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Symmetric iterated Betti numbers
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Journal of Combinatorial Theory, Series A 105 (2004) 233–254
Symmetric iterated Betti numbers$ Eric Babson, Isabella Novik, and Rekha Thomas Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA Received 3 March 2003 Communicated by Gil Kalai
Abstract We define a set of invariants of a homogeneous ideal I in a polynomial ring called the symmetric iterated Betti numbers of I: We prove that for IG ; the Stanley–Reisner ideal of a simplicial complex G; these numbers are the symmetric counterparts of the exterior iterated Betti numbers of G introduced by Duval and Rose, and that the extremal Betti numbers of IG are precisely the extremal (symmetric or exterior) iterated Betti numbers of G: We show that the symmetric iterated Betti numbers of an ideal I coincide with those of a particular reverse lexicographic generic initial ideal GinðIÞ of I; and interpret these invariants in terms of the associated primes and standard pairs of GinðIÞ: We close with results and conjectures about the relationship between symmetric and exterior iterated Betti numbers of a simplicial complex. r 2003 Elsevier Inc. All rights reserved. Keywords: Algebraic shifting; Generic initial ideals; Extremal Betti numbers; Standard pairs; Local cohomology
1. Introduction The goal of this paper is to define and study a set of invariants of a homogeneous ideal in a polynomial ring, called the symmetric iterated Betti numbers of the ideal. For a simplicial complex G; the symmetric iterated Betti numbers of the Stanley– Reisner ideal of G (also referred to as the symmetric iterated Betti numbers of G) are preserved by symmetric algebraic shifting. $
Research partially supported by NSF Grants DMS 0070571 and DMS 0100141. E-mail addresses: [email protected] (E. Babson), [email protected] (I. Novik), [email protected] (R. Thomas). 0097-3165/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jcta.2003.11.003
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We discuss two versions of algebraic shifting (both introduced by Kalai [6,16]) which given a simplicial complex G with vertex set ½n :¼ f1; 2; y; ng provide new simplicial complexes with the same vertex set. We denote these versions by DðGÞ for the symmetric shifting of G (see Definition 2.1) and by De ðGÞ for the exterior shifting of G (see Definition 7.1). There are instances for which DðGÞaDe ðGÞ [18]. It is known that: (P1) DðGÞ and De ðGÞ are shifted. (A simplicial complex K is shifted if for every F AK and joiAF ; the set ðF \figÞ,f jg is also in K:) (P2) If G is shifted, then DðGÞ ¼ De ðGÞ ¼ G: (P3) G; DðGÞ and De ðGÞ have the same f -vector, that is, they have the same number of i-dimensional faces for every i: (P4) If G0 is a subcomplex of G; then DðG0 ÞDDðGÞ and De ðG0 ÞDDe ðGÞ: Both versions were studied extensively from the algebraic point of view in a series of recent papers by Aramova, Herzog, Hibi and others (surveyed in [13], see also [1–4]). Consider the polynomial ring S ¼ k½y1 ; y; yn where k is a field of characteristic zero. Let N denote the set of non-negative integers. If AD½n then write yA ¼ Q ½n the monomials of S by identifying a function f : ½n-N in aAA ya : Denote by N Q f ðiÞ ½n N with the monomial iA½n yi and consider N½n as a multiplicative monoid. Thus f0; 1g½n ¼ y2 is the set of squarefree monomials. If GD2½n is a simplicial complex then the Stanley–Reisner ideal of G [20, Definition II.1.1] is the squarefree monomial ideal ½n
IG :¼ /y2
½n
G
SCS:
The (bi-graded) Betti numbers of a homogeneous ideal ICS are the invariants bi; j ðIÞ that appear in the minimal-free resolution of I as an S-module: -?
M j
Sð jÞbi; j ðIÞ -?-
M j
Sð jÞb1;j ðIÞ -
M
Sð jÞb0;j ðIÞ -I-0
j
Here Sð jÞ denotes S with grading shifted by j: We say that bi;iþj ðIÞ; is extremal P if 0abi;iþj ðIÞ ¼ i0 Xi;j 0 Xj bi0 ;i0 þj0 ðIÞ: (This is equivalent to having 0abi;iþj ðIÞ and 0 ¼ bi0 ;i0 þj0 ðIÞ for every i0 Xi and j 0 Xj; ði0 ; j 0 Þaði; jÞ:) Since DðGÞ and De ðGÞ are shifted complexes, their combinatorial structures are simpler than that of G: Nonetheless, the two shifting operations preserve many combinatorial and topological properties. 1. D and De preserve topological Betti numbers (see [6, Theorem 3.1; 1, Proposition 8.3] for exterior shifting and [13, Corollary 8.25] for symmetric shifting). Moreover, exterior algebraic shifting preserves the exterior iterated Betti numbers of a simplicial complex. (There are two versions of exterior iterated Betti numbers—one due to Kalai [17, Corollary 3.4] and another due to Duval and Rose [9]. Both sets of numbers are preserved under exterior shifting.)
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2. D and De preserve Cohen–Macaulayness: a simplicial complex G is Cohen– Macaulay if and only if DðGÞ (De ðGÞ) is Cohen–Macaulay, which happens if and only if DðGÞ (De ðGÞ) is pure (see [17, Theorem 5.3], [1, Proposition 8.4] for exterior shifting and [16, Theorem 6.4] for symmetric shifting). 3. D and De preserve extremal Betti numbers: bi;iþj ðIG Þ is an extremal Betti number of IG if and only if bi;iþj ðIDðGÞ Þ is extremal for IDðGÞ ; in which case bi;iþj ðIG Þ ¼ bi;iþj ðIDðGÞ Þ: The same assertion holds for De : (See [4] for symmetric shifting and [1, Theorem 9.7] for both versions.) Property 3 is a far-reaching generalization of property 2, while property 1 played a crucial role in Kalai’s proof of property 2 for exterior shifting. This suggests that there might be a connection between the iterated Betti numbers of a simplicial complex G on the one hand and the extremal Betti numbers of the ideal IG on the other. This is one of the connections we establish in this paper. Consider the action of GLðS1 Þ on S and choose uAGLðS1 Þ to be generic. Denote by m ¼ /S1 S the irrelevant ideal of S: If I is a homogeneous ideal in S then define the S-modules Mi ðIÞ as follows: M0 ðIÞ :¼ S=uI and Mi ðIÞ :¼ Mi 1 ðIÞ=ðyi Mi 1 ðIÞ þ H 0 ðMi 1 ðIÞÞÞ for 1pipn; where H 0 ð Þ denotes the 0th local cohomology with respect to the irrelevant ideal m: We now come to the central definition of this paper. Definition 1.1. The symmetric iterated Betti numbers of a homogeneous ideal I in S are bi;r ðIÞ :¼ dim H 0 ðMi ðIÞÞr
for 0pi; rpn;
where H 0 ð Þr stands for the rth component of the 0th local cohomology. If G is a simplicial complex with vertex set ½n; define the symmetric iterated Betti numbers of G to be bi;r ðGÞ :¼ bi;r ðIG Þ; 0pi; rpn: Our first result gives a combinatorial interpretation of the symmetric iterated Betti numbers of a simplicial complex G and shows that they are invariant under symmetric algebraic shifting. Let maxðGÞ denote the set of facets (maximal faces) of G: Write dimðGÞ ¼ maxfjF j 1 : F AGg: Theorem 4.1. Let G be a simplicial complex. Then jfF A maxðDðGÞÞ : jF j ¼ i; ½i rDF ; i r þ 1eF gj if rpi; bi;r ðGÞ ¼ 0 otherwise: In particular, since DðDðGÞÞ ¼ DðGÞ; it follows that the symmetric iterated Betti numbers of G are invariant under symmetric shifting. Theorem 4.1 implies that bi;r ðGÞ ¼ 0 unless 0prpipdimðGÞ þ 1: The exterior iterated Betti numbers of G; bei;r ðGÞ; defined by Duval and Rose have precisely the same combinatorial formula (up to a slight change in indices), except that in their definition, one replaces DðGÞ by De ðGÞ [9, Theorem 4.1]. In fact, the operation of ‘‘peeling off’’ the ith variable at step i we are using to define the symmetric iterated
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Betti numbers is analogous to the ‘‘deconing’’ of the shifted complex used in the Duval–Rose definition. The extremal Betti numbers of an ideal I ¼ IG are the extremal iterated Betti numbers (symmetric or exterior) of the simplicial complex G in the following sense. Theorem 5.3 and 7.4. Let G be a simplicial complex. The extremal Betti numbers of IG form a subset of the symmetric as well as of the exterior iterated Betti numbers of G: More precisely, bj 1;iþj ðIG Þ is an extremal Betti number of IG if and only if bn j0 ;i0 ðGÞ ¼ 0
8ði0 ; j 0 Þaði; jÞ; i0 Xi; j 0 Xj;
and
bn j;i ðGÞa0:
In such a case bj 1;iþj ðIG Þ ¼ bn j;i ðGÞ: The same assertion holds for be ; : Let GinðIÞ denote the reverse lexicographic generic initial ideal of a homogeneous ideal I in S with variables ordered as yn gyn 1 g?gy1 : It follows from [4, Corollary 1.7] that the symmetric iterated Betti numbers of I coincide with those of GinðIÞ: We provide an alternate proof of this fact in Section 4 (see Corollary 4.6). Our next result—Theorem 6.6—interprets the symmetric iterated Betti numbers bi;r ðIÞ in terms of the associated primes of GinðIÞ: This paper is organized as follows. In Section 2 we recall the basics of symmetric shifting. Section 3 defines and interprets certain monomial sets that are at the root of all our proofs. In Sections 4–7 we prove the theorems stated above. We conclude in Section 7 with some results and conjectures on the relationship between the exterior and symmetric iterated Betti numbers of a simplicial complex.
2. Algebraic shifting In this section we recall the basics of symmetric algebraic shifting. (The description of exterior shifting is deferred to Section 7.) For further details on symmetric and exterior shifting see the survey articles by Herzog [13] and Kalai [18]. Let Ns denote the set of all finite degree monomials in the variables yi with iAs and Nsr denote the set of elements of degree r in Ns : In particular, if ½n ¼ ½1; n ¼ f1; y; ng then N½n is the set of all monomials in S and f0; 1gs is the set of all square free finite degree monomials in Ns : In this paper we fix the reverse lexicographic order g on NZ with yi gyi 1 for all iAZ extending the partial ordering by degree. (Thus, for example, y2n gyn 1 yn gy2n 1 gyn 2 yn gyn 2 yn 1 g?:) We also define the square free map F : NZ -f0; 1gZ by Fðyi0 yi1 ?yik Þ :¼ yi0 k yi1 ðk 1Þ ?yij ðk jÞ ?yik ; where i0 pi1 p?pik :
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In other Q words, F is the unique degree and order preserving bijection for which Fðyn0 Þ ¼ noip0 yi : For each homogeneous ideal ICS there exists a Zariski open set UðIÞC GLðS1 Þ such that the ideal Ing ðuIÞ; (the initial ideal of uI with respect to the monomial order g on S), is independent of the choice of uAUðIÞ: The ideal Ing ðuIÞ is called the generic initial ideal of I with respect to g and is denoted by GinðIÞ ¼ Ging ðIÞ (see [10, Chapter 15]). If I is a homogeneous ideal in S then one way to explicitly and uniformly construct an element aAUðIÞ is to consider the extension K ¼ kðfai; j gi; jA½n Þ=k and then for any ideal I in S the element n X ai; j yj a : SK ¼ S#k K-SK given by ayi ¼ j¼1
is generic for KI as an ideal of SK : For a homogeneous ideal I in S and a generic linear map uAUðIÞ define BðIÞ ¼ fmAN½n : m is not in the linear span of fnjmgng,uIg: Note that BðIÞ is a basis of the vector space M0 ðIÞ ¼ S=uI and hence BðIÞ ¼ N½n GinðIÞ: Definition 2.1. The symmetric algebraic shifting of a simplicial complex GD2½n is DðGÞ where yDðGÞ ¼ FðBðIG ÞÞ-N½1;N Df0; 1g½n : Note that this means that IDðGÞ ¼ /FðN½n BðIG ÞÞS: The fact that DðGÞ is a simplicial complex satisfying conditions (P1)–(P4) was proved in [16, Theorem 6.4; 3] by using certain properties of BðIÞ: We list some of them below: (B1) BðIÞ is a basis of S=uI; as well as of S=GinðIÞ: (B2) BðIÞ is an order ideal—if mABðIÞ and m0 jm; then m0 ABðIÞ: (B3) BðIÞ is shifted—if joi and yi mABðIÞ then yj mABðIÞ: Condition (B1) was discussed above while (B2) follows from the fact that GinðIÞ is an ideal. Condition (B3) is a consequence of the fact that generic initial ideals are Borel fixed [10, Theorem 15.20]. In characteristic 0, this is equivalent to GinðIÞ being strongly stable [10, Theorem 15.23], which means that if joi and yj mAGinðIÞ then yi mAGinðIÞ: In the case when I ¼ IG ; BðIG Þ has another fundamental property: (B4) If mABðIG Þ-N½k;n and rXk then mNfkg DBðIG Þ as well. r This is due to Kalai [16, Lemma 6.3] and implies that y1 ; y; yn is an almost regular M0 ðIG Þ-sequence (a notion introduced by Aramova and Herzog [1]; it played a crucial role in their proof that extremal Betti numbers are preserved by algebraic shifting).
