Cohen-Macaulay Modifications of Squarefree Monomial Ideals GC

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Cohen–Macaulay squarefree monomial ideals admit nontrivial Cohen–Macaulay modi- .... Here the support of a monomial u is the set supp(u) = {i| xi divides u}.
Cohen-Macaulay Modifications of Squarefree Monomial Ideals

Since 1864

Name

:

Safyan Ahmad

Year of Admission :

2005

Registration No.

30-GCU-PHD-SMS-05

:

Abdus Salam School of Mathematical Sciences

GC University Lahore, Pakistan

Cohen-Macaulay Modifications of Squarefree Monomial Ideals

Submitted to Abdus Salam School of Mathematical Sciences GC University Lahore, Pakistan

in the partial fulfillment of the requirements for the award of degree of

Doctor of Philosophy in

Mathematics By

Name

:

Safyan Ahmad

Year of Admission

:

2005

Registration No.

:

30-GCU-PHD-SMS-2005

Abdus Salam School of Mathematical Sciences GC University Lahore, Pakistan

i

DECLARATION I, Mr. Safyan Ahmad Registration No. 30-GCU-Ph.D.-SMS-05 student at Abdus Salam School of Mathematical Sciences GC University in the subject of Mathematics session (2005-2009), hereby declare that the matter printed in this thesis titled “Cohen-Macaulay Modifications of Squarefree Monomial Ideals” is my own work and that

(i)

I am not registered for the similar degree elsewhere contemporaneously.

(ii)

No direct major work had already been done by me or anybody else on this topic; I worked on for the Ph.D. degree.

(iii)

The work, I am submitting for the Ph.D. degree has not already been submitted elsewhere and shall not in future be submitted by me for obtaining similar degree from any other institution.

Dated: -------------------------

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Signature

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RESEARCH COMPLETION CERTIFICATE Certified that the research work contained in this thesis titled “Cohen-Macaulay Modifications of Squarefree Monomial Ideals” has been carried out and completed by Mr. Safyan Ahmad Registration No. 30GCU-Ph.D.-SMS-2005 under my supervision.

-----------------------------

-------------------------------

Date

Supervisor Prof. Juergen Herzog

Submitted Through

Prof. Dr. A. D. Raza Choudary

--------------------------------

Director General

Controller of Examination

Abdus Salam School of Mathematical Sciences

GC University

GC University Lahore, Pakistan.

Lahore, Pakistan.

iii

To my .parents

In this dissertation we study Cohen–Macaulay monomial ideals with a given radical. Among this set of ideals are the so-called Cohen–Macaulay modifications. Not all Cohen–Macaulay squarefree monomial ideals admit nontrivial Cohen–Macaulay modifications. It is shown that if there exists one such modification, then there exist indeed infinitely many. We also present classes of Cohen–Macaulay squarefree monomial ideals with infinitely many nontrivial Cohen–Macaulay modifications. We also study ideals which are the intersections of irreducible monomial ideals of height 2, whose radical is Cohen–Macaulay. Such ideals are naturally associated to a graph. For these ideals we give sufficient conditions in terms of the graph to be Cohen– Macaulay.

All praise unto Allah, Lord of all the worlds. The most Affectionate, The Merciful. Firstly, I am deeply grateful to my supervisor, Prof. Juergen Herzog for his quality supervision, encouragement and guidance. I would like to thank Prof. Dorin Popescu for his help and encouragement during the research. I would also like to thank Dr. A. D. Raza Chaudhry for providing us world class faculty at our door steps and facilitating us with best research environment. Special thanks to Dr. Asif Naseer who introduced me to Algebra, without his help this work was not possible. Secondly, I would like to express my gratitude to my parents, my sisters and my brother for their prayers, steady support and love. Specially, I am very grateful to my bother for facilitating me with advanced technology and encouraging me. Finally, I wish to thank the following: Higher education commission of Pakistan for partial financial support, Muhammad Naeem for his help during course work and research, Awais and Nauman for the solutions of official and computer related problems respectively and all classmates for their nice company.

Contents Introduction 1 Preliminaries 1.1 Some monomial algebra . . . . . . . . . . . . . . . . 1.1.1 Monomial ideals . . . . . . . . . . . . . . . . 1.1.2 Primary decomposition . . . . . . . . . . . . . 1.1.3 Cohen–Macaulay rings and modules . . . . . . 1.1.4 Polarization . . . . . . . . . . . . . . . . . . . 1.1.5 Complete intersections . . . . . . . . . . . . . 1.1.6 Graded free resolutions . . . . . . . . . . . . . 1.2 Basic facts on graphs . . . . . . . . . . . . . . . . . . 1.2.1 Graphs and Trees . . . . . . . . . . . . . . . . 1.2.2 The marriage problem . . . . . . . . . . . . . 1.2.3 Chordal graphs . . . . . . . . . . . . . . . . . 1.2.4 Cohen–Macaulay graphs . . . . . . . . . . . . 1.3 Stanley-Reisner and facet ideal of a simplicial complex 1.3.1 Basic concepts . . . . . . . . . . . . . . . . . 1.3.2 The Alexander dual . . . . . . . . . . . . . . . 1.3.3 The Eagon-Reiner theorem . . . . . . . . . . . 1.3.4 Simplicial trees . . . . . . . . . . . . . . . . .

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2 Cohen–Macaulay monomial ideals with given radical 2.1 A Characterization of Monomial Complete Intersections . . . 2.2 On the existence of nontrivial Cohen–Macaulay modifications 2.3 Classes which admit nontrivial Cohen–Macaulay modifications 2.3.1 Simplicial complexes of dim ≤ 1 . . . . . . . . . . . 2.3.2 Simplicial trees . . . . . . . . . . . . . . . . . . . . . 2.3.3 Monomial ideals of codimension 2 . . . . . . . . . .

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3 Cohen-Macaulay intersections 3.1 Cohen–Macaulay intersections associated to a graph . . . . . . . . . . . 3.2 Constructions of chordal graphs and the proof of the main theorem . . . .

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Bibliography

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Introduction Monomials give a link between Commutative Algebra and Combinatorics. Monomial algebras are studied in the books of Bruns-Herzog [4], Stanley [26], Hibi [20] and Villarreal [28]. In this thesis we concentrate only on squarefree monomial ideals. With a simplicial complex ∆ one can associate two squarefree monomial ideals: the Stanley-Reisner ideal I∆ whose generators correspond to the non-faces of ∆, and the facet ideal I(∆) whose generators correspond to the facets of ∆. The work of Stanley [26] has demonstrated that there are deep relations between the combinatorial properties of ∆ and algebraic properties of I∆ . Edge ideals associated to graphs were firstly considered by Villareal [28], which are in fact generated by monomials corresponding to the edges of the graph and can be seen as special case of facet ideals. Villarreal shows some relations among algebraic properties of edge ideals and combinatorial properties of graphs. Trees are the simplest graphs. In [12] Faridi, generalizes the notion of trees for simplicial complexes of any dimension. In the first chapter of the thesis we recall the basic notions concerning monomial ideals, graphs and simplicial complexes. This chapter is divided into three parts: in the first part, we recall basic properties of monomial ideals, for instance, primary decomposition and polarization are discussed and we describe some properties of polarization which can be found in [11]. A Cohen-Macaulay ring is a particular type of commutative ring, possessing some of the algebraic-geometric properties of a collection of nonsingular points. They are named after Francis Sowerby Macaulay, who proved the unmixedness theorem for polynomial rings in Macaulay (1916), and after Irvin S. Cohen, who proved the unmixedness theorem for formal power series rings in Cohen (1946). (All Cohen-Macaulay rings have the unmixedness property.) Let R is Noetherian and M a finite R module, if the “algebraic” invariant depth(M) equals the “geometric” invariant dim(M), then M is called a Cohen–Macaulay module. Complete intersections are the ideals which are generated by regular sequences and are in particular Cohen–Macaulay. Minimal graded free resolutions are also introduced here. In the second part of Chapter 1 we introduce some basic facts from graphs theory and the well known marriage problem (Theorem 6.1.8, [28]) is discussed. One of the fascinating results in classical graph theory is Dirac’s theorem [6] on chordal graphs appeared in 1960 and which states that, a finite graph G is chordal if and only it it has a perfect elimination ordering. In commutative algebra, the chordal graph first appeared in the work of Fr¨oberg in about 1990. Herzog, Hibi and Zheng [17] classify all Cohen–Macaulay chordal graphs. Also the notion of Cohen–Macaulay graphs is introduced here. Third part of Chapter 1 deals with combinatorics. Here we recall the definition of simplicial complex and study the Stanley-Reisner and facet ideals associated to a simplicial complex. The notion of simplicial trees is also described. The Alexander dual of a

simplicial complex plays an essential role in combinatorics and commutative algebra. We recall the well known Hochster’s formula which can be used to compute the graded Betti numbers of the Stanley-Reisner ideal of a simplicial complex and it can also be used to prove a famous result by Eagon–Reiner [7]. In order to describe the results of this dissertation, let FI be the set of monomial ideals √ J ⊂ S with the property that the radical J of J coincides with I. We are interested in the set of monomial ideals J ∈ FI such that S/J is Cohen–Macaulay. There is a natural subset GI ⊂ FI . In order to describe this set, we denote as usual the unique minimal system of monomial generators of I by G(I). Let G(I) = {u1 , . . . , um }. Then we call a monomial ideal J a modification of I, if G(J) = {v1 , . . . , vm } and supp(vi ) = supp(ui ) for all i. Here the support of a monomial u is the set supp(u) = {i| xi divides u}. Obviously, any modification of I belongs to FI , and we denote the set of modifications of I by GI . A monomial ideal J is called a trivial modification of I, if there exist nonnegative integers a1 , . . . , an such that J is obtained from I by the substitution xi 7→ xai i for i = 1, . . . , n. If J is a trivial modification of I, then J is Cohen–Macaulay, since J = ϕ (I)S where ϕ : S → S is the flat K-algebra homomorphism with ϕ (xi ) = xai i for all i. Thus the questions discussed here can be specified as follows: For which squarefree Cohen–Macaulay monomial ideals does there exist at least one nontrivial Cohen– Macaulay modification, or do exist infinitely many nontrivial Cohen–Macaulay modifications, and for which Cohen–Macaulay monomial ideals I are all monomial ideals J ∈ GI or J ∈ FI Cohen–Macaulay? Related questions have been studied in [19] and [1]. The last of these questions is answered in Section 1 of Chapter 2 where it is shown in Theorem 2.1.1 that all modifications of I are Cohen–Macaulay (respectively unmixed), if and only if I is a complete intersection. As a consequence one obtains that all J ∈ FI are Cohen–Macaulay if and only if I = (x1 , . . . , xn ), see Corollary 2.1.6. In the Section 2 of Chapter 2 we show that the Stanley–Reisner ideal of the natural triangulation of the real projective plane has only trivial Cohen–Macaulay modifications. The proof of this fact is based on Theorem 2.2.4 where it is shown that a squarefree Cohen–Macaulay monomial ideal has a nontrivial Cohen–Macaulay modification if and only if it has infinitely many nontrivial Cohen–Macaulay modifications, and that this property can be checked in a finite number of steps. For the proof of this theorem we use partial polarization as well as a theorem of Takayama [27, Theorem 1] which allows to compute local cohomology of arbitrary monomial ideals in terms of simplicial cohomology of certain simplicial complexes. Theorem 2.2.4 is used to show that the Stanley–Reisner ideal of the natural triangulation of the real projective plane has only trivial Cohen–Macaulay modifications, but in contrast to complete intersections not all modifications are trivial. Finally in Section 3 of Chapter 2 we consider classes of squarefree Cohen–Macaulay monomial ideals which admit infinitely many nontrivial Cohen–Macaulay modifications. First we show (Proposition 2.3.1 and Theorem 2.3.2) that if ∆ is a simplicial complex of

dimension ≤ 1 which is not a complete intersection, then I∆ has this property. This shows that with respect to dimension the Stanley–Reisner ideal of the natural triangulation of the real projective plane is the smallest example which has only trivial Cohen–Macaulay modifications. We also show that the facet ideal of a Cohen–Macaulay simplicial tree (Corollary 2.3.4), and any squarefree Cohen-Macaulay monomial ideal of codimenion 2 have infinitely many nontrivial Cohen–Macaulay modifications. In the third chapter, we complement results of Herzog, Takayama and Terai [19] where they consider the following question: for a subset F ⊂ [n], let PF be the prime ideal generated by the xi with i ∈ F. The minimal prime ideals of a squarefree monomial ideal I in the polynomial ring S = K[x1 , . . . , xn ] over a field K are all of this form, and since T I is a radical ideal it is the intersection of its minimal prime ideals, say, I = ri=1 PFi with Fi ⊂ [n]. Suppose I is Cohen-Macaulay (which by convention means that S/I is T a Cohen–Macaulay). For which exponents ai j is the ideal J = ri=1 (x j i j : j ∈ Fi ) again Cohen-Macaulay? In that paper the authors describe all squarefee monomial ideals I with the property that J is Cohen–Macaulay for all choices of the numbers ai j . Here we consider a special case of the above question and find a partial answer in that case. We restrict our attention to subsets F ⊂ [n] with |F| = 2, thus we consider T Cohen–Macaulay ideals of the form I = {vi ,v j }∈E(G) (xi , x j ), where G is a simple graph on the vertex set V (G) = {v1 , v2 , . . ., vn } and edge set E(G). In the main theorem of this chapter (Theorem 3.1.3) we give conditions in terms of G when for suitable choices of the ai j the resulting ideal J is Cohen–Macaulay, the conditions in Theorem 3.1.3 are only sufficient. Related to our theorem are results of Sarfraz Ahmad and Dorin Popescu in [2] where they show that certain classes of ideals in the polynomial ring in 4 variables are T pretty clean. We associate to G the ideal IG = {vi ,v j }∈E(G) (xi , x j ) and I(G) = ({xi x j } : {vi , v j } ∈ E(G)). By using the Alexander duality and a result of Eagon and Reiner [7], as well as a result of Fr¨oberg [13] we translate the above problem to combinatorics i.e. the ideal IG is Cohen-Macaulay if and only if the complementary graph G is chordal. For the proof of Theorem 3.1.3 we describe constructions of new chordal graphs from chordal graphs and prove Proposition 3.2.8 which is the essential step in the proof of theorem 3.1.3. The results of Chapter 2 is a joint work with M.Naeem.

