Simplicial Trees are Sequentially Cohen-Macaulay

Transcription

1 Simplicial Trees are Seqentiall Cohen-Macala Sara Faridi Agst 27, 2003 Abstract This paper ses dalities between facet ideal theor and Stanle-Reisner theor to show that the facet ideal of a simplicial tree is seqentiall Cohen-Macala. The proof involves showing that the Aleander dal (or the cover dal, as we call it here) of a simplicial tree is a componentwise linear ideal. We conclde with additional combinatorial properties of simplicial trees. The main reslt of the this paper is that the facet ideal of a simplicial tree is seqentiall Cohen-Macala. Seqentiall Cohen-Macala modles were introdced b Stanle [S] (following the introdction of nonpre shellabilit b Björner and Wachs [BW]) so that a nonpre shellable simplicial comple had a seqentiall Cohen-Macala Stanle-Reisner ideal. Herog and Hibi ([HH]) then defined the notion of a componentwise linear ideal, which etended a criterion of Eagon and Reiner ([ER]) for Cohen-Macalaness of an ideal to a criterion for seqential Cohen-Macalaness. Simplicial trees, on the other hand, were introdced in [F1] in the contet of Rees rings, and their facet ideals were stdied frther in [F2] for their Cohen-Macala properties, and in [Z] for their resoltions. The facet ideal of a given simplicial comple is a sqare-free monomial ideal where ever generator is the prodct of the vertices of a facet of the comple. If the simplicial comple is a tree (Definition 3.5), it trns ot that its facet ideal has man interesting algebraic and combinatorial properties. Given a sqare-free monomial ideal, one cold consider it as the facet ideal of one simplicial comple, and the Stanle-Reisner ideal of another. This in a sense gives two langages to std a sqare-free monomial ideal. Below we provide a dictionar which makes it eas to move from one langage to the other. We se this dictionar to translate eisting criteria for Cohen-Macalaness and seqential Cohen-Macalaness into the langage of facet ideals, and then finall se these criteria to show that the facet ideal of a simplicial tree is seqentiall Cohen-Macala (Corollar 5.6). There are several bprodcts. An immediate one is that the facet ideal of an nmied simplicial tree (Definition 1.5) is Cohen-Macala (Corollar 5.8). This is discssed at length and proved independentl in [F2], where we introdce the concept of grafting a simplicial comple. As it trns ot, an nmied tree is grafted, and an grafted simplicial comple is Cohen-Macala. This fact, in addition to proving the statement of Corollar 5.8, gives the precise combinatorial strctre of a Cohen-Macala tree. Université d Qébec à Montréal, Laboratoire de combinatoire et d informatiqe mathématiqe, Case postale 8888, sccrsale Centre-Ville, Montréal, QC Canada H3C 3P8. faridi@math.qam.ca. This research was spported b a NSERC Postdoctoral Fellowship. 1

2 Another otcome is that the Stanle-Reisner comple corresponding to a Cohen-Macala tree is shellable. This was known in the case of graphs ([V]). In general, shellabilit is onl a necessar condition for Cohen-Macalaness. The paper is organied as follows: Section 1 reviews the basics of facet ideal theor, introdcing cover complees. In Section 2 we discss how facet ideal theor relates to Stanle-Reisner theor. In Section 3 we define simplicial trees and discss their localiation. In Section 4 we define seqentiall Cohen-Macala and componentwise linear ideals, and introdce a criterion for an ideal to be seqentiall Cohen-Macala, which we se in Section 5 to prove that trees are seqentiall Cohen-Macala. For the convenience of the reader, we have inclded a table of notation at the end of the paper (Figre 2). We wold like to thank Jürgen Herog for raising the qestion of whether simplicial trees are seqentiall Cohen-Macala, and for an earlier reading of this manscript. 1 Basic Definitions This section is a review of the basic definitions and notations in facet ideal theor. Mch of the material here appeared in more detail in [F1] and [F2], ecept for the discssion on the cover comple. Definition 1.1 (simplicial comple, facet, sbcollection and more). A simplicial comple over a set of vertices V = {v 1,..., v n } is a collection of sbsets of V, with the propert that {v i } for all i, and if F then all sbsets of F are also in (inclding the empt set). An element of is called a face of, and the dimension of a face F of is defined as F 1, where F is the nmber of vertices of F. The faces of dimensions 0 and 1 are called vertices and edges, respectivel, and dim = 1. The maimal faces of nder inclsion are called facets of. The dimension of the simplicial comple is the maimal dimension of its facets. We denote the simplicial comple with facets F 1,..., F q b = F 1,..., F q and we call {F 1,..., F q } the facet set of. A simplicial comple with onl one facet is called a simple. B a sbcollection of we mean a simplicial comple whose facet set is a sbset of the facet set of. Definition 1.2 (connected simplicial comple). A simplicial comple = F 1,..., F q is connected if for ever pair i, j, 1 i < j q, there eists a seqence of facets F t1,..., F tr of sch that F t1 = F i, F tr = F j and F ts F ts+1 for s = 1,..., r 1. Definition 1.3 (facet ideal, facet comple). Let k be a field and 1,..., n be a set of indeterminates, and R = k[ 1,..., n ] be a polnomial ring. Let be a simplicial comple over n vertices labeled v 1,..., v n. We define the facet ideal of, denoted b F( ), to be the ideal of R generated b sqare-free monomials i1... is, where {v i1,..., v is } is a facet of. Let I = (M 1,..., M q ) be an ideal in R, where M 1,..., M q are sqare-free monomials in 1,..., n that form a minimal set of generators for I. We define the facet comple of I, denoted b δ F (I), to be the simplicial comple over a set of vertices v 1,..., v n with facets F 1,..., F q, where for each i, F i = {v j j M i, 1 j n}. 2

