Breaking up Simplicial Homology and Subadditivity of Syzygies
aa r X i v : . [ m a t h . A C ] F e b B reaking up S implicial H omology and S ubadditivity of S yzygies Sara Faridi ∗ Mayada Shahada † March 3, 2020
Abstract
We consider the following question: if a simplicial complex ∆ has d -homology, then doesthe corresponding d -cycle always induce cycles of smaller dimension that are not boundariesin ∆ ? We provide an answer to this question in a fixed dimension. We use the breaking ofhomology to show the subadditivity property for the maximal degrees of syzygies of monomialideals in a fixed homological degree. The motivation for this paper is investigating whether the subadditivity property holds for the max-imal degrees of syzygies of monomial ideals in polynomial rings. These syzygies are known tobe characterized as dimensions of homology modules of topological objects, and as a result, thesubadditivity question can be reduced to this general type of question: if the topological object O has i-homology, and i = a + b, does O have “large enough” sub-objects that have a-homologyand b-homology? The phrase “large enough” guarantees the degrees of the corresponding syzygiesbeing large enough to satisfy subadditivity and will be explained in detail in the next section.This approach was taken by the first author in [F1], where the topological objects were atomiclattices (lcm lattices of monomial ideals); see Question 2.1 and Question 2.2 below. In this paper,using Hochster’s formula (Equation (1)), we examine this problem from the point of view of theStanley-Reisner complex, and we can show that if our topological object O above is a simplicialcomplex, and i + O , then there are “large enough”induced subcomplexes of O that have a -homology and b -homology. As a result we show that sub-additivity holds in a fixed homological degree for all monomial ideals. ∗ Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada, [email protected] † Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada, [email protected] Setup
Throughout the paper, let S = k [ x , . . . , x n ] be a polynomial ring over a field k . If I is a graded idealof S with minimal free resolution0 → ⊕ j ∈ N S ( − j ) β p , j → ⊕ j ∈ N S ( − j ) β p − , j → · · · → ⊕ j ∈ N S ( − j ) β , j → S , then for each i and j , the rank β i , j ( S / I ) of the free S -modules appearing above are called the gradedBetti numbers of the S -module S / I .If we let t a = max { j | β a , j ( S / I ) , } , a question is whether the t a satisfy the subadditivity property : t a + b t a + t b ?The answer is known to be negative for a general homogeneous ideal [ACI], and unknown inthe case of monomial ideals. For the case of monomial ideals, there are special cases that areknown [HS, AN, FG, F1, BH, A].In the case of monomial ideals, Betti numbers can be interpreted as the homology of objects indiscrete topology: simplicial complexes, order complexes of lattices, etc.; see for example [P] fora survey of this approach. As a result, the subadditivity question can be viewed as a question ofbreaking up homology in these objects. This idea was explored in [F1] by the first author, where thesubadditivity problem was solved for facet ideals of simplicial forests using homology of lattices.By a method called polarization [Fr], one can reduce questions regarding Betti numbers ofmonomial ideals to the class of square-free monomial ideals.If u ⊂ [ n ] = { , . . . , n } , then we define m u = Π i ∈ u x i to be the square-free monomial with support u .For our purposes it is useful to consider a finer grading of the Betti numbers