Chapter 1 Preliminaries 1.1 Some monomial algebra 1.1.1 Monomial ideals In this section, we will introduce some basic notions of commutative algebra. Let K be a field, and let S = K[x1 , . . ., xn ] the polynomial ring in n variables over K. Any product of the form xa11 . . . xann with ai ∈ Z+ is called a monomial. If u = xa11 . . . xann is a monomial, then we write u = xa with a = (a1 , . . . , an ) ∈ Z+ n . The set Mon(S) of all monomials of S forms a canonical K-basis of S. In other words, any polynomial f ∈ S is a unique K-linear combination of monomials. f=

∑ u∈Mon(S)

au u

with au ∈ K.

Then we call the set supp( f ) = {u ∈ Mon(S) : au 6= 0} the support of f . Definition 1.1.1. An ideal I ⊂ S is called a monomial ideal if it is generated by monomials. An important property of monomial ideals is given in the following: For monomial ideals we have Proposition 1.1.2. Each monomial ideal has a unique minimal monomial set of generators. Usually, we denote by G(I), the unique minimal set of monomial generators of the monomial ideal I.

1.1.2 Primary decomposition The decomposition of an ideal into primary ideals is a traditional pillar of ideal theory. It provides the geometric foundation for decomposing an algebraic variety into irreducible components. We can see primary decomposition as a generalization of factorization of an integer as a product of prime powers. By a chain of prime ideals of a ring R we mean a finite strictly increasing sequence of prime ideals P0 ⊂ P1 ⊂ . . . ⊂ Pn , the integer n is called the length of the chain. The Krull dimension of R, denoted by dim(R), is the supremum of the lengths of all chains of prime ideals in R. Let P be a prime ideal of R, the height of P, denoted by height(P) is the supremum of the lengths of all chains of prime ideals P0 ⊂ P1 ⊂ . . . ⊂ Pn = P, which end at P. Note that dim(RP ) = height(P). If I is an ideal of R, then height(I), the height of I, is defined as height(I) = min{height(P) | I ⊂ P and P ∈ Spec(R)}. In general, we have the following inequality, dim(R/I) + height(I) ≤ dim(R). The difference dim(R) − dim(R/I) is called the codimension of I. Let M be an R-module. The annihilator of M is given by AnnR (M) = {x ∈ R|xM = 0}, if m ∈ M the annihilator of M is AnnR (m) = AnnR (Rm). Recall that the dimension of an R-module M is dim(M) = dim(R/ AnnR (M)) and the codimension of M is codim(M) = dim(R) − dim(M). Definition 1.1.3. Let I be an ideal of a ring S. The radical of I is rad(I) = (x ∈ S : xn ∈ I for some n > 0). p p The radical of an ideal I is often denoted by (I). In particular, (0), denoted by NS , is the set of nilpotent elements of S and is called as the nilradical of S. A ring is reduced if its nilradical is zero.

Proposition 1.1.4. If I is a proper ideal of a ring R, then the rad(I) is the intersection of all prime ideals containing I. Definition 1.1.5. Let M be a module over a ring R, the set of associated primes of M, denoted by AssR (M), is the set all prime ideals P of R with the property that there is a monomorphism ϕ of R-modules: ϕ

0 → R/P − → M. Note that P = AnnR (ϕ (1)). If M = R/I it is usual to say that an associated prime ideal of R/I is an associated prime ideal of I and to set AssR (I) = AssR (R/I). Definition 1.1.6. Let M be an R-module, the support of M, denoted by supp(M), is the set of all prime ideals P of R such that MP 6= 0. The support of a module M has the following property: Lemma 1.1.7. If 0 → M ′ → M → M ′′ → 0 is a short exact sequence of modules over a ring R, then supp(M) = supp(M ′ ) ∪ supp(M ′′ ) and AssR (M) ⊂ AssR (M ′ ) ∪ AssR (M ′′ ).

Corollary 1.1.8. Let R be Noetherian, M is finitely generated. Then AssR (M) is a finite set. Definition 1.1.9. Let M be an R-module. A submodule N of M is said to be a primary submodule if AssR (M/N) = {P}. An ideal Q of a ring R is a primary ideal if AssR (R/Q) = {P}, thus Q is a primary ideal if and only if xy ∈ Q and x ∈ / Q implies yn ∈ Q for some n ≥ 1. Definition 1.1.10. Let M be an R-module. A submodule N of M is said to be irrreducible if N cannot be written as an intersection of two submodules of M that properly contain N. Every irreducible submodule of a module M is primary. Definition 1.1.11. Let M be an R-module and N, N1 , . . . , Nr submodules of M. A decomposition N = N1 ∩ . . . ∩ Nr of N is said to be irredundant if N 6= N1 ∩ . . . ∩ Ni−1 ∩ Ni+1 ∩ . . . ∩ Nr

for all i.

Theorem 1.1.12. Let M be an R-module. If N is a submodule of M, then N has an irredundant primary decomposition N = N1 ∩ . . . ∩ Nr so that:

• AssR (M/Ni ) = {Pi } for all i. • N 6= N1 ∩ . . . ∩ Ni−1 ∩ Ni+1 ∩ . . . ∩ Nr

for all i.

• Pi 6= Pj if Ni 6= N j . Remark 1.1.13. If N 6= M and N = N1 ∩ . . . ∩ Nr is an irredundant primary decomposition of N with AssR (M/Ni ) = {Pi }, then AssR (M/N) = {P1 , . . ., Pr }. Corollary 1.1.14. If R is a Noetherian ring and I a proper ideal of R, then I has an irredundant primary decomposition I = Q1 ∩ . . . ∩ Qr such that Qi is a Pi -primary ideal and AssR (R/I) = {P1 , . . . , Pr }.

1.1.3 Cohen–Macaulay rings and modules After dimension, depth is the most fundamental invariant of a Noetherian local ring R or a finite R-module M. Depth is defined in terms of regular sequences. Regular sequences Let M be an R-module. We say that x ∈ R is an M-regular element if xz = 0 for z ∈ M, implies z = 0, i.e. x is a nonzero divisor for M. A sequence x = x1 , x2 , . . . , xn of elements of R is a weak M-regular sequence if xi is a nonzero divisor on M/(x1 , . . ., xi−1 ) for all 1 ≤ i ≤ n. We call n, the length of the sequence x. The sequence x is called M-regular if it is weak M-regular and (x)M 6= M. If (R, m) is a local ring and (x) ⊂ m, then by Nakayama lemma weak M-regular are same as Mregular. Grade and depth Let R be a Noetherian ring and M an R-module. An M-sequence x = x1 , . . . , xn (contained in an ideal I) is maximal (in I) if x1 , . . . , xn+1 is not an M-sequence for any xn+1 , where xn+1 ∈ I. Theorem 1.1.15 (Rees). Let R be a Noetherian ring, M a finite R-module, and I an ideal such that IM 6= M. Then all maximal M-sequences in I have the same length n given by n = min{i : ExtR (R/I, M) 6= 0}.

This result allows us to introduce the fundamental notions of grade and depth. Definition 1.1.16. Let R be a Noetherian ring, M a finite R-module, and I an ideal such that IM 6= M. Then the common length of the maximal M-sequences in I is called the grade of I on M, denoted by grade(I, M). We complement this definition by setting grade(I, M) = ∞ if IM = M. A special situation will occur so often that it merits a special notion: Definition 1.1.17. Let (R, m, k) be a Noetherian local ring, and M a finite R-module. Then the grade of m on M is called the depth of M, denoted by depth(M). Depth and dimension Let (R, m, K) be Noetherian local and M a finite R-module. All the minimal elements of supp(M) belongs to AssR (M). Therefore, if x ∈ M is an M-regular element, then x ∈ /P for all minimal elements of supp(M), and induction yields dim(M/xM) = dim(M) − n if x = x1 , . . ., xn is an M-sequence. In fact, we proved: Proposition 1.1.18. Let (R, m, K) be a Noetherian local ring and M 6= 0 a finite Rmodule.Then depth(M) ≤ dim(M). A nicer inequality is: Proposition 1.1.19. With the above notions, one has depth(M) ≤ dim(R/P) for all P ∈ AssR (M). Cohen–Macaulay rings and modules Let R be a Noetherian local ring and M a finite R-module. If the ”algebraic” invariant depth(M) equals the ”geometric” invariant dim(M), then M is called a Cohen–Macaulay module: Definition 1.1.20. Let R be a Noetherian local ring. A finite R-module M 6= 0 is a Cohen– Macaulay module if depth(M) = dim(M). If R itself is a Cohen–Macaulay module, then it is called a Cohen–Macaulay ring. A maximal Cohen–Macaualy module is a Cohen– Macaulay module M such that dim(M) = dim(R). Definition 1.1.21. Let R be Notherian ring and M an R-module. We say that M is a Cohen–Macualy R-module if Mm is a Cohen–Macaulay Rm -module for all maximal ideals m ∈ supp(M).

Some basic properties of Cohen–Macaulay modules are described in the following theorem: Theorem 1.1.22. Let (R, m, K) be a Noetherian local ring and M 6= 0 a Cohen–Macaualy R-module. Then 1. dim(R/P) = depth(M) for all P ∈ Ass(M). 2. grade(I, M) = dim(M) − dim(M/IM) for all ideals I ⊂ m. 3. x = x1 , . . ., xr is an M-sequence if and only if dim(M/xM) = dim(M) − r. 4. x is an M-sequence if and only if it is part of a system of parameters of M. Definition 1.1.23. An ideal I of a ring R is unmixed or height unmixed if height(I) = height(P) for all P ∈ AssR (R/I), that is the associated prime ideals of R/I are the minimal prime ideals of I of the same height. Theorem 1.1.24. A Noetherian ring R is Cohen–Macaulay if and only if every ideal I generated by height(I) elements is unmixed.

1.1.4 Polarization Given a monomial u = xa11 xa22 · · · xann we define the following monomial in a new set of variables p

n

ai

u = ∏ ∏ xi j . i=1 j=1

Now let I ⊂ S be an arbitrary monomial ideal with minimal set of monomial generators {u1 , . . ., um }. Then we set p I p = (u1 , . . ., ump ). This ideal is called the polarization of I. If we choose an arbitrary set {v1 , . . . , vr } of monomial generators of I, then we have p

I p = I p T = (v1 , . . . , vrp )T, where T is the polynomial ring over K in the variables which are needed to polarize the monomials vi . We shall also need the following rule: Suppose I = I1 ∩ I2 ∩ . . . ∩ Ir where each I j is a monomial ideal, then I p = I pT = (I1pT ∩ I2p T ∩ · · · ∩ Irp T ),

(1.1)

where T is again the polynomial ring over K in the variables which are needed to polarize all the monomials involved.

Proposition 1.1.25 (See [8]). Let I be a monomial ideal. The following are equivalent: • I • Ip

is Cohen–Macaulay. is Cohen–Macaulay.