3 Throghot this paper we often se a letter to denote both a verte of and the corresponding variable appearing in F( ), and i1... ir to denote a facet of as well as a monomial generator of F( ). Eample 1.4. If is the simplicial comple,, v drawn below, v then F( ) = (,, v) is its facet ideal. Facet ideals give a one-to-one correspondence between simplicial complees and sqarefree monomial ideals. Net we define the notion of a verte cover. The combinatorial idea here comes from graph theor. In algebra, it corresponds to prime ideals ling over the facet ideal of a given simplicial comple. Definition 1.5 (verte cover, verte covering nmber, nmied). Let be a simplicial comple with verte set V. A verte cover for is a sbset A of V that intersects ever facet of. If A is a minimal element (nder inclsion) of the set of verte covers of, it is called a minimal verte cover. The smallest of the cardinalities of the verte covers of is called the verte covering nmber of and is denoted b α( ). A simplicial comple is nmied if all of its minimal verte covers have the same cardinalit. Eample 1.6. If is the simplicial comple in Eample 1.4, then the verte covers of are: {, }, {, }, {, v}, {, }, {, v}, {,, }, {,, }, {,, v},.... The first five verte covers above (highlighted in bold), are the minimal verte covers of. In all the argments in this paper, nless otherwise stated, k denotes a field. Given a sqare-free monomial ideal I in a polnomial ring k[ 1,..., n ], the vertices of δ F (I) are those variables that divide a monomial in the generating set of I; this set ma not inclde all elements of { 1,..., n }. The fact that some etra variables ma appear in the polnomial ring has little effect on the algebraic or combinatorial strctre of δ F (I). On the other hand, if is a simplicial comple, being able to consider the facet ideals of its sbcomplees as ideals in the same ambient ring simplifies man of or discssions. For this reason we make the following definition. Definition 1.7 (variable cover). Let I be a sqare-free monomial ideal in a polnomial ring R = k[ 1,..., n ]. A sbset A of the variables { 1,..., n } is called a (minimal) variable cover of = δ F (I) (or of I) if A is the generating set for a (minimal) prime ideal of R containing I. If 1,..., n are all vertices of = δ F (I), then a variable cover of is eactl the same as a verte cover of. In general ever variable cover of contains a verte cover of. For eample for the ideal I = (, ) k[,,, ], {, } is a variable cover bt not a verte cover. The minimal verte covers of, however, are alwas the same as the minimal variable covers of. We now constrct a new simplicial comple sing the minimal verte covers of a given simplicial comple. 3

