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arXiv:1108.3667v1 [math.AC] 18 Aug 2011

DETERMINANTAL FACET IDEALS

VIVIANA ENE, J¨URGEN HERZOG, TAKAYUKI HIBI AND FATEMEH MOHAMMADI

Abstract. We consider ideals generated by general sets of m-minors of an m×n-matrix

of indeterminates. The generators are identified with the facets of an (m−1)-dimensional

pure simplicial complex. The ideal generated by the minors corresponding to the facets

of such a complex is called a determinantal facet ideal. Given a pure simplicial complex

∆, we discuss the question when the generating minors of its determinantal facet ideal

J∆ form a Gr¨ obner basis and when J∆ is a prime ideal.

Introduction

Let K be a field, X = (xij) be an m×n-matrix of indeterminates and S = K[X] be the

polynomial ring over K in the indeterminates xij. We assume that m ≤ n. Classically the

ideals It(X) generated by all t-minors of X have been considered. Hochster and Eagon

[15] proved that the rings S/It(X) are normal Cohen–Macaulay domains. A standard

reference on the classical theory of determinantal ideals, including the study of the powers

of It(X) is the book [4] of Bruns and Vetter. Motivated by geometrical considerations the

more general class of ladder determinantal ideals have been considered as well, [6]. A new

aspect to the theory of determinantal ideals was introduced by Sturmfels [17] and Caniglia

et al. [5] who showed that the t-minors of X form a Gr¨ obner basis of It(X) with respect

to any monomial order which selects the diagonals of the minors as leading terms. This

technique provides a new proof of the Cohen–Macaulayness of the determinantal rings

S/It(X) and was subsequently also used to compute important numerical invariants of

these rings, including the a-invariant, the multiplicity and the Hilbert function, see [2], [7]

and [13]. An excellent survey on the theory of determinantal ideals regarding the Gr¨ obner

basis aspect and with many references to more recent work is the article [1] by Bruns and

Conca.

Applications in algebraic statistics prompted the study of determinantal ideals generated

by quite general classes of minors, including ideals generated by adjacent 2-minors, see

[16] and [11], or ideals generated by an arbitrary set of 2-minors in a 2 × n-matrix [12].

Thus one may raise the following questions: given an arbitrary set of minors of X, what

can be said about the ideal they generate? When is such an ideal a radical ideal, when is

it a prime ideal, what is its primary decomposition, when is it Cohen–Macaulay, what is

its Gr¨ obner basis? Apart from the classical cases mentioned before, satisfying answers to

some of these questions are known for ideals generated by arbitrary sets of 2-minors of a

2 × n-matrix of indeterminates. All these ideals are radical, their primary decomposition

and their Gr¨ obner basis are known, see [12].

1991 Mathematics Subject Classification. 13C40, 13H10, 13P10, 05E40.

Key words and phrases. Determinantal rings and ideals, Gr¨ obner bases, Simplicial complexes.

This paper was written while J. Herzog and F. Mohammadi were staying at Osaka University. They

were supported by the JST (Japan Science and Technology Agency) CREST (Core Research for Evolutional

Science and Technology) research project Harmony of Gr¨ obner Bases and the Modern Industrial Society in

the frame of the Mathematics Program “Alliance for Breakthrough between Mathematics and Sciences.”

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The purpose of this paper is to extend some of the results shown in [12] to ideals gener-

ated by an arbitrary set of maximal minors of an m×n-matrix of indeterminates. For any

sequence of integers 1 ≤ a1< a2< ··· < am≤ n we denote by [a1a2...am] the maximal

minor of X with columns a1,a2,...,am. The set of integers {a1,a2,...,am} may be viewed

as a facet of a simplex on the vertex set [n]. This leads us to the following definition: let

∆ be a pure simplicial complex on the vertex set [n] = {1,...,n} of dimension m − 1.

With each facet F = {a1< a2< ··· < am} we associate the minor µF = [a1a2...am],

and call the ideal

J∆= (µF: F ∈ F(∆))

the determinantal facet ideal of ∆. Here F(∆) denotes the set of facets of ∆.

When m = 2, ∆ may be identified with a graph G and the m-minors are binomials. In

that case the determinantal facet ideal coincides with the binomial edge ideal of [12].

In the first section of this paper we answer the question when the maximal minors

generating J∆form a Gr¨ obner basis of J∆. In order to explain this result, we have to

introduce some notation. Let Γ be a simplicial complex. We denote by Γ(i)the i-skeleton

of Γ. The simplicial complex Γ(i)is the collection of all simplices of Γ whose dimension is

at most i.

Now let ∆ be a pure (m − 1)-dimensional simplicial complex on the vertex set [n] =

{1,2,...,n}. Then there exist uniquely determined simplices Γ1,...,Γr of dimension

≥ m − 1 such that for any simplex Γ with Γ(m−1)⊂ ∆, one has Γ ⊂ Γi for some i.

Let ∆i= Γ(m−1)

i

. Then ∆ = ∆1∪ ∆2∪ ··· ∪ ∆r. The simplicial complex whose facets

are the Γi is called the clique complex of ∆, the ∆i are called the cliques of ∆ and

∆ = ∆1∪ ∆2∪ ··· ∪ ∆rthe clique decomposition of ∆.

The complex ∆ is called closed (with respect to the given labeling) if for any two facets

F = {a1< ··· < am} and G = {b1< ··· < bm} with ai= bifor some i, the (m − 1)-

skeleton of the simplex on the vertex set F ∪ G is contained in ∆. In terms of its clique

decomposition, the property of ∆ of being closed can be expressed in the following ways:

(i) ∆ is closed, if and only if for all i ?= j and all F = {a1< a2< ··· < am} ∈ ∆iand

G = {b1< b2< ··· < bm} ∈ ∆jwe have aℓ?= bℓfor all ℓ.

(ii) ∆ is closed, if and only if for all i ?= j and all {a1,...,am} ∈ ∆iand {b1,...,bm} ∈

∆j, the monomials in<[a1...am] and in<[b1...bm] are relatively prime, where < is the

lexicographical order induced by the natural order of indeterminates

x11> x12> ··· > x1n> x21> ··· > x2n> ··· > xmn,

row by row from left to right.

The main result (Theorem 1.1) of Section 1 states that the minors generating the facet

ideal J∆form a quadratic Gr¨ obner basis with respect to the lexicographic order induced by

the natural order of the variables, if and only if ∆ is closed. We also show that whenever

∆ is closed, then J∆is Cohen–Macaulay and the K-algebra generated by the minors which

generate J∆is Gorenstein, see Corollary 1.3 and Corollary 1.4.

In Section 2 we discuss when a determinantal facet ideal is a prime ideal. As a main

result we show in Theorem 2.2 that if ∆ is closed and J∆is a prime ideal, then the clique

complexes ∆i of ∆ satisfy the following intersection properties: for all 2 ≤ t ≤ m =

dim∆ + 1 and for any pairwise distinct cliques ∆i1,...,∆itone has

|V (∆i1) ∩ ··· ∩ V (∆it)| ≤ m − t.

