we have

regI₁=max{j∈Z: β_h,j(R/I )=0} −h+1

=max{deg lcm(m1, . . . ,m_i, . . . , m_h, m_h+1) : 1≤i≤h} −h+1

= h

i=1

degAi+ h

j=1

degBj−min{degAi :1≤i≤h} −h+1,

as required. Moreover,r(R/I )=

j∈Zβh,j(R/I )=p.

In the caseI=I3,I4orI5, one can also prove the formula by a similar argument

as above. So we omit the proof here.

As an application of our classification, we consider the analytic spread. The an- alytic spread ofI is defined by (I ):=(I_m)=dim

n≥0I_mⁿ/mI_mⁿ, and satisfies araI≤(I )≤μ(I ).

Corollary 4.6 LetI be an almost complete intersection squarefree monomial ideal.

Then it is of linear type, that is,(I )=μ(I ).

Proof SetI=(m₁, . . . , m_h+1). It suffices to show that the kernel of the natural map R[Y1, . . . , Y_h+1] →R[m1t, . . . , m_h+1t]is generated bym_iY_j−m_jY_i, 1≤i < j ≤ h+1. One can easily reduce the proof to the casep=hinI1(resp.h=2 inI3,I4

orI₅). Then it is easy to check it.

From Theorem4.4, we can also classify almost complete intersection (not nec- essarily squarefree) monomial ideals. ForA=x_i^j1

1 · · ·x_i^jm

m (j1, . . . , jm>0), we set

√A:=x_i1· · ·x_im.

Corollary 4.7 LetI be an almost complete intersection monomial ideal. ThenI is one of the following types, whereC1, C2, . . . , D1, D2, . . .are monomials no two of which have common factor:

(1) (D1, . . . , Dp, Cp+1Dp+1, . . . , CqDq, Cq+1, . . . , Ch, Ch+1D1· · ·Dq), where p≥0,q≥1,p≤q,√

Di=

Difor each ofi=1, . . . , qandDiis not divisi- ble byD_i for each ofi=1, . . . , p,C_p+1, . . . , C_h, D₁, . . . , D_p=1. Moreover, if p=0, thenq≥2 orC_h+1=1.

(2) (C₁D₁D₂, C₂D₁D₃, C₃D₂D₃, C₄, . . . , C_h+1), where√ D_i=

D_ifor each of i=1,2,3,C4, . . . , Ch+1, D1, D2, D3=1.

When this is the case,√

I is a complete intersection if and only ifp≥1 in(1).

Proof As h=heightI =height√

I ≤μ(√

I )≤μ(I )=h+1, √

I is a complete intersection or an almost complete intersection.

Case 1: √

I is a complete intersection

By the assumption, the idealI can be written asI =(M₁, . . . , M_h+1), whereM_i are monomials such that√

Mh+1∈(√

M1, . . . ,√

Mh). PutBi =gcd(√ Mi,√

Mh+1).

By renumbering the monomials, we may assume thatp≥1, and thatB_i=1 if and only if 1≤i≤qand

Mi=

Bi if 1≤i≤p; AiBi ifp+1≤i≤q,

whereAi=1 is a squarefree monomial fori=p+1, . . . , q. Thus we can write I=(D₁, . . . , D_p, C_p+1D_p+1, . . . , C_qD_q, C_q+1, . . . , C_h, C_h+1D₁· · ·D_p), whereC_i,D_j (resp.C_i,D_j) are coprime monomials and√

C_i=A_i and

D_j=B_j (resp.

D_j=Bj). Conversely, ifI can be written in the above form, then

√I =(B1, . . . , Bp, Ap+1Bp+1, . . . , AqBq, Aq+1, . . . , Ah)

is a complete intersection.

Case 2: √

I is an almost complete intersection

By the assumption, the idealI can be written asI =(M1, . . . , M_h+1), whereM_i are monomials such that√

I=(√

M₁, . . . ,√ M_h+1).

Suppose that√

I=(A₁B₁, . . . , A_qB_q, A_h+1, . . . , A_h, A_h+1B₁· · ·B_q), whereA_i, Bj are non-trivial monomials fori=1, . . . , h, j =1, . . . , q, andAh+1 is a mono- mial; see Theorem4.4(1),(2). ThenI can be written as:

(C₁D₁, . . . , C_qD_q, C_q+1, . . . , C_h, C_h+1D₁· · ·D_q), where√

Ci =Ai,√

Di=Bi, andq≥2 whenCh+1=1. This is the form described in (1) withp=0.

