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A study on monomial ideals and Specht ideals

March 2022

Kosuke Shibata

Graduate School of Natural Science and Technology (Doctor’s Course)

OKAYAMA UNIVERSITY

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Contents

Preface 2

1 Strongly stable ideals 5

1.1 Introduction . . . 5

1.2 The construction of the Alexander duality for strongly stable ideals 7 1.3 The Hilbert series ofHmi(S/I) . . . 13

1.4 Relation to squarefree strongly stable ideals . . . 17

1.5 The irreducible components ofI and b-pol(I) . . . 19

1.6 Remarks on irreducible components of strongly stable ideals . . . . 24

2 Edge ideals 28 2.1 Introduction . . . 28

2.2 Preliminaries . . . 29

2.3 Edge-weighted edge ideals of very well-covered graphs . . . 30

2.4 Examples . . . 36

3 Specht ideals 38 3.1 Introduction . . . 38

3.2 Main theorem and related arguments . . . 40

3.3 The initial monomials of Specht polynomials . . . 41

3.4 The proof of the main theorem and some examples . . . 45  

1

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Preface

Combinatorial commutative algebra lies at the intersection of two more es- tablished fields, commutative algebra and combinatorics. In commutative alge- bra, Cohen-Macaulay and Gorenstein properties, local cohomologies, Castelnuovo- Mumford regularities and Hilbert series are important objects. One of the purposes of this field is to investigate the relationship between these commutative algebraic properties and combinatorial objects. For example, the important events in com- binatorial commutative algebra was R. Stanley’s proof([36]) of the upper bound conjecture for the simplicial sphere, based on early work by M. Hoschster and G.

Reisner. The problem can be defined in combinatorial and geometric terms, but the method of the proof makes full use of commutative ring theory. In this proof, Stanley uses the Stanley-Reisner ring of the simplicial complex.

Monomial ideals are an important subject in combinatorial commutative alge- bra. Standard methods in combinatorial commutative algebra for studying homo- logical and enumerative problems about arbitrary monomial ideals are to reduce to squarefree or Borel fixed cases. Borel fixed ideals are monomial ideals of a poly- nomial ring fixed under the action of upper triangular matrices, and it play an important role in Gr¨obner basis theory and many related areas, since they appear as the generic initial ideals of homogeneous ideals. Moreover, Borel fixed ideals are strongly stable ideals, when characteristic is 0. On the other hand, any monomial ideal is reduced to a squarefree monomial ideal by (standard) polarization.

Edge ideals are also known to be an important object in combinatorial commu- tative algebra. The study of edge ideals was started by Villarreal in [43]. An edge ideal is a quadratic squarefree monomial ideal associated with a finite simple graph.

By using edge ideals, the relationship between ring-theoretic and graph-theoretic properties has been actively investigated. These studies include the characterization of Cohen–Macaulay and Gorenstein properties.

A Specht module is one of the important representations of symmetric groups.

It is studied by W. Specht in 1935. The Specht modules form a complete set of irreducible representations of the symmetric group, in characteristic 0. Such modules are vector spaces spanned by Specht polynomials, which can be constructed combinatorially. We can also consider ideals generated by Specht polynomials. This is called the Specht ideal. Such ideals are known to be related to combinatorial commutative algebra, subspace arrangements, equivariant cohomologies of Springer

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fibers and symmetric system of equations.

The organization of this doctoral thesis is as follows; it consists of three chapters.

In Chapter 1, we study the Alexander duality for strongly stable ideals. In [46], Yanagawa constructed the alternative polarization b-pol(I) of a strongly stable ideal I. Note that b-pol(I) is the squarefree monomial ideal. On the other hand, the Alexander duality for squarefree monomial ideals is a very powerful tool in the Stanley–Reisner ring theory.

In this thesis, we construct the Alexander dual of strongly stable ideal, and as one of its applications, we describe the formula of the Hilbert series of the local cohomology modules of the quotient ring by a strongly stable ideal using its irreducible decomposition. On the other hand, we show that strongly stable property is characterized by its irreducible decomposition.