4 Definition 1.8 (cover comple). Given a simplicial comple, the simplicial comple M called the cover comple of, is the simplicial comple whose facets are the minimal verte covers of. Eample 1.9. In Eample 1.6, =,, v and M =,, v,, v. It is worth observing that being nmied is eqivalent to M being pre (meaning that all facets of M are of the same dimension). This fact becomes sefl in or discssions below. For eample the simplicial comple in Eample 1.6 is nmied, and M is pre. The following fact is known in hpergraph theor (see, for eample, [B]). We otline a proof below. Proposition 1.10 (The cover comple is a dal). If is a simplicial comple, then M is a dal of ; i.e. MM =. Proof. Sppose that = F 1,..., F q and M = G 1,..., G p. Sppose that for i = 1,..., p, q i is the prime ideal generated b the elements of G i, so that we have I = F( ) = q 1... q p. We first show that ever facet of is a verte cover of M. Consider the facet F 1. Since the monomial F 1 q i for all i, it follows that F 1 contains at least one verte of each of the G i. This proves that F 1 is a verte cover of M. Sppose now that F is an minimal verte cover of M. Since F contains a verte of each of the G i, it belongs to all the ideals q i (if we consider F as a monomial), and therefore F I. So some generator F j of I mst divide F. This means that F j F, bt since F j is alread a verte cover of M, it follows that F = F j. This shows that F 1,..., F q are all the minimal verte covers of M. 2 Relations to Stanle-Reisner theor We begin b the basic definitions from Stanle-Reisner theor. For a detailed coverage of this topic, we refer the reader to [BH]. Definition 2.1 (nonface ideal, nonface comple). Let k be a field and 1,..., n be a set of indeterminates, and R = k[ 1,..., n ] be a polnomial ring. Let be a simplicial comple over n vertices labeled v 1,..., v n. We define the nonface ideal or the Stanle-Reisner ideal of, denoted b N ( ), to be the ideal of R generated b sqare-free monomials i1... is, where {v i1,..., v is } is not a face of. Let I = (M 1,..., M q ) be an ideal in R, where M 1,..., M q are sqare-free monomials in 1,..., n that form a minimal set of generators for I. We define the nonface comple or the Stanle-Reisner comple of I, denoted b δ N (I), to be the simplicial comple over a set of vertices v 1,..., v n, where {v i1,..., v is } is a face of δ N (I) if and onl if i1... is / I. Notation 2.2. To simplif notation, we se N to mean the nonface comple of F( ) for a given simplicial comple. In other words, we set N = δ N (F( )). 4

5 Given an ideal I k[v ] where V = { 1,..., n }, if there is no reason for confsion, we se and N to denote δ F (I) and δ N (I), respectivel. If F is a face of = F 1,..., F q, we let the complements of F and be F c = V \ F and c = F 1 c,..., F q c. Definition 2.3 (Aleander dal). Let I be a sqare-free monomial ideal in the polnomial ring k[v ] with V = { 1,..., n }. Then the Aleander dal of N is the simplicial comple N = {F V F c / N }. It is eas to see that N = N. We now focs on the relations between and N for a given sqare-free monomial ideal I. The first qestion we tackle is how to constrct N from. Proposition 2.4. Given a simplicial comple, we have (a) N = M c ; (b) N = M N = c. Proof. (a) This is eas to check. See, for eample, [BH] Theorem (b) The last eqalit follows from Proposition 1.10 and Part (a), since M N = M M c = c. We translate both sides of the first eqation sing the notations in 2.2 and Definition 2.3: N = {F c F / N } and c = F c F is a facet of. Sppose that F c N. Then F / N, and therefore if f denotes the monomial that is the prodct of the vertices of F, and I = N ( N ), then f I. It follows that for some generator g of I, g f. If G is the facet of corresponding to g, we have G F, which implies that F c G c ; so F c c. Conversel, let G c. Then G F c, where F is a facet of, so f I which implies that F / N. So F c N, which implies that G N. Proposition 2.4 is basicall saing that the relationship between M and N is the same as the relationship between and N. The eample below clarifies this point. Eample 2.5. Let I = (, ) k[,,, ]. Then the dal ideal of I, which is the facet ideal of M, or eqivalentl the nonface ideal of N, is the ideal J = (,, ). The relationship between the for simplicial complees and the two ideals is shown in Figre 1. Proposition 2.4 jstifies the following definition. Definition 2.6 (dal of an ideal). Given a sqare-free monomial ideal I in a polnomial ring and = δ F (I), we define the dal of I, denoted b I, to be the facet ideal of M, or eqivalentl, the nonface ideal of N. So I = F( M ) = N ( N ). 5