The following example describes how polarization works. Example 1.1.26. Let J = (x21 , x1 x2 , x32 ) ⊂ S = K[x1 , x2 ]. Then J p = (x11 x12 , x11 x21 , x21 x22 x23 )

is the polarization of J in the polynomial ring T = K[x11 , x12 , x21 , x22 , x23 ]. Note that by identifying each xi with xi1 , one can consider S as a polynomial extension of R. Below are some basic properties of polarization: Proposition 1.1.27 (See [11]). Suppose that S = K[x1 , . . . , xn ] is a polynomial ring over a field K, I and J are two monomial ideals of R. Then 1. (I + J) p T = I pT + J p T ; 2. For two monomials u and v in S, u|v if and only if u p |v p ; 3. (I ∩ J) p T = I pT ∩ J p T ; 4. If p = (xi1 , . . . , xir ) is a (minimal) prime containing I, then p p is a (minimal) prime containing I pT ; 5. height(I) = height(I pT ) Where T is the polynomial ring over K in the variables which are needed to polarize all the monomials.

1.1.5 Complete intersections The nomenclature “complete intersection” comes from algebraic geometry. Definition 1.1.28. Let R be a ring and let I be an ideal of R. If I is generated by a regular sequence, we say that I is a complete intersection. Complete intersections are very important in the study of Cohen–Macaulay modules. Proposition 1.1.29. Let (R, m, K) be a Cohen–Macaulay local ring and let I be an ideal of R. If I is a complete intersection, then R/I is Cohen–Macaulay and I is unmixed. Suppose R is the coordinate ring of an affine variety over an algebraically closed field K. Then R has the form R = S/I, where S is a polynomial ring over K, and R is called a complete intersection if I is generated by the least possible number of elements, namely height(I) = codim(V ). Then V is the intersection of codim(V ) hypersurfaces, and I is generated by an S-sequence.

1.1.6 Graded free resolutions Let K be a field and S = K[x1 , . . ., xn ] be the polynomial ring in n variable over K. The Zn -grading Let a ∈ Zn , then f ∈ S is called homogeneous of degree a if f is of the form cxa with c ∈ K. An S-module M is called Zn -graded if M = ⊕a∈Zn Ma and Sa Mb ⊂ Ma+b for all a, b ∈ Zn . Let M, N be Zn -graded S-modules. A module homomorphism ϕ : N → M is called homogeneous if ϕ (Na ) ⊂ Ma for all a ∈ Zn , and N is called a Zn -graded submodule of M if N ⊂ M and the inclusion map is homogeneous. In this case, the factor module M/N inherits a natural Zn -grading with components (M/N)a = Ma /Na for all a ∈ Zn . An ideal I ⊂ S is a Zn -graded submodule of S if and only if it is a monomial ideal, in which case S/I is also canonically Zn -graded. Minimal graded free resolutions Let M be an S-module, a long exact sequence of the form F : . . . → F2 → F1 → F0 → M → 0 of graded S-modules with Fi = j S(− j)βi j , is called a graded free S- resolution of M. The graded free resolution need not to be unique. L

Definition 1.1.30. Let M be a finitely generated graded S-module. A graded free Sresolution F of M is called minimal , if for all i, the image of Fi+1 → Fi is contained in mFi . The minimal graded free resolution always exists and is unique up to isomorphism. The next result shows that the numerical data given by a graded minimal free S-resolution of M depend only on M and not on the particular chosen resolution. Proposition 1.1.31. Let M be a finitely generated graded S-module and F : . . . → F2 → F1 → F0 → M → 0 a minimal graded free S-resolution of M with Fi =

βi j = dimk Tori (K; M) j

L

j S(− j)

βi j

for all i. Then

for all i and j.

The numbers βi j = dimK Tori (K, M) j are called graded Betti numbers of M, and βi = ∑ j βi j is called the ith Betti number of M. There are two important invariants attached to a graded ideal I ⊂ S defined in terms of the minimal graded free resolution of S/I, 1. the projective dimension of S/I pd(S/I) = max{i : βi j (S/I) 6= 0 for some j}, and 2. the regularity of I reg(I) = max{ j : βi,i+ j (I) 6= 0 for some i}. The following theorem gives us an effective relation between projective dimension and depth of a module. Theorem 1.1.32 (Auslander-Buchsbaum). Let (R, m, K) be a Noetherian local ring, and M 6= 0 a finite R-module. If pd(M) < ∞, then pd(M) + depth(M) = depth(R). Definition 1.1.33. Let S = K[x1 , . . ., xn ] be the polynomial ring, and M a graded S-module. We say that M has a d-linear resolution if the graded minimal free resolution of M is of the form 0 → S(−d − t)βt → S(−d − t + 1)βt−1 → . . . → S(−d − 1)β1 → S(−d)β0 → M → 0.

1.2 Basic facts on graphs In this chapter, we will recall some notions on graphs and basic facts of graph theory which will be used throughout the thesis.

1.2.1 Graphs and Trees A simple graph G consists of a finite set V of vertices and a collection E of subsets of V called edges such that every edge of G is a pair {vi , v j } for some vi , v j in V . We will write V (G) and E(G) for the vertex set and edge set respectively. If e = {vi , v j } is an edge of G one says that the vertices vi and v j are adjacent or connected be e. In this case one also says that the edge e is incident with the vertex vi or v j , or the edge e has the ends vi and v j . The degree of a vertex v in V , denoted be deg(v), is the number of edges which are incident with v. A vertex of degree one (resp. zero) is called an end (resp. isolated vertex). If all the vertices of G are isolated, G is called a discrete graph.

Definition 1.2.1. The edge ideal I(G) associated to the graph G is the ideal of S = k[x1 , . . ., xn ] generated by the set of squarefree monomials xi x j such that vi is adjacent to v j , that is, I(G) = ({xi x j : {vi , v j } ∈ E(G)}) If all the vertices of G are isolated we set I(G) = (0). Let H and G be two graphs, it is said that H is a subgraph of G if V (H) ⊂ V (G) and E(H) ⊂ E(G). A walk of length n in G is an alternating sequence of vertices and edges w = {v0 , z1 , v1 , . . . , vn−1 , zn , vn }, where zi = {vi−1 , vi } is the edge joining vi−1 and vi . A walk may also be written {v0 , . . ., vn } with the edges understood, or z1 , z2 , . . . , zn with the vertices understood. if v0 = vn , the walk w is called a closed walk. A path is a walk with all its vertices distinct. We say that G is connected if for every pair of vertices v1 and v2 there is a path from Sp v1 to v2 . Note that G is a vertex disjoint decomposition G = i=1 Gi , where G1 , . . . , G p are the maximal (with respect to inclusion) connected subgraphs of G, the Gi are called the connected components of G. A graph is connected if and only if it has only one connected component. If e is an edge, we denote by G \ {e} the spanning subgraph of G obtained by deleting e and keeping all the vertices of G. The removal of a vertex v from a graph G results in the subgraph G \ {v} of G consisting of all the vertices in G except v and all the edges not incident with v. A cycle of length n is a closed walk {v0 , . . . , vn } in which n ≥ 3 and the vertices v1 , . . . , vn are distinct. A cycle is even (resp. odd) if its length is even (resp. odd). We denote by Cn the graph consisting of a cycle with n vertices. In particular, C3 will be called a triangle, C4 a square and so on. A tree is a connected graph without cycle, and forest is graph whose connected components are trees. The complete graph Kn has every pair of its n vertices adjacent. A graph G is bipartite if its vertex set V can be partitioned into disjoint subsets V1 and V2 such that every edge of G joins V1 with V2 . For example, V1 = {u, v, w, x} and V2 = {a, b, c, d} u ◦

v ◦

w ◦

x ◦

◦ a

◦ b

◦ c

◦ d

The following characterization of bipartite graphs is due to K¨onig: Proposition 1.2.2. Let G be a graph. The following conditions are equivalent:

1. G is bipartite; 2. all the cycles of G are even. Proof. (i) ⇒ (ii): Let V1 and V2 be a partition of V (G) such that every edge of G joins V1 with V2 . If {v0 , . . . , vn } is a cycle of G, one may assume that v0 ∈ V1 . Note v1 ∈ V2 , it follows immediately that vi ∈ V1 if and only if i is even, thus n must be even. (ii)⇒(i): It is enough to show that each connected component of G is bipartite, thus one may assume G connected. Pick a vertex v0 ∈ V . Set V1 = {v ∈ V (G) |

d(v, v0) is even} and V2 = V (G) \V1.

It follows that no two vertices of Vi are adjacent for i = 1, 2, otherwise G would contain an odd cycle. Therefore G is bipartite. Because any forest has no cycle, we obtain: Corollary 1.2.3. If T is a forest, then T is a bipartite graph.

1.2.2 The marriage problem In this section we shall state the famous marriage problem of graph theory. Definition 1.2.4. A set of edges of a graph G is called a matching if no two of them have a vertex in common. The matching number of a graph G, denoted by β1 (G), is the size of the largest matching in G. A vertex v is saturated by a matching M if some edge of M is incident with v. A matching which saturates all vertices of G is called perfect matching. Note that not every graph has a perfect matching. The marriage problem theorem guarantees when a bipartite graph has a perfect matching. Let G be a graph with the vertex set V (G). Given a subset U ⊆ V , the neighbor set of U , denoted by NG (U ) or simply N(U ), is defined as N(U ) = {v ∈ V | v is adjacent to some vertex in U }. Obviously, if C is a cycle in G with vertex set VC , then VC ⊂ N(VC ). Definition 1.2.5. Let G be a graph with vertex set V (G). A subset C ⊆ V (G) is called a minimal vertex cover of G, if the following conditions are satisfied: 1. every edge of G is incident with one vertex in C; 2. there is no proper subset of C with the the property 1.

If C satisfies condition 1 only, then C is called a vertex cover of G. The vertex covering number of G, denoted by α0 (G) is

α0 (G) = min{|C| : where C is a minimal vertex cover of G}. For a discrete graph G, one takes the empty set as a minimal vertex cover of G. Definition 1.2.6. A set of edges in a graph G is independent if no two of them have a vertex in common. Remark 1.2.7. Let G be a graph with vertex set V . A subset U of V is a maximal independent set of G if and only if V \U is a minimal vertex cover of G. Lemma 1.2.8. For any graph G, we have β1 (G) ≤ α0 (G). For a bipartite graph we have the following equality: Theorem 1.2.9 (K¨onig). If G is a bipartite graph, then β1 (G) = α0 (G). Proof. See, [28]. By using the K¨onig theorem we have Theorem 1.2.10 (Marriage problem). Let G be a bipartite graph with the vertex set V (G). Then the following conditions are equivalent: 1. G has a perfect matching; 2. |A| ≤ |N(A)| for all independent set A ⊂ V (G).

1.2.3 Chordal graphs In commutative algebra, the chordal graph first appeared in the work of Fr¨oberg in about 1990. Definition 1.2.11. A chord of a cycle C is an edge {i, j} of G such that i and j are vertices of C with {i, j} ∈ / E(C). A chordal graph is a finite graph each of whose cycles of length > 3 has a chord. Every induced subgraph of a chordal graph is again chordal. A subset C of [n] is called a clique of G if for all i and j belonging to C with i 6= j one has {i, j} ∈ E(G). The clique complex of a finite graph G on [n] is the simplicial complex ∆(G) on [n] whose faces are the cliques of G. For example,

G

∆(G)

A graph G is complete, if it is a clique. Definition 1.2.12. The complementary graph of a finite graph G on vertex set V (G) is a finite graph G on V (G) whose edge set E(G) consists of those 2-element subsets {i, j} of V (G) for which {i, j} ∈ / E(G). The following theorem is a milestone in combinatorial commutative algebra: Theorem 1.2.13 (Fr¨oberg, [13]). The edge ideal I(G) of a finite graph G has linear resolution if and only if the complementary graph G of G is chordal. Definition 1.2.14. Let ∆ be a simplicial complex on [n], a vertex i of ∆ is called a free vertex of ∆ if i belongs to exactly one face. Recently, Herzog, Hibi and Zheng in [17] classify all Cohen–Macaulay chordal graphs: Theorem 1.2.15 (Herzog-Hibi-Zheng). Let K be a field, and let G be a chordal graph on the vertex set V (G). Let F1 , . . . , Fm be the facets of ∆(G) which admit a free vertex. Then the following conditions are equivalent: 1. G is Cohen–Macaulay; 2. G is Cohen–Macaulay over K; 3. G is unmixed; 4. V (G) is the disjoint union of F1 , . . ., Fm . One of the fascinating results in classical graph theory is Dirac’s theorem [6] on chordal graphs. In 1961 Dirac proved this theorem. Let G be a finite graph on V (G). A perfect elimination ordering of G is an ordering in , . . ., i2 , i1 of the vertices 1, 2, . . ., n of G such that, for each 1 < j ≤ n, Ci j = {ik ∈ V (G) : 1 ≤ k < j, {ik , i j } ∈ E(G)} is a clique of G. Theorem 1.2.16 (Dirac). Given a finite graph G on V (G), the following conditions are equivalent: • G is chordal; • G has a perfect elimination ordering.