In Chapter 2, we study the edge-weighted edge ideals. In this chapter, we consider a finite simple graph. The edge-weighted edge ideal of an edge-weighted graph was introduced in [28]. They also investigated unmixedness and Cohen- Macaulayness of these ideals, in the case that a graph is a cycle, a tree or a com- plete graph. The purpose of this thesis is to continue this research on a Cohen–

Macaulay very-well covered graph. In particular, we characterize unmixed and Cohen-Macaulay properties of edge-weighted edge ideals of Cohen–Macaulay very well-covered graphs. Our results can be seen as generalizations of the results con- cerning the Cohen-Macaulay property of usual edge ideals of very well-covered graphs.Another kind of generalization of edge ideals is considered in [17, 29, 30].

Indeed, [29] introduced the vertex-weighted edge ideal of an oriented graph. In this paper, we provide the counterexample of the conjecture[29, Conjecture 53].

In Chapter 3, we study the (Castelnuovo-Mumford) regularity and the Hilbert series of Specht ideals for some partitions. A Specht idealIλSpfor a partitionλ is an ideal generated by the Specht polynomials ofλ. For the partition λ= (n−d, d) or (d, d,1), Yanagawa show that IλSp is a radical ideal over any field, and the quotient ring of these ideals are Cohen–Macaulay using by a result of Etingof et al. [13], which concerns the characteristic 0 case. In addition, in [22], results on the Cohen–

Macaulay property of R/I(nSpd,d) are proved without using the results of Etingof et al. The paper [44] computes the Betti numbers of Specht ideals for hook type partitions, it means that we know its Hilbert series in this case.

In this thesis, we compute the Hilbert series of a quotient ring by a Specht ideal of (n −d, d) and (d, d,1). We also prove that the Hilbert series of these Specht ideals is independent of the characteristic of the field, using the theory of Gr¨obner basis. The main tool in this calculation is the recursive formulas between Specht ideals when considering the number of variables. As an application, we compute the regularity reg(R/IλSp), whenR/IλSp is Cohen–Macaulay.

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Acknowledgments. I would like to express my deep gratitude to my Super- visor Naoki Terai, and Kohji Yanagawa for their careful guidance and warm en- couragement. I am also grateful to Yuji Yoshino, Murai Satoshi, Gunnar Fløystad, Siamak Yassemi, Seyed Amin Seyed Fakhari for stimulating discussion and helpful comments. In addition, I would like to thank Takeshi Suzuki and Masao Ishikawa for their help. Finally, I would also like to express my gratitude to my family for their support.

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Chapter 1

Strongly stable ideals

1.1 Introduction

This chapter is based on the author’s paper [38] with Kohji Yanagawa. Strongly stable ideals are monomial ideals defined by a simple condition, and they appear as the generic initial ideals of homogeneous ideals in the characteristic 0 case (so they are also calledBorel fixed ideals in this case). In a positive characteristic case, the generic initial ideal for any homogeneous ideal is the Borel fixed ideal, but a Borel fixed ideal is not necessarily strongly stable. However, any strongly stable ideal is always Borel fixed.

One of standard methods in combinatorial commutative algebra for treating ho- mological and combinatorics problems about arbitrary monomial ideals is to reduce to the squarefree or Borel-fixed case. In particular, (standard) polarization is often used as a method to reduce general monomial ideals to squarefree monomial ideals.

Extending an idea of [26], Yanagawa([46]) constructed the alternative polariza- tion b-pol(I) of a strongly stable ideal I. We briefly explain this notion here. Let S = K[x1, . . . , xn] be a polynomial ring over a field K. For a monomial ideal I, G(I) denotes the set of minimal monomial generators of I. If I ⊂S is a strongly stable ideal with deg(m)≤dfor allm∈G(I), we consider a larger polynomial ring Se = K[xi,j | 1 i n,1 j d] with the surjection f : Se xi,j 7−→ xi S.

Then we can construct a squarefree monomial ideal b-pol(I) Se (if there is no danger of confusion, we will simply write Iefor b-pol(I)) satisfying the conditions f(I) =e I and βi,jSe(Ie) = βi,jS (I) for all i, j, where βi,j stands for the graded Betti number. The alternative polarization is much more compatible with operations for strongly stable ideals than the standard polarization.