by indexing theBetti numbers with monomials of the polynomial ring S . A multigraded Betti number of S / I is ofthe form β i , m ( S / I ) where m is a monomial in S and β i , j ( S / I ) = X u ⊆ [ n ] and | u | = j β i , m u ( S / I ) . A simplicial complex Γ on a set W is a set of subsets of W with the property that if F ∈ Γ then forevery subset G ⊆ F we have G ∈ Γ . Every element of Γ is called a face , the maximal faces underinclusion are called facets , and a simplicial complex contained in Γ is called a subcomplex of Γ .The set of all v ∈ W such that { v } ∈ Γ is called the vertex set of Γ , and is denoted by V ( Γ ). The setof facets of Γ is denoted by Facets( Γ ). If Facets( Γ ) = { F , . . . , F q } , then we denote Γ by Γ = h F , . . . , F q i . A ⊂ V ( Γ ), then the induced subcomplex Γ A is defined as Γ A = { F ∈ Γ | F ⊆ A } . The
Alexander dual Γ ∨ of Γ , if we set F c = V ( Γ ) \ F , is defined as Γ ∨ = { F ⊆ V ( Γ ) | F c < Γ } = { V ( Γ ) \ F | F < Γ } . The link of a face F of Γ islk Γ ( F ) = { G ∈ Γ | F ∩ G = ∅ and F ∪ G ∈ Γ } . If I is a square-free monomial ideal in S , it corresponds uniquely to a simplicial complex N ( I ) = { u ⊂ [ n ] | m u < I } called the Stanley-Reisner complex of I . Conversely, if Γ is a simplicial complex whose verticesare labelled with x , . . . , x n , then one can associate to it its unique Stanley-Reisner ideal N ( Γ ) = { m u | u ⊂ [ n ] and u < Γ } . The uniqueness of the Stanley-Reisner correspondence implies that N ( Γ ) = I ⇐⇒ N ( I ) = Γ . A lattice is a partially ordered set where every two elements have a greatest lower bound called their meet and a lowest upper bound called their join . A bounded lattice has an upper and a lower bounddenoted by ˆ1 and ˆ0, respectively.If L is a lattice with r elements, then the order complex of L is the simplicial complex on r vertices, where the elements of each chain in L form a face.If I is a monomial ideal, then the lcm lattice of I , denoted by LCM( I ), is a bounded latticeordered by divisibility, whose elements are the generators of I and their least common multiples,and the meet of two elements is their least common multiple.Two elements of a lattice are called complements if their join is ˆ1 and their meet is ˆ0. If thelattice is LCM( I ), then it was observed in [F1] that two monomials in LCM( I ) are complements iftheir gcd is not in I and their lcm is the lcm of all the generators of I .Gasharov, Peeva and Welker [GPW, P] showed that multigraded Betti numbers of S / I can becalculated from the homology of (the order complex of) the lattice LCM( I ): if m is a monomial in L = LCM( I ), then β i , m ( S / I ) = dim k e H i − ((1 , m ) L ; k )where (1 , m ) L refers to the subcomplex of the order complex consisting of all nontrivial monomialsin L strictly dividing m .On the other hand, in a 1977 paper, Baclawski [B] showed that if L is a finite lattice whoseproper part has nonzero homology, then every element of L has a complement.The following question was raised in [F1] as a potential way to answer the subadditivity ques-tion. 3 uestion 2.1. If I is a square-free monomial ideal in variables x , . . . , x n , and β i , n ( S / I ) , a , b > i = a + b , are there complements m and m ′ in LCM( I ) with β a , m ( S / I ) , β b , m ′ ( S / I ) , n , in thiscase) [EF1, F1], a positive answer to Question 2.1 will establish the subadditivity property for allmonomial ideals, since t a + t b > deg( m ) + deg( m ′ ) > n = t i . Question 2.1 can be written more generally as a question about the homology of the lcm lattice,or in fact, any finite lattice.