Next suppose that√

I =(A₁B₁B₂, A₂B₁B₃, A₃B₂B₃, A₄, . . . , A_h+1), that is, it is isomorphic toIi for somei=3,4,5,6. Then one can easily see thatI can be written

as the ideal in (2).

Remark 4.8 One can prove this corollary using polarization (see [13, p. 107, Chap- ter II Section 1]).

5 Arithmetical rank of the case arithdegI=regI

In this section, using Alexander duality, we consider the arithmetical rank of square- free monomial ideals with arithdegI=regI. Before stating our result, we recall the definition and fundamental properties of Alexander duality.

PutV = [n]. For⊆2^V,is called a simplicial complex on the vertex setV if (a){i} ∈for everyi∈V and (b)F ∈,G⊆F impliesG∈. For a simplicial complexonV, the Alexander dual complex ofis defined by^∗:= {F ⊂V : V\F /∈}.

LetI be a squarefree monomial ideal ofR. Then there exists a simplicial com- plex on V such that I =I, where I is the Stanley–Reisner ideal of : I=(xi₁· · ·xi_p : 1≤i1<· · ·< ip≤n, {i1, . . . , ip}∈/ )R. Now suppose that heightI ≥2. Then we set I^∗:=I^∗ and call it the Alexander dual ideal of I. Then it is easy to see that I^∗∗=I. Let I =Q₁∩Q₂∩ · · · ∩Q_q be the irre- dundant primary decomposition of I, and let m be the product of all variables which appear inQ for each=1, . . . , q. ThenI^∗=(m1, m2, . . . , mq). This im- plies that heightI =indegI^∗ andμ(I^∗)=arithdegI. Moreover, it is known that pd_RR/I=regI^∗(see, e.g., [14, Corollary 1.6]). Considering Alexander dual of the relation (2.1), we have

indegI≤regI≤arithdegI. (5.1)

See [7,8].

Schenzel–Vogel [11] and Schmitt–Vogel [12] showed that araI=pd_RR/I for the squarefree monomial idealI with indegI=arithdegI. We generalize it as follows:

Theorem 5.1 LetI be a squarefree monomial ideal. If arithdegI =regI, then we have araI=pd_RR/I.

From now on, we prove this theorem. When heightI =1,I can be written in the form uI0, whereu is a squarefree monomial and heightI0≥2. In order to prove Theorem5.1, we may assume that heightI≥2 by replacingI withI₀. Then we have following:

Lemma 5.2 Any squarefree monomial idealI with arithdegI=regIcan be written (by renumbering the variables)in the form

I=(y₁, x_t11, . . . , x_t1j

1)∩(y₂, x_t21, . . . , x_t2j

2)∩ · · · ∩(y_q, x_tq1, . . . , x_tqjq), wherey,x_tij are variables inRwithy=x_tij,y_i=y_ifori=i, andx_tij =x_tij for j=j.

Proof Since a squarefree monomial ideal has no embedded associated primes, the

assertion follows from [8, Theorem 2.6].

Lemma 5.3 Using the same notation as in Lemma5.2, we have pd_RR/I={x_t11, x_t12, . . . , x_t1j

1, x_t21, x_t22, . . . , x_t2j

2, . . . , x_tq1, x_tq2, . . . , x_tqjq} +1.

Proof Since the Taylor resolution ofI^∗is minimal, we have regR/I^∗=deg lcm

m₁/y₁, . . . , m_q/y_q .

Since pd_RR/I=regI^∗=regR/I^∗+1, we obtain the required assertion.

Proof of Theorem5.1 By Lemma5.2,I can be written as follows:

I=Q1∩Q2∩ · · · ∩Q_q, Q_i=(y_i, x_ti1, . . . , x_tiji), i=1, . . . , q,

whereq=arithdegI. Set{x₁, x₂, . . . , x_r} = {x_tij}. By Lemma5.3, to prove the theo- rem, it suffices to findr+1 elementsg0, . . . , gr ∈I such that√

I=√

(g0, . . . , gr).

Note that any squarefree monomial with respect tox1, . . . , x_r can be written as follows:

x_i=x_i1,...,i=x_i1· · ·x_i (1≤i₁<· · ·< i≤r; =0,1, . . . , r).