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3. Special monomial subsets In this section we identify and interpret certain subsets of monomials in the basis BðIÞ of M0 ðIÞ that are at the root of all our proofs. Definition 3.1. Let I be a homogeneous ideal in S: For iA½0; n define n o / BðIÞ ; Ai;r ðIÞ :¼ Ai ðIÞ-NN Ai :¼ mAN½iþ1;n : mNfig DBðIÞ; mNfiþ1g D r : Several remarks are in order. Since BðIÞ is shifted (B3), mNfig DBðIÞ iff mN½i DBðIÞ: Since BðIG Þ satisfies (B4), Ai;r ðIG Þ ¼ |
if r4i
and hence
Ai ðIG Þ ¼
i [
Ai;r ðIG Þ:
ð1Þ
r¼0
Also if mAN½r;n then mABðIG Þ iff mN½r DBðIG Þ: Hence r n o i r i rþ1 Ai;r ðIG Þ ¼ mAN½iþ1;n : y
mABðI Þ; y
meBðI Þ : G G r i iþ1
ð2Þ
In [21], Sturmfels, Trung and Vogel introduced a decomposition of the standard monomials of an arbitrary monomial ideal M; called its standard pair decomposition, in order to study the multiplicities of associated primes and degrees of M: We study these quantities for the monomial ideals IG ; IDðGÞ ; and GinðIÞ: In Section 6 we show their relationship to the symmetric iterated Betti numbers of G and I; respectively. These results rely on the fact that the sets of monomials Ai ðIÞ defined above index the standard pairs of GinðIÞ: For a monomial mANZ ; let suppðmÞ :¼ fi: yi j mgCZ be called the support of m: Thus supp : f0; 1gs -2s is a bijection. Definition 3.2 (Sturmfels et al. [21]). Let M ¼ /M-N½n SDS be a monomial ideal. A standard monomial of M is an element of N½n M: An admissible pair of M is a subset mNs DN½n M with mAN½n s or equivalently if we take Zs to be Laurent monomials then an admissible pair is a subset mZs -N½n with mZs -M ¼ |: A standard pair of M is a(n inclusion) maximal admissible pair. Lemma 3.3. If IDS is an ideal then the standard pairs of GinðIÞ are faN½i : aAAi ðIÞg: (Here ½0 ¼ |:) Proof. We first argue that all standard pairs of GinðIÞ are of the form aN½i for some iA½0; n: Suppose mNs is an admissible pair of GinðIÞ with k ¼ maxðsÞ: Since BðIÞ is shifted (B3) and mNfkg DBðIÞ we obtain that mN½k DBðIÞ and hence mNs DmZ½k -N½n DBðIÞ: If mNs is standard (maximal) this implies that mNs ¼ mZ½k -N½n and thus that s ¼ ½k: If mN½i DBðIÞ is standard then by the above argument mNfiþ1g D / BðIÞ so mAAi ðIÞ:
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Finally, if mAAi ðIÞ; then mNfig DBðIÞ and hence mN½i DBðIÞ is admissible. If 0 mN½i DBðIÞ is not standard then mN½i Cm0 N½i DBðIÞ so i0 4i and mNfiþ1g DBðIÞ contradicting the choice of m: & We remark that Lemma 3.3 and Eq. (1) imply that if mN½i is a standard pair of GinðIG Þ then the degree of m is at most i: The standard pairs of monomial ideals of moderate size can be computed using the computer algebra package Macaulay 2 [12] (see the chapter Monomial Ideals in [11] for details). This gives a method for computing the sets Ai ðIÞ for small examples— see Example 3.7 below. In the case when I ¼ IG there is another interpretation of the monomials in Ai ðIG Þ that relates them to the shifted complex DðGÞ; and is useful for the proofs of Theorems 4.1 and 6.6. Lemma 3.4. There is a bijection between the sets Ai;r ðIG Þ and fF AmaxðDðGÞÞ : jF j ¼ i; ½i rDF ; i r þ 1eF g given by F with Ai;r ðIG Þ{m/½i r,suppðFðmÞÞ ¼ suppðFðmyi r i ÞÞ: Proof. For r4i the assertion follows from the fact that both sets are empty (see (1)). To deal with the case rpi; note that byð2Þ
fFðmÞ : mAAi;r ðIG Þg ¼ (
mABðIG Þ-N½iþ1;n ; r
FðmÞ :
)
yi r
mABðIG Þ; yi rþ1
meBðIG Þ i iþ1 Def: 2:1
¼
GADðGÞ :
jGj ¼ r; G-½i r þ 1 ¼ |; G,½i rADðGÞ; G,½i r þ 1eDðGÞ
¼ fF \½i r: F AmaxðDðGÞÞ; jF j ¼ i; ½i rDF ; i r þ 1eF g;
ð3Þ
where in the last equality we used the fact that DðGÞ is shifted. Indeed, if G,½i r þ 1eDðGÞ; then G,½i r,f jgeDðGÞ for every j4i r þ 1; jeG; implying that G,½i r is a facet of DðGÞ for every element G of the set (3). & Corollary 3.5. The standard pairs of GinðIG Þ are in bijection with the facets of DðGÞ: mN½i is a standard pair of GinðIG Þ if and only if ½i r,suppðFðmÞÞ is a facet of DðGÞ of size i: In Section 4 we verify that bi;r ðIÞ ¼ jAi;r ðIÞj for all i; rA½0; n; which via Lemma 3.4 proves Theorem 4.1. (Hence, in particular, it follows from Theorem 4.1 that Ai ðIG Þ ¼ | for all i4dimðGÞ þ 1:) Thus the sets Ai;r ðIG Þ and their cardinalities bi;r ðGÞ carry important information about G; and we record them in the following triangles.
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Definition 3.6. The b-triangle and monomial b-triangle of a simplicial complex G are the lower triangular matrices whose respective ði; rÞth entries are bi;r ðGÞ and Ai;r ðIG Þ for 0piprpdimðGÞ þ 1: Example 3.7. Let G be the simplicial complex whose facets are maxðGÞ ¼ ff1; 2; 4g; f1; 2; 6g; f1; 3; 4g; f1; 3; 7g; f1; 5; 6g; f1; 5; 7g; f2; 3; 5g; f2; 3; 7g; f2; 4; 5g; f2; 6; 7g; f3; 4; 6g; f3; 5; 6g; f4; 5; 7g; f4; 6; 7gg: Then it can be checked from Fig. 1 that the Stanley–Reisner ideal of G in the ring S :¼ k½a; b; c; d; e; f ; g is IG ¼ /efg; cfg; afg; ceg; beg; cdg; bdg; adg; abg; def ; bef ; bdf ; adf ; bcf ; acf ; cde; ade; ace; abe; bcd; abcS: Using Macaulay 2 one can compute that under the reverse lexicographic order g with ggf g?gbga; GinðIG Þ ¼ /gf 2 ; f 3 ; f 2 e; g2 f ; gfe; fe2 ; gfd; f 2 d; fed; g2 e; ge2 ; e3 ; ged; e2 d; fd 2 ; g3 ; g2 d; gd 2 ; ed 2 ; g2 c; gfc; d 4 S: Applying the map F to the generators of GinðIG Þ we get IDðGÞ ¼ /gea; gfa; ecb; fcb; gcb; edb; fdb; gdb; feb; geb; gfb; edc; fdc; gdc; fec; gec; gfc; fed; ged; gfd; gfe; dcbaS; which shows that the shifted complex DðGÞ has facets: maxðDðGÞÞ ¼ ff1; 2; 3g; f1; 2; 4g; f1; 2; 5g; f1; 2; 6g; f1; 2; 7g; f1; 3; 4g; f1; 3; 5g; f1; 3; 6g; f1; 3; 7g; f1; 4; 5g; f1; 4; 6g; f1; 4; 7g; f1; 5; 6g; f2; 3; 4g; f5; 7g; f6; 7gg:
3
5
7
2
7
4
6
1
3
4
3
5
7
Fig. 1. The simplicial complex G in Example 3.7. Here parallel boundary regions are identified.
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The 1-skeleton of DðGÞ is (like that of G) the complete graph on its 7 vertices. The triangles are obtained by coning 1 with all edges involving the vertices 2; 3; 4; 5; 6 and 7 except for f5; 7g and f6; 7g and adding the triangle f2; 3; 4g: Thus all the triangles and the edges f5; 7g and f6; 7g are facets. We compute the b-triangle and the monomial b-triangle of G by first computing the standard pairs of GinðIG Þ using Macaulay 2. b-Triangle of G 0
1
2
0
0
1
0
0
2
0
0
2
3
1
4
8
Monomial b-triangle of G 3
1
0
1
2
3
0
|
1
|
|
2
|
|
fg2 ; gf g
3
f1g
fg; f ; e; dg
fge; gd; f 2 ; fe; fd; e2 ; ed; d 2 g
fd 3 g
The standard pairs of IG ; GinðIG Þ and IDðGÞ are shown in the following table. Columns 2, 3, and 4 illustrate Lemma 3.4 and Corollary 3.5.
StdPairs(IG ) form: Ns
StdPairs(GinðIG Þ) form: mN½i
FðmÞ -suppðFðmÞ)
StdPairs(IDðGÞ ) form: Ns
Nf4;6;7g Nf2;6;7g Nf4;5;7g Nf1;5;7g Nf2;3;7g Nf1;3;7g Nf3;5;6g Nf1;5;6g Nf3;4;6g Nf1;2;6g Nf2;4;5g Nf2;3;5g Nf1;3;4g Nf1;2;4g
Nf1;2;3g gNf1;2;3g geNf1;2;3g gdNf1;2;3g f Nf1;2;3g f 2 Nf1;2;3g feNf1;2;3g fdNf1;2;3g eNf1;2;3g e2 Nf1;2;3g edNf1;2;3g dNf1;2;3g d 2 Nf1;2;3g d 3 Nf1;2;3g
1-| g-f7g gd-7; 4 gc-f7; 3g f -6 fe-f6; 5g fd-f6; 4g fc-6; 3 e-f5g ed-f5; 4g ec-f5; 3g d-f4g dc-f4; 3g dcb-f4; 3; 2g
Nf1;2;3 Nf1;2;7g Nf1;4;7g Nf1;3;7g Nf1;2;6g Nf1;5;6g Nf1;4;6g Nf1;3;6g Nf1;2;5g Nf1;4;5g Nf1;3;5g Nf1;2;4g Nf1;3;4g Nf2;3;4g
g2 Nf1;2g gf Nf1;2g
gf -f7; 6g ge-f7; 5g
Nf6;7g Nf5;7g
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4. Local cohomology In this section we prove Theorem 4.1, which provides a simple combinatorial formula for the symmetric iterated Betti numbers of a simplicial complex. Theorem 4.1. For a simplicial complex G bi;r ðGÞ ¼
jfF AmaxðDðGÞÞ : jF j ¼ i; ½i rDF ; i r þ 1eF gj
if rpi;
0
otherwise:
The symmetric iterated Betti numbers bi;r ðGÞ were defined as the dimensions of the vector spaces H 0 ðMi ðIG ÞÞr ; where for a homogeneous ideal I in S and a generic linear map uAUðIÞ; M0 ðIÞ ¼ S=uI and Mi ðIÞ ¼ Mi 1 ðIÞ=ðyi Mi 1 ðIÞ þ H 0 ðMi 1 ðIÞÞÞ for 1pipn: Thus at step i we ‘‘peel off’’ the ith variable. This is similar to the ‘‘deconing’’ of the shifted complex De ðGÞ used in the definition of the exterior iterated Betti numbers of G by Duval and Rose [9]. In view of Lemma 3.4, it suffices to show that jAi;r ðIÞj ¼ dim H 0 ðMi ðIÞÞr for all i; rX0 in order to prove Theorem 4.1. We establish this in Lemma 4.3 below. However, we first digress briefly to derive and illustrate certain facts needed in the proof of Lemma 4.3. Recall that if M is an S-module, N is a submodule and I is an ideal in S then ðN : I N ÞM ¼ fmAMj for some rAN; I r mDNg and if I ¼ /f S it is typical to write ðN : /f SN Þ ¼ ðN : f N Þ: For an S-module M; the 0th local cohomology of M with respect to the irrelevant ideal m ¼ Sþ ¼ /y1 ; y; yn S is defined as H 0 ðMÞ ¼ fmAM : mk m ¼ 0 for some kg ¼ ð0 : mN ÞM : In particular H 0 ðMÞ is graded when M is graded. Hence the equivalent definition of Mi ðIÞ is Mi ðIÞ ¼ S=Ji ðIÞ; where J0 ðIÞ :¼ uI
and
Ji ðIÞ :¼ yi S þ ðJi 1 ðIÞ : mN Þ:
Fix an i such that 1pion: Then for all 1pkoi; ðJi 1 ðIÞ : yN k Þ ¼ S since yk AJi 1 ðIÞ: For a j such that 1piojpn; consider the family of automorphisms ga AGLðS1 Þ such that ga ðyi Þ ¼ ayi þ ð1 aÞyj ; ga ðyj Þ ¼ ð1 aÞyi þ ayj and ga ðyr Þ ¼ yr otherwise, parameterized by all aAk: By induction ga ðJi 1 ðIÞÞ ¼ Ji 1 ðIÞ: N Thus ga ðJi 1 ðIÞ : yN i Þ ¼ ðJi 1 ðIÞ : ðayi þ ð1 aÞyj Þ Þ; and so the two colon ideals are isomorphic. If IDS is a fixed ideal and f AS varies then the ideal ðI : f N Þ depends only on which associated primes of I contain f : Thus if for a family of f ’s over k all the colon ideals ðI : f N Þ are isomorphic they must in fact be equal. Hence 8ioj;
N N we have ðJi 1 ðIÞ : yN i Þ ¼ ðJi 1 ðIÞ : yj Þ ¼ ðJi 1 ðIÞ : m Þ:
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Fix I and write B ¼ BðIÞ; Ai ¼ Ai ðIÞ; Ji ¼ Ji ðIÞ and Mi ¼ Mi ðIÞ: For the proof of Lemma 4.3 we introduce the sets ! [ Ci ¼ B Aj -N½iþ1;n : jpi
Lemma 4.2. The sets Ai and Ci have the following properties: (1) (2) (3) (4)
Ci 1 is the disjoint union Ci 1 ¼ Ai ,C ’ i ,y ’ i Ci 1 : Ci and Ci ,Ai are shifted order ideals in N½iþ1;n : kCi -Ji ¼ f0g: BDkCi þ ðJi : yN iþ1 Þ:
Proof. (1) If mN½ j is a standard pair of GinðIÞ with jpi; then no monomial in mN½ j lies in Ci : Thus Ci is the set of all monomials in B-N½iþ1;n that lie in standard pairs of the form N½ j where j4i: In other words, Ci ¼ fmAB-N½iþ1;n : mNfiþ1g DBg: This implies that yiþ1 Ci DCi and Ci 1 ¼ fmAB-N½i;n : mNfig DBg ’ ¼ fmAB-N½iþ1;n : mNfiþ1g DBg, / Bg,y ’ i Ci 1 fmAB-N½iþ1;n : mNfig DB; mNfiþ1g D ¼ Ci ,A ’ i ,y ’ i Ci 1 (2) The definitions of Ci and Ai and the fact that B is shifted imply via an induction that Ci and Ci ,Ai are shifted order ideals in N½iþ1;n : (3) We establish this fact by induction on i: Note that kC0 -J0 ¼ f0g since all elements of C0 DB are standard monomials of GinðIÞ ¼ P Ing ðJ0 Þ: Assume kCi 1 -Ji 1 ¼ f0g; but there exists 0af AkCi -Ji : Since f ¼ bm mAkCi ; each mACi CN½iþ1;n ; and so me/y1 ; y; yi S: Therefore, f AJi ¼ yi S þ ðJi 1 : yN i Þ implies k Þ—i.e., fy AJ for some k: Since kC -J ¼ f0g; fyki ¼ that f AðJi 1 : yN i 1 i 1 i 1 i i P bm m yki ekCi 1 and we infer that at least one of the monomials m is not in Ci 1 (since yi Ci 1 DCi 1 ). However, Ci DCi 1 ; and hence this m is also not in Ci ; which is a contradiction. Thus kCi -Ji ¼ f0g: S (4) Since B-N½iþ1;n ¼ ðCi , jpi Ai Þ-N½iþ1;n it suffices to show that if jpi then S Aj DkCi þ ðJi : yN iþ1 Þ: For every aA jpi Aj ; there exists some t40 such that aytiþ1 AGinðIÞ ¼ Ing ðJ0 ÞDIng ðJi Þ (since J0 DJi ). Therefore there exists f AJi such P that f ¼ aytiþ1 am m where aytiþ1 is the leading term of f with respect to g; am Ak; and the monomials m are standard monomials of Ing ðJi Þ: Hence mAB-N½iþ1;n (since all standard monomials of Ing ðJi Þ are in B and y1 ; y; yi AIng ðJi Þ). Further, since aytiþ1 gm for each m and since P aytiþ1 ; mAN½iþ1;n ; it follows that ytiþ1 j m: Thus a ¼ f =ytiþ1 þ am ðm=ytiþ1 Þ: Finally, since each m=ytiþ1 !a in the above sum, we are done by induction. &
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Lemma 4.3. jAi;r ðIÞj ¼ dim H 0 ðMi ðIÞÞr for all i; rX0: Proof. Note that dim H 0 ðMi Þr ¼ dimðfmAðS=Ji Þr : mAðJi : mN ÞgÞ ¼ dimðfmAðS=Ji Þr : mAðJi : yN iþ1 ÞgÞ ¼ dimðJi : yN iþ1 Þr dimðJi Þr : Hence, to prove the lemma it suffices to show that kAi "Ji ¼ ðJi : yN iþ1 Þ: We do this by establishing the following set of equalities: ½1
½2