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We expect that this intersection property actually characterizes closed simplicial complexes

whose determinantal facet ideal is prime, but could not prove it yet. In Theorem 2.4 we

can give only a partial converse of Theorem 2.2.

We show in Example 2.5 that primality of determinantal facet ideals satisfying the

above intersection condition can only be expected for closed simplicial complexes. For

non-closed simplicial complexes the primality problem seems to be pretty hard.

In Section 3 we study primality of J∆for a closed simplicial complex under the following

very strict intersection condition: let ∆ = ∆1∪ ... ∪ ∆rbe the clique decomposition of

∆. We require that

(i) |V (∆i) ∩ V (∆j)| ≤ 1 for all i < j;

(ii) V (∆i) ∩ V (∆j) ∩ V (∆k) = ∅ for all i < j < k.

For m = 3, this is exactly the necessary condition for primality formulated in Theorem 2.2.

Assuming (i) and (ii), we let G∆be the simple graph with the vertices v1,...,vr, and

edges {vi,vj} for all i ?= j with V (∆i) ∩ V (∆j) ?= ∅. The question arises for which graphs

G∆the determinantal facet ideal J∆is a prime ideal. This is the case when ∆ is closed

and G∆ is a forest or a cycle, see Theorem 3.2 and Theorem 3.3. Finally we show in

Theorem 3.4 that J∆is a prime ideal for any graph G∆when ∆ is closed and each clique

of ∆ is a simplex. It is also shown in this theorem that for any graph G there is a closed

simplicial complex ∆ with G = G∆whose cliques are all simplices.

1. Determinantal facet ideals whose generators form a Gr¨ obner basis

In this section we intend to classify those ideals generated by maximal minors of a

generic m × n-matrix X whose generating minors form a Gr¨ obner basis. As explained

in the introduction we identify each m-minor [a1a2...am] of X with the (m − 1)-simplex

F = {a1,a2,...,am}. Thus an arbitrary collection of m-minors of X can be indexed by the

facets of a pure (m−1)-dimensional simplicial complex ∆ on the vertex set [n]. The ideal

generated by these minors will be denoted J∆, and is called the determinantal facet ideal

of ∆. In other words, if F(∆) denotes the set of facets of ∆, then J∆= (µF: F ∈ F(∆)),

where µF= [a1a2···am] for F = {a1,a2,...,am}.

In analogy to the case of 2-minors, as considered in [12], we say that ∆ is closed (with

respect to the given labeling) if for any two facets F = {a1< ··· < am} and G = {b1<

··· < bm} with ai= bifor some i, the (m − 1)-skeleton of the simplex on the vertex set

F ∪ G is contained in ∆.

In what follows it will be useful to characterize closed simplicial complexes in terms

of the clique decomposition of ∆. We first observe that there exist uniquely determined

simplices Γ1,...,Γrof dimension ≥ m − 1 such that for any simplex Γ with Γ(m−1)⊂ ∆,

one has Γ ⊂ Γifor some i. Let ∆i= Γ(m−1)

i

. Then ∆ = ∆1∪∆2∪···∪∆r. The simplicial

complexes ∆iare called the cliques of ∆ and ∆ = ∆1∪ ∆2∪ ··· ∪ ∆ris called the clique

decomposition of ∆. By referring to this decomposition one has:

(i) ∆ is closed, if and only if for all i ?= j and all F = {a1< a2< ··· < am} ∈ ∆iand

G = {b1< b2< ··· < bm} ∈ ∆jwe have aℓ?= bℓfor all ℓ.

(ii) ∆ is closed, if and only if for all i ?= j and all {a1,...,am} ∈ ∆iand {b1,...,bm} ∈

∆j, the monomials in<[a1...am] and in<[b1...bm] are relatively prime, where <

is the lexicographical order induced by the natural order of indeterminates

x11> x12> ··· > x1n> x21> ··· > x2n> ··· > xmn,

row by row from left to right.

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The main result of this section is

Theorem 1.1. The set G = {[a1...am] : {a1,...,am} ∈ ∆} is a Gr¨ obner basis of J∆

with respect to the lexicographical order induced by the natural order of indeterminates if

and only if ∆ is closed.

For the proof of Theorem 1.1 we need the following technical result.

Lemma 1.2. For any m−1 rows c1,c2,...,cm−1and m+1 columns d1,d2,...,dm−2,e1,e2,e3

of X one has the following identity:

(−1)k[c1...cm−1|d1...dm−2e3][d1...dm−2e1e2]

+(−1)j[c1...cm−1|d1...dm−2e2][d1...dm−2e1e3]

+(−1)i[c1...cm−1|d1...dm−2e1][d1...dm−2e2e3] = 0,

provided that d1< d2< ··· < di−1< e1< di< ··· < dj−2< e2< dj−1< ··· < dk−3<

e3< dk−2< ··· < dm−2.

Proof. Our assumption on the sequence of integers means that e1is the ithterm, e2the

jthterm, and e3the kthterm of the above sequence.

Now consider the matrix

gd1

...gdi−1

ge1

...

M =

x1d1

...

xmd1

...x1di−1

...

xmdi−1

x1e1

...

xme1

...x1e2

...

xme2

ge2

...x1e3

...

xme3

ge3

...x1dm−2

...

xmdm−2

gdm−2

...... ...

...

...

...

,

where gℓ= [c1...cm−1|d1...dm−2ℓ] for each ℓ ∈ {d1,d2,...,dm−1,e1,e2.e3}. Expanding

gℓby the last column we get

gℓ=

m−1

?

i=1

(−1)m−1+i[c1...ci−1ci+1...cm−1|d1...dm−2]xciℓ

for each ℓ. Therefore the last row of M is a linear combination of the rows c1,...,cm−1of

M. Hence the determinant of M is zero. On the other hand, gℓ= 0 for ℓ = d1,...,dm−2,

since for these ℓ the polynomial gℓis the determinant of a matrix with two equal columns.

Now computing the determinant of M by expanding its last row we obtain the desired

identity.

?

Proof of Theorem 1.1. Assume that ∆ is closed. We show that all S-pairs,

S([a1...am],[b1...bm])

reduce to zero. If ai?= bifor all i, then in<[a1...am] and in<[b1...bm] have no common

factor. Therefore S([a1...am],[b1...bm]) reduces to zero.

Let ai= bifor some i. Since ∆ is closed, all m-subsets of {a1,...,am}∪{b1,...,bm} be-

long to ∆. Therefore S([a1...am],[b1...bm]) reduces to zero with respect to the m-subsets

of {a1,...,am} ∪ {b1,...,bm}, and hence with respect to G. Then by using Buchberger’s

criterion, it follows that G is a Gr¨ obner basis of J∆.