Question 2.2. If L = LCM( I ) and e H i − ((1 , x · · · x n ) L ; k ) , a , b > i = a + b , are therecomplements m and m ′ in LCM( I ) with e H a − ((1 , m ) L ; k ) , e H b − ((1 , m ′ ) L ; k ) , Let I = ( m , . . . , m q ) be a square-free monomial ideal in the polynomial ring S = k [ x , . . . , x n ].Hochster’s formula (see for example [HH, Cor. 8.1.4 and Prop. 5.1.8]) states that if I = N ( Γ ) and m u a monomial, then β i , m u ( S / I ) = dim k ˜ H i − (lk Γ ∨ ( u c ) , k ) = dim k ˜ H | u |− i − ( Γ u , k ) (1)where u c = [ n ] \ u is the set complement of u . We would now like to reinterpret Question 2.1 in thelanguage of Hochster’s formula. To begin with, since we are dealing with square-free monomials,we can consider a monomial m u equivalent to the set u and use intersections for gcd, unions forlcm, and m cu for u c .Suppose β i , x ... x n ( S / I ) = dim k ˜ H i − (lk Γ ∨ ( ∅ ) , k ) = dim k ˜ H i − ( Γ ∨ , k ) , i = a + b where a , b >
0. We would like to know if there are complements m , m ′ ∈ LCM( I )such that β a , m ( S / I ) , β b , m ′ ( S / I ) , . First observe that, Γ ∨ = h m c , . . . , m cq i (e.g. [HH] or [F2, Prop. 2.4]).We have m ∈ LCM( I ) ⇐⇒ m = m i ∪ m i ∪ · · · ∪ m i s for some 1 i < i < · · · < i s q ⇐⇒ m c = m ci ∩ m ci ∩ · · · ∩ m ci s for some 1 i < i < · · · < i s q ⇐⇒ m c is the intersection of some facets of Γ ∨ . m , m ′ ∈ LCM( I ), then m , m ′ are complements ⇐⇒ m ∪ m ′ = [ n ] and m ∩ m ′ < I ⇐⇒ m c ∩ m ′ c = ∅ and m ∩ m ′ ∈ Γ ⇐⇒ m c ∩ m ′ c = ∅ and ( m ∩ m ′ ) c < Γ ∨ ⇐⇒ m c ∩ m ′ c = ∅ and m c ∪ m ′ c < Γ ∨ . So we are looking for subsets A , B ⊆ [ q ] such that1. m c = T j ∈ A m cj and m ′ c = T j ∈ B m cj m c ∩ m ′ c = ∅ m c ∪ m ′ c < Γ ∨
4. ˜ H a − (lk Γ ∨ ( m c ) , k ) , H b − (lk Γ ∨ ( m ′ c ) , k ) , Question 2.3. If ∆ = h F , . . . , F q i is a simplicial complex with ˜ H i − ( ∆ , k ) , i = a + b where a , b >
0, can we find subsets A , B ⊆ [ q ] such that1. F = T j ∈ A F j and G = T j ∈ B F j F ∩ G = ∅ F ∪ G < ∆
4. ˜ H a − (lk ∆ ( F ) , k ) , H b − (lk ∆ ( G ) , k ) , Example 2.4. If N ( I ) ∨ = Γ = h acd , ace , ade , bcd , bce , bde , ab i , dec ab then I = ( ac , bc , ad , bd , ae , be , cde ) has Betti table0 1 2 3 4total : 1 7 11 6 10 : 1 . . . . . . . β i , abcde , i = ,
4, which corresponds to nonvanishing of homology of links of facesof Γ in dimensions 1 ,
2. We consider each case separately:1. i = a = b =
2. Then ˜ H ( Γ , k ) ,
0. Let F = ab and G = ade ∩ bde = de , then F ∩ G = ∅ , F ∪ G = abde < Γ , and˜ H a − (lk Γ ( F ) , k ) = ˜ H − ( h∅i , k ) , H b − (lk Γ ( G ) , k ) = ˜ H ( h a , b i , k ) , . i = a = b =
3. Then ˜ H ( Γ , k ) ,
0. Let F = bcd and G = acd ∩ ade ∩ ace ∩ ab = a , then F ∩ G = ∅ , F ∪ G = abcd < Γ , and˜ H a − (lk Γ ( F ) , k ) = ˜ H − ( h∅i , k ) , H b − (lk Γ ( G ) , k ) = ˜ H ( h cd , de , ce , b i , k ) , . i = a = b =
2. Then ˜ H ( Γ , k ) ,
0. Let F = bcd ∩ bde = bd and G = acd ∩ ace = ac ,then F ∩ G = ∅ , F ∪ G = abcd < Γ , and˜ H a − (lk Γ ( F ) , k ) = ˜ H ( h c , e i , k ) , H b − (lk Γ ( G ) , k ) = ˜ H ( h d , e i , k ) , . A dual version of Question 2.3 can be stated as follows (see Corollary 3.6 for the justification).