S ¼i kCi "kAi "Ji ¼i kCi "ðJi : yN iþ1 Þ
for all iX0:
For i ¼ 0 the first equality, ½10 ; follows from the facts that kB ¼ kA0 "kC0 and kB"J0 ¼ S: We now show that ½1i implies ½2i and ½1iþ1 : Assume that S ¼ kCi "kAi "Ji : Since Ai DkCi þ ðJi : yN iþ1 Þ (Lemma 4.2(4)), we obtain that S ¼ kCi þ ðJi : yN iþ1 Þ:
ð4Þ
In order to verify ½2i we must show that kCi -ðJi : yN iþ1 Þ ¼ f0g: This follows from the facts that yiþ1 Ci DCi and that kCi -Ji ¼ f0g (Lemma 4.2(3)). Thus ½2
S ¼i kCi "ðJi : yN iþ1 Þ Lemma 4:2 ð1Þ
¼
kCiþ1 "kAiþ1 "kyiþ1 Ci "ðJi : yN iþ1 Þ ¼ kCiþ1 "kAiþ1 "Jiþ1 ;
where in the last step we used Eq. (4) along with the definition of Jiþ1 : Equality ½1iþ1 follows. & We close this section with several remarks. Remark 4.4. Lemma 4.3 was verified in the special case of Stanley–Reisner ideals of Buchsbaum complexes in [19]. Remark 4.5. Recall that by property (B1), BðIÞ is a basis of S=uI as well as of S=GinðIÞ: Thus the proof of Lemma 4.3 implies also that jAi;r ðIÞj ¼ dim H 0 ðMi ðGinðIÞÞr
for all i; rX0;
and we recover the following fact (originally due to Bayer et al. [4, Corollary 1.7]). Corollary 4.6. Modules H 0 ðMi ðIÞÞ and H 0 ðMi ðGinðIÞÞÞ have the same Hilbert function ( for i ¼ 0; 1; y; n). In other words, the symmetric iterated Betti numbers of I are identical to those of GinðIÞ:
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Remark 4.7. Similar to the proof of Theorem 4.1, one can show that dim H 0 ðS=/uIG ; y1 ; y; yi SÞr ¼ jfGADðGÞ : jGj ¼ r; ½i r þ 1-G ¼ |; ½i r þ 1,GeDðGÞgj: Thus for the shifted complex G; dimensions of the modules H 0 ðS=/uIG ; y1 ; y; yi SÞr coincide with Kalai’s (exterior) iterated Betti numbers of G (see [17, Section 3]). For that reason we refer to the numbers b%i;r ðGÞ :¼ dim H 0 ðS=/uIG ; y1 ; y; yi SÞr as Kalai’s symmetric iterated Betti numbers. Remark 4.8. Another fact worth mentioning is that bi;i ðGÞ ¼ b%i;i ðGÞ are just reduced (topological) Betti numbers of G; that is, bi;i ðGÞ ¼ b%i;i ðGÞ ¼ bi 1 ðGÞ 80pipdimðGÞ þ 1; where bi 1 ðGÞ ¼ dim H˜ i 1 ðG; kÞ: This result is a consequence of Theorem 4.1 together with the fact [6] that for a shifted complex K bi 1 ðKÞ ¼ jfF AmaxðKÞ : jF j ¼ i; 1eF gj; and the fact that symmetric shifting preserves topological Betti numbers [13]. Remark 4.9. Finally, we note that if G is a Buchsbaum complex (i.e., a pure simplicial complex all of whose vertices have Cohen–Macaulay links), then bi;r ðGÞ ¼ i 1 r 1 br 1 ðGÞ for every 0prpipdimðGÞ; where br 1 ðGÞ are reduced (topological) Betti numbers of G: This follows from Lemma 4.3 and [19, Lemma 4.1].
5. Extremal Betti numbers This section is devoted to the proof of Theorem 5.3, which relates the graded algebraic Betti numbers of IG to the symmetric iterated Betti numbers of G: Every homogeneous ideal ICS admits a graded free S-resolution of the form -?
M j
Sð jÞbi; j -?-
M j
Sð jÞb1;j -
M
Sð jÞb0;j -I-0;
j
where Sð jÞ denotes S with grading shifted by j: Moreover, there exists a unique (up to isomorphism) resolution in which all the exponents bi; j are simultaneously minimized, called the minimal graded free S-resolution of I: The numbers bi; j appearing in this minimal free resolution of I are called the graded Betti numbers of I: A Betti number of I; bi;iþj ðIÞ; is extremal if bi;iþj ðIÞa0; but bi0 ;i0 þj 0 ðIÞ ¼ 0 for all i0 Xi; j 0 Xj; ði0 ; j 0 Þaði; jÞ: This terminology comes from the Betti diagram of I output by the program Macaulay 2 in which the Betti numbers are arranged in a rectangular array whose columns are indexed by i and rows by j and the ði; jÞth entry is the Betti
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number bi;iþj : Thus bi;iþj is an extremal Betti number of I if it lies in a south-east corner of the Macaulay 2 Betti diagram of I: Let G be a simplicial complex on the vertex set ½n: The Alexander dual of G is the simplicial complex G ¼ fF D½n : ½n\F eGg: The next two results (both due to Bayer et al. [4], see [1] also for the second theorem) provide connections between the extremal Betti numbers of the Stanley– Reisner ideals of G and G ; and the shifted complex DðGÞ: Theorem 5.1. Let G be a simplicial complex and G be its Alexander dual. The Stanley–Reisner ideals IG and IG have the same extremal Betti numbers. More precisely, bi;iþj ðIG Þ is extremal if and only if bj 1;iþj ðIG Þ is extremal. Also, in such a case bi;iþj ðIG Þ ¼ bj 1;iþj ðIG Þ: Theorem 5.2. Extremal Betti numbers are preserved by algebraic shifting: for a simplicial complex G; bi;iþj ðIG Þ is extremal if and only if bi;iþj ðIDðGÞ Þ is extremal. Moreover, in such a case bi;iþj ðIG Þ ¼ bi;iþj ðIDðGÞ Þ: We are now in a position to prove the main theorem of this section. Theorem 5.3. The extremal Betti numbers of IG are contained among the symmetric iterated Betti numbers of G: They are precisely the extremal entries in the b-triangle of G: bj 1;iþj ðIG Þ is an extremal Betti number of IG if and only if bn j0 ;i0 ðGÞ ¼ 0
8ði0 ; j 0 Þaði; jÞ; i0 Xi; j 0 Xj; and bn j;i ðGÞa0:
Moreover, in this case, bj 1;iþj ðIG Þ ¼ bn j;i ðGÞ: Example 3.7 (continued). The minimal free resolution and Betti diagram of IG (computed by Macaulay 2) are given below. Note that the entries in the southeast corners of the Betti diagram of IG (the extremal Betti numbers of IG ) are precisely the entries in the north-east corners of the b-triangle of G from Section 3. 0-S 2 -S15 -S 42 -S 49 -S21 -S-0 total : 1
21
0: 1:
1 :
: :
2: 3:
: :
21 :
49 42 15 : :
: :
: :
49 42 14 : : 1
2 : : 2 :
The proof of Theorem 5.3 relies on the following lemma, which is a consequence of [14, Theorem 2.1(b)] (see also [15, Proposition 12]) and [8, Corollary 6.2]. For completeness we provide a different self-contained proof.
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Lemma 5.4. The symmetric iterated Betti numbers of G are related to the graded Betti numbers of the Stanley–Reisner ideal IDðG Þ as follows: X n r j bi;iþj ðIDðG Þ Þ ¼ bn j;n r j ðGÞ: i r Proof. Let G be a simplicial complex and let G be its Alexander dual. Recall that the Stanley–Reisner ideal IDðG Þ CS is a squarefree monomial ideal whose minimal generators correspond to minimal non-faces of DðG Þ: Let G be the set of minimal ½n generators of IDðG Þ ; let Gj ¼ G-Nj ; and let minðgÞ ¼ minfi: yi jgg for a monomial gAG: Since DðG Þ is a shifted complex, it follows that the ideal IDðG Þ is squarefree strongly stable, which means that for a monomial mAIDðG Þ ; if yi jm; iojpn; and yj is not a divisor of m; then myj =yi AIDðG Þ as well. Hence the graded Betti numbers of IDðG Þ are given by the following formula [13, Corollary 3.4] (which is the analog of the Eliahou–Kervaire formula for strongly stable ideals): X n minðgÞ þ 1 j bi;iþj ðIDðG Þ Þ ¼ i gAGj X n r j ð5Þ ¼ jfgAGj : minðgÞ ¼ r þ 1gj: i r It is well known and is easy to prove that DðG Þ ¼ DðGÞ : Thus, gAGj if and only if s ¼ suppðgÞ is a minimal non-face of DðG Þ of size j; which happens if and only if ½n\s is a facet of DðGÞ of size n j: Moreover, r þ 1 ¼ minfi : iAsg if and only if ½rD½n\s; but r þ 1e½n\s: Hence jfgAGj : minðgÞ ¼ r þ 1gj ¼ jfF AmaxðDðGÞÞ : jF j ¼ n j; ½rDF ; but r þ 1eF gj Theorem 4:1
¼
bn j;n r j ðGÞ:
Substituting (6) in (5) gives the result.
ð6Þ &
Proof of Theorem 5.3. The theorem is an easy consequence of Lemma 5.4. Indeed, since n r j is positive for rpn i j and is zero otherwise, it follows from the i lemma that bi0 ;i0 þj0 ðIDðG Þ Þ ¼ 0 iff bn j0 ;n j0 r ðGÞ ¼ 0 for all rpn i0 j 0 : Thus, bi;iþj ðIDðG Þ Þa0 is extremal 3 bi0 ;i0 þj 0 ðIDðG Þ Þ ¼ 0
for all i0 Xi; j 0 Xj; ði; jÞaði0 ; j 0 Þ
3 bn j 0 ;i0 ðGÞ ¼ 0 for all i0 Xi; j 0 Xj; ði; jÞaði0 ; j 0 Þ:
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Moreover, if this is the case, then all except the first summand in X n r j i bn j;i ðGÞ þ bi;iþj ðIDðG Þ Þ ¼ bn j;n r j ðGÞ i i ron i j vanish, implying that bi;iþj ðIDðG Þ Þ ¼ bn j;i ðGÞ: The result then follows from Theorems 5.1 and 5.2. & We remark that one can use [1, Corollary 1.2] and certain properties of sets Ai ðIÞ to provide a different proof of the following more general result. We omit the details. Theorem 5.5. Let I be a homogeneous ideal. The extremal Betti numbers of I form a subset of the symmetric iterated Betti numbers of I: More precisely, bj 1;iþj ðIÞ is an extremal Betti number of I if and only if bn j0 ;i0 ðIÞ ¼ 0
8ði0 ; j 0 Þaði; jÞ; i0 Xi; j 0 Xj;
and
bn j;i ðIÞa0:
Moreover, in this case, bj 1;iþj ðIÞ ¼ bn j;i ðIÞ:
6. Associated primes and standard pairs The associated primes of a homogeneous ideal ICS with a primary decompffiffiffiffiffi position I ¼ Q1 -Q2 -?-Qt are the prime ideals Pi :¼ Qi ; i ¼ 1; y; t; pffiffiffiffiffi where Qi denotes the radical of Qi : The set of associated primes of I; customarily denoted as AssðIÞ; is independent of the primary decomposition of I: The minimal elements of AssðIÞ with respect to inclusion are called the minimal primes of I; and the non-minimal ones are called the embedded primes of I: We denote the set of minimal primes of I as MinðIÞ: Recall that the irreducible (isolated) components of V ðIÞ; the variety of I in kn ; are the varieties V ðPÞ for PAMinðIÞ: Let Zi :¼ V ðPi Þ be the variety of Pi in kn : The finite invariant degðZi Þ; called the degree of Zi ; is the cardinality of Zi -L for almost all linear subspaces L of dimension equal to the codimension of Zi : Definition 6.1 (Bayer and Mumford [5], Sturmfels et al. [21]). (1) If P is a homogeneous prime ideal in S then the multiplicity of P (with respect to I), denoted as multI ðPÞ is the length of the largest ideal of finite length in the ring SP =ISP : P (2) The degree of I; degðIÞ :¼ fdimðZi Þ¼dimðIÞg multI ðPi ÞdegðZi Þ: P (3) The geometric degree of I; geomdegðIÞ :¼ fPi AMinðIÞg multI ðPi ÞdegðZi Þ: P (4) The arithmetic degree of I; arithdegðIÞ :¼ fPi AAssðIÞg multI ðPi ÞdegðZi Þ: The invariant multI ðPÞ40 if and only if PAAssðIÞ: Our main goal in this section is to prove Theorem 6.6. We first specialize Definition 6.1 to monomial ideals. If M is a monomial ideal, then every associated prime of M is
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of the form Ps :¼ /yj : jesS for some set sD½n: Hence V ðPs Þ is the jsj-dimensional linear subspace spanned by fej : jAsg and degðV ðPs ÞÞ ¼ 1: The three degrees of M from Definition 6.1 are therefore appropriate sums of multiplicities of ideals in AssðMÞ with respect to M: For a monomial ideal M the multiplicities of associated primes as well as all the degrees referred to in Definition 6.1 can be read off from the standard pairs of M (see Definition 3.2) as shown in the following lemma. The statements in this lemma are either stated or can be derived easily from the results in [21]. Lemma 6.2. Let M be a monomial ideal. Then, the set of standard pairs of M is well defined, Ns is a standard pair of M if and only if Ps AAssðMÞ; Ns is a standard pair of M if and only if Ps AMinðMÞ; the dimension of M is the maximal size of a set s such that Ns is a standard pair of M; (5) if Ps AAssðMÞ; then multM ðPs Þ is the number of standard pairs of M of the form Ns and (6) (a) degðMÞ is the number of standard pairs Ns of M such that jsj ¼ dimðMÞ; (b) geomdegðMÞ is the number of standard pairs Ns of M such that Ns is a standard pair of M and (c) arithdegðMÞ is the total number of standard pairs of M:
(1) (2) (3) (4)
Lemma 3.3 showed that mNs is a standard pair of GinðIÞ if and only if s ¼ ½i and mAAi ðIÞ for some 0pipn: Combining this fact with Lemma 6.2 we obtain the following. Corollary 6.3 (See also Eisenbud [10, Corollary 15.25]). (i) P½d ; d ¼ dim ðIÞ; is the unique minimal prime of GinðIÞ (if I ¼ IG then d ¼ dim G þ 1), and (ii) all embedded primes of GinðIÞ are of the form P½k for some kod: Thus the submonoids in the standard pairs of GinðIÞ are initial intervals of ½n while the cosets can be complicated. On the other hand, for the square free monomial ideals IG and IDðGÞ ; the cosets of the standard pairs are trivial and the submonoids determine the ideals (cf. Example 3.7). T Corollary 6.4. If G is a simplicial complex then IG ¼ sAmaxðGÞ Ps is the irredundant prime decomposition of IG : In particular, IG has no embedded primes and its standard pairs are fNs : sAmaxðGÞg: By Corollary 3.5, mN½i is a standard pair of GinðIG Þ if and only if ½i r,suppðFðmÞÞ is a facet of DðGÞ of size i: Combining this fact with Corollary 6.4 we get the following bijection as well.