Assume that G is a Gr¨ obner basis for the ideal J∆. Let [a1a2...am] with a1< a2<

··· < amand [b1b2...bm] with b1< b2< ··· < bmbelong to G, and assume that ai= bi

for some i.

We will show that ∆ is closed. The proof is by descending induction on

k = |{a1,...,am} ∩ {b1,...,bm}|.

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First assume that k = m−1. Then there exists an integer ℓ such that a1= b1,...,aℓ−1=

bℓ−1and aℓ?= bℓ. We may assume bℓ< aℓ. Then

{b1< ... < bm} = {a1< a2< ··· < aℓ−1< bℓ< aℓ< ··· < aℓ′−1< aℓ′+1< ··· < am}

for some ℓ′≥ ℓ.

In order to prove that in this case ∆ is closed we have to show that

{a1,...,am,bℓ} \ {ar} ∈ ∆

for all r.

Since ai= bifor some i we have that either ℓ′< m or 1 < ℓ. We first assume that

ℓ′< m, and choose an integer r with ℓ′< r ≤ m.

Then we use the determinantal identity of Lemma 1.2 for {d1< ··· < dm−2} equal to

{a1< ··· < aℓ−1< aℓ< ··· < aℓ′−1< aℓ′+1< ··· < ar−1< ar+1< ··· < am}

and {e1< e2< e3} = {bℓ< aℓ′ < ar}, and obtain

(−1)ℓ′+1

+(−1)r+1

+(−1)ℓ

[1...m − 1|a1...ˆ aℓ′ ...am][a1...aℓ−1bℓaℓ...ˆ ar...am]

[1...m − 1|a1...ˆ ar...am][b1...bm]

[1...m − 1|a1...aℓ−1bℓaℓ...ˆ aℓ′ ...ˆ ar...am][a1...am] = 0.

Since the last two terms are in J∆and G is a Gr¨ obner basis for J∆, the initial monomial

of the first term is divisible by the initial monomial of a minor in G.

The initial monomial of the first term is

u = (x1a1

···

xℓ′−1aℓ′−1xℓ′aℓ′+1···xm−1am)

×(x1a1···xℓ−1aℓ−1xℓbℓxℓ+1aℓxℓ+2aℓ+1···xrar−1xr+1ar+1···xmam).

It follows that in<[a1...aℓ−1bℓaℓ...ˆ ar...am] is the only initial monomial of a maximal

minor of X which divides the above monomial.

of a maximal minor which divides u we need to choose an increasing subsequence of

a1< ··· < aℓ−1< bℓ< aℓ< aℓ+1< ··· < amwith m elements. Note that for the first

ℓ − 1 and the last m − r elements we have a unique choice, namely a1< ··· < aℓ−1and,

respectively, ar+1 < ··· < am. Hence we have to choose a subsequence with r − ℓ + 1

elements of bℓ< aℓ< aℓ+1< ··· < ar. Now we observe that xrardoes not divide u, hence

we cannot keep arin the above sequence. Therefore, the unique choice of the subsequence

is bℓ< aℓ< aℓ+1< ··· < ar−1.

Hence we deduce that

Indeed, to find the initial monomial

[a1...aℓ−1bℓaℓ...ˆ ar...am] ∈ G

and so {a1,...,aℓ−1,bℓ,aℓ,...,ˆ ar,...,am} is in ∆ for all r > ℓ′.

Next we assume that 1 < ℓ. Then we deduce as above that

{a1,...,ˆ ar,...,aℓ−1,bℓ,aℓ,...,am}

is in ∆ for r < ℓ. More precisely, we use again Lemma 1.2, but for [c1...cm−1] = [2...m]

and get the following identity:

(−1)r

+(−1)ℓ−1

+(−1)ℓ′−1

[2...m|a1...ˆ ar...am][b1...bm]

[2...m|a1...ˆ ar...bℓaℓ...ˆ aℓ′ ...am][a1...am]

[2...m|a1...ˆ aℓ′ ...am][a1...ˆ ar...bℓaℓ...am] = 0.

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The last term in this identity belongs to J∆, thus its initial monomial is divisible by the

initial monomial of a minor in G. By using similar arguments as before, we get the claim.

Finally we show that for all r we have {a1,...,am,bℓ} \ {ar} ∈ ∆. To this end we

may assume that ℓ′< m and choose r = ℓ′+ 1, to obtain by the above arguments that

{a1,...,aℓ−1,bℓ,aℓ,...,ˆ aℓ′+1,...,am} is a facet of ∆. Comparing this facet with the facet

{a1,...,aℓ−1,bℓ,aℓ,...,ˆ aℓ′,...,am} of ∆ it follows from the above considerations that

{a1,...,aℓ−1,bℓ,aℓ,...,am} \ {ar} ∈ ∆ for all r ≤ ℓ′.

Assume now that |{a1,...,am} ∩ {b1,...,bm}| = k < m − 1. Let s be the number of

integers i such that ai= bi. By our assumption, s ≥ 1, and of course s ≤ k. Assume that

a1= b1,...,as= bsand as+1< bs+1. Then

in<([s + 1...m|bs+1...bm][a1...am] − [s + 1...m|as+1...am][b1...bm])

(xs+1bs+1···xmbm)(x1a1···xs−1as−1xsas+1xs+1asxs+2as+2···xmam) = u,=

because the monomials bigger than u in the expression whose initial monomial we compute

cancel. Therefore there exists a minor [c1...cm] in G with c1< c2< ··· < cmsuch that

in<[c1...cm] divides the monomial

(xs+1bs+1···xmbm)(x1a1···xs−1as−1xsas+1xs+1asxs+2as+2···xmam),

and we have

c1= a1,...,cs−1= as−1,cs= as+1,cs+1= bs+1, and cℓ∈ {aℓ,bℓ} for ℓ ≥ s + 2.

First consider the case s = k. Then cmis either amor bm, and we may assume that

cm= am. Therefore |{c1,...,cm} ∩ {a1,...,am}| > k. Applying the inductive hypothesis

for the facets {c1,...,cm} and {a1,...,am} of ∆, we conclude that all m-subsets of

{a1,...,am} ∪ {c1,...,cm}

belong to ∆.

Note that there exists some cisuch that ci?∈ {a1,...,am}, since as?∈ {c1,...,cm}. It

follows that ci= bi, and consequently bi?∈ {a1,...,am}. Moreover, since k < m − 1 there

exist two integers j1and j2such that

aj1,aj2?∈ {b1,...,bm}.