Question 2.5. If Γ is a simplicial complex on the vertex set { x , . . . , x n } , and ˜ H i − ( Γ , k ) ,
0, and n − i + = a + b , where a and b are positive integers, are there nonempty subsets C , D ⊆ { x , . . . , x n } such that1. C ∪ D = { x , . . . , x n } C ∩ D ∈ Γ
3. ˜ H | C |− a − ( Γ C , k ) , H | D |− b − ( Γ D , k ) , Example 2.6.
Let N ( I ) = Γ = h adx , cdx , bcx , abx , aby , bcy , cdy , ady i . a b cd xy bd xy a b cd Γ Γ C Γ D I = ( xy , ac , bd ) has Betti table 0 1 2 3total : 1 3 3 10 : 1 . . . . . . . . . . . . β , abcdxy ( S / I ) , Γ in dimension 2 (i.e.˜ H ( Γ , k ) , a = b =
2. Choose C = { x , y } and D = { a , b , c , d } . Then C ∪ D = { a , b , c , d , x , y } , C ∩ D = ∅ ∈ Γ and˜ H | C |− a − ( Γ C , k ) = ˜ H ( h x , y i , k ) , H | D |− b − ( Γ D , k ) = ˜ H ( h ab , bc , cd , ad i , k ) , The following lemma is an easy exercise.
Lemma 3.1. Γ simplicial complex and A ∈ Γ and B ∈ lk Γ ( A ) , then lk lk Γ ( A ) ( B ) = lk Γ ( A ∪ B ) . In a simplicial complex Γ we say a d -cycle Σ is supported on faces F , . . . , F q if Σ = a F + · · · + a q F q for nonzero scalars a , . . . , a q ∈ k . We say that Σ is a face-minimal cycle or minimallysupported on F , . . . , F q if additionally no proper subset of F , . . . , F q is the support of a d -cycle.If Σ is supported on F . . . , F q , we call the simplicial complex h F , . . . , F q i the support complexof Σ .A variation of the following result appears as Theorem 4.2 of [RW]. Theorem 3.2.
Let k be a field, Γ a d-dimensional simplicial complex, and Σ = a F + · · · + a q F q a , . . . , a q ∈ ka d-cycle in Γ supported on F , . . . , F q , so that ˜ H d ( Γ , k ) , . Suppose A is a face of the supportcomplex of Σ such that for some s q we haveA ⊆ F ∩ . . . ∩ F s , and A * F j if j > sand | A | d + . Then1. there are ǫ i ∈ {± } for i = , . . . , s such that Σ A = ǫ a ( F \ A ) + · · · + ǫ s a s ( F s \ A ) is a ( d − | A | ) -cycle in lk Γ ( A ) that is not a boundary in lk Γ ( A ) ; . ˜ H d −| A | (lk Γ ( A ) , k ) , ;3. A = F ∩ . . . ∩ F s .Proof. The case | A | = d + Γ ( A ) = {∅} which has ( − | A | d . To prove Statement 1 we will proceed using induction on a = | A | . If a =
0, thenlk Γ ( A ) = Γ , Σ A = Σ and there is nothing to prove.Suppose a > A = { v , . . . , v a } , A ′ = { v , . . . , v a − } (or A ′ = ∅ when a =