6 facet I = (,) nonface cover dal complements Aleander dal facet J = (,,) nonface Figre 1: Diagram of Eample 2.5 We now state a criterion for the Cohen-Macalaness of a sqare-free monomial ideal that is de to Eagon and Reiner ([ER]) in the langage stated above. First we define an ideal with a linear resoltion. Definition 2.7 (linear resoltion). An ideal I in a polnomial ring R = k[ 1,..., n ] over a field k, with the standard grading deg( i ) = 1 for all i, is said to have a linear resoltion if R/I has a minimal free resoltion sch that for all j > 1 the nonero entries of the matrices of the maps R β j R β j 1 are of degree 1. Theorem 2.8 ([ER] Theorem 3). Let I be a sqare-free monomial ideal in a polnomial ring R. Then R/I is Cohen-Macala if and onl if I has a linear resoltion. 3 Simplicial Trees Considering simplicial complees as higher dimensional graphs, one can define the notion of a tree b etending the same concept from graph theor. Simplicial trees were first introdced in [F1] in order to generalie reslts of [SVV] on facet ideals of graph-trees. The constrction trned ot to have interesting additional combinatorial and algebraic properties. Before we define a tree, we determine what removing a facet from a simplicial comple means. We define this idea so that it corresponds to dropping a generator from its facet ideal. 6

7 Definition 3.1 (facet removal). Sppose is a simplicial comple with facets F 1,..., F q and F( ) = (M 1,..., M q ) its facet ideal in R = k[ 1,..., n ]. The simplicial comple obtained b removing the facet F i from is the simplicial comple \ F i = F 1,..., ˆF i,..., F q. Note that F( \ F i ) = (M 1,..., ˆM i,..., M q ). Also note that the verte set of \ F i is a sbset of the verte set of. Eample 3.2. Let be a simplicial comple with facets F = {,, }, G = {,, } and H = {, v}. Then \ F = G, H is a simplicial comple with verte set {,,, v}. In graph theor, a tree is defined as a connected ccle-free graph. An eqivalent definition is that a tree is a connected graph whose ever sbgraph has a leaf, where a leaf is a verte that belongs to onl one edge. We make an analogos definition for simplicial complees b etending (and slightl changing) the definition of a leaf. Definition 3.3 (leaf). A facet F of a simplicial comple is called a leaf if either F is the onl facet of, or for some facet G \ F we have F ( \ F ) G. Eqivalentl, the facet F is a leaf of if F ( \ F ) is a face of \ F. Eample 3.4. Let I = (,, v). Then F = is a leaf, bt H = is not, as one can see in the pictre below. v = v = Definition 3.5 (tree, forest). A connected simplicial comple is a tree if ever nonempt sbcollection of has a leaf. If is not necessaril connected, bt ever sbcollection has a leaf, then is called a forest. Eample 3.6. The simplicial complees in eamples 1.4 and 3.4 are both trees, bt the one below is not becase it has no leaves. It is an eas eercise to see that a leaf mst contain a free verte, where a verte is free if it belongs to onl one facet. An effective wa to make algebraic argments on trees is sing localiation. It trns ot that the minimal generating set of a localiation of the facet ideal of a tree corresponds to a forest. As we shall see below, this fact makes it eas to se indction on the nmber of vertices of a tree. For details on the localiation of s simplicial comple see [F2]. Here we give an eample to clarif what we mean b localiation. 7