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Corollary 6.5. There is a bijection between the standard pairs of GinðIG Þ and those of IDðGÞ given by: mN½i is a standard pair of GinðIG Þ with degðmÞ ¼ r if and only if N½i r,suppðFðmÞÞ is a standard pair of IDðGÞ : The following theorem is a corollary of the results in Sections 3 and 4, and those stated thus far in this section. Theorem 6.6. The iterated Betti numbers of a homogeneous ideal I are related to the ideal GinðIÞ: Those of an ideal IG are related to the ideals GinðIG Þ; IDðGÞ ; and the shifted complex DðGÞ: The relationships are as follows. (1) The multiplicity of P½i with respect to GinðIÞ is X bi;r ðIÞ: multGinðIÞ ðP½i Þ ¼ r
If I ¼ IG then multGinðIG Þ ðP½i Þ ¼
X
bi;r ðGÞ ¼ jfF AmaxðDðGÞÞ : jF j ¼ igj:
r
(2) The degree, geometric degree, and arithmetic degree of GinðIG Þ and IDðGÞ have the following interpretations: P ðiÞ degðGinðIG ÞÞ ¼ geomdegðGinðIG ÞÞ ¼ r bd;r ðIG Þ ði0 Þ ¼ degðIDðGÞ Þ ¼ jfF AmaxðDðGÞÞ : jF j ¼ dgj; P ðiiÞ arithdegðGinðIG ÞÞ ¼ i;r bi;r ðIG Þ ðii0 Þ ¼ arithdegðIDðGÞ Þ ¼ jmaxðDðGÞÞj: Eqs. (i) and (ii) also hold for arbitrary homogeneous ideals I in S: Proof. (1) By Lemmas 6.2(5) multGinðIG Þ ðP½i Þ ¼ jfstandard pairs of GinðIÞ of the form N½i gj ¼ jAi ðIÞj ðby Lemma 3:3Þ X bi;r ðIÞ ðby Theorem 4:3Þ: ¼ r
In particular, P½i is an associated prime of GinðIÞ if and only if bi;r ðIÞ40 for some r: For a simplicial complex G on ½n; by Lemma 3.4, X multGinðIG Þ ðP½i Þ ¼ jAi ðIG Þj ¼ bi;r ðGÞ ¼ jfF AmaxðDðGÞÞ : jF j ¼ igj: r
In particular, P½i is an associated prime of GinðIG Þ if and only if DðGÞ has a facet of size i: (2) The same lemmas along with Definition 6.1 yield these results. & We now establish certain further facts about AssðGinðIG ÞÞ:
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Definition 6.7. For any ideal I in a polynomial ring, its poset of associated primes AssðIÞ has the chain property if whenever PAAssðIÞ is an embedded prime, then there exists QAAssðIÞ such that P*Q and dimðQÞ ¼ dimðPÞ þ 1: Corollary 6.8. The poset AssðGinðIG ÞÞ possesses the chain property if and only if maxðDðGÞÞ has the property that whenever DðGÞ has a facet of size kpdimðGÞ then it also has a facet of size k þ 1: Proof. By Corollary 6.3, the poset AssðGinðIG ÞÞ has the chain property if whenever P½k AAssðGinðIG ÞÞ for some kpdim G then P½kþ1 AAssðGinðIG ÞÞ: By Theorem 6.6(1), this is equivalent to the condition that whenever DðGÞ has a facet of size kpdim G then it also has a facet of size k þ 1: & Corollary 6.9. If G is a Buchsbaum complex (cf. Remark 4.9) then AssðGinðIG ÞÞ has the chain property. Proof. It was shown in [19] that if G is a ðd 1Þ-dimensional Buchsbaum complex i 1 br 1 for iod; where br 1 is the reduced (topological) Betti then bi;r ðGÞ ¼ r 1 P P i 1 number of G: Hence for iod; multGinðIG Þ ðP½i Þ ¼ r bi;r ¼ r r 1 br 1 : Therefore if some bk a0 then multGinðIG Þ ðP½i Þ40 for all iXk; which implies that AssðGinðIG ÞÞ has the chain property. &
7. Iterated Betti numbers: exterior versus symmetric We close the paper with several remarks and conjectures on connections between symmetric iterated Betti numbers and exterior iterated Betti numbers of a simplicial complex. The superscript e is used to denote exterior shifting. We start of exterior algebraic shifting extracted from [13]. V with a brief description V Let E ¼ ðk½y1 ; y; yn 1 Þ ¼ S1 be the exterior algebra over the n-dimensional vector space S1 : A monomial in E is an expression of the form m ¼ yi1 4yi2 4?4yik ; where 1pi1 oi2 o?oik pn; the set fi1 ; i2 ; y; ik g is called the support of m; and is denoted by suppðmÞ: The exterior Stanley–Reisner ideal of a simplicial complex G on ½n is JG :¼ /mAE : m is a monomial; suppðmÞeGS: Definition 7.1. The exterior algebraic shifting of G; De ðGÞ; is the simplicial complex defined by JDe ðGÞ :¼ GinðJG Þ; where GinðJG Þ is the generic initial ideal of JG with respect to the reverse lexicographic order with yn gyn 1 g?gy1 : The exterior iterated Betti numbers of a simplicial complex G were introduced by Duval and Rose [9]. They have the following combinatorial description (up to a slight change in the indexing), which we adopt as their definition.
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Definition 7.2. The exterior iterated Betti numbers of a simplicial complex G are bei;r ðGÞ ¼ jfF A maxðDe ðGÞÞ: jF j ¼ i; ½i rDF ; i r þ 1eF gj: Since De ðDðGÞÞ ¼ DðGÞ and DðDe ðGÞÞ ¼ De ðGÞ; the above definition and Theorem 4.1 imply that bi;r ðGÞ ¼ bei;r ðDðGÞÞ;
bei;r ðGÞ ¼ bi;r ðDe ðGÞÞ:
Hence we infer the following corollary from Lemma 5.4. Corollary 7.3. For a simplicial complex G X n r j e bi;iþj ðID ðG Þ Þ ¼ ben j;n r j ðGÞ: i r Since exterior shifting preserves extremal Betti numbers [1], the same proof as in Theorem 5.3 yields Theorem 7.4. bj 1;iþj ðIG Þ is an extremal Betti number of IG if and only if ben j0 ;i0 ðGÞ ¼ 0
8ði0 ; j 0 Þaði; jÞ; i0 Xi; j 0 Xj; and ben j;i ðGÞa0:
Moreover, if this is the case, then bj 1;iþj ðIG Þ ¼ bj 1;iþj ðIDe ðGÞ Þ ¼ bj 1;iþj ðIDðGÞ Þ ¼ ben j;i ðGÞ ¼ bn j;i ðGÞ: We remark that there are no known connections between general (non-extremal) graded Betti numbers of IDðGÞ and IDe ðGÞ : However it is conjectured (see [13, Conjecture 8.9]) that the following holds. Conjecture 7.5. For every simplicial complex G; bi; j ðIDðGÞ Þpbi; j ðIDe ðGÞ Þ: In analogy, we propose the following. Conjecture 7.6. For every simplicial complex G; bi; j ðGÞpbei; j ðGÞ for all ipdimðGÞ: Note that since all the coefficients in the expression of graded Betti numbers in terms of iterated Betti numbers (see Lemma 5.4 and Corollary 7.3) are non-negative, Conjecture 7.6 if true would imply Conjecture 7.5 except for the case j i ¼ n dimðGÞ 1 (that is, except for the last column of the Betti diagram of IDðGÞ and IDe ðGÞ output by Macaulay 2). Conjecture 7.5 was verified by Aramova et al. [2] in the case when G is the Alexander dual of a sequentially Cohen–Macaulay complex (a notion introduced by Stanley [20, Definition II.2.9]): they showed that in such a case bi; j ðIDðGÞ Þ ¼ bi; j ðIDe ðGÞ Þ ¼ bi; j ðIG Þ: We have the following related result.
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Proposition 7.7. Conjecture 7.6 holds for all sequentially Cohen–Macaulay complexes. More precisely, if G is sequentially Cohen–Macaulay, then bi;r ðGÞ ¼ bei;r ðGÞ ¼ hi;r ðGÞ; where ðhi;r ðGÞÞ0prpipdimðGÞþ1 is the h-triangle of G: The notion of f - and h-triangles was introduced by Bjo¨rner and Wachs [7]. We recall the definition. For a simplicial complex G set fi; j ðGÞ :¼ jfF AG : jF j ¼ j; dimðst F Þ ¼ i 1gj; where st F denotes the star of F in G: The h-triangle of G; ðhi; j ðGÞÞ0p jpipdimðGÞþ1 ; is defined by j X i s hi; j ðGÞ ¼ ð 1Þ j s fi;s ðGÞ: j s s¼0 Proof of Proposition 7.7. Duval [8, Theorem 5.1] showed that if G is a sequentially Cohen–Macaulay simplicial complex, then hi; j ðGÞ ¼ hi; j ðDe ðGÞÞ: In his proof he relied only on properties (P3) and (P4) of the operator De ; and the fact that G is Cohen–Macaulay if and only if De ðGÞ is pure. Since operator D (symmetric shifting) possesses all these properties as well, it follows that for a sequentially Cohen– Macaulay complex G; hi; j ðGÞ ¼ hi; j ðDe ðGÞÞ ¼ hi; j ðDðGÞÞ:
ð7Þ
Another result due to Duval [8, Corollary 6.2] is that for a shifted complex K; bei; j ðKÞ ¼ hi; j ðKÞ: Thus hi; j ðDe ðGÞÞ ¼ bei; j ðDe ðGÞÞ ¼ bei; j ðGÞ
ð8Þ
hi; j ðDðGÞÞ ¼ bei; j ðDðGÞÞ ¼ bi; j ðGÞ:
ð9Þ
and
Eqs. (7)–(9) imply the proposition.
&
We close the paper with one additional conjecture for the special case of Buchsbaum complexes. Conjecture 7.8. If G is a Buchsbaum complex, then bi; j ðGÞ ¼ bei; j ðGÞ; and hence bi; j ðIDðGÞ Þ ¼ bi; j ðIDe ðGÞ Þ:
References [1] A. Aramova, J. Herzog, Almost regular sequences and Betti numbers, Amer. J. Math. 122 (2000) 689–719. [2] A. Aramova, J. Herzog, T. Hibi, Ideals with stable Betti numbers, Adv. Math. 152 (1) (2000) 72–77.