Since {a1,...,ˆ aj1,...,am,bi} and {a1,...,ˆ aj2,...,am,bi} are m-subsets of

{a1,...,am} ∪ {c1,...,cm},

these sets belong to ∆. Now applying the inductive hypothesis to the sets {b1,...,bm}

and {a1,...,ˆ aj1,...,am,bi} which intersect in k +1 elements, we will get all m-subsets of

{a1,...,ˆ aj1,...,am,bi} ∪ {b1,...,bm}

in ∆. By the same argument we deduce that all m-subsets of

{a1,...,ˆ aj2,...,am,bi} ∪ {b1,...,bm}

belong to ∆.

Now assume that F is an arbitrary subset of {a1,...,am} ∪ {b1,...,bm} such that

aj1,aj2∈ F and bj?∈ F for some j. By the above statements we have (F \ {aj1}) ∪ {bj}

and (F \ {aj2}) ∪ {bj} in ∆. Then comparing these two facets we deduce that F ∈ ∆,

since their intersection has cardinality m − 1.

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We remark that in the more general case that aℓ1= bℓ1,...,aℓs= bℓs, the proof is

similar. We just consider the minor

[1...ˆℓ1...ˆℓs...m|a1...ˆ aℓ1...ˆ aℓs...am]

instead of [s + 1...m|as+1...am] and the minor

[1...ˆℓ1...ˆℓs...m|b1...ˆbℓ1...ˆbℓs...bm]

instead of [s + 1...m|bs+1...bm] to get the desired minors in G.

Therefore, the assertion of the theorem is proved if s = k.

Now assume that s < k, and for every two sets in ∆ with k common elements which

at least s + 1 of them have the same position in both sets, the result holds. Let aℓ1=

bt1,...,aℓk−s= btk−sfor some integers ℓ1< ··· < ℓk−sand t1< ··· < tk−s, where tr?= ℓr

for r = 1,...,k − s. Assume that

{aℓσ1,...,aℓσp} ⊂ {cs+2,...,cm}, and {aℓτ1,...,aℓτq} ?⊂ {cs+2,...,cm},

for {σ1,...,σp,τ1,...,τq} = {ℓ1,...,ℓk−s}. First assume that p = k − s. Note that there

exists some index j with j ?∈ {1,...,s + 1,ℓ1,...,ℓk−s}, since k < m − 1. If cj = aj

for some j ?∈ {1,...,s + 1,ℓ1,...,ℓk−s}, then |{a1,...,am} ∩ {c1,...,cm}| > k, and by

the inductive hypothesis we will get all m-subsets of {a1,...,am} ∪ {c1,...,cm} in ∆.

Otherwise |{b1,...,bm}∩{c1,...,cm}| > k, and by the inductive hypothesis all m-subsets

of {b1,...,bm} ∪ {c1,...,cm} belong to ∆. In both cases applying the same argument as

in the case s = k, we deduce that all desired m-subsets are in ∆.

Now assume that p < k − s. We claim that

cℓr= bℓr

forr = τ1,...,τq,

in particular, {bℓτ1,...,bℓτq} ⊂ {c1,...,cm}.

Indeed, suppose that aℓr?∈ {cs+2,...,cm}. Therefore cℓr= bℓrand ctr= atr.

Since as+1< bs+1< ··· < bm, we have ℓr> s + 1 for all r. Therefore

c1= b1,...,cs−1= bs−1,cs+1= bs+1,cℓτ1= bℓτ1,...,cℓτq= bℓτq,

cℓσ1= aℓσ1= btσ1,...,cℓσp= aℓσp= btσp,

which shows that {c1,...,cm} and {b1,...,bm} have at least k common elements and

s+q ≥ s+1 of them have the same position in both sets. Now applying the result of the

first case to these two sets, we deduce that all m-subsets of

{b1,...,bm} ∪ {c1,...,cm}

are in ∆. Now the same argument as in case k = s, for {b1,...,bm}∪{c1,...,cm} instead

of {a1,...,am} ∪ {c1,...,cm}, implies that all desired m-subsets belong to ∆.

?

For determinantal facet ideals of closed simplicial complexes we may compute important

numerical invariants.

Corollary 1.3. Let ∆ be a closed simplicial complex of dimension (m − 1) and let ∆ =

∆1∪ ∆2∪ ··· ∪ ∆rits clique decomposition. Then:

(a) heightJ∆=?r

(b) J∆is Cohen-Macaulay.

7

ℓ=1heightJ∆ℓ=?r

ℓ=1nℓ− (m − 1)r.

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(c) The Hilbert series of S/J∆has the form

HS/J∆(t) =

?r

ℓ=1Qℓ(t)

ℓ=1nℓ+(m−1)r,

(1 − t)mn−?r

where

Qℓ(t) = [det(

?

k

?m − i

k

??nℓ− j

k

?

)1≤i,j≤m−1]/t(m−1

2)

for 1 ≤ ℓ ≤ r.

(d) The multiplicity of S/J∆is

e(S/J∆) =

r?

ℓ=1

?

nℓ

m − 1

?

.

Proof. It follows from the characterization (ii) of closed simplicial complexes that the initial

ideals in<(J∆ℓ) are monomial ideals in disjoint sets of variables, therefore the first equality

in (a) is obvious. The second equality follows from the known formula of the height of

determinantal ideals, see for instance [8, Theorem 6.35]. By using the known formulas for

the Hilbert series and multiplicity for determinantal rings defined by maximal minors [7],

we straightforward get (c) and (d). For (b) we use the fact that the simplicial complex

attached to the squarefree monomial ideal in<(J∆ℓ) is shellable (see [13] or [1]). This

implies that each in<(J∆ℓ) is a Cohen–Macaulay ideal. By using again the characterization

(ii) of closed simplicial complexes it follows that in<(J∆) = in<(J∆1) + ··· + in<(J∆r) is

also Cohen–Macaulay. This implies that J∆ is Cohen–Macaulay, see for example [10].

?

Corollary 1.4. Suppose that ∆ is closed with clique decomposition ∆ = ∆1∪ ... ∪ ∆r.

Then the K-algebra

A = K[{[a1...am] : {a1,...,am} ∈ ∆}]

is Gorenstein of dimension r +?r

set of ∆i.

i=1m(ni− m), where niis the cardinality of the vertex

Proof. We first observe that

B := K[{in<[a1...am]: {a1,...,am} ∈ ∆}]∼=

r

?

i=1

K[{in<[a1...am]: {a1,...,am} ∈ ∆i}].

We use the Sagbi basis criterion (see [8, Theorem 6.43]) which asserts that the minors

[a1...am] with {a1,...,am} ∈ ∆ form a Sagbi basis of A, that is, the monomials [a1...am]

with {a1,...,am} ∈ ∆ generate the initial algebra in<(A), if a generating set of binomial

relations of the algebra B can be lifted. It follows from the tensor presentation of B that a

set of binomial relations of B is obtained as the union of the binomial relations of each of

the algebras K[{in<[a1...am]: {a1,...,am} ∈ ∆i}]. For these algebras it is known that

they admit a set of liftable relations. Thus it follows that B = in<(A).