On the other hand, the Alexander duality for squarefree monomial ideals is a very powerful tool in the Stanley–Reisner ring theory. For a squarefree monomial ideal I S, I ⊂S denotes its Alexander dual. There is a one to one correspon- dence between the elements of G(I) and the irreducible components of I. Let Se = K[yi,j | 1 i d,1 j n] be a polynomial ring with the isomorphism

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()t:Se∋xi,j 7−→yj,i ∈Se. For a strongly stable idealI, there is a strongly stable ideal I ⊂K[y1, . . . , yd] with b-pol(I) = (b-pol(I))t. Clearly, the correspondence I ←→I should be considered as the Alexander duality for strongly stable ideals.

After we finished an earlier version of [38], we were informed that, in Fløystad [15, §6], the above duality has been constructed using the notion of generalized (co-)letterplace ideals. Each approach has each advantage. The paper [15] treats the duality in a much wider context, but if one starts from the generator setG(I), our construction is more direct (Proposition 1.31 and Theorem 1.23 give a simple procedure to computeI fromG(I)). We will give a complete proof of the existence of the duality, since we will re-use ideas of the proof in later sections.

The outline of the paper is as follows. Section 2 is mainly devoted to the proof of the existence of the dual I. If I is a Cohen–Macaulay strongly stable ideal, S/e Ieis the Stanley–Reisner ring of a ball or a sphere (a ball in most cases), and its canonical module can be easily described. In Section 3, we show the formula

H(Hmi(S/I), λ1) = ∑

j∈Z

βij,nj(Ij (1−λ)j

on the Hilbert series of the local cohomology moduleHmi(S/I). This is more or less a consequence of a classical result [10], and we will improve this formula later.

In Section 4, we discuss the relation to the notion of asquarefree strongly stable ideal, which is a squarefree analog of a strongly stable ideal. For a strongly sta- ble ideal I S, Aramova et al [1] constructed a squarefree strongly stable ideal Iσ T = K[x1, . . . , xN] with N 0. The class of squarefree strongly stable ideals is closed under the (usual) Alexander duality ofT, so our duality can be con- structed throughIσ. However, without b-pol(I), it is hard to compare the algebraic properties ofI with those ofI.

In Section 5, we give a procedure to construct the irreducible decomposition of b-pol(I) from that of a strongly stable ideal I. As corollaries, we will give formu- las on the arithmetic degree adeg(S/I) and H(Hmi(S/I), λ) from the irredundant irreducible decomposition

I = ∩

aE

ma

with E Z>0 (Z>0)2 ∪ · · · ∪(Z>0)n. Here, for a = (a1, . . . , at) (Z>0)t with t n, ma denotes the irreducible ideal (xa11, . . . , xatt) of S. In this situation, set t(a) :=t,e(a) :=at, and w(a) := n−t

i=1ai. Then we have adeg(S/I) =∑

aE

e(a) and

H(Hmi(S/I), λ1) =



aE, t(a)=ni

w(a)+λw(a)+1+· · ·+λw(a)+e(a)1)



/(1−λ)i.

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Section 6 gives additional results on the irreducible decompositions of strongly stable ideals. While a strongly stable ideal I is characterized by the “left shift property” on G(I), Theorem 1.35 states that it is also characterized by the “right shift property” on the irreducible components of I.

1.2 The construction of the Alexander duality for strongly stable ideals

In this section, we define the Alexander duality for strongly stable ideals using the alternative polarization. As applications, we show that the alternative polar- ization of a Cohen-Macaulay strongly stable ideal is the Stanley–Reisner ideal of a ball or a sphere, and give a description of its canonical module.

First, we introduce the convention and notation used throughout the paper. For a positive integer n, set [n] :={1, . . . , n}. Let S:=K[x1, . . . , xn] be a polynomial ring over a field K, and m = (x1, . . . , xn) the unique graded maximal ideal of S.

For a monomial idealI ⊂S, G(I) denotes the set of minimal monomial generators of I. We say an ideal I S is strongly stable, if it is a monomial ideal, and the condition that m∈G(I), xi|m and j < i imply (xj/xi)·m∈I is satisfied.

Let d be a positive integer, and set

Se:=K[xi,j|1≤i≤n,1≤j ≤d].

Note that

Θ := {xi,1 −xi,j|1≤i≤n, 2≤j ≤d} ⊂Se

forms a regular sequence with the isomorphismS/(Θ)e =S induced bySe∋xi,j 7−→

xi ∈S.