Fori=(i1, . . . , i), we put(i)= {j :1≤j ≤q, xi∈/Qj}. Set Pr−=

xi₁· · ·xi j∈(i)

yj

: 1≤i1<· · ·< i≤r

for every=0,1, . . . , r. In particular, P0= {x₁· · ·x_r}andPr = {y₁· · ·y_q}. Then the following are satisfied:

(SV-1) _r

=0P=P contains all minimal monomial generators ofI. (SV-2) P0=1.

(SV-3) For any (0≤ < r)and anya, a∈Pr−(a=a), there is( < ≤r) anda∈Pr− such thata·a∈(a).

Let us check (SV-3) only. Fori=(i₁, . . . , i),i=(i₁, . . . , i)withi=i, we have that ({i} ∪ {i})≥+1. We also see that ({i} ∪ {i})⊆({i}). The assertion immediately follows from here.

If we set

g=

i=(i1,...,i)

xi₁· · ·xi j∈(i)

yj

for every=0,1, . . . , r ,

then we have√ I =√

(a:a∈P)R=√

(g₀, . . . , g_r)R by virtue of Schmitt–Vogel

lemma (see [12, Lemma, p. 249]).

Example 5.4 Let us consider

I =(y1, x1, x2)∩(y2, x2, x3)∩(y3, x4)

=(x₁x₃x₄, x₁x₃y₃, x₁x₄y₂, x₂x₄, x₃x₄y₁, x₁y₂y₃, x₂y₃, x₃y₁y₃, x₄y₁y₂, y₁y₂y₃).

Theng’s in the proof of Theorem5.1are given by the following:

g₀=x₁x₂x₃x₄,

g₁=x₁x₂x₃y₃+x₁x₂x₄+x₁x₃x₄+x₂x₃x₄,

g₂=x₁x₂y₃+x₁x₃y₃+x₁x₄y₂+x₂x₃y₃+x₂x₄+x₃x₄y₁, g₃=x₁y₂y₃+x₂y₃+x₃y₁y₃+x₄y₁y₂,

g4=y1y2y3.

6 Alexander dual of almost complete intersection squarefree monomial ideals In this section, we consider squarefree monomial ideals with arithdegI=indegI+1.

For such an idealIwith heightI≥2, the Alexander dualJofIis an almost complete intersection. Utilizing this fact, we determine araI.

Theorem 6.1 IfIis a squarefree monomial ideal with arithdegI=indegI+1, then araI=pd_RR/I.

Proof We may assume that heightI ≥2. Puth=indegI ≥2, and let J =I^∗ de- note the Alexander dual ofI. Then sinceJ is an almost complete intersection, it is isomorphic to one ofI1, . . . , I6; see Theorem4.4. Noting thatI=J^∗, we get Lemma 6.2 LetI be a squarefree monomial ideal with arithdegI=indegI+1 and heightI ≥2. ThenI is isomorphic to one of the following ideals, wherex_ij,yare variables that are different from each other:

•I₁ =(x11, x12, . . . , x1j₁)∩(x21, x22, . . . , x2j₂)∩ · · · ∩(xh1, xh2, . . . , xhj_h)

∩(x11, x12, . . . , x1i₁, x21, x22, . . . , x2i₂, . . . , xp1, xp2, . . . , x_pip), where 2≤p≤h, 1≤i< j(=1,2, . . . , p),j_p+1, . . . , j_h≥1.

•I₂=(x11, x12, . . . , x1j₁)∩(x21, x22, . . . , x2j₂)∩ · · · ∩(xh1, xh2, . . . , x_hjh)

∩(x_h+11, x_h+12, . . . , x_h+1j_h+1,

x11, x12, . . . , x1i₁, x21, x22, . . . , x2i₂, . . . , xp1, xp2, . . . , x_pip),

where 1≤p≤h, 1≤i< j(=1,2, . . . , p),j_p+1, . . . , j_h, j_h+1≥1.

•I₃=(x11, x12, . . . , x1i₁, y1, y2, . . . , y_p)∩(x21, x22, . . . , x2i₂, y1, y2, . . . , y_p)

∩(x₃₁, x₃₂, . . . , x_3j3)∩ · · · ∩(x_h1, x_h2, . . . , x_hjh)

∩(x₁₁, x₁₂, . . . , x_1i1, x₂₁, x₂₂, . . . , x_2i2), whereh≥2,p, i₁, i₂, j₃, . . . , j_h≥1.