1.2.4 Cohen–Macaulay graphs In this section we introduce the so called edge ideal of a graph and recall some results of Cohen–Macaulay graphs. Let R be a Noetherian local ring. A finite R-module M 6= 0 is a Cohen–Macaulay module if depth(M) = dim(M). If R itself is a Cohen–Macaulay module, then it is called a Cohen–Macaulay ring. Let I be an ideal of R. If the quotient ring R/I is Cohen–Macaulay, then we say I is Cohen–Macaulay ideal. Let G be a graph over the vertex set V (G) = {1, . . ., n} and S = K[x1 , . . . , xn ] the polynomial ring in n variables over the field K. Definition 1.2.17. The edge ideal I(G) of a graph G is the ideal of S generated by the monomials xi x j such that {i, j} ∈ E(G). The edge ring is the subalgebra of R which is generated by the monomials {xi x j : {i, j} ∈ E(G)} over K, denoted by K[G], i.e, K[G] = K[xi x j : {i, j} ∈ E(G)]. If all the vertices of G are isolated, we set I(G) = 0. Since G has no loop, the edge ideal I(G) is a squarefree monomial ideal generated in degree 2. Definition 1.2.18. A graph G is said to be Cohen–Macaulay if the edge ideal I(G) is a Cohen–Macaulay ideal. Remark 1.2.19. In general, the Cohen–Macaulay property of a graph G depend on the field K. The next result establishes a one to one correspondence between the minimal vertex covers of a graph and the minimal primes of the corresponding edge ideal. Proposition 1.2.20. Let G be a graph over the vertex set V (G) and S = K[x1 , . . . , xn ] a polynomial ring over a field K. If P is an ideal generated by {xi1 , . . . , xis }, then the following conditions are equivalent: 1. P is a minimal prime of I(G); 2. {i1 , . . . , is } is a minimal vertex cover of G. Proof. (1) ⇒ (2) : is obvious. (2) ⇒ (1) : Since I(G) is a monomial ideal, every associated prime of I(G) is an ideal generated by some variables. Since I(G) ⊂ P and for every proper subset B of {i1 , . . . , is}, B is not a vertex cover of G. We have P is a minimal prime of I(G). Corollary 1.2.21. Let G be a graph and I(G) its edge ideal. Then α0 (G) = height(I(G)). Definition 1.2.22. A graph G is unmixed if all minimal vertex covers of G have same cardinality.

Since any Cohen–Macaulay ideal in a polynomial ring is unmixed, we have: Proposition 1.2.23. If G is a Cohen–Macaulay graph, then G is unmixed. The class of Cohen–Macaualy graphs is huge. In [28], Villarreal gave several constructions of Cohen–Macaulay graphs. In particular, he gave the following effective description of Cohen–Macaulay trees and presented an interesting family of graphs containing all Cohen–Macaulay trees. Theorem 1.2.24 (Villarreal). Let T be a tree with vertex set V (G) and edge set E(G). Then T is Cohen–Macaulay if and only if |V | ≤ 2 or 2 < |V | = 2r and there are vertices a1 , . . . , ar , b1 , . . . , br so that deg(bi ) ≥ 2, and {ai , bi } ∈ E(G) for i = 1, . . ., r. Recently, Herzog and Hibi classified all Cohen–Macaulay bipartite graphs in [16] by using the Alexander dual of some special simplicial complex. Their main result is Theorem 1.2.25 (Herzog-Hibi). Let G be a bipartite graph with vertex set V (G) and edge set E(G). Then G is Cohen–Macaulay if and only if after a suitable labeling of the vertices the following conditions hold: • V = V1 ∪V2 where V1 = {x1 , . . ., xn } and V2 = {y1 , . . ., yn }; • {xi , yi } ∈ E(G) for all i ∈ [n]; • if {xi , yi } ∈ E(G), then i ≤ j; • if {xi , y j }, {x j , yk } ∈ E(G), with i < j < k, then {xi , yk } ∈ E(G).

1.3 Stanley-Reisner and facet ideal of a simplicial complex In this section we will introduce concept of simplicial complexes and some ideal such as Stanley-Reisner ideal and facet ideal, and define simplicial trees which is natural generalization of trees. We also put some well known results of combinatorial commutative algebra, for example, Hochster’s formula and Eagon-Reiner theroem.

1.3.1 Basic concepts Here we will recall the definition of simplicial complex and study two squarefree monomial ideals (Stanley-Reisner and facet ideals) associated to a simplicial complex.

Definition 1.3.1. Let [n] = {1, ..., n} be the vertex set and ∆ a simplicial complex on [n]. Then ∆ is a collection of subsets of [n] such that if F ∈ ∆ and G ⊂ F, then G ∈ ∆. Often it is also required that {i} ∈ ∆ for all i ∈ [n], however we will not assume this condition. Each element F ∈ ∆ is called a face of ∆. The dimension of a face F is |F| − 1. Let d = max{|F| : F ∈ ∆} and define the dimension of ∆ to be dim(∆) = d − 1. An edge of ∆ is a face of dimension 1. A vertex of ∆ is a face of dimension 0. A facet is a maximal face of ∆ (with respect to inclusion). Let F (∆) denote the set of facets of ∆. When F (∆) = {F1 , . . ., Fm }, we write ∆ = hF1 , . . ., Fm i. We say that a simplicial complex is pure if all facets have the same cardinality. A nonface of ∆ is a subset F of [n] with F ∈ / ∆. let N (∆) denote the set of minimal nonfaces of ∆. Let fi = fi (∆) denote the number of faces of ∆ of dimension i. In particular, f0 = n. The sequence f (∆) = ( f0 , f1 , . . ., fd−1 ) is called the f-vector of ∆. Letting f−1 = 1, we define the h-vector h(∆) = {h0 , h1 , . . ., hd } of ∆ by the formula d



i=0

d

fi−1 (t − 1)d−i = ∑ hit d−i . i=0

We visualize a simplicial complex by using its geometric realization. For example,

pure

non-pure

Definition 1.3.2. Let S = K[x1 , . . ., xn ] be the polynomial ring in n variables over a field K and ∆ a simplicial complex on [n]. For each subset F ⊂ [n] we set xF = ∏ xi . i∈F

The Stanley-Reisner ideal of ∆ is the ideal I∆ of S which is generated by those squarefree monomials xF with F ∈ / ∆. In other words, I∆ = (xF : F ∈ N (∆)). The Stanley-Reisner ring of ∆ (with respect to the field K) is the homogeneous k-algebra K[∆] = K[x1 , . . . , xn ]/I∆ . The facet ideal of ∆ is the ideal of S which is generated by those squarefree monomials xF with F ∈ F (∆) and is denoted by I(∆). Thus if ∆ = hF1 , . . . , Fm i, then I(∆) = (xF1 , . . . , xFm ). Let G be the graph. If G contains no isolated vertex, then the facet ideal of G coincides with the so called edge ideal of G.

1.3.2 The Alexander dual The Alexander dual of a simplicial complex plays an essential role in combinatorics and commutative algebra. Given a simplicial complex ∆ on [n], we define ∆∨ by ∆∨ = {[n] \ F : F ∈ / ∆}. Lemma 1.3.3. The collection of sets ∆∨ is a simplicial complex and (∆∨ )∨ = ∆. This simplicial complex ∆∨ is called the Alexander dual of ∆. The facets of ∆∨ are, F (∆∨ ) = {[n] \ F : F ∈ N (∆)}. For each subset F ⊂ [n] we set F c = [n] \ F. The simplicial complex ∆c = hF c : F ∈ F (∆)i. is called the complement of ∆. Lemma 1.3.4. One has I∆∨ = I(∆c). For each subset F ⊂ [n] we set, PF = (xi : i ∈ F). Theorem 1.3.5. The standard primary decomposition of I∆ is I∆ =

\

PF c .

F∈F (∆)

In particular, dim K[∆] = dim∆ + 1. Corollary 1.3.6. Let I∆ = PF1 ∩ . . . ∩ PFm be the standard primary decomposition of I∆ , where each Fj ⊂ [n]. Then G(I∆∨ ) = {xF1 , . . . , xFm }. Similar with the graph case, we have Definition 1.3.7. A vertex cover of ∆ is a set G ⊂ [n] such that G∩F 6= 0/ for all F ∈ F (∆). A vertex cover G of ∆ is minimal, if any proper subset of G is not a vertex cover of ∆. We denote by C (∆) the set of minimal vertex covers of ∆. If all the minimal vertex covers of ∆ have the same cardinality, then we say ∆ is unmixed. Proposition 1.3.8. Let ∆ be a simplicial complex on the vertex set [n] and I(∆) the facet ideal of ∆. Then an ideal P = (xi1 , . . . , xis ) is a minimal prime of I(∆) if and only if {i1 , . . ., is } is a minimal vertex cover of ∆.

1.3.3 The Eagon-Reiner theorem A very useful result to compute the graded Betti numbers of the Stanley-Reisner ideal of a simplicial complex is the so-called Hochster formula. To state the formula we need some notations and terminologies. Definition 1.3.9. Let ∆ be a simplicial complex on [n]. For a face F of ∆, the link of F in ∆ is the subcomplex link∆ (F) = {G ∈ ∆ : F ∪ G ∈ ∆, F ∩ G = 0}. / In particular, link∆ (0) / = ∆. For a subset W of [n], the restriction of ∆ on W is the subcomplex ∆W = {F ∈ ∆ : F ⊂ W }. The notion H˜ q (∆, K) stands for the qth reduced homology group of ∆ with coefficients in K, where K is a field. Hochster’s formula The following fundamental theorem of Hochster gives a very useful description of the Zn -graded Betti numbers of a Stanley-Reisner ideal. Theorem 1.3.10 (Hochster). Let ∆ be a simplicial complex and a ∈ Zn . Then we have: 1. TorSi (K; I∆ )a = 0 if a is not squarefree; 2. if a is squarefree and W = supp(a), then TorSi (K; I∆)a ∼ = H˜ |W |−i−2 (∆W ; K) for all i. We can compute graded Betti numbers of I∆ using: Corollary 1.3.11. Let ∆ be a simplicial complex, a ∈ Zn be squarefree and F = [n] \ supp(a). Then ToriS (K; I∆ )a ∼ = H˜ i−1 (link∆∨ (F); K) for all i. In particular it follows that the graded Betti number βi j (I∆ ) of I∆ can be computed by the formula βi j (I∆ ) = ∑ dimK H˜ i−1(link∆∨ ; K). F∈∆∨ ,|F|=n− j

Reisner’s criterion The K-algebra K[∆] = S/I∆ is called the Stanley–Reisner ring of ∆. We say that ∆ is Cohen–Macaualy over K if K[∆] is Cohen–Macaulay. Lemma 1.3.12. Every Cohen–Macaulay simplicial complex is pure. The following result is the well known Reisner’s criterion for the Cohen–Macaulay property of the Stanley-Reinser ring. Theorem 1.3.13 (Reisner). A simplicial complex ∆ is Cohen–Macaulay over K if and only if, for all faces F of ∆ including the empty face 0/ and for all i < dim (link∆ (F)), one has H˜ i (link∆ F; K) = 0. Now the well known Eagon-Reiner [7] theorem states: Theorem 1.3.14 (Eagon-Reiner). Let ∆ be a simplicial complex on [n] and let K be a field. Then the Stanley–Reisner ideal I∆ ⊂ K[x1 , . . . , xn ] has a linear resolution if and only if K[∆∨ ] is Cohen–Macaulay. More precisely, I∆ has q-linear resolution if and only if K[∆∨ ] is Cohen–Macaulay of dimension n − q − 1.