Definition 1.1. For a monomial ideal I S, a polarization of I is a squarefree monomial ideal J ⊂Se satisfying the following conditions.

(1) Through the isomorphism S/(Θ)e =S, we have S/(Θ)e SeS/Je =S/I. (2) Θ forms aS/Je -regular sequence.

For a = (a1, . . . , an) Nn, xa denotes the monomial ∏n

i=1xaii S. For a monomial xa ∈S with deg(xa)≤d, set

pol(xa) := ∏

1≤i≤n

xi,1xi,2· · ·xi,ai ∈S.e

IfI ⊂Sis a monomial ideal with deg(m)≤dfor allm∈G(I), then it is well-known that

pol(I) := ( pol(m)|m∈G(I) )

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is a polarization of I, which is called the standard polarization.

Any monomial m∈S has a unique expression m=

e i=1

xαi with 1≤α1 ≤α2 ≤ · · · ≤αe≤n. (1.2.1) Ife(= deg(m))≤d, we set

b-pol(m) :=

e i=1

xαi,i ∈S.e (1.2.2)

As another expression, for a monomial xa ∈S with deg(xa)≤d, set bi :=∑i

j=1aj for each i≥1 and b0 = 0. Then

b-pol(xa) = ∏

1≤i≤n bi1+1jbi

xi,j ∈S.e

For a monomial ideal I S with deg(m) d for all m G(I) (in the sequel, we always assume this condition), set

b-pol(I) := ( b-pol(m)|m∈G(I) )⊂S.e See the beginning of Example 1.6 below.

In [46], Yanagawa showed the following.

Theorem 1.2([46, Theorem 3.4]). IfI ⊂S is a strongly stable ideal, then b-pol(I) gives a polarization of I.

In the rest of the paper, the next fact is frequently used without comment.

Lemma 1.3. Let I S be a strongly stable ideal. For a monomial m S with deg(m)≤d, m∈I if and only if b-pol(m)b-pol(I).

Proof. The necessity is shown in [46, Lemma 3.1], and the sufficiency is an easy exercise.

Anirreducible monomial idealof S is an ideal of the form (xaii|ai >0) for some a Nn. A presentation of a monomial ideal I as an intersection I = ∩r

i=1Qi of irreducible monomial ideals is called an irreducible decomposition. An intersection I =∩r

i=1Qi is irredundant, if none of the ideals Qi can be omitted in this presen- tation. Any monomial ideal has a unique irredundant irreducible decomposition I =∩r

i=1Qi. In this case, each Qi is called an irreducible component of I. IfI is a squarefree monomial ideal, then the irreducible components are nothing other than the associated primes.

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IfI ⊂Sis a squarefree monomial ideal (equivalently,S/I is the Stanley–Reisner ring of some simplicial complex), then the irreducible components of I are of the formmF := (xi |i∈F) for someF [n], and the ideal

I :=( ∏

iF

xi |mF is an irreducible component of I )

called the Alexander dual of I. Then we have I∨∨ = I. This duality is very important in the Stanley–Reisner ring theory. See, for example, [10, 23].

Lemma 1.4. For a monomial idealI ⊂S, the following conditions are equivalent.

(1) I is strongly stable.

(2) b-pol(I)⊂Sehas an irreducible decompositionr

s=1Pssatisfying the following property.

(∗) For each s, there is a positive integerts, and integers γis for1≤i≤ts such that Ps= (xi,γ⟨s⟩

i |1≤i≤ts) and 1≤γ1s≤γ2s ≤ · · · ≤γtss. Proof. (1) (2): This is shown already in [46, Remark 3.3].