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[3] A. Aramova, J. Herzog, T. Hibi, Shifting operations and graded Betti numbers, J. Algebraic Combin. 12 (3) (2000) 207–222. [4] D. Bayer, H. Charalambous, S. Popescu, Extremal Betti numbers and applications to monomial ideals, J. Algebra 221 (2) (1999) 497–512. [5] D. Bayer, D. Mumford, What can be computed in algebraic geometry? in: D. Eisenbud, L. Robbiano (Eds.), Computational Algebraic Geometry and Commutative Algebra, Proceedings Cortona 1991, Cambridge University Press, Cambridge, 1993, pp. 1–48. [6] A. Bjo¨rner, G. Kalai, An extended Euler-Poincare´ theorem, Acta Math. 161 (1988) 279–303. [7] A. Bjo¨rner, M. Wachs, Shellable nonpure complexes and posets I, Trans. Amer. Math. Soc. 348 (1996) 1299–1327. [8] A. Duval, Algebraic shifting and sequentially Cohen–Macaulay simplicial complexes, Electron. J. Combin. 3 (1996) #R21. [9] A. Duval, L. Rose, Iterated homology of simplicial complexes, J. Algebraic Combin. 12 (2000) 279–294. [10] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Text in Mathematics, Springer, New York, 1995. [11] D. Eisenbud, D. Grayson, M. Stillman, B. Sturmfels (Eds.), Computations in Algebraic Geometry with Macaulay 2, Algorithms and Computation in Mathematics, Vol. 8, Springer, Berlin, 2000. [12] D. Grayson, M. Stillman, Macaulay 2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2. [13] J. Herzog, Generic initial ideals and graded Betti numbers, in: T. Hibi (Ed.), Computational Commutative Algebra and Combinatorics, Advanced Studies in Pure Mathematics, Vol. 33, Mathematical Society of Japan, Tokyo, 2002, pp. 75–120. [14] J. Herzog, T. Hibi, Componentwise linear ideals, Nagoya Math. J. 153 (1999) 141–153. [15] J. Herzog, V. Reiner, V. Welker, Componentwise linear ideals and Golod rings, Michigan Math. J. 46 (2) (1999) 211–223. [16] G. Kalai, The diameter of graphs of convex polytopes and f-vector theory, in: P. Gritzmann, B. Sturmfels (Eds.), Applied Geometry and Discrete Mathematics—The Victor Klee Festschrift, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 4, Amer. Math. Soc. 1991, pp. 387–411. [17] G. Kalai, Algebraic shifting, Manuscript 1993. [18] G. Kalai, Algebraic shifting, in: T. Hibi (Ed.), Computational Commutative Algebra and Combinatorics, Advanced Studies in Pure Mathematics, Vol. 33, Mathematical Society of Japan, Tokyo, 2002, pp. 121–163. [19] I. Novik, Upper bound theorems for homology manifolds, Israel J. Math. 108 (1998) 45–82. [20] R. Stanley, Combinatorics and Commutative Algebra, 2nd Edition, Birkha¨user, Boston, 1996. [21] B. Sturmfels, N. Trung, W. Vogel, Bounds on degrees of projective schemes, Math. Ann. 302 (1995) 417–432.
Journal of Combinatorial Theory, Series A 105 (2004) 233–254
Symmetric iterated Betti numbers$ Eric Babson, Isabella Novik, and Rekha Thomas Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA Received 3 March 2003 Communicated by Gil Kalai
Abstract We define a set of invariants of a homogeneous ideal I in a polynomial ring called the symmetric iterated Betti numbers of I: We prove that for IG ; the Stanley–Reisner ideal of a simplicial complex G; these numbers are the symmetric counterparts of the exterior iterated Betti numbers of G introduced by Duval and Rose, and that the extremal Betti numbers of IG are precisely the extremal (symmetric or exterior) iterated Betti numbers of G: We show that the symmetric iterated Betti numbers of an ideal I coincide with those of a particular reverse lexicographic generic initial ideal GinðIÞ of I; and interpret these invariants in terms of the associated primes and standard pairs of GinðIÞ: We close with results and conjectures about the relationship between symmetric and exterior iterated Betti numbers of a simplicial complex. r 2003 Elsevier Inc. All rights reserved. Keywords: Algebraic shifting; Generic initial ideals; Extremal Betti numbers; Standard pairs; Local cohomology
1. Introduction The goal of this paper is to define and study a set of invariants of a homogeneous ideal in a polynomial ring, called the symmetric iterated Betti numbers of the ideal. For a simplicial complex G; the symmetric iterated Betti numbers of the Stanley– Reisner ideal of G (also referred to as the symmetric iterated Betti numbers of G) are preserved by symmetric algebraic shifting. $
Research partially supported by NSF Grants DMS 0070571 and DMS 0100141. E-mail addresses: [email protected] (E. Babson), [email protected] (I. Novik), [email protected] (R. Thomas). 0097-3165/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jcta.2003.11.003
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We discuss two versions of algebraic shifting (both introduced by Kalai [6,16]) which given a simplicial complex G with vertex set ½n :¼ f1; 2; y; ng provide new simplicial complexes with the same vertex set. We denote these versions by DðGÞ for the symmetric shifting of G (see Definition 2.1) and by De ðGÞ for the exterior shifting of G (see Definition 7.1). There are instances for which DðGÞaDe ðGÞ [18]. It is known that: (P1) DðGÞ and De ðGÞ are shifted. (A simplicial complex K is shifted if for every F AK and joiAF ; the set ðF \figÞ,f jg is also in K:) (P2) If G is shifted, then DðGÞ ¼ De ðGÞ ¼ G: (P3) G; DðGÞ and De ðGÞ have the same f -vector, that is, they have the same number of i-dimensional faces for every i: (P4) If G0 is a subcomplex of G; then DðG0 ÞDDðGÞ and De ðG0 ÞDDe ðGÞ: Both versions were studied extensively from the algebraic point of view in a series of recent papers by Aramova, Herzog, Hibi and others (surveyed in [13], see also [1–4]). Consider the polynomial ring S ¼ k½y1 ; y; yn where k is a field of characteristic zero. Let N denote the set of non-negative integers. If AD½n then write yA ¼ Q ½n the monomials of S by identifying a function f : ½n-N in aAA ya : Denote by N Q f ðiÞ ½n N with the monomial iA½n yi and consider N½n as a multiplicative monoid. Thus f0; 1g½n ¼ y2 is the set of squarefree monomials. If GD2½n is a simplicial complex then the Stanley–Reisner ideal of G [20, Definition II.1.1] is the squarefree monomial ideal ½n
IG :¼ /y2
½n
G
SCS:
The (bi-graded) Betti numbers of a homogeneous ideal ICS are the invariants bi; j ðIÞ that appear in the minimal-free resolution of I as an S-module: -?
M j
Sð jÞbi; j ðIÞ -?-
M j
Sð jÞb1;j ðIÞ -
M
Sð jÞb0;j ðIÞ -I-0
j
Here Sð jÞ denotes S with grading shifted by j: We say that bi;iþj ðIÞ; is extremal P if 0abi;iþj ðIÞ ¼ i0 Xi;j 0 Xj bi0 ;i0 þj0 ðIÞ: (This is equivalent to having 0abi;iþj ðIÞ and 0 ¼ bi0 ;i0 þj0 ðIÞ for every i0 Xi and j 0 Xj; ði0 ; j 0 Þaði; jÞ:) Since DðGÞ and De ðGÞ are shifted complexes, their combinatorial structures are simpler than that of G: Nonetheless, the two shifting operations preserve many combinatorial and topological properties. 1. D and De preserve topological Betti numbers (see [6, Theorem 3.1; 1, Proposition 8.3] for exterior shifting and [13, Corollary 8.25] for symmetric shifting). Moreover, exterior algebraic shifting preserves the exterior iterated Betti numbers of a simplicial complex. (There are two versions of exterior iterated Betti numbers—one due to Kalai [17, Corollary 3.4] and another due to Duval and Rose [9]. Both sets of numbers are preserved under exterior shifting.)
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2. D and De preserve Cohen–Macaulayness: a simplicial complex G is Cohen– Macaulay if and only if DðGÞ (De ðGÞ) is Cohen–Macaulay, which happens if and only if DðGÞ (De ðGÞ) is pure (see [17, Theorem 5.3], [1, Proposition 8.4] for exterior shifting and [16, Theorem 6.4] for symmetric shifting). 3. D and De preserve extremal Betti numbers: bi;iþj ðIG Þ is an extremal Betti number of IG if and only if bi;iþj ðIDðGÞ Þ is extremal for IDðGÞ ; in which case bi;iþj ðIG Þ ¼ bi;iþj ðIDðGÞ Þ: The same assertion holds for De : (See [4] for symmetric shifting and [1, Theorem 9.7] for both versions.) Property 3 is a far-reaching generalization of property 2, while property 1 played a crucial role in Kalai’s proof of property 2 for exterior shifting. This suggests that there might be a connection between the iterated Betti numbers of a simplicial complex G on the one hand and the extremal Betti numbers of the ideal IG on the other. This is one of the connections we establish in this paper. Consider the action of GLðS1 Þ on S and choose uAGLðS1 Þ to be generic. Denote by m ¼ /S1 S the irrelevant ideal of S: If I is a homogeneous ideal in S then define the S-modules Mi ðIÞ as follows: M0 ðIÞ :¼ S=uI and Mi ðIÞ :¼ Mi 1 ðIÞ=ðyi Mi 1 ðIÞ þ H 0 ðMi 1 ðIÞÞÞ for 1pipn; where H 0 ð Þ denotes the 0th local cohomology with respect to the irrelevant ideal m: We now come to the central definition of this paper. Definition 1.1. The symmetric iterated Betti numbers of a homogeneous ideal I in S are bi;r ðIÞ :¼ dim H 0 ðMi ðIÞÞr
for 0pi; rpn;
where H 0 ð Þr stands for the rth component of the 0th local cohomology. If G is a simplicial complex with vertex set ½n; define the symmetric iterated Betti numbers of G to be bi;r ðGÞ :¼ bi;r ðIG Þ; 0pi; rpn: Our first result gives a combinatorial interpretation of the symmetric iterated Betti numbers of a simplicial complex G and shows that they are invariant under symmetric algebraic shifting. Let maxðGÞ denote the set of facets (maximal faces) of G: Write dimðGÞ ¼ maxfjF j 1 : F AGg: Theorem 4.1. Let G be a simplicial complex. Then jfF A maxðDðGÞÞ : jF j ¼ i; ½i rDF ; i r þ 1eF gj if rpi; bi;r ðGÞ ¼ 0 otherwise: In particular, since DðDðGÞÞ ¼ DðGÞ; it follows that the symmetric iterated Betti numbers of G are invariant under symmetric shifting. Theorem 4.1 implies that bi;r ðGÞ ¼ 0 unless 0prpipdimðGÞ þ 1: The exterior iterated Betti numbers of G; bei;r ðGÞ; defined by Duval and Rose have precisely the same combinatorial formula (up to a slight change in indices), except that in their definition, one replaces DðGÞ by De ðGÞ [9, Theorem 4.1]. In fact, the operation of ‘‘peeling off’’ the ith variable at step i we are using to define the symmetric iterated
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Betti numbers is analogous to the ‘‘deconing’’ of the shifted complex used in the Duval–Rose definition. The extremal Betti numbers of an ideal I ¼ IG are the extremal iterated Betti numbers (symmetric or exterior) of the simplicial complex G in the following sense. Theorem 5.3 and 7.4. Let G be a simplicial complex. The extremal Betti numbers of IG form a subset of the symmetric as well as of the exterior iterated Betti numbers of G: More precisely, bj 1;iþj ðIG Þ is an extremal Betti number of IG if and only if bn j0 ;i0 ðGÞ ¼ 0
8ði0 ; j 0 Þaði; jÞ; i0 Xi; j 0 Xj;
and
bn j;i ðGÞa0:
In such a case bj 1;iþj ðIG Þ ¼ bn j;i ðGÞ: The same assertion holds for be ; : Let GinðIÞ denote the reverse lexicographic generic initial ideal of a homogeneous ideal I in S with variables ordered as yn gyn 1 g?gy1 : It follows from [4, Corollary 1.7] that the symmetric iterated Betti numbers of I coincide with those of GinðIÞ: We provide an alternate proof of this fact in Section 4 (see Corollary 4.6). Our next result—Theorem 6.6—interprets the symmetric iterated Betti numbers bi;r ðIÞ in terms of the associated primes of GinðIÞ: This paper is organized as follows. In Section 2 we recall the basics of symmetric shifting. Section 3 defines and interprets certain monomial sets that are at the root of all our proofs. In Sections 4–7 we prove the theorems stated above. We conclude in Section 7 with some results and conjectures on the relationship between the exterior and symmetric iterated Betti numbers of a simplicial complex.
2. Algebraic shifting In this section we recall the basics of symmetric algebraic shifting. (The description of exterior shifting is deferred to Section 7.) For further details on symmetric and exterior shifting see the survey articles by Herzog [13] and Kalai [18]. Let Ns denote the set of all finite degree monomials in the variables yi with iAs and Nsr denote the set of elements of degree r in Ns : In particular, if ½n ¼ ½1; n ¼ f1; y; ng then N½n is the set of all monomials in S and f0; 1gs is the set of all square free finite degree monomials in Ns : In this paper we fix the reverse lexicographic order g on NZ with yi gyi 1 for all iAZ extending the partial ordering by degree. (Thus, for example, y2n gyn 1 yn gy2n 1 gyn 2 yn gyn 2 yn 1 g?:) We also define the square free map F : NZ -f0; 1gZ by Fðyi0 yi1 ?yik Þ :¼ yi0 k yi1 ðk 1Þ ?yij ðk jÞ ?yik ; where i0 pi1 p?pik :
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In other Q words, F is the unique degree and order preserving bijection for which Fðyn0 Þ ¼ noip0 yi : For each homogeneous ideal ICS there exists a Zariski open set UðIÞC GLðS1 Þ such that the ideal Ing ðuIÞ; (the initial ideal of uI with respect to the monomial order g on S), is independent of the choice of uAUðIÞ: The ideal Ing ðuIÞ is called the generic initial ideal of I with respect to g and is denoted by GinðIÞ ¼ Ging ðIÞ (see [10, Chapter 15]). If I is a homogeneous ideal in S then one way to explicitly and uniformly construct an element aAUðIÞ is to consider the extension K ¼ kðfai; j gi; jA½n Þ=k and then for any ideal I in S the element n X ai; j yj a : SK ¼ S#k K-SK given by ayi ¼ j¼1
is generic for KI as an ideal of SK : For a homogeneous ideal I in S and a generic linear map uAUðIÞ define BðIÞ ¼ fmAN½n : m is not in the linear span of fnjmgng,uIg: Note that BðIÞ is a basis of the vector space M0 ðIÞ ¼ S=uI and hence BðIÞ ¼ N½n GinðIÞ: Definition 2.1. The symmetric algebraic shifting of a simplicial complex GD2½n is DðGÞ where yDðGÞ ¼ FðBðIG ÞÞ-N½1;N Df0; 1g½n : Note that this means that IDðGÞ ¼ /FðN½n BðIG ÞÞS: The fact that DðGÞ is a simplicial complex satisfying conditions (P1)–(P4) was proved in [16, Theorem 6.4; 3] by using certain properties of BðIÞ: We list some of them below: (B1) BðIÞ is a basis of S=uI; as well as of S=GinðIÞ: (B2) BðIÞ is an order ideal—if mABðIÞ and m0 jm; then m0 ABðIÞ: (B3) BðIÞ is shifted—if joi and yi mABðIÞ then yj mABðIÞ: Condition (B1) was discussed above while (B2) follows from the fact that GinðIÞ is an ideal. Condition (B3) is a consequence of the fact that generic initial ideals are Borel fixed [10, Theorem 15.20]. In characteristic 0, this is equivalent to GinðIÞ being strongly stable [10, Theorem 15.23], which means that if joi and yj mAGinðIÞ then yi mAGinðIÞ: In the case when I ¼ IG ; BðIG Þ has another fundamental property: (B4) If mABðIG Þ-N½k;n and rXk then mNfkg DBðIG Þ as well. r This is due to Kalai [16, Lemma 6.3] and implies that y1 ; y; yn is an almost regular M0 ðIG Þ-sequence (a notion introduced by Aramova and Herzog [1]; it played a crucial role in their proof that extremal Betti numbers are preserved by algebraic shifting).