1.3.4 Simplicial trees In [9], Faridi introduced the notion of trees for higher dimensional simplicial complexes. As we have seen in the graph case, a connected graph G is a tree if it has no cycle, in other words, any subgraph of G has a leaf (an edge with a free vertex). Let us recall the definition of a leaf for an arbitrary simplicial complex. Definition 1.3.15. Let ∆ be a simplicial complex. A facet F of ∆ is called a leaf if either F is the only facet of ∆, or there exists a facet G 6= F in ∆, such that F ∩ H ⊂ F ∩ G for any H ∈ F (∆), H 6= F. A facet G with this property is called a branch of F in ∆. Let ∆ be a simplicial complex with the vertex set [n] and F ∈ F (∆). If i is a vertex of F and i does not belong to any other facets of ∆, then we call i a free vertex of F in ∆. It is clear that if F is a leaf of ∆, then F has at least one free vertex. But the converse is not true. Example 1.3.16. Let ∆ = h{1, 2, 3}, {3, 4, 5}, {5, 6, 7}i is a pure simplicial complex, the facet {3, 4, 5} has a free vertex 4, but it is not a leaf of ∆. Let us now recall the definition of the simplicial tree. Definition 1.3.17. Let ∆ be a connected simplicial complex. Then ∆ is called a simplicial tree if every nonempty subcomplex of ∆ has a leaf. A simplicial complex ∆ with the property that every connected component is a tree is called a simplicial forest.

Note the following key result, Lemma 1.3.18. Let ∆ be a simplicial tree of two or more facets. Then ∆ has at least two leaves. An important property of simplicial trees is given in Corollary 8.3 of [10], namely: Theorem 1.3.19. Let ∆ be a simplicial tree over a set of vertices x1 , . . ., xn , and let K be a field. Then the quotient ring K[x1 , . . . , xn ]/I(∆) is Cohen–Macaulay if and only if ∆ is unmixed.

Chapter 2 Cohen–Macaulay monomial ideals with given radical 2.1 A Characterization of Monomial Complete Intersections Let I be a squarefree monomial ideal in S = K[x1 , . . ., xn ]. As usual we denote by G(I) the unique minimal set of monomial generators of I. Let G(I) = {u1 , . . . , um }. We consider the set GI of all monomial ideals J with G(J) = {v1 , . . . , vm } and supp(vi ) = supp(ui ) for i = 1, . . ., m. This set of monomial ideals is contained in the set FI of all monomial ideals √ J whose radical J coincides with I. Given an integer vector a = (a1 , . . . , an ) we let M a be the set of monomials xb11 · · · xbnn in S with the property that bi ≤ ai for i = 1, . . . , n, and set GIa = GI ∩ M a . Then we have GIa ⊂ GI ⊂ FI . Let J ⊂ S be a graded ideal. We say that S/J (or J) is unmixed, if all associated prime ideals of S/J has the same height. In particular, an unmixed ideal has no embedded prime ideals. In this section we want to prove the following Theorem 2.1.1. Let I ⊂ S be a squarefree monomial ideal with G(I) = {u1 , . . ., um }, and let a = (2, 2, . . ., 2). The following conditions are equivalent: (a) I is a complete intersection. (b) S/J is Cohen–Macaulay for all J ∈ GI . (c) S/J is unmixed for all J ∈ GI .

(d) S/J is Cohen–Macaulay for all J ∈ GIa . (e) S/J is unmixed for all J ∈ GIa . For the proof of the theorem we need the following simple lemmata as well as the marriage theorem: Lemma 2.1.2. Let I be a monomial ideal. Then I is unmixed, if and only if I p is unmixed. Proof. Let I = Q1 ∩ . . . ∩ Qr , be the unique irredundant presentation of I as an intersection of irreducible monomial ideals Qi , see [28]. By using the above property of polarization, we get p

I pT = (Q1 T ∩ . . . ∩ Qrp T ). √ Suppose I is unmixed. Then height(I) = height( Qi ) = height(Qi ) for all i = 1, . . . , r. On the other hand, each Qi is Cohen–Macaulay. Hence the ideals Qip T are Cohen–Macaulay, p and hence unmixed for i = 1, . . ., r. Since height(Qi T ) = height(Qi ) = height(I) for all i, it follows that I p T is unmixed. This implies that I p is unmixed. Lemma 2.1.3. Let ∆ be a simplicial complex on the vertex set [n], and let C be a minimal vertex cover of ∆. Then (a) |C| ≤ |F (∆)|; (b) |C| < |F (∆)|, if ∆ is unmixed but not a complete intersection. Proof. (a) For any subset D ⊂ [n] we let PD ⊂ S be the monomial prime ideal with G(PD ) = {xi : i ∈ D}. Then D is a minimal vertex cover of ∆ if and only if PD is a minimal prime ideal of I(∆). Let I be a graded ideal in S or an ideal in a Noetherian local ring. We denote by µ (I) the number of a minimal set of generators of I. Then for the given minimal vertex cover C of ∆ we have |F (∆)| = µ (I(∆)) ≥ µ (I(∆)SPC ) = µ (PC ) = |C|, since I(∆)SPC = PC SPC , and this is the case because I(∆) = C PC , where the intersection is taken over all minimal vertex covers of ∆. (b) If ∆ is unmixed, suppose all minimal vertex covers have cardinality r. Now (a) implies that r ≤ |F (∆)|. T

But if r = |F (∆)|, then I is a complete intersection, a contradiction. Hence r < |F (∆)|.

If ∆ is a simplicial complex on the vertex set [n] and v ∈ [n], let ∆\{v} be the simplicial complex obtained from ∆ by removing those facets from ∆ which contain v. Lemma 2.1.4. Let C be a minimal vertex cover for ∆ and v ∈ C then C\{v} is a minimal vertex cover for ∆\{v}. Proof. Suppose C\{v} is not a vertex cover of ∆\{v}, then there exist a facet F ∈ F (∆\{v}) such that F ∩C\{v} = 0. / But then C is not a vertex cover of ∆, because F ∈ F (∆) and F ∩C = 0, / since v 6∈ F. This is a contradiction. Suppose C\{v} is not minimal. Then there exists a vertex cover D of ∆\{v} with D ( C\{v} . As all facets of ∆ which are not facets of ∆\{v}, contain v as a vertex, it follows that D ∪ {v} is a vertex cover of ∆ with D ( C, a contradiction. The following result is known as the marriage theorem, see for example [28]. Proposition 2.1.5. Let G be a bipartite graph with vertex sets V1 ,V2 having m and n vertices respectively. If |A| ≤ |N(A)| for all A ⊂ V1 , then there are m independent edges in G. Now we are ready to prove our main theorem. Proof of Theorem 2.1.1: (a) ⇒ (b): Since I is a complete intersection, it follows that gcd(ui , u j ) = 1 for all i 6= j. Hence if v1 , . . . , vm are monomials with supp(vi ) = supp(ui ) for i = 1, . . . , m, then gcd(vi , v j ) = 1 for all i 6= j, as well. Therefore the monomial ideal J with G(J) = {v1 , . . . , vm } is also a complete intersection, and hence S/J is Cohen– Macaulay for all J ∈ GI . The implications (b) ⇒ (d) and (d) ⇒ (e) are obvious. (e) ⇒ (a): Suppose that I is not a complete intersection. We have to find a J ∈ GIa such that S/J is not unmixed. Since I is squarefree, there exists a simplicial complex ∆ on [n] such that I = I(∆). As I ∈ GIa , our hypothesis implies that I unmixed, and hence all minimal vertex covers of ∆ have same cardinality, say r. The following claim is the crucial step in our proof: r + 1 pairwise different facets F1 , . . . , Fr+1 ∈ F (∆) and r + 1 pairwise different vertices v1 , . . . , vr+1 can be chosen such that vi ∈ Fi for all i = 1, . . . , r + 1. In fact, after a suitable renumbering of the vertices, we may assume that C = {1, . . ., r} is a minimal vertex cover for ∆. Let F (∆) = {F1 , . . . , Fs }, and consider the bipartite graph B with the vertex set V (B) = {1, . . ., r} ∪ {F1 , . . . , Fs},

and {i, Fj } is an edge of B if and only if i ∈ Fj . We shall now see that B satisfies the conditions of the marriage theorem. Let V ⊂ C and consider the neighbor set of V , N(V ) = {F ∈ F (∆) : F ∩V 6= 0}. / After a suitable renumbering of the elements of C, we may assume that V = {1, . . .,t}, where t ≤ r. Lemma 2.1.4 says that C\{t + 1} is a minimal vertex cover for ∆\{t + 1} and recursively C\{t + 1,t + 2 . . . , r} = V is a minimal vertex cover for ∆0 = ∆\{t + 1,t + 2, . . . , r}. By Lemma 2.1.3, |V | ≤ |F (∆0)| ≤ |N(V )|. Hence by the marriage theorem we have r independent edges in B, say {{1, F1}, . . . , {r, Fr }}. In order to complete the proof of our claim it remains to show that there exists a vertex v ∈ [n], v > r and a facet G of ∆ with G 6= Fi for i = 1, . . ., r and with v ∈ G. Lemma 2.1.3 implies that F (∆) = {F1 , . . . , Fm } with m > r. For the further discussions we have to distinguish two cases: Case 1: There exists some Fi with i > r and Fi * C. In this case we choose t ∈ Fi \C, and we may take {t, Fi} as (r + 1)-th pair. Case 2: Fi ⊂ C for all i = r + 1, . . ., m. We first notice that |Fi | > 1 for i = r + 1, . . ., m. Indeed, if Fi = { j}, then j ∈ C and hence j ∈ Fj , contradiction the fact that Fi 6= Fj and that Fj is a facet. We have that I(∆)SPC = PC SPC = (x1 , . . . , xr )SPC . Hence since Fi ⊂ C and |Fi | > 1 for all i = r +1, . . . , m, it follows that xFi ∈ (x1 , . . ., xr )2 SPC for i = r + 1, . . . , m. This implies that (x1 , . . . , xr )SPC = (xF1 , . . . , xFr )SPC , and hence Fi ∩C = {i} for

i = 1, . . ., r.

Now let j ∈ Fr+1 ; then j ∈ Fj because Fr+1 ⊂ C. If |Fj | = 1, then Fj is a proper subset of Fr+1 , a contradiction because Fj is a facet. Hence since |Fj | ≥ 2 and Fj ∩ C = { j}, there exists an element k ∈ Fj with k ∈ / C. Then in the list {{1, F1}, . . . , {r, Fr } we replace { j, Fj } by {k, Fj } and add the new list the pair { j, Fr+1 }. This completes the proof of the claim.

After a suitable relabeling of the vertices we may assume that i ∈ Fi for i = 1, . . ., r +1. Let J = (x1 xF1 , . . . , xr+1 xFr+1 , xFr+2 , . . . , xFm ). Then J ∈ GIa . After polarization, replacing the squares x2i in the generators of J by xi yi for i = 1, . . . , r + 1, we obtain the polarized ideal J p in T == S[y1 , . . ., yr ], namely J p = (y1 xF1 , . . . , yr+1 xFr+1 , xFr+2 , . . . , xFm ) = (y1 , . . . , yr+1 ) ∩ IT . Thus we see that (y1 , . . ., yr+1 ) is a minimal prime ideal of J p . Since height J p = height J = height I = r, it follows that J p is not unmixed. Hence by Lemma 2.1.2, J also not unmixed, a contradiction. Corollary 2.1.6. Let I be squarefree monomial ideal. The following conditons are quivalent: (a) S/J is Cohen–Macaulay for all J ∈ FI , that is, for all monomial ideals J with √ J = I. (b) I is the graded maximal ideal m = (x1 , . . . , xn ) of S. √ √ Proof. (a) ⇒ (b): Since I 2 ⊂ mI ⊂ I, it follows that I = I 2 ⊂ mI ⊂ I, and hence √ J := mI = I. For any u ∈ I \ J, we have mu ∈ J. This shows that depth S/J = 0. Since S/J is Cohen– Macaualy by assumption, it follows that dim S/I = dim S/J = 0. The only squarefree monomial ideal I with dim S/I = 0 is the ideal m. (b) ⇒ (a): If I = m, then dim S/J = 0 for all J ∈ FI . In particular, S/J is Cohen– Macaulay.

2.2 On the existence of nontrivial Cohen–Macaulay modifications of squarefree monomial ideals √ Let FI be the set of monomial ideals J ⊂ S with the property that the radical J of J coincides with I. We are interested in the set of monomial ideals J ∈ FI such that S/J is Cohen–Macaulay. There is a natural subset GI ⊂ FI . In order to describe this set, we denote as usual the unique minimal system of monomial generators of I by G(I). Let G(I) = {u1 , . . . , um }. Then we call a monomial ideal J a modification of I, if G(J) = {v1 , . . . , vm } and supp(vi ) = supp(ui ) for all i. Here the support of a monomial u is the set supp(u) = {i| xi divides u}. Obviously, any modification of I belongs to FI , and we denote the set of modifications of I by GI . A monomial ideal J is called a trivial modification of I, if there exist nonnegative integers a1 , . . . , an such that J is obtained from I by the substitution xi 7→ xai i for i = 1, . . . , n. If J is a trivial modification of I, then J is Cohen–Macaulay, since J = ϕ (I)S where ϕ : S → S is the flat K-algebra homomorphism with ϕ (xi ) = xai i for all i.