(2) (1): For a contradiction, assume thatIe:= b-pol(I) satisfies the condition () but I is not strongly stable. Then it is easy to see that there is some m = xa G(I) such that xj+1|m and (xj/xj+1)·m̸∈ I for some j < n. Then we have b-pol((xj/xj+1)·m) ̸∈ b-pol(I), and it implies that b-pol((xj/xj+1)·m) ̸∈ Ps = (x1,γ⟨s⟩

1

, x2,γ⟨s⟩

2

, . . . , xt

sts⟨s⟩) for some s. As before, set b0 := 0 and bi :=∑i

j=1aj for i≥1. Since

b-pol(m) = ∏

1in bi1+1jbi

xi,j,

we have γis ̸∈ {bi1+ 1, . . . , bi} for all i ̸=j, j + 1, γjs ̸∈ {bj1+ 1, . . . , bj+ 1}, and γj+1s ̸∈ {bj + 2, . . . , bj+1}. Here we have b-pol(m) b-pol(I) Ps, and it impliesγj+1s =bj+ 1. Sinceγjs ≤γj+1s (= bj+ 1) andγjs ̸∈ {bj1+ 1, . . . , bj+ 1}, we have γjs bj1. If j 2, combining γj−1s γjs( bj1) with γj−1s ̸∈ {bj2+ 1, . . . , bj1}, we have γjs1 bj2. Repeating this argument, we have γ1s b0. Since γ1s1 and b0 = 0, this is a contradiction.

Let Se := K[yi,j|1 i d,1 j n] be a polynomial ring with the ring isomorphism ()t:Se→Se defined by Se∋xi,j 7−→yj,i ∈Se.

Theorem 1.5 (c.f. [15]). Let I ⊂S be a strongly stable ideal. Then there exists a strongly stable ideal I ⊂S :=K[y1, . . . , yd] such that b-pol(I) = (b-pol(I))t.

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Proof. As before, set Ie:= b-pol(I). There is a one to one correspondence between the irreducible components of Ie and the elements of G(Ie). If the irrdundant irreducible decomposition ofIeis given by

Ie=

r s=1

(xi,γ⟨s⟩

i |1≤i≤ts)⊂S,e then we have

(Ie)t= (∏ts

i=1

yγs

i ,i |1≤s ≤r

)⊂Se.

Since γ1⟨s⟩≤γ2⟨s⟩ ≤ · · · ≤γt⟨s⟩s by Lemma 1.4, we have b-pol(I) = (Ie)t for I =

(∏ts

i=1

yγs

i |1≤s≤r

)⊂S.

There also exists a one to one correspondence between the irreducible compo- nents of Ie and the elements of G(Ie), equivalently, the elements of G(I). If the monomial m in (1.2.1) belongs to G(I), the irreducible component of Ie given by m is of the form (xα1,1, xα2,2, . . . , xαe,e) by the expression (1.2.2). Then the cor- responding irreducible component of (Ie)t(= b-pol(I)) is (y1,α1, . . . , ye,αe) Se. Since α1 ≤ · · · ≤αe,I is strongly stable by Lemma 1.4.

The above theorem gives a duality between strongly stable ideals I S = K[x1, . . . , xn] whose generators have degree at most d and strongly stable ideals I ⊂S =K[y1, . . . , yd] whose generators have degree at most n.

Example 1.6. For a strongly stable ideal I = (x21, x1x2, x1x3, x22, x2x3), we have b-pol(I) = (x1,1x1,2, x1,1x2,2, x1,1x3,2, x2,1x2,2, x2,1x3,2)

= (x1,1, x2,1)(x1,1, x2,2, x3,2)(x1,2, x2,2, x3,2) b-pol(I) = (x1,1x2,1, x1,1x2,2x3,2, x1,2x2,2x3,2)

(b-pol(I))t = (y1,1y1,2, y1,1y2,2y2,3, y2,1y2,2y2,3), hence the dual strongly stable ideal is given by

I = (y12, y1y22, y23).

On the other hand, if we use the standard polarization, we have pol(I) = (x1,1x1,2, x1,1x2,1, x1,1x3,1, x2,1x2,2, x2,1x3,1)

= (x1,1, x2,1)(x1,1, x2,2, x3,1)(x1,2, x2,1, x3,1) pol(I) = (x1,1x2,1, x1,1x2,2x3,1, x1,2x2,1x3,1).

Here (pol(I))t = (y1,1y1,2, y1,1y1,3y2,2, y1,2y1,3y2,1) can not be the standard or al- tarnative polarization of any ideal.

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The next two results are implicitly contained in Fløystad [15]. However they are stated in the context of the preceding papers [16, 8], where the words “letterplace ideal” and “coletterplace ideals” are used in the narrow sense (see Remark 1.8 below).

Proposition 1.7. If I S is a strongly stable ideal with

I = m, then b-pol(I) (more precisely, b-pol(I)t) is the letterplace ideal L(J;d,[n]) in the sense of [8].