Next we observe that for each i, the K-algebra K[{in<[a1...am] : {a1,...,am} ∈ ∆i}]

is the Hibi ring associated to the distributive lattice Liof all maximal m-minors [a1...am]

with {a1,...,am} ∈ ∆iwhose partial order is given by

[a1...am] ≤ [b1,...,bm]

⇔

ai≤ bi

fori = 1,...,m.

The distributive lattice Liis graded, which by a theorem of Hibi [14] implies that

K[{in<[a1...am] : {a1,...,am} ∈ ∆i}]

8

Page 9

is Gorenstein. It follows that A is Gorenstein, see [1, Theorem 3.16]

Finally we notice that

dimA = dimin<(A)=

r

?

i=1

r

?

i=1

dimK[{in<[a1...am]: {a1,...,am} ∈ ∆i}]

=dimK[{[a1...am]: {a1,...,am} ∈ ∆i}].

The desired formula for the dimension of A follows, because K[{[a1...am]: {a1,...,am} ∈

∆i}] is the algebra of all maximal minors of an m×ni-matrix of indeterminates, and hence

its dimension is m(ni− m) + 1, see for example [8, Theorem 6.45].

?

2. Primality of determinantal facet ideals

In this and the following section we want to discuss when a determinantal facet ideal is

a prime ideal. In general J∆need not to be a prime ideal even if ∆ is closed. For exam-

ple, if ∆ is the simplicial complex with facets F(∆) = {{1,2,3},{2,3,4}} or F(∆) =

{{1,2,3},{2,3,6},{3,4,5}}, then J∆ is not a prime ideal.

heightJ∆= 2 and P = (x2y3− x3y2,x2z3− x3z2,y2z3− y3z2) is a prime ideal of height

2 which obviously strictly contains J∆. In the second case, we get heightJ∆ = 3 and

J∆? (x3,y3,z3), hence clearly J∆is not prime. Even in this rather simple examples we

see, by using Singular [9], that the primary decomposition of determinantal facet ideals

looks much more complicated than that for binomial edge ideals.

The main result of this section, Theorem 2.2, explains why J∆is not a prime ideal in

the above examples.

The proofs of primality that follow depend on localization with respect to nonzero

divisors. This technique allows to use induction arguments. Indeed, suppose we want to

show that J ⊂ S is a prime ideal. Then we are trying to find an element f ∈ S which is

regular modulo J. This implies that the natural map S/J → (S/J)fis injective. Now, if

we can find a prime ideal L ⊂ S such that (S/L)f∼= (S/J)f, we conclude that (S/J)f,

and consequently S/J, is a domain which implies that J is a prime ideal. This procedure

often allows us to use inductive arguments, as in many cases L is of a simpler structure.

The next lemma helps us to understand the effect of localization when we are dealing

with ideals generated by minors of a matrix.

Indeed, in the first case,

Lemma 2.1. Let K be a field, X be an m×n-matrix of indeterminates and I ⊂ S = K[X]

an ideal generated by a set G of minors. Furthermore, let xijbe an entry of X. We assume

that for each minor [a1...at|b1...bt] ∈ G there exists ℓ such that aℓ= i.

Then (S/I)xij∼= (S/J)xijwhere J is generated by the minors [a1...at|b1...bt] ∈ G with

bℓ?= j for all ℓ, and the minors [a1...ˆ aℓ...at|b1...ˆbk...bt] where [a1...at|b1...bt] ∈ G

and aℓ= i and bk= j.

Proof. For simplicity we may assume that i = 1 and j = 1. We apply the automorphism

ϕ: Sx11→ Sx11with

?

xij,

Let I′⊂ Sx11be the ideal which is the image of ISx11under the automorphism ϕ. Then

(S/I)x11∼= Sx11/I′. The ideal I′is generated in Sx11by the elements ϕ(µM) where µM∈ G.

Note that if µM= [a1...at|b1...bt], then ϕ(µM) = det(x′

xij?→ x′

ij=

xij+ xi1x−1

11x1j, if i ?= 1 and j ?= 1,

if i = 1 or j = 1.

aibj)i,j=1,...,t.

9

Page 10

In the following we may assume that a1< a2< ··· < atand b1< b2< ··· < btfor

µM = [a1...at|b1...bt] ∈ G. Then our assumption implies that a1 = 1. Let us first

consider the case that b1?= 1. Then ϕ(µM) is the determinant of the matrix

xatb1+ xat1x−1

xatb2+ xat1x−1

By subtracting suitable multiples of the first row from the other rows we see that

x1b1

x1b2

···

···

x1bt

xa2b1+ xa21x−1

11x1b1

xa2b2+ xa21x−1

11x1b2

xa2bt+ xa21x−1

11x1bt

...

...

···

···

...

11x1b1

11x1b2

xatbt+ xat1x−1

11x1bt

ϕ(µM) = det(xaibj)i,j=1,...,t= µM.

In the case that b1= 1, the element ϕ(µM) is the determinant of the matrix

xat1

xatb2+ xat1x−1

Applying suitable row operations we obtain the matrix

0xatb2

It follows that ϕ(µM) = det(xaibj)i,j=2,...,t. These calculations show that I′= JSx11, as

desired.

x11

xa21

...

x1b2

···

···

x1bt

xa2b2+ xa21x−1

11x1b2

xa2bt+ xa21x−1

11x1bt

...

···

···

...

11x1b2

xatbt+ xat1x−1

11x1bt

1

0

...

x−1

11x1b2

xa2b2

...

···

···

x−1

11x1bt

xa2bt

...

xatbt

···

···

?

Now we are ready to prove

Theorem 2.2. Let ∆ be a pure (m − 1)-dimensional closed simplicial complex on the

vertex set [n] and let ∆ = ∆1∪...∪∆rbe the clique decomposition of ∆. If J∆is a prime

ideal, then for all 2 ≤ t ≤ m and for any pairwise distinct cliques ∆i1,...,∆itwe have

|V (∆i1) ∩ ··· ∩ V (∆it)| ≤ m − t.

Proof. We make induction on m. The initial step, m = 2, is already known [12].

Let us make the induction step. We first consider t < m. Let us assume that there

exist ∆i1,...,∆itsuch that |V (∆i1) ∩ ··· ∩ V (∆it)| > m − t. Without loss of generality

we may assume that V (∆1) ∩ ··· ∩ V (∆t) = {a1,a2,...,aℓ} with ℓ ≥ m − t + 1 and

1 ≤ a = a1< ··· < aℓ≤ n. We may further assume that there exists s ≥ t such that

a ∈ V (∆i) for 1 ≤ i ≤ s and a / ∈ V (∆i) for s + 1 ≤ i ≤ r. Since J∆is prime, it follows

that xmais regular on J∆and J∆Sxmais also a prime ideal in the localization Sxmaof

S. Thus (S/J∆)xmais a domain. By Lemma 2.1, it follows that (S/J∆)xma∼= (S/L)xma,

where L = L1+?r

dimensional simplicial complex ∆′with the clique decomposition ∆′= ∆′

∆′

vertices, by induction, it follows that L1is not a prime ideal which will imply, as we are

going to show, that L is not a prime ideal. But this is a contradiction, since (S/L)xma

must be a domain.