8 Eample 3.7. Let be the simplicial comple below with I = (,, v) its facet ideal in the polnomial ring R = k[,,,, v]. Let p = (,, ) be a prime ideal of R. Then I p = (,, ) = (, ) is the facet ideal of the forest below on the left. If q = (,, v) then I q = (,, v) = (, v) corresponds to the tree on the right. Localiation at p: Localiation at q: Eample 3.7 is an eample of the following general fact. Lemma 3.8 (Localiation of a tree is a forest). Let I k[ 1,..., n ] be the facet ideal of a tree, where k is a field, and sppose that p is a prime ideal of k[ 1,..., n ]. Then for an prime ideal p of R, δ F (I p ) is a forest. Proof. See [F2] Lemma Seqentiall Cohen-Macala simplicial complees The notion of a seqentiall Cohen-Macala ideal was introdced b Stanle following the introdction of nonpre shellabilit b Björner and Wachs [BW]. It was known that ever shellable simplicial comple (which was b definition pre) was Cohen-Macala, bt what abot nonpre shellable simplicial complees? As it trns ot, seqentiall Cohen-Macala is the correct notion to fill in the gap here. On the other hand, the criterion of Eagon and Reiner ([ER]) stated that a simplicial comple is Cohen-Macala if and onl if its Aleander dal has a linear resoltion. Herog and Hibi ([HH]) developed the definition of a componentwise linear ideal so that the above criterion etended to seqentiall Cohen-Macala ideals: a simplicial comple is seqentiall Cohen-Macala if and onl if its Aleander dal is componentwise linear. In or setting, we se an eqivalent characteriation of seqentiall Cohen-Macala given b Dval, along with Theorem 2.8 and the relationship between Aleander dalit and cover comple dalit discssed in Section 2, to prove that simplicial trees are seqentiall Cohen-Macala. In fact, we show that if I is the facet ideal of a simplicial tree, then the dal I of I has sqare-free homogeneos components with linear qotients. This propert is slightl stronger than what we need, and it shows that if I is a Cohen-Macala ideal to begin with, then N is shellable (which was known for the case where is a graph; Theorem of [V]). Another otcome is the fact that an nmied tree is Cohen-Macala (Corollar 5.8), which was shown in [F2] sing ver different tools. Definition 4.1 ([S] Chapter III, Definition 2.9). Let M be a finitel generated Z- graded modle over a finitel generated N-graded k-algebra, with R 0 = k. We sa that M is seqentiall Cohen-Macala if there eists a finite filtration 0 = M 0 M 1... M r = M of M b graded sbmodles M i satisfing the following two conditions. 8 v v

9 (a) Each qotient M i /M i 1 is Cohen-Macala. (b) dim (M 1 /M 0 ) < dim (M 2 /M 1 ) <... < dim (M r /M r 1 ), where dim dimension. denotes Krll A simplicial comple is said to be seqentiall Cohen-Macala if its Stanle-Reisner ideal has a seqentiall Cohen-Macala qotient. The following characteriation of a seqentiall Cohen-Macala simplicial comple given b Dval ([D] Theorem 3.3) is what we se in this paper. Theorem 4.2 ([D] seqentiall Cohen-Macala). Let I be sqare-free monomial ideal I in a polnomial ring R over a field k, and let N = δ N (I). Then R/I is seqentiall Cohen-Macala if and onl if for ever i, 1 i dim N, if N, i is the pre i- dimensional sbcomple of N, then R/N ( N, i ) is Cohen-Macala. Eample 4.3. let I = (, ) be the ideal of Eample 2.5 in the diagram above. Then for i = 0, 1, 2, we have the following three simplicial complees, respectivel, which are, respectivel, the nonface complees of the ideals I 0 = (,,,,, ), I 1 = (,, ) and I 2 = (). One can verif that all three of these ideals have Cohen- Macala qotients, so I is seqentiall Cohen-Macala. We define a componentwise linear ideal in the sqare-free case sing [HH] Proposition 1.5. Definition 4.4 (sqare-free homogeneos component, componentwise linear). Let I be a sqare-free monomial ideal in a polnomial ring R. For a positive integer k, the k-th sqare-free homogeneos component of I, denoted b I [k] is the ideal generated b all sqare-free monomials in I of degree k. The ideal I above is said to be componentwise linear if for all k, the sqare-free homogeneos component I [k] has a linear resoltion. Let = F 1,..., F q be a simplicial comple with F( ) k[v ], V = { 1,..., n }, and let M = G 1,..., G p be its cover comple. Then b Proposition 2.4 we know that N = G 1 c,..., G p c. For a given i, consider the pre i-dimensional sbcomple of N N, i = H 1,..., H. B Theorem 2.8 showing that I i = N ( N, i ) is a Cohen-Macala ideal is eqivalent to showing that I i has a linear resoltion. B Proposition 2.4, I i is the facet ideal of N, i c. 9