HereJ is an order ideal ofHom([n],[d]). Conversely, any letterplace idealL(J;d,[n]) arises in this way from a strongly stable ideal I with

I =m.

Proof. If I S is a strongly stable ideal with

I = m, then the dual I S = K[y1, . . . , yd] is a strongly stable ideal whose minimal generators all have degree n.

As shown in [16, §6.1], b-pol(I) is a co-letterplace idealL([n], d;J) for some order ideal J ⊂ Hom([n],[d]). Then the Alexander dual of b-pol(I), which coincides with b-pol(I)t, is the letterplace idealL(J;d,[n]) by definition.

The second assertion follows from the fact that any co-letterplace idealL([n], d;J) is the b-pol() of some strongly stable ideal whose generators all have degreen.

Remark 1.8. In [15], Fløystad generalized the notions of a (co-)letterplace ideal so that b-pol(I) of any strongly stable ideal I belongs to these classes (one of the crucial points is considering an order ideal J in Hom([n],N), not in Hom([n],[d])).

Through this idea, he gave the duality.

For a monomial xa S with a = (a1, . . . , an) Nn, set ν(xa) := max{i |ai >

0}. It is well-known that ifI is strongly stable, then

projdimS(S/I) = max{ν(m)|m∈G(I)} and ht(I) = max{i|xi ∈√ I}. Hence, for a strongly stable ideal I with ht(I) =c,S/I is Cohen–Macaulay if and only if ν(m)c for all m G(I), if and only if m K[x1, . . . , xc] for all m G(I).

Of course,S/e b-pol(I) is Cohen–Macaulay if and only if so isS/I.

Corollary 1.9. Let (0) ̸= I S be a Cohen–Macaulay strongly stable ideal, and setIe:= b-pol(I). ThenS/e Ieis the Stanley–Reisner ring of a ball or a sphere. More precisely, if n≥2, then S/e Ieis the Stanley–Reisner ring of a ball.

If n = 1, thenI = (xe) for somee ≤d. Hence Ie= (x1,1x1,2· · ·x1,e), andS/e Ieis the Stanley–Reisner ring of a sphere (resp. ball) if e=d (resp. e < d).

Proof. First, assume that

I =m. In this case, Ieis a letterplace ideal L(J;d,[n]) by Proposition 1.7, and the assertion follows from [8, Theorem 5.1] (note that the poset [n] is an antichain if and only ifn = 1).

If

I ̸=m (equivalently, c := ht(I) < n), then we have I = J S for a strongly stable ideal J K[x1, . . . , xc] with

J = (x1, . . . , xc). Moreover, the simplicial complex associated with Ieis the cone over the one associated with b-pol(J). So the assertion can be reduced to the first case.

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For xa ∈S with deg(xa)≤d and l :=ν(xa), set µ(xa) :=

(∏l1

i=1

xi,bi+1

)·b-pol(xa),

where bi := ∑i

j=1aj for each i as before. In [27], R. Okazaki and Yanagawa con- structed a minimal S-free resolutione Pe of b-pol(I) of a strongly stable ideal I. If S/I is a Cohen-Macaulay ring of codimension c, the “last” term Pec of the minimal free resolution is isomorphic to

mG(I) ν(m)=c

S(e deg(µ(m))).

We also set

Xe := ∏

1in 1jd

xi,j

and

ω(m) :=X/µ(m)e for m∈G(I).

Corollary 1.10. Let (0) ̸=I ⊂S be a Cohen–Macaulay strongly stable ideal with ht(I) =c, and set Ie:= b-pol(I). Then the canonical module ωS/e Ieis isomorphic to the ideal of S/e Iegenerated by (the image of ) {ω(m)|m∈G(I), ν(m) =c}.