In this section we use a theorem of Takayama [27] to study Cohen–Macaulay modifications of squarefree monomial ideal. Let us first recall the result of Takayama which describes the local cohomology of a ring with monomial relations. For a monomial u = xa11 . . . xann , we set supp u = {i | ai 6= 0}. Furthermore for any a ∈ Zn , we set Ga = {i | ai < 0}. Let I ⊂ S be a monomial ideal, and let ∆ be the simplicial complex on the vertex set √ [n] with I∆ = I. Now for any a ∈ Zn , we define the simplicial complex ∆a (I), whose faces are the sets F \ Ga with Ga ⊂ F and F ∈ ∆, and such that for all u ∈ G(I) there exists j 6∈ F with ν j (u) > a j ≥ 0. Notice that we may have ∆a (I) = 0/ for some a ∈ Z. With the notation introduced one has Theorem 2.2.1 (Takayama). Let I ⊂ S = K[x1 , . . ., xn ] be a monomial ideal. Then the multi-graded Hilbert series of the local cohomology modules of R = S/I with respect to the Zn -graded is given by Hilb(Hmi (R),t) =

∑ ∑ dimK H˜ i−|F|−1(∆a(I); K)t a,

F∈∆ a

where t a = t1a1 · · ·tnan . The second sum runs over a ∈ Zn such that Ga = F and a j ≤ ρ j (I) − 1, j = 1, . . ., n, with ρ j (I) = max{ν j (u) | u ∈ G(I)} for√j = 1, . . . , n, and ∆ is the simplicial complex corresponding to the Stanley-Reisner ideal I. Now consider the following construction: let I = (u1 , . . . , um ) be a squarefree CohenMacaulay monomial ideal in S = K[x1 , . . . , xn ], and J = (v1 , . . . , vm ) a monomial ideal with supp(ui ) = supp(vi ) for all i = 1, . . ., m such that ν j (vi ) ≤ 2 for all j = 1, . . ., n. For j = 1, . . . , n let k j ≥ 0 be any integers, and set n

wi =

∏ j=1

k

x j j vi

for

i = 1, . . . , m,

ν j (vi )=2

and L = (w1 , . . . , wm ). Thus L is obtained from J by raising those variables x j to a power ≥ 2 in all monomial generators of I where they appear as a square. For example, if we take I = (x1 x2 x3 , x2 x4 x5 , x1 x4 , x2 x5 x6 ) and J = (x21 x2 x3 , x22 x4 x5 , x1 x24 , x22 x5 x6 ), then L could be any of the ideals L = (xa1 x2 x3 , xb2 x4 x5 , x1 xc4 , xb2 x5 x6 ), where a, b, c are any integers ≥ 2. Notice that each variable which appears with a power ≥ 2 in some monomial generator of L, then this variable appears with the same power in all other monomial generators of L where its power is ≥ 2. With the notation introduced we have Lemma 2.2.2. L is Cohen–Macaulay, if J is Cohen–Macaulay.

Proof. Let R′ = S/L. Then by Theorem 2.2.1 we have either Hmi (R′ )a = 0 or Hmi (R′ )a = H˜ i−|Ga|−1 (∆a (L); K), if Ga ∈ ∆ and a j ≤ ρ j (L)−1, j = 1, . . ., n, with ρ j (L) = max{ν j (u)| u ∈ G(L)} for j = 1, . . . , n; otherwise Hmi (R′ )a = 0. To prove that L is Cohen-Macaulay we have to show that Hmi (R′ )a = 0 for all a ∈ Zn and i < d, where d is the dimension of R′ . If Ga 6∈ ∆, then Hmi (R′ )a = 0 . Thus we may assume from now that Ga ∈ ∆. Let R = S/J. Since by assumption R is Cohen–Macaulay, it follows that Hmi (R)b = 0 for all i < d and all b ∈ Zn . We now will show that for any a ∈ Zn with a j ≤ ρ j (L) − 1, j = 1, . . ., n, there exists a b ∈ Zn with b j ≤ ρ j (J) − 1, j = 1, . . ., n, such that Ga = Gb and ∆a (L) = ∆b (J). Then for all i < d, Hmi (R′ )a = H˜ i−|Ga|−1 (∆a (L); K) = H˜ i−|Gb|−1 (∆b (J); K) = Hmi (R)b = 0. Suppose a ∈ Zn with a j ≤ ρ j (L) − 1, j = 1, . . ., n. Take b ∈ Zn with bi = 1 if ai > 0, and bi = ai , otherwise. By definition Ga = Gb . From the construction of L, it follows that if ν j (vi ) = 2 then ν j (wi ) = ν j (vi ) +k j , and if ρ j (J) = 2 then ρ j (L) = ρ j (J) +k j , otherwise ν j (vi ) = ν j (wi ) and ρ j (L) = ρ j (J). Thus we see that b j ≤ ρ j (J) − 1. Let F \ Ga ∈ ∆a (L); then F ∈ ∆ and Ga ⊂ F. We want to show that F \ Gb ∈ ∆b (J). Since Ga = Gb it follows that Gb ⊂ F. Let vi ∈ G(J). Since F \ Ga ∈ ∆a (L), there exists j 6∈ F such that ν j (wi ) > a j ≥ 0. If ν j (wi ) ≥ 2, then ν j (vi ) = ν j (wi ) − k j > b j ≥ 0 because ν j (vi ) = 2 and 0 ≤ b j ≤ 1. Otherwise ν j (wi ) = ν j (vi ) and a j ≥ b j ≥ 0, which shows that we have again ν j (vi ) > b j ≥ 0. Thus F \ Gb ∈ ∆b (J), and hence ∆a (L) ⊂ ∆b (J). Now let F \ Gb ∈ ∆b (J). Then for any vi ∈ G(J) there exists j ∈ / F such that ν j (vi ) > b j ≥ 0. The number ν j (vi ) can be only 1 or 2. If ν j (vi ) = 1, then b j = 0, hence a j = 0. Therefore in this case we have ν j (wi ) = ν j (vi ) > a j = b j ≥ 0. If ν j (vi ) = 2, then ν j (wi ) = ν j (vi ) + k j = ρ j (L) > ρ j (L) − 1 ≥ a j ≥ b j ≥ 0. Thus F \ Gb ∈ ∆a (L), and hence ∆b (J) ⊂ ∆a (L). Conversely, if L is Cohen–Macaulay, then the subsequent Lemma 2.2.3 implies that J is Cohen–Macaulay as well. Lemma 2.2.3. Let L ⊂ S be a Cohen-Macaulay monomial ideal with G(L) = {u1 , . . . , um }. Fix a number 1 ≤ j ≤ n and assume for simplicity that ν j (u1 ) ≤ ν j (u2 ) ≤ . . . ≤ ν j (um ). Let 0 ≤ a1 ≤ a2 ≤ . . . ≤ am be a sequence of integers with ai ≤ ν j (ui ) for all i and ai − ai−1 ≤ ν j (ui ) − ν j (ui−1) for i = 2, . . . , m, and let J be the monomial ideal with G(J) = {v1 , . . . , vm }, where  νk (ui ), if k 6= j, νk (vi ) = ai , if k = j. Then J is Cohen–Macaulay.

Proof. Without loss of generality we may assume that j = 1. We set bi = ν1 (ui ) for i = 1, . . . , m. Then ui = xb1i vi , and x1 does not divide the monomial vi . We prove the assertion by induction on bm − am . If bm − am = 0, then ai = bi for all i, and the assertion is trivial. Now assume that bm − am > 0, and let t be the smallest integer such that bt − at > 0. Polarizing the first variable we obtain an L′ with G(L′ ) = {u′1 , . . . , u′m }, where u′i = x11 · · · x1bi vi for i = 1, . . . , m. Substituting for j = at + 1, . . . , b1 the variable x1 j by 1, we obtain a monomial ideal L′′ ⊂ K[x11 , . . . , x1at , x1bt +1 , . . ., x1bm , x2 , . . ., xn ] with G(L′′ ) = {u′′1 , . . . , u′′m } where u′′i = x11 · · · x1at x1bt +1 · · · x1bm vi . The ideal L′′ is, up to a flat extension, the localization of L′ with respect to variables x1at +1 , . . . , x1bt . Since the Cohen–Macaulay property under partial polarization and localization is preserved, we see that L′′ is Cohen–Macaulay. Now we observe that L′′ may be obtained by polarizing the first variable of the ideal b′ b′ L∗ = (x11 v1 , . . . , x1m vm ), where b′i

=



bi = ai , bi − (bt − at )

for i < t, for i ≥ t.

Since L′′ is Cohen–Macaulay, it follows that L∗ is Cohen–Macaulay as well, and since L∗ satisfies all conditions of the lemma with b′m − am < bm − am , we may apply our induction hypothesis to obtain the desired conclusion. Now we come to the main result of this section which is an immediate consequence of Lemma 2.2.2 and Lemma 2.2.3. Theorem 2.2.4. Let I ⊂ S be a squarefree Cohen-Macaulay monomial ideal with G(I) = {u1 , . . ., um }. The following conditions are equivalent: (a) I admits infinitely many nontrivial Cohen–Macaulay modifications. (b) I admits one nontrivial Cohen–Macaulay modification. (c) Let G j (I) be the set of monomials of G(I) which are divisible by x j . Then for some j = 1, . . ., n and some proper nonempty subset U ⊂ G j (I), the monomial ideal J = (v1 , . . . vm ) with  uℓ , if uℓ 6∈ U, vℓ = x j uℓ , if uℓ ∈ U is Cohen–Macaulay.

Proof. (a) ⇒ (b) is trivial and (c) ⇒ (a) follows from Lemma 2.2.2. (b) ⇒ (c): Let L = (v1 , . . ., vm ) be the nontrivial Cohen-Macaulay modification of I. Then there exists an integer j such that G j (L) contains two monomials ur , us with ν j (ur ) > ν j (us ) > 0. Let k 6= j with ρk (L) 6= 0. Applying Lemma 2.2.3 for the index k with ai = 1 whenever νk (vi ) > 0, we obtain an ideal L′ = (v′1 , . . . , v′m ) which is again a Cohen–Macaulay modification of I with νk (v′i ) = νk (ui ) for all i. Applying Lemma 2.2.3 similarly for all all k 6= j, we may assume without loss of generality that for L itself we have νk (vi ) = νk (ui ) for all i and all k 6= j. After a relabeling of the generators of L we may assume that 0 = ν j (v1 ) = · · · = ν j (vi ) < ν j (vi+1 ) ≤ . . . ≤ ν j (vℓ ) < ν j (vℓ+1 ) ≤ . . . ≤ ν j (vm ). Applying Lemma 2.2.3 to the ideal L with respect to the sequence   0, if t ≤ i, at = 1, if i < t ≤ ℓ,  2, if ℓ > t. yields the desired ideal Cohen–Macaulay ideal J.

All modifications of a monomial complete intersection ideal are trivial. The next example shows that a squarefree Cohen–Macaulay monomial ideal with nontrivial modifications may have only trivial Cohen–Macaulay modifications. Example 2.2.5. We use Theorem 2.2.4 to show that the Stanley–Reisner ideal associated to triangulation of the real projective plane, I = (x1 x2 x4 , x1 x2 x6 , x1 x3 x5 , x1 x3 x4 , x1 x5 x6 , x2 x4 x5 , x2 x3 x6 , x2 x3 x5 , x3 x4 x6 , x4 x5 x6 ) in the ring S = k[x1 , x2 , x3 , x4 , x5 , x6 ] has only trivial Cohen–Macaulay modifications. Consider, G1 (I) = {x1 x2 x4 , x1 x2 x6 , x1 x3 x4 , x1 x3 x5 , x1 x5 x6 }. There are 30 non-empty proper subsets of G1 (I) and for each proper non-empty subset U ⊂ G1 (I), the ideal J = (v1 , . . . , vm ) with  uℓ , if uℓ 6∈ U, vℓ = x1 uℓ , if uℓ ∈ U is not Cohen–Macaulay. This can easily be verified by using CoCoA. The same can be shown for all G j (I) with j = 2, 3, 4, 5, 6. Hence by Theorem 2.2.4, I has only trivial Cohen–Macaulay modifications.

2.3 Classes of simplicial complexes which admit nontrivial Cohen–Macaualy modifications In this section we will consider big classes of Cohen–Macaulay simplicial complexes with infinitely many nontrivial Cohen–Macaulay modifications.