Since L1is not prime, there exist polynomials f,g in S such that fg ∈ L1and f,g / ∈ L1.

We claim that f,g / ∈ L. Let us assume, for instance, that f ∈ L. Then we may write

10

i=s+1J∆i, and L1is the determinantal facet ideal of the closed (m−2)-

1∪···∪∆′s, where

i= ?F \ {a} : F ∈ F(∆i),a ∈ F? for 1 ≤ i ≤ s. As ∆′

1,...,∆′tintersect in ℓ − 1 ≥ m − t

Page 11

f =?

over all G ∈

mapping the indeterminates xmjto zero for all j ?= a and xmato 1, we get f =?

for some polynomials h′

ideal.

It remains to consider the case t = m. We may assume that |V (∆1)∩···∩V (∆m)| ≥ 1.

Let a ∈ V (∆1) ∩ ··· ∩ V (∆m). It is clear that J∆⊂ (J∆′,x1a,...,xma) where ∆′= {F ∈

∆ : a / ∈ F}. Since ∆ is closed, it follows that ∆′is closed as well and, moreover, by using

Corollary 1.3,

GhGγG+?

FhFµF for some polynomials hG,hF∈ S where the first sum is taken

i=1F(∆′

?s

i), and the second one over all F ∈

?r

i=s+1F(∆i). Then, by

Gh′

GγG

G∈ S, thus f ∈ L1, a contradiction. Therefore, L is not a prime

heightJ∆′ =

m

?

i=1

((ni− 1) − m + 1) +

r

?

i=m+1

(ni− m + 1) = heightJ∆− m.

Since x1a,...,xmais obviously a regular sequence on S/J∆′, we have

height(J∆′,x1a,...,xma) = heightJ∆′ + m = heightJ∆.

Let P be a minimal prime of (J∆′,x1a,...,xma) of height equal to height(J∆). Since

J∆and P are prime ideals of the same height, we must have J∆= P. But P contains

the indeterminates x1a,...,xma, which do not belong to J∆. Therefore, we have got a

contradiction.

?

The proofs of primality that follow depend on localization with respect to nonzero

divisors. The next result tells us that in our situation all variables are nonzero divisors.

Lemma 2.3. Let ∆ be closed (m − 1)-dimensional simplicial complex with the property

that any m pairwise distinct cliques of ∆ have an empty intersection. Assume further that

K is infinite. Then each of the variables xijis regular modulo J∆.

Proof. In order to show that xijis regular modulo J∆we consider the ideal

I = (J∆,x1j,...,xmj).

Let ∆′be the simplicial complex whose facets are those of ∆ which do not contain j.

Observe that ∆′is again closed, and that I = (J∆′,x1j,...,xmj). We use the formula in

Corollary 1.3 to compare the height of I with that of J∆. If ∆ = ∆1∪ ··· ∪ ∆ris the

clique decomposition of ∆ with ni= |∆i|. Then heightJ∆=?r

We may assume that ∆icontains the vertex j for i = 1,...,s. Our assumptions implies

that s ≤ m−1. Note that the clique decomposition of ∆′= ∆′

each ∆′

for i = 1,...,s and ∆′

i= ∆ifor i > s. Hence we get

i=1(ni− m + 1).

1∪···∪∆′rwhere the facets of

i| = |∆i|−1 = ni−1

iare those facets of ∆iwhich do not contain j. It follows that |∆′

heightI= heightJ∆′ + m =

s

?

i=1

(ni− 1 − m + 1) +

r

?

i=s+1

(ni− m + 1) + m

= heightJ∆− s + m > heightJ∆.

Our considerations show that I/J∆⊂ S/J∆has positive height. Since S/J∆is Cohen–

Macaulay and K is infinite, it follows that a generic linear combination a1x1j+ a2x2j+

··· +amxmjof the variables x1j,...,xmj(whose residue classes generate I/J∆) is regular

modulo J∆. Since the above linear combination is generic, we may assume that ai= 1.

Now we consider the linear automorphism ϕ: S → S with ϕ(xik) = a1x1k+a2x2k+···+

amxmkfor k = 1,...,n and ϕ(xℓk) = xℓkfor ℓ ?= i and all k. Let X′be the matrix whose

11

Page 12

entries are the elements ϕ(xℓk) for ℓ = 1,...,m and k = 1,...,n. Then X′is obtained

from X by elementary row operations. It follows that ϕ(J∆) = J∆.

By our choice of ϕ we have that yij= ϕ(xij) is regular modulo J∆. Since J∆= ϕ(J∆)

it follows that xij= ϕ−1(yij) is regular modulo ϕ−1(J∆) = ϕ−1(ϕ(J∆)) = J∆, as desired.

?

We do not know whether for a closed simplicial complex ∆ the necessary condition for

J∆to be a prime ideal given in Theorem 2.2 is also sufficient. For the moment we can

only present a partial converse of this result.

Proposition 2.4. Let ∆ be a simplicial complex with clique decomposition ∆ = ∆1∪∆2∪

··· ∪ ∆r. Assume that all cliques are simplices of dimension m − 1 and that

(1) |V (∆r) ∩ ··· ∩ V (∆r−s+1)| ≤ m − s for s = 2,...,r;

(2) V (∆i1) ∩ ··· ∩ V (∆is) ⊂ V (∆r) ∩ ··· ∩ V (∆r−s+1) for all subsets {i1,...,is} ⊂ [r]

of cardinality s with 2 ≤ s ≤ r.

Then J∆is a prime ideal.

Proof. We make induction on m. The initial step, m = 2, is already known [12]. Assume

that |V (∆1) ∩ ··· ∩ V (∆r)| = k. We consider the following labeling on the vertices of ∆

such that

V (∆ℓ) = {aℓ1< ··· < aℓ,m−k−ℓ+1< b1< ··· < bk< cℓ1< ··· < cℓ,ℓ−1}

for all ℓ = 1,...,r, where the numbers aijare pairwise distinct, and for each s = 2,...,r

we choose cijsuch that

crj= cr−1,j= ··· = cr−s+1,j,

for j = 1,...,|V (∆r) ∩ ··· ∩ V (∆r−s+1)| − k.

Then with respect to this labeling, ∆ is closed and so xmb1is a regular element modulo

J∆by Lemma 2.3. It follows from Lemma 2.1 that (S/J∆)xmb1∼= (S/L)xmb1where L =

?r

prime ideal. Here Liis generated by the minor

i=1Li. Since xmb1is regular modulo J∆, J∆is a prime ideal if and only if Lxmb1is a

[1...m − 1|ai1...ai,m−k−i+1b2...bkci1...ci,i−1].