Proof. By Corollary 1.9,S/e Ieis the Stanley–Reisner ring of a ball or a sphere. Recall that, for the Stanley–Reisner ring K[∆] of a simplicial sphere ∆, K[∆] itself is the multigraded canonical module ofK[∆] (see [4, Corollary 5.6.5]). If ∆ is a simplicial ball, then the boundary ∂∆ is a sphere. Hence the ideal of K[∆] generated by all squarefree monomials associated with the faces ∆\∂∆ is a canonical module of K[∆] by [4, Theorem 5.7.2]. Anyway, the canonical module ωS/ee I is isomorphic to a multigraded ideal of S/e I. Sincee ωS/ee I = ExtcSe(S/e I, ωe Se) and ωSe is isomorphic to the principal ideal (X) ofe S,e ωS/ee I is a quotient of

HomSe(Pec, ωSe)= ⊕

mG(I) ν(m)=c

S(e deg(ω(m))).

So we are done.

For a Cohen–Macaulay strongly stable idealI, the canonical moduleωS/IofS/I itself is isomorphic toωS/ee ISeS/(Θ) and Θ forms a (ωe S/ee I)-regular sequence, where

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Θ = {xi,1−xi,j|1 i n, 2 j d}. However, ωS/I is not isomorphic to an ideal of S/I in general.

We also remark that [8, Corollary 4.3] gives a description of the canonical mod- ule of the quotient ring of a letterplace ideal, and it also works in the case of Corollary 1.10. However, our description is much simpler in this case.

1.3 The Hilbert series of H

mi

(S/I )

In this section, for a strongly stable ideal I, we show that the Hilbert series of Hmi(S/I) can be described by the irreducible decomposition of b-pol(I).

Let R = K[x1, . . . , xm] be a polynomial ring. For a Z-graded R-module M, H(M, λ) denotes the Hilbert series

i∈Z(dimKMii of M. Let ωR denote the graded canonical moduleR(−m) of R.

The following must be a fundamental formula on the Alexander duality of Stanley–Reisner ring theory, but we cannot find any reference.

Lemma 1.11. LetR =K[x1, . . . , xm]be a polynomial ring, and I ⊂R a squarefree monomial ideal. Then we have

H(ExtmRi(R/I, ωR), λ) = ∑

j0

βij,mj(Ij (1−λ)j .

Here I ⊂R is the Alexander dual of I, and βp,q(I) is the graded Betti number of I, that is, the dimension of [TorRp(I, K)]q.

Proof. Fora= (a1, . . . , am)Nm, the vector a= (a1, . . . , am)Nm is defined by ai =

{1 if ai 1, 0 if ai = 0.

By [45, Theorem 2.6], ExtiR(R/I, ωR) is a squarefree module. Hence we have [ExtiR(R/I, ωR)]a = 0 for all aZm\Nm, and

[ExtiR(R/I, ωR)]a = [ExtiR(R/I, ωR)]a

for all aNm. Furthermore, it is well-known (cf., [45, Theorem 3.4]) that [ExtiR(R/I, ωR)]a = [TorRm−|a|−i(Ie, K)]1a.

Here we set 1:= (1, . . . ,1)Nm, and |b| :=∑m

i=1bi for b = (b1, . . . , bm)Nm. It is also well-known that [TorRi (Ie, K)]a ̸= 0 for aZm impliesa is a 0-1 vector.

So we have

dimK[ExtmRi(R/I, ωR)]0 =βi,m(I)

(15)

and

dimK[ExtmRi(R/I, ωR)]l =

l j=1

a∈Nm

|a|=l,|a|=j

dimK[ExtmRi(R/I, ωR)]a

=

l j=1

a∈Nm a=a,|a|=j

(l−1 l−j

)

dimK[ExtmRi(R/I, ωR)]a

=

l j=1

a∈Nm a=a,|a|=j

(l−1 l−j

)

dimK[TorRij(I, K)]1a

=

l j=1

(l−1 l−j

)

βij,mj(I)

for l >0. So the assertion follows from the following computation

j0

βij,mj(Ij

(1−λ)j = βi,m(I) +∑

j1

{

βij,mj(Ij·

p0

(j +p−1 p

) λp

}

= βi,m(I) +∑

l1

{ l

j=1

(l−1 l−j )

βij,mj(I) }

λl

= dimK[ExtmRi(R/I, ωR)]0+∑

l1

dimK[ExtmRi(R/I, ωR)]l·λl, where l:=j+p.

Corollary 1.12. For a strongly stable ideal I ⊂S with Ie:= b-pol(I), we have H(Extndei

S (S/e I, ωe Se), λ) =∑

j0

βij,ndj(Ij (1−λ)j .