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3. Special monomial subsets In this section we identify and interpret certain subsets of monomials in the basis BðIÞ of M0 ðIÞ that are at the root of all our proofs. Definition 3.1. Let I be a homogeneous ideal in S: For iA½0; n define n o / BðIÞ ; Ai;r ðIÞ :¼ Ai ðIÞ-NN Ai :¼ mAN½iþ1;n : mNfig DBðIÞ; mNfiþ1g D r : Several remarks are in order. Since BðIÞ is shifted (B3), mNfig DBðIÞ iff mN½i DBðIÞ: Since BðIG Þ satisfies (B4), Ai;r ðIG Þ ¼ |
if r4i
and hence
Ai ðIG Þ ¼
i [
Ai;r ðIG Þ:
ð1Þ
r¼0
Also if mAN½r;n then mABðIG Þ iff mN½r DBðIG Þ: Hence r n o i r i rþ1 Ai;r ðIG Þ ¼ mAN½iþ1;n : y
mABðI Þ; y
meBðI Þ : G G r i iþ1
ð2Þ
In [21], Sturmfels, Trung and Vogel introduced a decomposition of the standard monomials of an arbitrary monomial ideal M; called its standard pair decomposition, in order to study the multiplicities of associated primes and degrees of M: We study these quantities for the monomial ideals IG ; IDðGÞ ; and GinðIÞ: In Section 6 we show their relationship to the symmetric iterated Betti numbers of G and I; respectively. These results rely on the fact that the sets of monomials Ai ðIÞ defined above index the standard pairs of GinðIÞ: For a monomial mANZ ; let suppðmÞ :¼ fi: yi j mgCZ be called the support of m: Thus supp : f0; 1gs -2s is a bijection. Definition 3.2 (Sturmfels et al. [21]). Let M ¼ /M-N½n SDS be a monomial ideal. A standard monomial of M is an element of N½n M: An admissible pair of M is a subset mNs DN½n M with mAN½n s or equivalently if we take Zs to be Laurent monomials then an admissible pair is a subset mZs -N½n with mZs -M ¼ |: A standard pair of M is a(n inclusion) maximal admissible pair. Lemma 3.3. If IDS is an ideal then the standard pairs of GinðIÞ are faN½i : aAAi ðIÞg: (Here ½0 ¼ |:) Proof. We first argue that all standard pairs of GinðIÞ are of the form aN½i for some iA½0; n: Suppose mNs is an admissible pair of GinðIÞ with k ¼ maxðsÞ: Since BðIÞ is shifted (B3) and mNfkg DBðIÞ we obtain that mN½k DBðIÞ and hence mNs DmZ½k -N½n DBðIÞ: If mNs is standard (maximal) this implies that mNs ¼ mZ½k -N½n and thus that s ¼ ½k: If mN½i DBðIÞ is standard then by the above argument mNfiþ1g D / BðIÞ so mAAi ðIÞ:
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Finally, if mAAi ðIÞ; then mNfig DBðIÞ and hence mN½i DBðIÞ is admissible. If 0 mN½i DBðIÞ is not standard then mN½i Cm0 N½i DBðIÞ so i0 4i and mNfiþ1g DBðIÞ contradicting the choice of m: & We remark that Lemma 3.3 and Eq. (1) imply that if mN½i is a standard pair of GinðIG Þ then the degree of m is at most i: The standard pairs of monomial ideals of moderate size can be computed using the computer algebra package Macaulay 2 [12] (see the chapter Monomial Ideals in [11] for details). This gives a method for computing the sets Ai ðIÞ for small examples— see Example 3.7 below. In the case when I ¼ IG there is another interpretation of the monomials in Ai ðIG Þ that relates them to the shifted complex DðGÞ; and is useful for the proofs of Theorems 4.1 and 6.6. Lemma 3.4. There is a bijection between the sets Ai;r ðIG Þ and fF AmaxðDðGÞÞ : jF j ¼ i; ½i rDF ; i r þ 1eF g given by F with Ai;r ðIG Þ{m/½i r,suppðFðmÞÞ ¼ suppðFðmyi r i ÞÞ: Proof. For r4i the assertion follows from the fact that both sets are empty (see (1)). To deal with the case rpi; note that byð2Þ
fFðmÞ : mAAi;r ðIG Þg ¼ (
mABðIG Þ-N½iþ1;n ; r
FðmÞ :
)
yi r
mABðIG Þ; yi rþ1
meBðIG Þ i iþ1 Def: 2:1
¼
GADðGÞ :
jGj ¼ r; G-½i r þ 1 ¼ |; G,½i rADðGÞ; G,½i r þ 1eDðGÞ
¼ fF \½i r: F AmaxðDðGÞÞ; jF j ¼ i; ½i rDF ; i r þ 1eF g;
ð3Þ
where in the last equality we used the fact that DðGÞ is shifted. Indeed, if G,½i r þ 1eDðGÞ; then G,½i r,f jgeDðGÞ for every j4i r þ 1; jeG; implying that G,½i r is a facet of DðGÞ for every element G of the set (3). & Corollary 3.5. The standard pairs of GinðIG Þ are in bijection with the facets of DðGÞ: mN½i is a standard pair of GinðIG Þ if and only if ½i r,suppðFðmÞÞ is a facet of DðGÞ of size i: In Section 4 we verify that bi;r ðIÞ ¼ jAi;r ðIÞj for all i; rA½0; n; which via Lemma 3.4 proves Theorem 4.1. (Hence, in particular, it follows from Theorem 4.1 that Ai ðIG Þ ¼ | for all i4dimðGÞ þ 1:) Thus the sets Ai;r ðIG Þ and their cardinalities bi;r ðGÞ carry important information about G; and we record them in the following triangles.
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Definition 3.6. The b-triangle and monomial b-triangle of a simplicial complex G are the lower triangular matrices whose respective ði; rÞth entries are bi;r ðGÞ and Ai;r ðIG Þ for 0piprpdimðGÞ þ 1: Example 3.7. Let G be the simplicial complex whose facets are maxðGÞ ¼ ff1; 2; 4g; f1; 2; 6g; f1; 3; 4g; f1; 3; 7g; f1; 5; 6g; f1; 5; 7g; f2; 3; 5g; f2; 3; 7g; f2; 4; 5g; f2; 6; 7g; f3; 4; 6g; f3; 5; 6g; f4; 5; 7g; f4; 6; 7gg: Then it can be checked from Fig. 1 that the Stanley–Reisner ideal of G in the ring S :¼ k½a; b; c; d; e; f ; g is IG ¼ /efg; cfg; afg; ceg; beg; cdg; bdg; adg; abg; def ; bef ; bdf ; adf ; bcf ; acf ; cde; ade; ace; abe; bcd; abcS: Using Macaulay 2 one can compute that under the reverse lexicographic order g with ggf g?gbga; GinðIG Þ ¼ /gf 2 ; f 3 ; f 2 e; g2 f ; gfe; fe2 ; gfd; f 2 d; fed; g2 e; ge2 ; e3 ; ged; e2 d; fd 2 ; g3 ; g2 d; gd 2 ; ed 2 ; g2 c; gfc; d 4 S: Applying the map F to the generators of GinðIG Þ we get IDðGÞ ¼ /gea; gfa; ecb; fcb; gcb; edb; fdb; gdb; feb; geb; gfb; edc; fdc; gdc; fec; gec; gfc; fed; ged; gfd; gfe; dcbaS; which shows that the shifted complex DðGÞ has facets: maxðDðGÞÞ ¼ ff1; 2; 3g; f1; 2; 4g; f1; 2; 5g; f1; 2; 6g; f1; 2; 7g; f1; 3; 4g; f1; 3; 5g; f1; 3; 6g; f1; 3; 7g; f1; 4; 5g; f1; 4; 6g; f1; 4; 7g; f1; 5; 6g; f2; 3; 4g; f5; 7g; f6; 7gg:
3
5
7
2
7
4
6
1
3
4
3
5
7
Fig. 1. The simplicial complex G in Example 3.7. Here parallel boundary regions are identified.
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The 1-skeleton of DðGÞ is (like that of G) the complete graph on its 7 vertices. The triangles are obtained by coning 1 with all edges involving the vertices 2; 3; 4; 5; 6 and 7 except for f5; 7g and f6; 7g and adding the triangle f2; 3; 4g: Thus all the triangles and the edges f5; 7g and f6; 7g are facets. We compute the b-triangle and the monomial b-triangle of G by first computing the standard pairs of GinðIG Þ using Macaulay 2. b-Triangle of G 0
1
2
0
0
1
0
0
2
0
0
2
3
1
4
8
Monomial b-triangle of G 3
1
0
1
2
3
0
|
1
|
|
2
|
|
fg2 ; gf g
3
f1g
fg; f ; e; dg
fge; gd; f 2 ; fe; fd; e2 ; ed; d 2 g
fd 3 g
The standard pairs of IG ; GinðIG Þ and IDðGÞ are shown in the following table. Columns 2, 3, and 4 illustrate Lemma 3.4 and Corollary 3.5.
StdPairs(IG ) form: Ns
StdPairs(GinðIG Þ) form: mN½i
FðmÞ -suppðFðmÞ)
StdPairs(IDðGÞ ) form: Ns
Nf4;6;7g Nf2;6;7g Nf4;5;7g Nf1;5;7g Nf2;3;7g Nf1;3;7g Nf3;5;6g Nf1;5;6g Nf3;4;6g Nf1;2;6g Nf2;4;5g Nf2;3;5g Nf1;3;4g Nf1;2;4g
Nf1;2;3g gNf1;2;3g geNf1;2;3g gdNf1;2;3g f Nf1;2;3g f 2 Nf1;2;3g feNf1;2;3g fdNf1;2;3g eNf1;2;3g e2 Nf1;2;3g edNf1;2;3g dNf1;2;3g d 2 Nf1;2;3g d 3 Nf1;2;3g
1-| g-f7g gd-7; 4 gc-f7; 3g f -6 fe-f6; 5g fd-f6; 4g fc-6; 3 e-f5g ed-f5; 4g ec-f5; 3g d-f4g dc-f4; 3g dcb-f4; 3; 2g
Nf1;2;3 Nf1;2;7g Nf1;4;7g Nf1;3;7g Nf1;2;6g Nf1;5;6g Nf1;4;6g Nf1;3;6g Nf1;2;5g Nf1;4;5g Nf1;3;5g Nf1;2;4g Nf1;3;4g Nf2;3;4g
g2 Nf1;2g gf Nf1;2g
gf -f7; 6g ge-f7; 5g
Nf6;7g Nf5;7g
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4. Local cohomology In this section we prove Theorem 4.1, which provides a simple combinatorial formula for the symmetric iterated Betti numbers of a simplicial complex. Theorem 4.1. For a simplicial complex G bi;r ðGÞ ¼
jfF AmaxðDðGÞÞ : jF j ¼ i; ½i rDF ; i r þ 1eF gj
if rpi;
0
otherwise:
The symmetric iterated Betti numbers bi;r ðGÞ were defined as the dimensions of the vector spaces H 0 ðMi ðIG ÞÞr ; where for a homogeneous ideal I in S and a generic linear map uAUðIÞ; M0 ðIÞ ¼ S=uI and Mi ðIÞ ¼ Mi 1 ðIÞ=ðyi Mi 1 ðIÞ þ H 0 ðMi 1 ðIÞÞÞ for 1pipn: Thus at step i we ‘‘peel off’’ the ith variable. This is similar to the ‘‘deconing’’ of the shifted complex De ðGÞ used in the definition of the exterior iterated Betti numbers of G by Duval and Rose [9]. In view of Lemma 3.4, it suffices to show that jAi;r ðIÞj ¼ dim H 0 ðMi ðIÞÞr for all i; rX0 in order to prove Theorem 4.1. We establish this in Lemma 4.3 below. However, we first digress briefly to derive and illustrate certain facts needed in the proof of Lemma 4.3. Recall that if M is an S-module, N is a submodule and I is an ideal in S then ðN : I N ÞM ¼ fmAMj for some rAN; I r mDNg and if I ¼ /f S it is typical to write ðN : /f SN Þ ¼ ðN : f N Þ: For an S-module M; the 0th local cohomology of M with respect to the irrelevant ideal m ¼ Sþ ¼ /y1 ; y; yn S is defined as H 0 ðMÞ ¼ fmAM : mk m ¼ 0 for some kg ¼ ð0 : mN ÞM : In particular H 0 ðMÞ is graded when M is graded. Hence the equivalent definition of Mi ðIÞ is Mi ðIÞ ¼ S=Ji ðIÞ; where J0 ðIÞ :¼ uI
and
Ji ðIÞ :¼ yi S þ ðJi 1 ðIÞ : mN Þ:
Fix an i such that 1pion: Then for all 1pkoi; ðJi 1 ðIÞ : yN k Þ ¼ S since yk AJi 1 ðIÞ: For a j such that 1piojpn; consider the family of automorphisms ga AGLðS1 Þ such that ga ðyi Þ ¼ ayi þ ð1 aÞyj ; ga ðyj Þ ¼ ð1 aÞyi þ ayj and ga ðyr Þ ¼ yr otherwise, parameterized by all aAk: By induction ga ðJi 1 ðIÞÞ ¼ Ji 1 ðIÞ: N Thus ga ðJi 1 ðIÞ : yN i Þ ¼ ðJi 1 ðIÞ : ðayi þ ð1 aÞyj Þ Þ; and so the two colon ideals are isomorphic. If IDS is a fixed ideal and f AS varies then the ideal ðI : f N Þ depends only on which associated primes of I contain f : Thus if for a family of f ’s over k all the colon ideals ðI : f N Þ are isomorphic they must in fact be equal. Hence 8ioj;
N N we have ðJi 1 ðIÞ : yN i Þ ¼ ðJi 1 ðIÞ : yj Þ ¼ ðJi 1 ðIÞ : m Þ:
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Fix I and write B ¼ BðIÞ; Ai ¼ Ai ðIÞ; Ji ¼ Ji ðIÞ and Mi ¼ Mi ðIÞ: For the proof of Lemma 4.3 we introduce the sets ! [ Ci ¼ B Aj -N½iþ1;n : jpi