1) and Γ ′ = lk Γ ( A ′ ),and suppose without loss of generality A ′ ⊆ F ∩ . . . ∩ F t and A ′ * F j for j > t > s . By the induction hypothesis, for some ǫ ′ i ∈ {± } there is a ( d − ( a − Σ A ′ = a ǫ ′ ( F \ A ′ ) + · · · + a t ǫ ′ t ( F t \ A ′ )in Γ ′ that is not a boundary in Γ ′ and ˜ H d − ( a − ( Γ ′ ( A ) , k ) ,
0. In particular, we must have t , s asotherwise the support complex of Σ A ′ would be a cone with every facet containing v a , a contradic-tion.We know that v a ∈ ( F i \ A ′ ) if and only if i s . Depending on the orientation of the faces of thecomplex Γ ′ , for some ǫ ′′ i ∈ {± } , we can write0 = ∂ ( Σ A ′ ) = ǫ ′ a ∂ ( F \ A ′ ) + · · · + ǫ ′ t a t ∂ ( F t \ A ′ ) = ǫ ′′ ǫ ′ a ( F \ A ) + · · · + ǫ ′′ s ǫ ′ s a s ( F s \ A ) + U + ∂ ( ǫ ′ s + a s + F s + \ A ′ + · · · + ǫ ′ t a t F t \ A ′ )where U consists of all the summands above which contain the vertex v a , and hence U = s X j = ǫ ′ j a j (cid:16) ∂ ( F j \ A ′ ) − ǫ ′′ j F j \ A (cid:17) = . If we set ǫ i = ǫ ′′ i ǫ ′ i and Σ A = ǫ a ( F \ A ) + · · · + ǫ s a s ( F s \ A ) it follows that Σ A = − ∂ ( ǫ ′ s + a s + ( F s + \ A ′ ) + · · · + ǫ ′ t a t ( F t \ A ′ ))and ∂ ( Σ A ) = − ∂ ( ǫ ′ s + a s + ( F s + \ A ′ ) + · · · + ǫ ′ t a t ( F t \ A ′ )) = . So Σ A is a ( d − a )-cycle in lk Γ ′ ( v a ) = lk Γ ( A ) by Lemma 3.1 (and since v a ∈ Γ ′ ). Since dim(lk Γ ( A )) = d − | A | , the ( d − | A | )-cycle Σ A is not a boundary in lk Γ ( A ). Therefore, ˜ H d −| A | ( Γ , k ) ,
0, provingStatement 2.To see Statement 3, note that if F . . . , F s all contain a vertex outside A , then lk Γ ( A ) would bea cone and hence have no homology, contradicting Statement 2. (cid:3) xample 3.3. Let
Γ = h acd , ace , ade , bcd , bce , bde , ab i and Γ ∨ as shown on the left and right in thepicture below, respectively. dec ab dec ab Γ Γ ∨ Then Γ ∨ has dimension d = u = , u Σ A lk Γ ∨ ( A ) Σ A ( u − | A | ) − homology?0 a − c a b ∅ ( − − homology0 a − c c h d , e i ∅ ( − − homology1 dc + ce + ed c h d , e i d − e − homology Corollary 3.4.
Let k be a field, Γ a d-dimensional simplicial complex with ˜ H d ( Γ , k ) , , and let Σ be a d-cycle in Γ which is not a boundary. Let A be a proper face of the support complex of Σ , andsuppose F . . . , F q are the facets of Γ that contain A. ThenA = q \ j = F j . Proof.
Since lk Γ ( A ) = h F \ A , . . . , F q \ A i , if there is a vertex of q \ j = F j which is not in A , thenlk Γ ( A ) would be a cone, and would therefore have no homology, contradicting Theorem 3.2. (cid:3) Theorem 3.5 ( Breaking up simplicial cycles on links).