10 So we focs on H c, where H is a facet of N, i. Since H belongs to a sbcomple of N, for some facet G j c of N, H G j c. This implies that G j = G j cc H c ; i.e. H c contains a minimal verte cover of, and so H c is a variable cover of of cardinalit n (i + 1). Similarl, if G is a variable cover of cardinalit n (i + 1) of, then one can see that G c is a facet of N, i. The discssion above shows that I i is generated b monomials corresponding to variable covers of cardinalit n i 1 of. In other words I i = I [n i 1], where I [j] denotes the j-th sqare-free homogeneos component of I, and showing that N, i is Cohen-Macala is eqivalent to showing that I [n i 1] has a linear resoltion. We have ths shown that: Proposition 4.5 (Criterion for being seqentiall Cohen-Macala). Let I be a sqare-free monomial ideal in a polnomial ring. Then I is a seqentiall Cohen-Macala ideal if and onl if I is componentwise linear. 5 Simplicial trees are Seqentiall Cohen-Macala This section contains the main reslts of the paper. Or goal here is to show that the facet ideal I of a simplicial tree is seqentiall Cohen-Macala. B Proposition 4.5 this is eqivalent to showing that the facet ideal I of the cover comple of a tree is componentwise linear (Definition 4.4). In fact, we show that I satisfies a stronger propert: for ever i, we show below that I [i] has linear qotients. This propert, defined b Herog and Takaama in [HT], implies that I [i] has a linear resoltion. It also implies additional combinatorial properties for I (see Corollar 5.9). Definition 5.1 (linear qotients ([HT])). If I k[ 1,..., n ] is a monomial ideal and G(I) is its niqe minimal set of monomial generators, then I is said to have linear qotients if there is an ordering M 1,..., M q on the elements of G(I) sch that for ever i = 2,..., q, the qotient ideal (M 1,..., M i 1 ) : M i is generated b a sbset of the variables 1,..., n. The following is a well-known fact. We reprodce an argment (almost identical to one given in [Z] for the case of trees). Lemma 5.2. If I = (M 1,..., M q ) is a monomial ideal in the polnomial ring R = k[ 1,..., n ] over the field k that has linear qotients and all the M i are of the same degree, then I has a linear resoltion. Proof. The proof is b indction on q. The case q = 1 is clear. Given that the ideal I = (M 1,..., M q 1 ) has linear qotients and therefore a linear resoltion, and that the degree of all the M i is eqal to d, we have that (see Section 5.5 of [BH]) for all i: Tor R i (k, R/I ) a = 0 nless a = i + d; Tor R i (k, R/I : I) a = 0 nless a = i + 1 (I : I is generated b degree 1 monomials). Consider the short eact seqence: 0 R/(I : I)( d)).mq R/I R/I 0 10

11 We obtain the long eact homolog seqence: Tor R i (k, R/(I : I)( d)) Tor R i (k, R/I ) Tor R i (k, R/I) For a given i, Tor R i Tor R i 1 (k, R/(I : I)( d)) (k, R/I) a = 0 nless Tor R i (k, R/I ) a 0 or Tor R i 1 (k, R/(I : I))( d) a 0. Either wa, this means that for an i, if Tor R i that R/I has a linear resoltion. (k, R/I) a 0, then a = i + d. This implies We now set ot to prove if I k[v ], V = { 1,..., n }, is the facet ideal of a tree (in fact, a forest), and i, α( ) i n, is a given integer, then I [i] has linear qotients. We se indction on n. If n = 1, can onl be the verte 1, and so the onl thing to check is if I [1] = ( 1 ) has linear qotients, which is obvios. Sppose that n > 1. We first deal with some special cases. If is a forest of singletons of the form = 1,..., j where j < n, then we can consider I = F( ) as an ideal in the polnomial ring R = k[ 1,..., n 1 ] (I and I have the same generating set, the onl live in two different rings). B the indction hpothesis, for ever i, I [i] has linear qotients. It is eas to see that for ever i, Sppose that I [i] = I [i] + n I [i 1]. I [i] = (A 1,..., A a ) and I [i 1] = (B 1,..., B b ) where the generators of both ideals are written in the correct order for linear qotients (recall that we are sing the notation A to mean {} A, since generall we are alwas thinking of sets as monomials). To see that I [i] = (A 1,..., A a ) + n (B 1,..., B b ) has linear qotients, we consider the case where for some monomial m in k[ 1,..., n ] (we can withot loss of generalit assme here that the prodcts are sqare-free), m n B j (A 1,..., A a, n B 1,..., n B j 1 ). If m n B j ( n B 1,..., n B j 1 ), since I [i 1] has linear qotients, it follows that for some variable dividing the monomial m, we have n B j ( n B 1,..., n B j 1 ) (note that m 1). If m n B j (A 1,..., A a ), then since B j is alread a variable cover of, for an variable not in B j, B j covers and is of cardinalit i, and hence B j {A 1,..., A a }. Therefore for an dividing m we can again conclde that n B j (A 1,..., A a ). This argment settles the case where = 1,..., j, and j < n. If = 1,..., n, then the onl ideal to consider is I [n] = ( 1... n ) which b definition has linear qotients. So now we can assme that is a forest containing a facet with more than one verte. We begin or discssion with the following simple observation. 11