2.3.1 Simplicial complexes of dim ≤ 1 In the previous section we have seen that the Stanley–Reisner ideal of the simplicial complex attached to the canonical triangulation of the real projective plane has only trivial Cohen–Macaulay modifications. Now we shall see that the Stanley–Reisner ideal of any simplicial complex of dimension ≤ 1 allows infinitely many nontrivial Cohen–Macaulay modifications. Proposition 2.3.1. Let ∆ be a 0-dimensional Cohen–Macaulay simplicial complex on the vertex set [n]. Then there exist infinitely many nontrivial Cohen–Macaulay modification of I∆ . Proof. Note that I = I∆ = ({xi x j for all i 6= j }). Let J be the ideal generated by x21 x2 and all the other generators xi x j 6= x1 x2 in I. Then J is a nontrivial modification of I. We show that J is Cohen–Macaulay. Then the desired result follows from Theorem 2.2.4. Thus it remain to show that depth S/J 6= 0. Suppose that depth S/J = 0. Then there exists f ∈ S \ J with m f ∈ J. Since J is a monomial ideal, we may assume that f is a monomial. The monomials not belonging to J are of the form xai or x1 xb2 . Since xai · xi = xa+1 , it follows i b+1 a b b that mxi 6∈ J, and since x1 x2 · x2 = x1 x2 , it follows that mx1 x2 6∈ J. Thus we have depth S/J > 0. In following theorem we shall assume that I∆ is not a complete intersection, because if it is a complete intersection then all the modifications are trivial. Theorem 2.3.2. Let ∆ be a 1-dimensional Cohen–Macaulay simplicial complex on the vertex set [n], such that I∆ is not a complete intersection. Then there exist infinitely many nontrivial Cohen–Macaulay modifications J of I∆ . Proof. Let G(I∆ ) = {u1 , . . ., um }. Since we assume that I∆ is not a complete intersection, we may assume that u1 and some ui for i > 1 have a common factor, say x1 . Let J be the monomial ideal with G(J) = {x1 u1 , u2 , . . ., um }. Then J is a nontrivial modification of I∆ . We will show that J is a Cohen–Macaulay ideal. Then Theorem 2.2.4 implies that I∆ has infinitely many Cohen–Macaulay modifications, as desired. Since I∆ is Cohen–Macaulay and not a complete intersection it follows that ∆ is connected and that n ≥ 4. Therefore there exist faces {1, i} and {i, j}. We may assume that i = 2 an j = 3. If {1, 3} is not a face, then x1 x3 is a minimal generator. Otherwise, x1 x2 x3 is a minimal generator. Therefore we may assume that u1 = x1 x3 or u1 = x1 x2 x3 . We set R′ = S/J. In order to prove that J is a Cohen–Macaulay ideal, we have show that Hmi (R′ ) = 0 for i = 0, 1. We show this by using the theorem of Takayama [27] which says that dimK Hmi (R′ )a = 0, or dimK Hmi (R′ )a = dimK H˜ i−|Ga|−1 (∆a (J); K),

(2.1)

where a = (a1, . . . , an ) ∈ Zn such that Ga ∈ ∆, and a1 ≤ 1 and ai ≤ 0 for all i = 2, . . ., n. Here Ga = {ai | ai < 0}. In case that ai ≤ 0 for all i, we have ∆a (J) = ∆a (I), and hence for such a it follows that dimK Hmi (R′ )a = dimK Hmi (K[∆])a. Thus, henceforth we may assume that a1 = 1 and ai ≤ 0 for i = 2, . . . , n. Since Ga ∈ ∆, it follows that |Ga | ≤ 2. If |Ga | = 2, then formula (2.1) implies that i Hm(R′ )a = 0 for i = 0, 1. Thus it suffices to consider the cases that |Ga | = 0 and |Ga | = 1. We first treat the case |Ga | = 0 and u1 = x1 x3 . In that case we claim that F (∆a (J)) ⊂ {F ∈ F (∆) | 1 ∈ F or 3 ∈ F}.

(2.2)

Therefore in this case ∆a (J) is a nonempty 1-dimensional connected simplicial complex, and hence Cohen–Macaulay, which implies that Hmi (R′ )a = 0 for i = 0, 1. It follows from the definition of ∆a (J) given before Theorem 2.2.1 that in our situation we have F (∆a(J)) = {F ∈ F (∆) | 1 ∈ F}∪{F ∈ F (∆)|1 6∈ F, F c \{1}∩supp(u) 6= 0/ for all u 6= u1 }. Hence for the proof of (2.2) it remains to be shown that {F ∈ F (∆)|1 6∈ F, F c \ {1} ∩ supp(u) 6= 0/ for all u 6= u1 } ⊂ {F ∈ F (∆) | 3 ∈ F}. Suppose this is not the case, then there exists a facet F = {i, j} of ∆ with 1, 3 6∈ {i, j} and F c \ {1} ∩ supp(u) 6= 0/ for all u 6= u1 . The last condition is equivalent to say that supp(u) 6⊂ {1, i, j} for all u 6= u1 . Since {i, j} is a facet, it follows that {1, i} or {1, j} or {1, i, j} is nonface, and hence the support of some u ∈ G(J) and since 3 6∈ {i, j}, thus u is different from u1 since its support is contained in {1, i, j} we arrive at a contradiction. In the case that |Ga | = 0 and u1 = x1 x2 x3 one shows in exactly the same way as above that F (∆a (J)) ⊂ {F ∈ F (∆) | 1 ∈ F} ∪ {2, 3}. Hence again ∆a (J) is a nonempty 1-dimensional connected simplicial complex, and we are done. Finally we have to consider the case the |Ga | = 1, say, Ga = {s}. We first treat the case that u1 = x1 x3 . Then it follows from the definition of Ga that s 6= 1. According to (2.1) we have dimK Hmi (R′ )a = dimK H˜ i−2 (∆a (J); K). Hence we can have Hmi (R′ )a 6= 0 only for i = 1, and this can only happen, if (∆a (J) = {0}. / Thus it remains to show that for Ga = {s} with s 6= 1 there exists a nonempty face of ∆a (J). We may assume ∆a (J) 6= 0, / because otherwise Hmi (R′ )a = 0. Thus 0/ is a face of ∆a (J) which implies that Ga = {s} ∈ ∆ and for all u ∈ G(J) there exists vertex j 6= s such that ν j (u) > a j ≥ 0. We now show that there exists F ∈ F (∆) such that F \ Ga ∈ ∆a (J).

Suppose s = 3. Then we can choose F = {2, 3}. Indeed, if u ∈ G(J) and u 6= u1 , then there exists j ∈ supp(u) with j 6∈ {1, 2, 3}, because otherwise u = x1 x2 or u = x2 x3 or u = x1 x2 x3 . The first two cases cannot happen, because {1, 2} and {2, 3} are edges of ∆ the third case is impossible, because {1, 3} is a minimal nonface. Thus for this j we have j 6∈ F and ν j (u) > a j ≥ 0, as desired. Suppose now that s 6= 3, then {1, s} ∈ F (∆). Indeed, if this is not the case then {1, s} is a minimal nonface of ∆. Then the monomial u = x1 xs belongs to G(J). But then 0/ 6∈ ∆a (J), because there is no j 6∈ Ga with ν j (u) > a j ≥ 0. Now for the facet F = {1, s} and all u ∈ G(J) we always can find j 6∈ F such that ν j (u) > a j ≥ 0. It follows that {1} = F \ Ga ∈ ∆a (J). In the case that |Ga | = 1 and u1 = x1 x2 x3 the arguments as above shows that {1} ∈ ∆a (J). This concludes the proof of the theorem.

2.3.2 Simplicial trees Suppose ∆ = hF1 , . . . , Fm i, m ≥ 2, is a simplicial tree on the vertex set [n]. So ∆ has at least two leaves. We are going to describe a method to modify ∆ to obtain a new simplicial tree ∆′ with the property that if ∆ is unmixed, then ∆′ is also unmixed. This construction will then be used to construct nontrivial Cohen–Macaulay modifications of I(∆). We may assume that F1 = {v1 , . . ., vs } is a leaf of ∆. As simplicial trees are connected and m ≥ 2, there exists at least one non-free vertex, say v1 , of F1 . To obtain ∆′ we replace the facet F1 by the facet F1′ = {v11 , v12 , . . . , v1k , v2 , . . . , vs }, k ≥ 2, where v1 = v11 and where v12 , . . . , v1k are new vertices. Proposition 2.3.3. Let ∆ be an unmixed simplicial tree. Then ∆′ is also an unmixed simplicial tree. Proof. We first show that ∆′ is again unmixed. It is enough to show that, for every minimal vertex cover C′ of ∆′ , there exist a minimal vertex cover C of ∆ such that |C′ | = |C|. We adopt the notation and the assumptions introduced in the definition of ∆′ and consider the following two cases: Case 1: v12 , . . ., v1k ∈ / C′ . In this case C′ is also a minimal vertex cover of ∆, and so we may choose C = C′ . Case 2: v1r ∈ C′ for some r ∈ {2, . . ., k}. In this case we choose C = {C′ \{v1r }}∪{vi }, where vi is a free vertex of F1 . Such a free vertex exists, since F1 is a leaf. Obviously, |C| = |C′ |. As v1r is free vertex of F1′ , C′ \ {v1r } is a minimal vertex cover of Γ = hF2 , . . . , Fm i. Therefore C = {C′ \ {v1r }} ∪ {vi } is a minimal vertex cover for ∆. It remains to be shown that ∆′ is a again a simplicial tree. Let Γ′ be a simplicial complex with F (Γ′ ) ⊂ F (∆′ ). We have to show that Γ′ has a leaf. If F1′ ∈ / Γ′ , then F (Γ′ ) ⊂ F (∆), and so Γ′ has a leaf, because ∆ is simplicial tree. Otherwise, Γ′ = hF1′ , Fj1 , . . . , Fjl i for certain integers jk ∈ {2, . . ., m}. In case, F1′ is a

leaf of Γ′ , we are done. If this is not case, we consider the simplicial complex Γ = hF1 , Fj1 , . . ., Fjl i. Then F1 is not a leaf of Γ, because hF1 i ∩ hFj1 , . . ., Fjl i = hF1′ i ∩ hFj1 , . . ., Fjl i. However, since ∆ is a simplicial tree, and since F (Γ) ⊂ F (∆), it follows that Γ has a leaf, say Fj p . Since F (Γ′ ) \ {F1′ } = F (Γ) \ {F1} and since F1 ∩ Fj p = F1′ ∩ Fj p , it follows that Fj p is also a leaf of Γ′ . Corollary 2.3.4. Let ∆ be a simplicial tree with at least two facets, and let be I = I(∆) the facet ideal of ∆. Suppose I is Cohen–Macaulay. Then there exist infinitely many nontrivial Cohen–Macaulay modification of I. Proof. Let ∆ = hF1 , . . . , Fm i, m ≥ 2, be a simplicial tree on the vertex set [n]. Then I = I(∆) = (xF1 , xF2 , . . ., xFm ). We may assume that F1 = {v1 , v2 , . . ., vs } is leaf of ∆ and that v1 is non-free vertex of F1 . We claim that for each integer k ≥ 0 the ideal Jk = (xk1 xF1 , xF2 , . . . , xFm ) is Cohen–Macaulay. From this the desired assertion follows, since each of the Jk is a modification of I. In order to prove the claim, we consider the polarization of Jk and obtain Jkp = (xF1′ , xF2 , . . ., xFm ), where F1′ = {v11 , v12 , . . . , v1k+1 , v2 , . . . , vs } with v11 = v1 . Thus we see that J = I(∆′), where ∆′ is constructed from ∆ as described above. Now Proposition 2.3.3 implies that ∆′ is an unmixed simplicial tree, and so Corollary 1.3.19 guarantees that Jk is Cohen– Macaulay.