Lemma 4.2. The sets Ai and Ci have the following properties: (1) (2) (3) (4)
Ci 1 is the disjoint union Ci 1 ¼ Ai ,C ’ i ,y ’ i Ci 1 : Ci and Ci ,Ai are shifted order ideals in N½iþ1;n : kCi -Ji ¼ f0g: BDkCi þ ðJi : yN iþ1 Þ:
Proof. (1) If mN½ j is a standard pair of GinðIÞ with jpi; then no monomial in mN½ j lies in Ci : Thus Ci is the set of all monomials in B-N½iþ1;n that lie in standard pairs of the form N½ j where j4i: In other words, Ci ¼ fmAB-N½iþ1;n : mNfiþ1g DBg: This implies that yiþ1 Ci DCi and Ci 1 ¼ fmAB-N½i;n : mNfig DBg ’ ¼ fmAB-N½iþ1;n : mNfiþ1g DBg, / Bg,y ’ i Ci 1 fmAB-N½iþ1;n : mNfig DB; mNfiþ1g D ¼ Ci ,A ’ i ,y ’ i Ci 1 (2) The definitions of Ci and Ai and the fact that B is shifted imply via an induction that Ci and Ci ,Ai are shifted order ideals in N½iþ1;n : (3) We establish this fact by induction on i: Note that kC0 -J0 ¼ f0g since all elements of C0 DB are standard monomials of GinðIÞ ¼ P Ing ðJ0 Þ: Assume kCi 1 -Ji 1 ¼ f0g; but there exists 0af AkCi -Ji : Since f ¼ bm mAkCi ; each mACi CN½iþ1;n ; and so me/y1 ; y; yi S: Therefore, f AJi ¼ yi S þ ðJi 1 : yN i Þ implies k Þ—i.e., fy AJ for some k: Since kC -J ¼ f0g; fyki ¼ that f AðJi 1 : yN i 1 i 1 i 1 i i P bm m yki ekCi 1 and we infer that at least one of the monomials m is not in Ci 1 (since yi Ci 1 DCi 1 ). However, Ci DCi 1 ; and hence this m is also not in Ci ; which is a contradiction. Thus kCi -Ji ¼ f0g: S (4) Since B-N½iþ1;n ¼ ðCi , jpi Ai Þ-N½iþ1;n it suffices to show that if jpi then S Aj DkCi þ ðJi : yN iþ1 Þ: For every aA jpi Aj ; there exists some t40 such that aytiþ1 AGinðIÞ ¼ Ing ðJ0 ÞDIng ðJi Þ (since J0 DJi ). Therefore there exists f AJi such P that f ¼ aytiþ1 am m where aytiþ1 is the leading term of f with respect to g; am Ak; and the monomials m are standard monomials of Ing ðJi Þ: Hence mAB-N½iþ1;n (since all standard monomials of Ing ðJi Þ are in B and y1 ; y; yi AIng ðJi Þ). Further, since aytiþ1 gm for each m and since P aytiþ1 ; mAN½iþ1;n ; it follows that ytiþ1 j m: Thus a ¼ f =ytiþ1 þ am ðm=ytiþ1 Þ: Finally, since each m=ytiþ1 !a in the above sum, we are done by induction. &
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Lemma 4.3. jAi;r ðIÞj ¼ dim H 0 ðMi ðIÞÞr for all i; rX0: Proof. Note that dim H 0 ðMi Þr ¼ dimðfmAðS=Ji Þr : mAðJi : mN ÞgÞ ¼ dimðfmAðS=Ji Þr : mAðJi : yN iþ1 ÞgÞ ¼ dimðJi : yN iþ1 Þr dimðJi Þr : Hence, to prove the lemma it suffices to show that kAi "Ji ¼ ðJi : yN iþ1 Þ: We do this by establishing the following set of equalities: ½1
½2
S ¼i kCi "kAi "Ji ¼i kCi "ðJi : yN iþ1 Þ
for all iX0:
For i ¼ 0 the first equality, ½10 ; follows from the facts that kB ¼ kA0 "kC0 and kB"J0 ¼ S: We now show that ½1i implies ½2i and ½1iþ1 : Assume that S ¼ kCi "kAi "Ji : Since Ai DkCi þ ðJi : yN iþ1 Þ (Lemma 4.2(4)), we obtain that S ¼ kCi þ ðJi : yN iþ1 Þ:
ð4Þ
In order to verify ½2i we must show that kCi -ðJi : yN iþ1 Þ ¼ f0g: This follows from the facts that yiþ1 Ci DCi and that kCi -Ji ¼ f0g (Lemma 4.2(3)). Thus ½2
S ¼i kCi "ðJi : yN iþ1 Þ Lemma 4:2 ð1Þ
¼
kCiþ1 "kAiþ1 "kyiþ1 Ci "ðJi : yN iþ1 Þ ¼ kCiþ1 "kAiþ1 "Jiþ1 ;
where in the last step we used Eq. (4) along with the definition of Jiþ1 : Equality ½1iþ1 follows. & We close this section with several remarks. Remark 4.4. Lemma 4.3 was verified in the special case of Stanley–Reisner ideals of Buchsbaum complexes in [19]. Remark 4.5. Recall that by property (B1), BðIÞ is a basis of S=uI as well as of S=GinðIÞ: Thus the proof of Lemma 4.3 implies also that jAi;r ðIÞj ¼ dim H 0 ðMi ðGinðIÞÞr
for all i; rX0;
and we recover the following fact (originally due to Bayer et al. [4, Corollary 1.7]). Corollary 4.6. Modules H 0 ðMi ðIÞÞ and H 0 ðMi ðGinðIÞÞÞ have the same Hilbert function ( for i ¼ 0; 1; y; n). In other words, the symmetric iterated Betti numbers of I are identical to those of GinðIÞ:
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Remark 4.7. Similar to the proof of Theorem 4.1, one can show that dim H 0 ðS=/uIG ; y1 ; y; yi SÞr ¼ jfGADðGÞ : jGj ¼ r; ½i r þ 1-G ¼ |; ½i r þ 1,GeDðGÞgj: Thus for the shifted complex G; dimensions of the modules H 0 ðS=/uIG ; y1 ; y; yi SÞr coincide with Kalai’s (exterior) iterated Betti numbers of G (see [17, Section 3]). For that reason we refer to the numbers b%i;r ðGÞ :¼ dim H 0 ðS=/uIG ; y1 ; y; yi SÞr as Kalai’s symmetric iterated Betti numbers. Remark 4.8. Another fact worth mentioning is that bi;i ðGÞ ¼ b%i;i ðGÞ are just reduced (topological) Betti numbers of G; that is, bi;i ðGÞ ¼ b%i;i ðGÞ ¼ bi 1 ðGÞ 80pipdimðGÞ þ 1; where bi 1 ðGÞ ¼ dim H˜ i 1 ðG; kÞ: This result is a consequence of Theorem 4.1 together with the fact [6] that for a shifted complex K bi 1 ðKÞ ¼ jfF AmaxðKÞ : jF j ¼ i; 1eF gj; and the fact that symmetric shifting preserves topological Betti numbers [13]. Remark 4.9. Finally, we note that if G is a Buchsbaum complex (i.e., a pure simplicial complex all of whose vertices have Cohen–Macaulay links), then bi;r ðGÞ ¼ i 1 r 1 br 1 ðGÞ for every 0prpipdimðGÞ; where br 1 ðGÞ are reduced (topological) Betti numbers of G: This follows from Lemma 4.3 and [19, Lemma 4.1].
5. Extremal Betti numbers This section is devoted to the proof of Theorem 5.3, which relates the graded algebraic Betti numbers of IG to the symmetric iterated Betti numbers of G: Every homogeneous ideal ICS admits a graded free S-resolution of the form -?
M j
Sð jÞbi; j -?-
M j
Sð jÞb1;j -
M
Sð jÞb0;j -I-0;
j
where Sð jÞ denotes S with grading shifted by j: Moreover, there exists a unique (up to isomorphism) resolution in which all the exponents bi; j are simultaneously minimized, called the minimal graded free S-resolution of I: The numbers bi; j appearing in this minimal free resolution of I are called the graded Betti numbers of I: A Betti number of I; bi;iþj ðIÞ; is extremal if bi;iþj ðIÞa0; but bi0 ;i0 þj 0 ðIÞ ¼ 0 for all i0 Xi; j 0 Xj; ði0 ; j 0 Þaði; jÞ: This terminology comes from the Betti diagram of I output by the program Macaulay 2 in which the Betti numbers are arranged in a rectangular array whose columns are indexed by i and rows by j and the ði; jÞth entry is the Betti
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number bi;iþj : Thus bi;iþj is an extremal Betti number of I if it lies in a south-east corner of the Macaulay 2 Betti diagram of I: Let G be a simplicial complex on the vertex set ½n: The Alexander dual of G is the simplicial complex G ¼ fF D½n : ½n\F eGg: The next two results (both due to Bayer et al. [4], see [1] also for the second theorem) provide connections between the extremal Betti numbers of the Stanley– Reisner ideals of G and G ; and the shifted complex DðGÞ: Theorem 5.1. Let G be a simplicial complex and G be its Alexander dual. The Stanley–Reisner ideals IG and IG have the same extremal Betti numbers. More precisely, bi;iþj ðIG Þ is extremal if and only if bj 1;iþj ðIG Þ is extremal. Also, in such a case bi;iþj ðIG Þ ¼ bj 1;iþj ðIG Þ: Theorem 5.2. Extremal Betti numbers are preserved by algebraic shifting: for a simplicial complex G; bi;iþj ðIG Þ is extremal if and only if bi;iþj ðIDðGÞ Þ is extremal. Moreover, in such a case bi;iþj ðIG Þ ¼ bi;iþj ðIDðGÞ Þ: We are now in a position to prove the main theorem of this section. Theorem 5.3. The extremal Betti numbers of IG are contained among the symmetric iterated Betti numbers of G: They are precisely the extremal entries in the b-triangle of G: bj 1;iþj ðIG Þ is an extremal Betti number of IG if and only if bn j0 ;i0 ðGÞ ¼ 0
8ði0 ; j 0 Þaði; jÞ; i0 Xi; j 0 Xj; and bn j;i ðGÞa0:
Moreover, in this case, bj 1;iþj ðIG Þ ¼ bn j;i ðGÞ: Example 3.7 (continued). The minimal free resolution and Betti diagram of IG (computed by Macaulay 2) are given below. Note that the entries in the southeast corners of the Betti diagram of IG (the extremal Betti numbers of IG ) are precisely the entries in the north-east corners of the b-triangle of G from Section 3. 0-S 2 -S15 -S 42 -S 49 -S21 -S-0 total : 1
21
0: 1:
1 :
: :
2: 3:
: :
21 :
49 42 15 : :
: :
: :
49 42 14 : : 1
2 : : 2 :
The proof of Theorem 5.3 relies on the following lemma, which is a consequence of [14, Theorem 2.1(b)] (see also [15, Proposition 12]) and [8, Corollary 6.2]. For completeness we provide a different self-contained proof.
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Lemma 5.4. The symmetric iterated Betti numbers of G are related to the graded Betti numbers of the Stanley–Reisner ideal IDðG Þ as follows: X n r j bi;iþj ðIDðG Þ Þ ¼ bn j;n r j ðGÞ: i r Proof. Let G be a simplicial complex and let G be its Alexander dual. Recall that the Stanley–Reisner ideal IDðG Þ CS is a squarefree monomial ideal whose minimal generators correspond to minimal non-faces of DðG Þ: Let G be the set of minimal ½n generators of IDðG Þ ; let Gj ¼ G-Nj ; and let minðgÞ ¼ minfi: yi jgg for a monomial gAG: Since DðG Þ is a shifted complex, it follows that the ideal IDðG Þ is squarefree strongly stable, which means that for a monomial mAIDðG Þ ; if yi jm; iojpn; and yj is not a divisor of m; then myj =yi AIDðG Þ as well. Hence the graded Betti numbers of IDðG Þ are given by the following formula [13, Corollary 3.4] (which is the analog of the Eliahou–Kervaire formula for strongly stable ideals): X n minðgÞ þ 1 j bi;iþj ðIDðG Þ Þ ¼ i gAGj X n r j ð5Þ ¼ jfgAGj : minðgÞ ¼ r þ 1gj: i r It is well known and is easy to prove that DðG Þ ¼ DðGÞ : Thus, gAGj if and only if s ¼ suppðgÞ is a minimal non-face of DðG Þ of size j; which happens if and only if ½n\s is a facet of DðGÞ of size n j: Moreover, r þ 1 ¼ minfi : iAsg if and only if ½rD½n\s; but r þ 1e½n\s: Hence jfgAGj : minðgÞ ¼ r þ 1gj ¼ jfF AmaxðDðGÞÞ : jF j ¼ n j; ½rDF ; but r þ 1eF gj Theorem 4:1
¼
bn j;n r j ðGÞ:
Substituting (6) in (5) gives the result.