12 Lemma 5.3. Let be a simplicial comple with F( ) k[v ], k a field, and V = { 1,..., n }. Sppose that V is sch that V \ {} is a variable cover for, and let p be the prime ideal generated b the set V \ {}. Then localiing at p corresponds, via the cover dalit, to removing all facets of M that contain. In other words, if = δ F (F( ) p ) and A 1,..., A t are the facets of M that contain, then M = M \ A 1,..., A t. Proof. Note that a facet of M is the generating set for a minimal prime of I = F( ) not containing, and therefore belongs to M as well. Conversel, if A is a facet of the right-hand-side, then it corresponds to a minimal prime of I not containing and hence to a minimal prime of I p. Now assme that the forest has a leaf F with positive dimension and a free verte (see Eample 3.6) = 1. We can write: M, [i] = δ F (I [i]) = A 1,..., A t B 1,..., B s where A 1,..., A t are all the variable covers of that have cardinalit i and do not contain, and B 1,..., B s are all the other variable covers of cardinalit i. Now let = δ F (F( ) p ) and = \ F. Both and are forests (b the definition of a tree, and b Lemma 3.8) whose verte sets are contained in { 2,..., n }. Also note that is a nonempt simplicial comple. With notation as above, b Lemma 5.3 Also notice that M, [i] = A 1,..., A t. M, [i 1] = B 1,..., B s. To see this last eqation, note that since for j = 1,..., s, B j covers, B j has to cover (as is a free verte of F and hence onl covers F ). On the other hand, if A is an variable cover of of cardinalit i 1, then A is in M, [i], and so A {B 1,..., B s }. Appling the indction hpothesis to the forests and we see that the ideals I [i] = (A 1,..., A t ) and I [i 1] = (B 1,..., B s ) of k[ 2,..., n ] both have linear qotients. Withot loss of generalit assme that the given orders on the A s and the B s are appropriate for taking qotients. We show that the ideal I [i] = (A 1,..., A t ) + (B 1,..., B s ) also has linear qotients. Here we assme that 1 < i < n, since I [n] = ( 1... n ) has linear qotients b definition, as does I [1] which is, if nonero, generated b a sbset of { 1,..., n }. The first case of interest is the ideal (A 1,..., A t ) : B 1. 12

13 Now B 1 is a variable cover of = \ F, so B 1 I [i] for an verte of F not in B 1. So if m is an monomial sch that mb 1 I [i], then for some monomial n and some j, assming withot loss of generalit that both prodcts below are sqare-free, we have mb 1 = na j. If B 1 alread contains a verte of F, then it is a variable cover of cardinalit i 1 for, and so for an m, B 1 {A 1,..., A t }. Otherwise, since there is some verte of F in A j, has to divide m, which again implies that B 1 (A 1,..., A t ). In general, for the ideal (A 1,..., A t, B 1,..., B j 1 ) : B j if for some monomial m, mb j (B 1,..., B j 1 ), then b the indction hpothesis on I [i 1] there is a variable that divides m sch that B j (B 1,..., B j 1 ). If mb j (A 1,..., A t ), then it follows from an argment identical to the case j = 1 above that there is a variable dividing m sch that B j (A 1,..., A t ). We have ths proved that: Theorem 5.4. If I k[ 1,..., n ] is the facet ideal of a simplicial tree (forest), then I [i] has linear qotients for all i = α( ),..., n. Theorem 5.4 along with Lemma 5.2 reslt in the following statement. Corollar 5.5. If is a simplicial tree (forest), then F( ) is a componentwise linear ideal. Ptting Corollar 5.5 together with Proposition 4.5, we arrive at or final goal. Corollar 5.6 (Trees are seqentiall Cohen-Macala). The facet ideal of a simplicial tree (forest) is seqentiall Cohen-Macala. Eample 5.7. The ideal I in Eample 4.3 is seqentiall Cohen-Macala becase it is the facet ideal of a tree. It follows easil that if the tree is nmied to begin with, then it mst be Cohen- Macala. This is becase in this case F( ) itself is a sqare-free homogeneos component, which has a linear resoltion. So b appling Theorem 2.8 we have Corollar 5.8 (An nmied tree is Cohen-Macala). If is an nmied simplicial tree, then F( ) has a Cohen-Macala qotient. Corollar 5.8 was proved in [F2] sing ver different tools. In particlar, in [F2] we show that a tree is nmied if and onl if it is grafted. The notion of grafting is what gives a Cohen-Macala tree its definitive combinatorial strctre. Another interesting fact that follows is that in the case of a simplicial tree, if is Cohen-Macala, then N is shellable (see [BH] for the definition). Given a sqarefree monomial ideal I, if δ N (I) is shellable, then I is Cohen-Macala (see [BH]), bt the converse is not tre in general. Corollar 5.9. If is a Cohen-Macala simplicial tree, then N is shellable. Proof. If I = F( ) is Cohen-Macala, then b Theorem 5.4, I has linear qotients (since it has generators of the same degree). The rest follows directl from the definitions of shellabilit and linear qotients; see [HHZ] Theorem 1.4, part (c). 13