2.3.3 Monomial ideals of codimension 2 Let I ⊂ S = K[x1 , x2 , . . . , xn ] be a squarefree Cohen–Macaulay monomial ideal of codimension 2. In this section we shall prove the existence of infinitely many Cohen-Macaulay modifications J for the ideal I. To describe the result we need to recall some notation and results from [24]. Definition 2.3.5. Let Γ be a tree on the vertex set [m] and i, j be two distinct vertices of Γ. Then there exist a unique path from i to j, that is, a sequence of numbers i = i0 , i1 , i2, . . . , ik−1 , ik = j such that {iℓ, iℓ+1 } ∈ E(Γ) for ℓ = 0, . . ., k − 1 are pairwise different edges. We set b(i, j) = i1 and e(i, j) = ik−1

For the proof of the main theorem of this section we shall need the following result (see [24, Theorem 1.5]): Theorem 2.3.6. (a) Let I ⊂ S = K[x1 , x2 , ..., xn] be a Cohen-Macaulay monomial ideal of codimension 2 generated by m + 1 elements. Then there exists a tree Γ with m + 1 vertices and for each edge {i, j} of Γ there exists monomials ui j and u ji in S such that (i) gcd(uib(i, j), u je(i, j)) = 1 for all i < j, and (ii) I = (u1 , . . . , um+1 ) with

u j = ∏m+1 i=1 uib(i, j)

for

j = 1, . . . , m + 1.

i6= j

(b) Conversely, if Γ is a tree with [m + 1] vertices and for each {i, j} ∈ E(Γ) we are given monomials ui j and u ji in S satisfying (a)(i). Then the ideal defined in (a)(ii) is CohenMacaulay of codimension 2. Now we are in a position to prove Theorem 2.3.7. Let I = (u1 , u2 , . . ., um+1 ) ⊂ S = K[x1 , x2 , . . . , xn ] be a squarefree Cohen– Macaulay monomial ideal of codimension 2 which is not a complete intersection. Then there exists infinitely many nontrivial Cohen-Macaulay modifications of I. Proof. Let Γ be a tree for I as described in Theorem 2.3.6(a). Then u j = ∏m+1 i=1 uib(i, j) for i6= j

j = 1, . . . , m + 1. Since I is not a complete intersection, the tree Γ has more that one edge, and hence at least two leaves, that is, edges with a free vertex. A vertex of Γ is called a free vertex, if it belongs to exactly one edge. After a relabeling of the vertices, we may assume that {1, 2} is a leaf with free vertex 1, and that {t, m + 1} is a leaf with free vertex m + 1. We may assume that x1 divides um+1,t . For a given integer k ≥ 1 we set u′21 = xk1 u21 and ′ u′i j = ui j if i 6= 2 or j 6= 1. Finally we set u′j = ∏m+1 i=1 uib(i, j) for j = 1, . . . , m + 1, and i6= j

J = (u′1 , . . ., u′m+1 ), and claim that J is a Cohen-Macaulay modification of I. To this end we have to show: (1) supp(u′i ) = supp(ui ) for i = 1, . . ., m + 1; (2) The monomials u′i j satisfy condition (a)(i) of Theorem 2.3.6.

Theorem 2.3.6(b) then implies that J is indeed Cohen-Macaulay. In order to prove (1) we first observe that u21 = u2b(2,1) is a factor of u1 , but does not coincide with any of the factors uib(i, j) in u j for j ≥ 2, because the only path from 2 to j with b(2, j) = 1 is the path from 2 to 1. This implies that u′1 = xk1 u1 and u′j = u j for j ≥ 2. On the other hand, the factor um+1,t = um+1,b(m+1,1) appears in u1 , so that x1 divides u1 . It follows that supp(u′1 ) = supp(u1 ).

For (2) we have to show that gcd(u′ib(i, j), u′je(i, j)) = 1 for all i < j. This is the case whenever (i, b(i, j)) 6= (2, 1) and ( j, e(i, j)) 6= (2, 1). Suppose first that (i, b(i, j)) = (2, 1). Then j = 1 and e(i, j) = 2. Thus in this case we need to show that gcd(xk1 u2 , u12 ) = 1, or equivalently that x1 does not divide u12 . To see this we observe that gcd(u12 , um+1,t ) = gcd(u1b(1,m+1) , um+1,e(1,m+1) ) = 1, and that x1 divides um+1,t . Finally suppose that ( j, e(i, j)) = (2, 1). Then i = 1 and b(i, j) = 2. Thus in this case we need to show that gcd(u12 , xk1 u2 ) = 1. This has already been shown in the first case.

Chapter 3 Cohen-Macaulay intersections 3.1 Cohen–Macaulay intersections associated to a graph Let G be a simple graph on the vertex set V (G) = {v1 , v2 , . . . , vn } with edge set E(G). It is assumed throughout the paper that G has no isolated vertices. We associate to G the ideal IG =

\

(xi , x j ),

{vi ,v j }∈E(G)

in the polynomial ring S = K[x1 , . . . , xn ] where K is an arbitrary field. Dually one defines the so-called edge ideal I(G) = ({xi x j } : {vi , v j } ∈ E(G)). These two ideals are related via Alexander duality. Indeed, let ∆G be the (unique) simplicial complex whose Stanley–Reisner ideal I∆G coincides with I(G). Then IG = I∆∨G , where ∆∨ G is the Alexander dual of ∆G , see [15]. By using the above duality and a result of Eagon and Reiner [7], as well as a result of Fr¨oberg [13] we immediately obtain Proposition 3.1.1. The ideal IG is Cohen-Macaulay if and only if the complementary graph G is chordal. Proof. The Eagon–Reiner theorem asserts that the Stanley–Reisner ideal I∆ is CohenMacaulay if and only if I∆∨ has a linear resolution, while Fr¨oberg’s theorem says that edge ideal of a graph G has a linear resolution if and only if the complementary graph G is chordal. Thus our assertion follows from the fact that IG = I∆∨G . Now let IG be Cohen-Macaulay and let A = (akl )k,l=1,...,n be a matrix of positive integers. We associate to the matrix A the monomial ideal JA =

\

a

a

(xi i j , x j ji ).

{vi ,v j }∈E(G)

√ Note that each unmixed monomial ideal L with L = IG is of this form. Thus the question is for which matrices A, the ideal JA is Cohen-Macaulay. Our first observation is the following: Proposition 3.1.2. The Cohen–Macaulay property of JA depends only on A and not on the characteristic of K. Now we are ready to prove Proposition 3.1.2. Proof of Proposition 3.1.2. By (Proposition 1.1.25), the ideal JA is Cohen–Macaulay if and only if JAp is Cohen–Macaulay. To compute JAp we apply the rule (1.1) and obtain p

\

JA =

a

a

(xi i j , x j ji ) p ,

{vi ,v j }∈E(G)

and a a (xi i j , x j ji ) p

ai j

a ji

= ( ∏ xik , ∏ x jl ) = k=1

l=1

\

(xik , x jl ).

k=1,...,ai j l=1,...,a ji

Thus we see that JAp is the intersection of monomial prime ideals of height 2, and hence there exists a unique graph G(A) such that p

JA = IG(A). Now we see that JAp is Cohen–Macaulay, if and only if the complementary graph G(A) of G(A) is chordal. This property does not depend on K. The proof of Proposition 3.1.2 shows that our problem of determining the Cohen√ Macaulay ideals J with J = IG reduces to the following combinatorial problem: given a graph G on the vertex set V (G) = {v1 , v2 , . . ., vn } whose complementary graph G is chordal, for which n × n-matrices A with positive integer entries is the graph G(A) again chordal? The set of adjacent vertices of vi will be denoted by AG (vi ). Theorem 3.1.3. Suppose that IG is Cohen–Macaulay. Let W = {vi1 , . . ., vir } be a set of pairwise distinct vertices of G with the property that each vik belongs to exactly one maximal independent set Tk of G, where Tk 6= Tl for k 6= l. For each vik ∈ W we choose a nonempty subset Ak ⊂ AG (vik ) with the property that, if some vik ∈ A j then vi j 6∈ Ak and Ak ∩ Al = 0/ for k 6= l, and let J=

\

{vi ,v j }

(xi , x j ) ∩

r \ \

aj

(xik , x j ),

k=1 v j ∈Ak

where the first intersection is taken over all edges {vi , v j } different from the edges {vik , v j } with v j ∈ Ak , and where each a j is a positive integer. Then J is Cohen–Macaulay.

The proof of the theorem will be postponed to the next section where we introduce certain constructions of graphs. The following examples demonstrate the theorem: Example 3.1.4. (a) Consider the graph shown in Figure 3.1. ◦ v1

◦ v2

◦ v3

◦ v4

Figure 3.1: The ideal associated to this graph is IG = (x1 , x2 ) ∩ (x2 , x3 ) ∩ (x3 , x4 ). The maximal independent sets are {v1 , v3 }, {v1 , v4 } and {v2 , v4 }. Now the elements which belongs to exactly one maximal independent set are v2 and v3 , hence W = {v2 , v3 }. In our notations AG (v2 ) = {v1 , v3 } and AG (v3 ) = {v2 , v4 }. We select A2 ⊂ AG (v2 ) and A3 ⊂ AG (v3 ) with A2 = {v1 , v3 } and A3 = {v4 } (note that v2 6∈ A3 as v3 ∈ A2 ). Then the ideal, J = (x1 a1 , x2 ) ∩ (x2 , x3 a2 ) ∩ (x3 , x4 a3 ) is Cohen-Macaulay for all choices of the positive integers a1 , a2 and a3 . We can also take A2 = {v1 } and A3 = {v2 , v4 }, in this case the ideal will be of the form J = (x1 a1 , x2 ) ∩ (x2 a2 , x3 ) ∩ (x3 , x4 a3 ) and will be Cohen-Macaulay for all choices of the positive integers a1 , a2 and a3 . On the other hand, v1 belongs to two maximal independent sets. Here AG (v1 ) = {v2 } and if we raise the power of x2 in (x1 , x2 ), the resulting ideal will not be Cohen–Macaulay anymore. Thus this example shows that in Theorem 3.1.3 we cannot drop the condition that the vertices vik belong to exactly one independent set of G. (b) Let G be a complete graph on the vertex set {v1 , . . . , vn }. Then the ideal associated to G is \ I= (xi , x j ). 1≤i< j≤n

In this case, the independent sets are exactly the singletons {vi } and therefore W = V (G). Fix a vi ∈ W we have AG (vi ) = {v1 , . . . , vi−1 , vi+1 , . . . , vn }. Take Ai = AG (vi ) and Ak = 0/ for all k 6= i. Then the ideal J=

[

1≤k 1 and suppose that the graph G′ = Gv1 ···vi−1 is chordal. Let w1 , . . ., wr be the new vertices attached to G in the construction of Gv1 ···vr . We claim that AG′ (vi ) = AG (vi ) ∪ {w1 , . . ., wi−1 }. It is clear that AG (vi ) ⊂ AG′ (vi ). On the other hand, since v j 6= vi for all j < i it follows from the construction of G′ that {vi , w j } ∈ E(G′ ) for all j = 1, . . ., i − 1. This implies that {w1 , . . . , wi−1 } ⊂ AG′ (vi ), and hence AG (vi ) ∪ {w1 , . . ., wi−1 } ⊂ AG′ (vi ). In order to prove the other inclusion, let v ∈ AG′ (vi ). If v 6= w j for all j < i, then v ∈ G and hence v ∈ AG (vi ), as desired. Now, since by assumption, cG (vi ) = 0, it follows that AG (vi ) is a clique. Also the set {w1 , . . ., wi−1 } is a clique in G′ . Since {vi , v j } 6∈ E(G) for all i 6= j, it follows that {v, w j } ∈ E(G′ ) for all v ∈ AG (vi ) and j = 1, . . ., i − 1. Thus we see that AG′ (vi ) is a

clique, which implies that cG′ (vi ) = 0. Hence, by Lemma 3.2.5, it follows that G′vi is chordal. The result for Gva1 ···var r follows by induction on ∑ri=1 ai from the following claim: 1 suppose that G b1 br is chordal for some positive integers b1 , . . ., br , and k is an integer v1 ···vr

with 1 ≤ k ≤ r. Then G

G Since (G

b

b −1

v11 ···vk k

Lemma 3.2.6.

b +1

b

v11 ...vk k

) ···vbr r vk

···vbr r

b +1

b

v11 ···vk k

=G

b

b

is again chordal. Indeed,

···vbr r

v11 ···vk k ···vbr r

= ((G

b

b −1

v11 ···vk k

) ) . ···vbr r vk vk

is chordal by assumption, the claim follows from

Now we are ready to prove our main theorem. aj

Proof of Theorem 3.1.3: Consider J = {vi ,v j } (xi , x j ) ∩ rk=1 v j ∈Ak (xik , x j ), be as in the theorem with ∑v j ∈Ak a j − |Ak | = bk . Since by assumption, IG is Cohen–Macaulay, it follows that G is chordal. Observe that a subset T ⊂ V (G) is a maximal independent set in G if and only if T is a maximal clique in G. Hence our hypothesis on W implies that vik ∈ W belongs to exactly one maximal clique in G, namely to Tk = {vik } ∪ AG (vik ), and hence cG (vik ) = 0 for all k. Condition Tk 6= Tl for all k 6= l guarantees that {vik , vil } 6∈ E(G) for ik 6= il . Therefore Proposition 3.2.8 implies that G b1 b2 br is chordal. T

Then

Jp

T

T

vi vi ···vir 1

2

= JH , where H is the complementary graph of G

b

b

1

2

vi 1 vi 2 ···vbirr

(in fact H is ob-

tained from G by joining bk new vertices with vik for all k). As explained in Section 1, this implies that J is Cohen–Macaulay.

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