ð6Þ &
Proof of Theorem 5.3. The theorem is an easy consequence of Lemma 5.4. Indeed, since n r j is positive for rpn i j and is zero otherwise, it follows from the i lemma that bi0 ;i0 þj0 ðIDðG Þ Þ ¼ 0 iff bn j0 ;n j0 r ðGÞ ¼ 0 for all rpn i0 j 0 : Thus, bi;iþj ðIDðG Þ Þa0 is extremal 3 bi0 ;i0 þj 0 ðIDðG Þ Þ ¼ 0
for all i0 Xi; j 0 Xj; ði; jÞaði0 ; j 0 Þ
3 bn j 0 ;i0 ðGÞ ¼ 0 for all i0 Xi; j 0 Xj; ði; jÞaði0 ; j 0 Þ:
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Moreover, if this is the case, then all except the first summand in X n r j i bn j;i ðGÞ þ bi;iþj ðIDðG Þ Þ ¼ bn j;n r j ðGÞ i i ron i j vanish, implying that bi;iþj ðIDðG Þ Þ ¼ bn j;i ðGÞ: The result then follows from Theorems 5.1 and 5.2. & We remark that one can use [1, Corollary 1.2] and certain properties of sets Ai ðIÞ to provide a different proof of the following more general result. We omit the details. Theorem 5.5. Let I be a homogeneous ideal. The extremal Betti numbers of I form a subset of the symmetric iterated Betti numbers of I: More precisely, bj 1;iþj ðIÞ is an extremal Betti number of I if and only if bn j0 ;i0 ðIÞ ¼ 0
8ði0 ; j 0 Þaði; jÞ; i0 Xi; j 0 Xj;
and
bn j;i ðIÞa0:
Moreover, in this case, bj 1;iþj ðIÞ ¼ bn j;i ðIÞ:
6. Associated primes and standard pairs The associated primes of a homogeneous ideal ICS with a primary decompffiffiffiffiffi position I ¼ Q1 -Q2 -?-Qt are the prime ideals Pi :¼ Qi ; i ¼ 1; y; t; pffiffiffiffiffi where Qi denotes the radical of Qi : The set of associated primes of I; customarily denoted as AssðIÞ; is independent of the primary decomposition of I: The minimal elements of AssðIÞ with respect to inclusion are called the minimal primes of I; and the non-minimal ones are called the embedded primes of I: We denote the set of minimal primes of I as MinðIÞ: Recall that the irreducible (isolated) components of V ðIÞ; the variety of I in kn ; are the varieties V ðPÞ for PAMinðIÞ: Let Zi :¼ V ðPi Þ be the variety of Pi in kn : The finite invariant degðZi Þ; called the degree of Zi ; is the cardinality of Zi -L for almost all linear subspaces L of dimension equal to the codimension of Zi : Definition 6.1 (Bayer and Mumford [5], Sturmfels et al. [21]). (1) If P is a homogeneous prime ideal in S then the multiplicity of P (with respect to I), denoted as multI ðPÞ is the length of the largest ideal of finite length in the ring SP =ISP : P (2) The degree of I; degðIÞ :¼ fdimðZi Þ¼dimðIÞg multI ðPi ÞdegðZi Þ: P (3) The geometric degree of I; geomdegðIÞ :¼ fPi AMinðIÞg multI ðPi ÞdegðZi Þ: P (4) The arithmetic degree of I; arithdegðIÞ :¼ fPi AAssðIÞg multI ðPi ÞdegðZi Þ: The invariant multI ðPÞ40 if and only if PAAssðIÞ: Our main goal in this section is to prove Theorem 6.6. We first specialize Definition 6.1 to monomial ideals. If M is a monomial ideal, then every associated prime of M is
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of the form Ps :¼ /yj : jesS for some set sD½n: Hence V ðPs Þ is the jsj-dimensional linear subspace spanned by fej : jAsg and degðV ðPs ÞÞ ¼ 1: The three degrees of M from Definition 6.1 are therefore appropriate sums of multiplicities of ideals in AssðMÞ with respect to M: For a monomial ideal M the multiplicities of associated primes as well as all the degrees referred to in Definition 6.1 can be read off from the standard pairs of M (see Definition 3.2) as shown in the following lemma. The statements in this lemma are either stated or can be derived easily from the results in [21]. Lemma 6.2. Let M be a monomial ideal. Then, the set of standard pairs of M is well defined, Ns is a standard pair of M if and only if Ps AAssðMÞ; Ns is a standard pair of M if and only if Ps AMinðMÞ; the dimension of M is the maximal size of a set s such that Ns is a standard pair of M; (5) if Ps AAssðMÞ; then multM ðPs Þ is the number of standard pairs of M of the form Ns and (6) (a) degðMÞ is the number of standard pairs Ns of M such that jsj ¼ dimðMÞ; (b) geomdegðMÞ is the number of standard pairs Ns of M such that Ns is a standard pair of M and (c) arithdegðMÞ is the total number of standard pairs of M:
(1) (2) (3) (4)
Lemma 3.3 showed that mNs is a standard pair of GinðIÞ if and only if s ¼ ½i and mAAi ðIÞ for some 0pipn: Combining this fact with Lemma 6.2 we obtain the following. Corollary 6.3 (See also Eisenbud [10, Corollary 15.25]). (i) P½d ; d ¼ dim ðIÞ; is the unique minimal prime of GinðIÞ (if I ¼ IG then d ¼ dim G þ 1), and (ii) all embedded primes of GinðIÞ are of the form P½k for some kod: Thus the submonoids in the standard pairs of GinðIÞ are initial intervals of ½n while the cosets can be complicated. On the other hand, for the square free monomial ideals IG and IDðGÞ ; the cosets of the standard pairs are trivial and the submonoids determine the ideals (cf. Example 3.7). T Corollary 6.4. If G is a simplicial complex then IG ¼ sAmaxðGÞ Ps is the irredundant prime decomposition of IG : In particular, IG has no embedded primes and its standard pairs are fNs : sAmaxðGÞg: By Corollary 3.5, mN½i is a standard pair of GinðIG Þ if and only if ½i r,suppðFðmÞÞ is a facet of DðGÞ of size i: Combining this fact with Corollary 6.4 we get the following bijection as well.
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Corollary 6.5. There is a bijection between the standard pairs of GinðIG Þ and those of IDðGÞ given by: mN½i is a standard pair of GinðIG Þ with degðmÞ ¼ r if and only if N½i r,suppðFðmÞÞ is a standard pair of IDðGÞ : The following theorem is a corollary of the results in Sections 3 and 4, and those stated thus far in this section. Theorem 6.6. The iterated Betti numbers of a homogeneous ideal I are related to the ideal GinðIÞ: Those of an ideal IG are related to the ideals GinðIG Þ; IDðGÞ ; and the shifted complex DðGÞ: The relationships are as follows. (1) The multiplicity of P½i with respect to GinðIÞ is X bi;r ðIÞ: multGinðIÞ ðP½i Þ ¼ r
If I ¼ IG then multGinðIG Þ ðP½i Þ ¼
X
bi;r ðGÞ ¼ jfF AmaxðDðGÞÞ : jF j ¼ igj:
r
(2) The degree, geometric degree, and arithmetic degree of GinðIG Þ and IDðGÞ have the following interpretations: P ðiÞ degðGinðIG ÞÞ ¼ geomdegðGinðIG ÞÞ ¼ r bd;r ðIG Þ ði0 Þ ¼ degðIDðGÞ Þ ¼ jfF AmaxðDðGÞÞ : jF j ¼ dgj; P ðiiÞ arithdegðGinðIG ÞÞ ¼ i;r bi;r ðIG Þ ðii0 Þ ¼ arithdegðIDðGÞ Þ ¼ jmaxðDðGÞÞj: Eqs. (i) and (ii) also hold for arbitrary homogeneous ideals I in S: Proof. (1) By Lemmas 6.2(5) multGinðIG Þ ðP½i Þ ¼ jfstandard pairs of GinðIÞ of the form N½i gj ¼ jAi ðIÞj ðby Lemma 3:3Þ X bi;r ðIÞ ðby Theorem 4:3Þ: ¼ r
In particular, P½i is an associated prime of GinðIÞ if and only if bi;r ðIÞ40 for some r: For a simplicial complex G on ½n; by Lemma 3.4, X multGinðIG Þ ðP½i Þ ¼ jAi ðIG Þj ¼ bi;r ðGÞ ¼ jfF AmaxðDðGÞÞ : jF j ¼ igj: r
In particular, P½i is an associated prime of GinðIG Þ if and only if DðGÞ has a facet of size i: (2) The same lemmas along with Definition 6.1 yield these results. & We now establish certain further facts about AssðGinðIG ÞÞ:
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Definition 6.7. For any ideal I in a polynomial ring, its poset of associated primes AssðIÞ has the chain property if whenever PAAssðIÞ is an embedded prime, then there exists QAAssðIÞ such that P*Q and dimðQÞ ¼ dimðPÞ þ 1: Corollary 6.8. The poset AssðGinðIG ÞÞ possesses the chain property if and only if maxðDðGÞÞ has the property that whenever DðGÞ has a facet of size kpdimðGÞ then it also has a facet of size k þ 1: Proof. By Corollary 6.3, the poset AssðGinðIG ÞÞ has the chain property if whenever P½k AAssðGinðIG ÞÞ for some kpdim G then P½kþ1 AAssðGinðIG ÞÞ: By Theorem 6.6(1), this is equivalent to the condition that whenever DðGÞ has a facet of size kpdim G then it also has a facet of size k þ 1: & Corollary 6.9. If G is a Buchsbaum complex (cf. Remark 4.9) then AssðGinðIG ÞÞ has the chain property. Proof. It was shown in [19] that if G is a ðd 1Þ-dimensional Buchsbaum complex i 1 br 1 for iod; where br 1 is the reduced (topological) Betti then bi;r ðGÞ ¼ r 1 P P i 1 number of G: Hence for iod; multGinðIG Þ ðP½i Þ ¼ r bi;r ¼ r r 1 br 1 : Therefore if some bk a0 then multGinðIG Þ ðP½i Þ40 for all iXk; which implies that AssðGinðIG ÞÞ has the chain property. &
7. Iterated Betti numbers: exterior versus symmetric We close the paper with several remarks and conjectures on connections between symmetric iterated Betti numbers and exterior iterated Betti numbers of a simplicial complex. The superscript e is used to denote exterior shifting. We start of exterior algebraic shifting extracted from [13]. V with a brief description V Let E ¼ ðk½y1 ; y; yn 1 Þ ¼ S1 be the exterior algebra over the n-dimensional vector space S1 : A monomial in E is an expression of the form m ¼ yi1 4yi2 4?4yik ; where 1pi1 oi2 o?oik pn; the set fi1 ; i2 ; y; ik g is called the support of m; and is denoted by suppðmÞ: The exterior Stanley–Reisner ideal of a simplicial complex G on ½n is JG :¼ /mAE : m is a monomial; suppðmÞeGS: Definition 7.1. The exterior algebraic shifting of G; De ðGÞ; is the simplicial complex defined by JDe ðGÞ :¼ GinðJG Þ; where GinðJG Þ is the generic initial ideal of JG with respect to the reverse lexicographic order with yn gyn 1 g?gy1 : The exterior iterated Betti numbers of a simplicial complex G were introduced by Duval and Rose [9]. They have the following combinatorial description (up to a slight change in the indexing), which we adopt as their definition.
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Definition 7.2. The exterior iterated Betti numbers of a simplicial complex G are bei;r ðGÞ ¼ jfF A maxðDe ðGÞÞ: jF j ¼ i; ½i rDF ; i r þ 1eF gj: Since De ðDðGÞÞ ¼ DðGÞ and DðDe ðGÞÞ ¼ De ðGÞ; the above definition and Theorem 4.1 imply that bi;r ðGÞ ¼ bei;r ðDðGÞÞ;
bei;r ðGÞ ¼ bi;r ðDe ðGÞÞ:
Hence we infer the following corollary from Lemma 5.4. Corollary 7.3. For a simplicial complex G X n r j e bi;iþj ðID ðG Þ Þ ¼ ben j;n r j ðGÞ: i r Since exterior shifting preserves extremal Betti numbers [1], the same proof as in Theorem 5.3 yields Theorem 7.4. bj 1;iþj ðIG Þ is an extremal Betti number of IG if and only if ben j0 ;i0 ðGÞ ¼ 0
8ði0 ; j 0 Þaði; jÞ; i0 Xi; j 0 Xj; and ben j;i ðGÞa0:
Moreover, if this is the case, then bj 1;iþj ðIG Þ ¼ bj 1;iþj ðIDe ðGÞ Þ ¼ bj 1;iþj ðIDðGÞ Þ ¼ ben j;i ðGÞ ¼ bn j;i ðGÞ: We remark that there are no known connections between general (non-extremal) graded Betti numbers of IDðGÞ and IDe ðGÞ : However it is conjectured (see [13, Conjecture 8.9]) that the following holds. Conjecture 7.5. For every simplicial complex G; bi; j ðIDðGÞ Þpbi; j ðIDe ðGÞ Þ: In analogy, we propose the following. Conjecture 7.6. For every simplicial complex G; bi; j ðGÞpbei; j ðGÞ for all ipdimðGÞ: Note that since all the coefficients in the expression of graded Betti numbers in terms of iterated Betti numbers (see Lemma 5.4 and Corollary 7.3) are non-negative, Conjecture 7.6 if true would imply Conjecture 7.5 except for the case j i ¼ n dimðGÞ 1 (that is, except for the last column of the Betti diagram of IDðGÞ and IDe ðGÞ output by Macaulay 2). Conjecture 7.5 was verified by Aramova et al. [2] in the case when G is the Alexander dual of a sequentially Cohen–Macaulay complex (a notion introduced by Stanley [20, Definition II.2.9]): they showed that in such a case bi; j ðIDðGÞ Þ ¼ bi; j ðIDe ðGÞ Þ ¼ bi; j ðIG Þ: We have the following related result.
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Proposition 7.7. Conjecture 7.6 holds for all sequentially Cohen–Macaulay complexes. More precisely, if G is sequentially Cohen–Macaulay, then bi;r ðGÞ ¼ bei;r ðGÞ ¼ hi;r ðGÞ; where ðhi;r ðGÞÞ0prpipdimðGÞþ1 is the h-triangle of G: The notion of f - and h-triangles was introduced by Bjo¨rner and Wachs [7]. We recall the definition. For a simplicial complex G set fi; j ðGÞ :¼ jfF AG : jF j ¼ j; dimðst F Þ ¼ i 1gj; where st F denotes the star of F in G: The h-triangle of G; ðhi; j ðGÞÞ0p jpipdimðGÞþ1 ; is defined by j X i s hi; j ðGÞ ¼ ð 1Þ j s fi;s ðGÞ: j s s¼0 Proof of Proposition 7.7. Duval [8, Theorem 5.1] showed that if G is a sequentially Cohen–Macaulay simplicial complex, then hi; j ðGÞ ¼ hi; j ðDe ðGÞÞ: In his proof he relied only on properties (P3) and (P4) of the operator De ; and the fact that G is Cohen–Macaulay if and only if De ðGÞ is pure. Since operator D (symmetric shifting) possesses all these properties as well, it follows that for a sequentially Cohen– Macaulay complex G; hi; j ðGÞ ¼ hi; j ðDe ðGÞÞ ¼ hi; j ðDðGÞÞ:
ð7Þ
Another result due to Duval [8, Corollary 6.2] is that for a shifted complex K; bei; j ðKÞ ¼ hi; j ðKÞ: Thus hi; j ðDe ðGÞÞ ¼ bei; j ðDe ðGÞÞ ¼ bei; j ðGÞ
ð8Þ
hi; j ðDðGÞÞ ¼ bei; j ðDðGÞÞ ¼ bi; j ðGÞ:
ð9Þ
and
Eqs. (7)–(9) imply the proposition.
&
We close the paper with one additional conjecture for the special case of Buchsbaum complexes. Conjecture 7.8. If G is a Buchsbaum complex, then bi; j ðGÞ ¼ bei; j ðGÞ; and hence bi; j ðIDðGÞ Þ ¼ bi; j ðIDe ðGÞ Þ:
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