Let ∆′be the (m − 2)-simplicial complex with the clique decomposition ∆′

where ∆′

a prime ideal by the inductive hypothesis.

1∪ ··· ∪ ∆′t,

i= ∆i\{b1}. Note that conditions (1) and (2) hold for ∆′. Therefore L = J∆′ is

?

Example 2.5. Let ∆ = ∆1∪···∪∆rwith the assumption of Theorem 2.4. Then we can

describe the vertices of each ∆iin a nice way as the ithrow of a simple matrix. As an

example let m = 6,r = 4, |V (∆4) ∩ V (∆3)| = 3, |?4

Then by the proof of the above theorem we get

which describes the labels of the ∆1,...,∆4.

i=2V (∆i)| = 3 and |?4

b1

b2

b2

c1

c1

c2

c3

c4

i=1V (∆i)| = 2.

1

5

8

2

6

9

b1

3

7

b1

b2

4

b1

b2

c1

10

Example 2.6. Let F(∆) = {{1,2,3},{1,4,5},{3,5,6},{2,4,6}}. Then J∆is not a prime

ideal, however the intersection condition of Theorem 2.2 holds for ∆. Thus for a converse

of Theorem 2.2 one should require that ∆ is a closed simplicial complex.

12

Page 13

This is also an example of a determinantal facet ideal whose initial ideal with respect

to the lexicographic order is not squarefree though J∆is a radical ideal.

3. Special classes of prime determinantal facet ideals

Let ∆ a pure simplicial complex of dimension m − 1 ≥ 2 and ∆ = ∆1∪ ... ∪ ∆rits

clique decomposition. In this section we pose the following intersection properties on the

cliques of ∆:

(i) |V (∆i) ∩ V (∆j)| ≤ 1 for all i < j;

(ii) V (∆i) ∩ V (∆j) ∩ V (∆k) = ∅ for all i < j < k.

Theorem 2.2 implies that for m = 3 the conditions (i) and (ii) are satisfied whenever

J∆is a prime ideal, and that for any m ≥ 3 these two conditions imply the intersection

conditions formulated in Theorem 2.2.

In this section we will show that whenever ∆ is closed, the conditions (i) and (ii) imply

primality of J∆under some additional assumptions depending on a graph which we are

going to define now.

For the simplicial complex with the properties (i) and (ii), we let G∆ be the simple

graph with vertex set V (G∆) = {v1,...,vr} and edge set

E(G∆) = {{vi,vj}: V (∆i) ∩ V (∆j) ?= ∅}.

In the following the phrase “∆ is a simplicial complex with graph G∆” will always imply

that ∆ satisfies the conditions (i) and (ii) (because otherwise G∆is not defined).

At present we are able to prove primality of J∆for certain classes of simplicial complexes

∆ only under the additional assumption that these complexes are closed. The next lemma

provides a necessary condition for a simplicial complex to be closed.

Lemma 3.1. Let ∆ be a closed simplicial complex with graph G∆. Then each vertex vi

of G∆has order at most min{|V (∆i)|,2dim(∆)}.

Proof. We say that a vertex ℓ ∈ ∆itakes the position s if there is an (m−1)-dimensional

face {a1< a2< ··· < am} of ∆isuch that ℓ = as. In the clique ∆ithere are exactly

min{|V (∆i)|,2dim(∆)} vertices which do not take all m positions. On the other hand the

assumption (ii) implies that each of these vertices can intersect with at most one clique

∆j, (where vjis a neighbor of vi) which completes the proof.

?

Now we are ready to consider primality of J∆for special classes of simplicial complexes.

Theorem 3.2. Let ∆ be simplicial complex such that G∆is a tree. Then

(a) J∆is a prime ideal, if ∆ is closed;

(b) ∆ is closed, if and only if each vertex of G∆has order at most min{|V (∆i)|,2dim(∆)}.

Let {i1 < ... < is} ⊂ [m] and {j1 < ... < jt} ⊂ [n]. We denote by Xj1j2...jt

submatrix of X with rows i1,...,is and columns j1,...,jt. Observe that Lemma 2.1

implies the well-known fact that if I is generated by all m-minors of Xj1...jt

generated by all (m − 1)-minors of the matrix Xj1...ˆjk...jt

i1...is

the

1...m, then Ixijkis

1...ˆi...m

.

Proof of Theorem 3.2. (a) We may assume that ∆ is a connected (m − 1)-dimensional

simplicial complex and that ∆ = ∆1∪∆2∪···∪∆ris the clique decomposition of ∆. The

proof is by induction on the number of cliques of ∆ (which is the number of vertices of

G∆). We may assume that v1is a vertex of degree one in G∆and that v2is its neighbor.

Then ∆1intersects with just one clique, namely ∆2.

13

Page 14

Let V (∆1) = {j1,...,jt} and V (∆2) = {ℓ1,...,ℓs} with m ≤ t,s. We may assume that

V (∆1) ∩ V (∆2) = {k} where k = j1 = ℓ1. Since ∆ is closed, by Lemma 2.3 we know

that xmkis regular modulo J∆. It follows from Lemma 2.1 that (S/J∆)xmk∼= (S/L)xmk

where L = L1+L2+?r

Xj2...jt

of L1are polynomials in a set of variables disjoint from those of L′= L2+?r

is known that L1 is a prime ideal, see [3, Theorem 7.3.1]. Thus L is a prime ideal if

and only L′is a prime ideal. To see this observe that (S/L′)xmk∼= (S/J∆′)xmkwhere

∆′is the closed simplicial complex with clique decomposition ∆′= ∆2∪ ··· ∪ ∆r. By

induction hypothesis, J∆′ is a prime ideal. Hence (S/L′)xmk∼= (S/J∆′)xmkwhich implies

that (J∆′)xmkis a prime ideal. Since the generators of L′are polynomials in variables

different from xmk, it follows that xmkis regular modulo L′. Consequently L′is a prime

ideal.

(b) Due to Lemma 3.1 it suffices to show that ∆ is closed if each vertex of G∆has order

at most min{|V (∆i)|,2dim(∆)}. We prove the assertion by induction on r. As before we

assume that ∆1intersects with just one clique, namely ∆2. By induction it follows that

∆′= ∆2∪ ··· ∪ ∆ris closed. Our assumption on the order of the vertices of G∆implies

that ∆2has at most min{|V (∆i)|,2dim(∆)} − 1 intersection points in ∆′.