Proof. The assertion follows from Lemma 1.11 (applying toIe⊂S) and the equalitye βp,q(Ie) =βp,q(I).

Theorem 1.13. Let I ⊂S be a strongly stable ideal. Then the Hilbert series of the local cohomology module Hmi(S/I) can be described as follows.

H(Hmi(S/I), λ1) =∑

j∈Z

βij,nj(Ij (1−λ)j .

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Proof. Set Θ := {xi,1−xi,j|1 i≤ n, 2 ≤j d}. By the full statement of [46, Theorem 3.4], if ExtiSe(S/e I,e S)e ̸= 0, then Θ forms an ExtiSe(S/e I,eS)-regular sequence.e Hence we have

[S/(Θ)e SeExtnei

S (S/e I, ωe Se)](nd−n)∼= ExtnSi(S/I, ωS) and

H(ExtnSi(S/I, ωS), λ) = λnnd·H(S/(Θ)e SeExtnei

S (S/e I, ωe Se), λ)

= λnnd(1−λ)ndn·H(Extnei

S (S/e I, ωe Se), λ)

= λnnd(1−λ)ndn

j0

βndn+ij,ndj(Ij (1−λ)j , where the last equality follows from Corollary 1.12. Replacing j bynd−n+j, we have

H(Hmi(S/I), λ−1) = H(ExtnSi(S/I, ωS), λ)

= λnnd(1−λ)ndn

jnnd

βij,nj(Indn+j (1−λ)ndn+j

= ∑

jnnd

βij,nj(Ij (1−λ)j .

Here the first equality follows from the fact thatHmi(S/I) is the graded Matlis dual of ExtnSi(S/I, ωS).

Corollary 1.14. Let I S be a strongly stable ideal. Then S/I is a Cohen- Macaulay ring if and only if I has a linear resolution.

Proof. Follows from Theorem 1.13, or from [10, Theorem 3].

Corollary 1.15. Let I be a strongly stable ideal. If the irredundant irreducible decomposition of b-pol(I) is of the form

b-pol(I) =

r s=1

(xi,γ⟨s⟩

i |1≤i≤ts)⊂S,e (1.3.1) then we have

H(Hmi(S/I), λ1) =

j1#{s∈[r]|ts=n−i, γtss =j}λij+1

(1−λ)i .

Proof. By the additivity of the statement, it suffices to compute how an irreducible component

Ps = (xi,γ⟨s⟩

i |1≤i≤ts)

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of b-pol(I) contributes to the Hilbert series H(Hmi(S/I), λ1). For simplicity, set γ =γt<s>s and t=ts. This component gives

t i=1

yγ<s>

i ∈G(I).

By the Eliahou-Kervaire formula ([12]), the contribution ofPs to the Betti numbers of I is

{

0 if j ̸=t, (γ1

i

) if j =t,

for βi,i+j(I), equivalently, {

0 if n−i̸=t, (γ1

ij

) if n−i=t,

for βij,nj(I). Hence, by Theorem 1.13, Ps concerns Hmi(S/I) if and only if i = n t. Moreover, if i = n −t, the contribution to H(Hmi(S/I), λ1) is the following

i j=iγ+1

(γ1

ij

)λj (1−λ)j =

i

j=iγ+1(1−λ)ij(γ1

ij

)λj (1−λ)i

=

γ1

k=0(1−λ)k(γ1

k

)λik

(1−λ)i (here k =i−j)

=

(∑γ1

k=0(1−λ)k(γ1

k

)λγ1k) λiγ+1 (1−λ)i

= ((1−λ) +λ)γ1λiγ+1 (1−λ)i

= λi−γ+1 (1−λ)i. So the proof is completed.

Example 1.16. For the strongly stable ideal I in Example 1.6, b-pol(I) has two height 3 irreducible components P2 = (x1,1, x2,2, x3,2) and P3 = (x1,2, x2,2, x3,2).

Clearly, γ32 =γ33 = 2 in the above notation. Hence we have H(Hm0(S/I), λ1) = 2λ−2+1= 2λ−1 by Corollary 1.14.

In Section 5, we will give a procedure to construct the irreducible decomposition of b-pol(I) from that of I itself. After this, we will return to the Hilbert series of Hmi(S/I). See Corollary 1.29 below.

参照

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