14 Notation Meaning First appearance F( ) facet ideal of Definition 1.3 δ F (I) facet comple of I Definition 1.3 α( ) verte covering nmber of Definition 1.5 M cover comple of Definition 1.8 N ( ) nonface ideal of Definition 2.1 δ N (I) nonface comple of I Definition 2.1 N δ N (F( )) Notation 2.2 F c, c complements of F and Notation 2.2 Aleander dal of Definition 2.3 I dal of I Definition 2.6 \ F removal of facet F from Definition 3.1 N, i pre i-dimensional sbcomple of N Theorem 4.2 I [i] i-th sqare-free homogeneos component of I Definition 4.4 p ideal generated b all variables bt Lemma 5.3 M, [i] facet comple of I [i] following Lemma 5.3 References Figre 2: Inde of Notation [B] [BH] [BW] [D] [ER] Berge, C. Hpergraphs, Combinatorics of finite sets, North-Holland Mathematical Librar, 45. North-Holland Pblishing Co., Amsterdam, Brns, W., Herog, J. Cohen-Macala rings, vol. 39, Cambridge stdies in advanced mathematics, revised edition, Björner, A., Wachs, M.L. Shellable nonpre complees and posets, I. Trans. Amer. Math. Soc. 348 (1996), no. 4, Dval, A.M. Algebraic shifting and seqentiall Cohen-Macala simplicial complees, Electron. J. Combin. 3 (1996), no. 1, Research Paper 21 Eagon J.A., Reiner, V. Resoltion of Stanle-Reisner rings and Aleander dalit, J. Pre Appl. Algebra 130 (1998), no. 3, [F1] Faridi, S. The facet ideal of a simplicial comple, Manscripta Mathematica 109 (2002), [F2] Faridi, S. Cohen-Macala properties of sqare-free monomial ideals, Preprint. [HH] Herog, J., Hibi, T. Componentwise linear ideals, Nagoa Math. J. 153 (1999), [HHZ] Herog, J., Hibi, T., Zheng, X. Dirac s theorem on chordal graphs and Aleander dalit, Preprint. 14

15 [HRW] Herog, J., Reiner, V., Welker, V. Componentwise linear ideals and Golod rings, Michigan Math. J. 46 (1999), no. 2, [HT] [S] Herog, J., Takaama, Y. Resoltions b mapping cones, The Roos Festschrift volme, 2. Homolog Homotop Appl. 4 (2002), no. 2, part 2, (electronic). Stanle, R.P. Combinatorics and commtative algebra, Second edition. Progress in Mathematics, 41. Birkhser Boston, Inc., Boston, MA, pp. ISBN: [SVV] Simis A., Vasconcelos W., Villarreal R., On the ideal theor of graphs, J. Algebra 167 (1994), no. 2, [V] [Z] Villarreal R., Monomial algebras, Monographs and Tetbooks in Pre and Applied Mathematics, 238. Marcel Dekker, Inc., New York, Zheng, X. Resoltions of facet ideals, Preprint. 15