Hence among the vertices of ∆2which are not intersection points in ∆′there is at least

one which does not take all m positions, say it misses the kthposition. By symmetry we

may assume this vertex is the intersection point with ∆1. Now we may label ∆1 such

that the vertex in the intersection point does not have position k for any facet of ∆ in

∆1.

i=3J∆i. Here L1is generated by all (m −1)-minors of the matrix

1...m−1, and L2is generated by all (m−1)-minors of the matrix Xℓ2...ℓs

1...m−1. The generators

i=3J∆i. It

?

Theorem 3.3. Let ∆ be a simplicial complex such that G∆is a cycle. Then J∆is a prime

ideal.

Proof. Let ∆ = ∆1∪...∪∆rbe the clique decomposition of ∆. We consider the labeling

on the vertices of ∆ such that

V (∆1) = {1,2,...,a1},V (∆2) = {a1,a1+ 1,...,a2},...,

V (∆r−1) = {ar−2,ar−2+ 1,...,ar−1},V (∆r) = {a1− 1,ar−1,ar−1+ 1,...,ar},

where 1 < a1< ··· < ar−1< ar= n. Then ∆ is closed with respect to the given labeling

and, by Lemma 2.3, x1a1is a regular element modulo J∆. It follows from Lemma 2.1

that (S/J∆)x1a1∼= (S/L)x1a1where L = L1+ L2+?r

all (m − 1)-minors of the matrix X1...a1−1

2...m

and L2is generated by all (m − 1)-minors of

the matrix Xa1+1...a2

2...m

. Therefore, J∆is a prime ideal, if Lx1a1is a prime ideal. Since the

generators of L are polynomials in variables different from x1a1, we conclude that x1a1is

regular modulo L. Hence J∆is a prime ideal if and only if L is a prime ideal.

We first show that the generators of L form a Gr¨ obner basis for L. In order to show

this, note that the generators of?r

since ∆3∪···∪∆ris closed. Also the generators of L1form a Gr¨ obner basis for JΓ1, where

Γ1is the pure (m−2)-dimensional simplicial complex on the vertices {1,...,a1−1}, and

the generators L2form a Gr¨ obner basis for JΓ2, where Γ2is the pure (m−2)-dimensional

simplicial complex on the vertices {a1+1,...,a2}. Finally, we note that the initial ideals

of L1, L2and, respectively,?r

disjoint sets of variables. Consequently, the generators of L form indeed a Gr¨ obner basis.

Next observe that the variable xm−1,a1−1does not appear in the support of the gener-

ators of in<(L). In particular, xm−1,a1−1is regular modulo L. By using Lemma 2.1 we

14

i=3J∆i. Here L1is generated by

i=3J∆i= J∆3∪···∪∆rform a Gr¨ obner basis for?r

i=3J∆i,

i=3J∆i, are minimally generated by monomials in pairwise

Page 15

get (S/L)xm−1,a1−1∼= (S/L′

(m−2)-minors of the matrix X1...a1−2

matrix Xar−1...ar

1...m−2,m.

Since the generators of L′= L′

ferent from xm−1,a1−1, we conclude that xm−1,a1−1is regular modulo L′. Hence Lxm−1,a1−1

is a prime ideal if and only if L′is a prime ideal.

Since L′

from the variables of the other summands, in order to show that L′is prime, it is enough

to show that C = L2+ Lr+?r−1

We define the pure (m−1)-simplicial complex ∆′to be the simplicial complex with clique

decomposition ∆′= ∆2∪···∪∆r. Since the associated graph of ∆′is a tree we know from

Theorem 3.2 that J∆′ is a prime ideal. Since (S/C)x1a1xm−1,a1−1∼= (S/J∆′)x1a1xm−1,a1−1

and since x1a1xm−1,a1−1is regular modulo C, the desired conclusion follows.

1+ L2+ Lr+?r−1

2...m−2,m, and Lris generated by all (m−1)-minors of the

i=3J∆i)xm−1,a1−1, where L′

1is generated by all

1+ L2+ Lr+?r−1

i=3J∆iare polynomials in variables dif-

1is a prime ideal and the generators of L′

1are polynomials in variables different

i=3J∆iis a prime ideal.

?

The next result describes the case when each clique of ∆ is a simplex.

Theorem 3.4. Let ∆ be a simplicial complex with graph G∆such that each clique of ∆

is a simplex. Then the following holds:

(a) If ∆ is closed, then J∆is a prime ideal generated by a regular sequence;

(b) ∆ is closed, if dim∆ + 1 is greater than or equal to the number of facets of ∆;

(c) Given a graph G and an integer m ≥ |V (G)|, there exists a closed simplicial

complex ∆ with G∆ = G such that each clique of ∆ is a simplex of dimension

m − 1.

Proof. (a) Let ∆ = ∆1∪ ··· ∪ ∆r be the clique decomposition of ∆. Since each clique

is a simplex, it follows that J∆i= (fi) for all i, where fi is a suitable m-minor, and

J∆ = (f1,...,fr).Since ∆ is closed the monomials in<(f1),...,in<(fr) are pairwise

relatively prime. This implies that f1,...,fris a regular sequence.

We prove primality of J∆by induction on r. Since ∆ is closed, there exists at least

one free vertex in ∆.We may assume that V (∆1) = {a1,...,as,as+1,...,am} and

a1,...,asare the free vertices of ∆1. By our assumption on ∆ there exist unique cliques

∆is+1,...,∆imsuch that V (∆ij) ∩ V (∆1) = {aj} for j = s + 1,...,m. By Lemma 2.3 we

have that y = xs+1,as+1···xmamis a regular element modulo J∆. Localizing at y amounts

to localize step by step at xs+1,as+1,...,xmam. Thus we may apply Lemma 2.1 and obtain

that (S/J∆)y∼= (S/L)y, where L = L1+?m

minor [1...s|a1...as], and Ljis generated by the (m − 1)-minor [1...ˆj ...m|bj2...bjm],

where ∆ij= {aj,bj2,...,bjm}. Finally, ∆′is the simplicial complex whose cliques are ∆k

for k ?= 1,is+1,...,im.

It remains to show that Lyis a prime ideal. This is the case if and only if L′yis a prime

ideal where L′=?m

L′have no variables in common.

In order to see that L′yis a prime ideal, we consider the pure (m − 1)-dimensional

simplicial complex Γ with clique decomposition Γ = ∆2∪ ··· ∪ ∆r. Since Γ is a closed

simplicial complex with fewer cliques than ∆ we know by the inductive hypothesis that

JΓis a prime ideal. Since y = xs+1,as+1···xmamis a regular element modulo JΓ, and JΓ

is a prime ideal, we conclude that L′yis a prime ideal, since (S/L′)y∼= (S/JΓ)y.

(c) We first assume that m = |V (G)|, and prove in this case the assertion by induction

on the number of vertices of G. The induction beginning is trivial. Now assume that

15

j=s+1Lj+ J∆′. Here L1is generated by the

j=s+1Lj+ J∆′. Indeed, L = L1+ L′, and the polynomials in L1and

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