Let k be a field and ∆ = h F , . . . , F r i be ad-dimensional simplicial complex such that ˜ H d ( ∆ , k ) , and d + = a + b for some a , b > . Suppose ∆ contains a d-dimensional cycle Σ = q X j = a j F j upported on the facets F , . . . , F q of ∆ , and Σ is not boundary in ∆ . Then there are subsets A , B ⊆ [ q ] ⊆ [ r ] with F = \ j ∈ A F j and G = \ j ∈ B F j such that1. F ∩ G = ∅ ;2. F ∪ G < ∆ ;3. ˜ H a − (lk ∆ ( F ) , k ) , and ˜ H b − (lk ∆ ( G ) , k ) , .Moreover, if a , b > , F and G and ǫ j , δ j ∈ {± } could be chosen to additionally satisfy:4. | F | = b and | G | = a;5. Σ F = X j ∈ A ǫ j a j (cid:16) F j \ F (cid:17) is an ( a − -cycle in lk ∆ ( F ) which is not a boundary ;6. Σ G = X j ∈ B δ j a j (cid:16) F j \ G (cid:17) is a ( b − -cycle in lk ∆ ( G ) which is not a boundary.Proof. Set i = d +
2. We first consider the case b = a = i −
1. If a =
1, then d = ∆ is disconnected. Let F and G be two facets each belonging to a distinct connected component of ∆ .Then we clearly have F ∩ G = ∅ and F ∪ G < ∆ . Moreover, lk ∆ ( F ) = lk ∆ ( G ) = {∅} and so˜ H a − (lk ∆ ( F ) , k ) = ˜ H b − (lk ∆ ( G ) , k ) = ˜ H − ( {∅} , k ) , b = a = i − >
1, then d = a + b − >
0. By Theorem 3.2, if we take a vertex v in thesupport complex of Σ , then ˜ H i − (lk ∆ ( v ) , k ) , ∆ has nonvanishing homology, it is not a cone, so not all facets of ∆ contain v . ByCorollary 3.4, F = { v } is the intersection of the facets of ∆ which contain { v } . Let G be a facet of ∆ that does not contain v . Then F ∩ G = ∅ and F ∪ G < ∆ (as G is a facet), and moreover˜ H a − (lk ∆ ( F ) , k ) = ˜ H i − (lk ∆ ( v ) , k ) , H b − (lk ∆ ( G ) , k ) = ˜ H − ( { , k ) ∅} , . Now suppose a , b > a = i − b . Suppose F = { w , v , . . . , v i − } . Then since F is in thesupport of the ( i − Σ , { w , v , . . . , v i − } must appear in another one of the F j in the supportof Σ , say F . Suppose F = { w , w , v , . . . , v i − } . Considering that a = i − b i −
2, let G = { v , . . . , v a } and F = { v a + , . . . , v i − , w , w } . Then | G | = a and | F | = i − + − a = b . Moreover F ∩ G = ∅ by construction, and if i − = d ,then F ∪ G < ∆ since | F ∪ G | = d + ∆ .By Theorem 3.2, and noting that i − − | G | = b − i − − | F | = a −
2, we have˜ H a − (lk ∆ ( F ) , k ) , H b − (lk ∆ ( G ) , k ) , , A = { j ∈ [ q ] | F ⊂ F j } and B = { j ∈ [ q ] | G ⊂ F j } then F = \ j ∈ A F j and G = \ j ∈ B F j . (cid:3) Another version of Theorem 3.5 below is one which gives lower-dimensional cycles in inducedsubcomplexes.
Corollary 3.6 ( Breaking up simplicial cycles).
Let Γ be a simplicial complex on the vertex set { x , . . . , x n } , and suppose ˜ H d − ( Γ , k ) , , where d is the smallest possible size of a nonface of Γ .Suppose n − d + = a + b, where a and b are positive integers. Then there are nonempty subsetsC , D ⊆ { x , . . . , x n } such that1. C ∪ D = { x , . . . , x n } ;2. C ∩ D ∈ Γ ;3. ˜ H | C |− a − ( Γ C , k ) , and ˜ H | D |− b − ( Γ D , k ) , .Proof. By Alexander duality - see Prop. 5.1.10 and the discussion preceding Prop. 5.1.8 in [HH]-we have that ˜ H n − d − ( Γ ∨ , k ) ,
0. Now d is the smallest possible size of a nonface of Γ , so by thedefinition of Alexander duals, dim( Γ ∨ ) = n − d − Γ ∨ = h F , . . . , F r i . If n − d + = a + b , then, by Theorem 3.5, there are subsets A and B of [ r ] such that F = \ j ∈ A F j and G = \ j ∈ B F j and(i) F ∩ G = ∅ ;(ii) F ∪ G < Γ ∨ ;(iii) ˜ H a − (lk Γ ∨ ( F ) , k ) , H b − (lk Γ ∨ ( G ) , k ) , C = F c = [ j ∈ A F cj and D = G c = [ j ∈ B F cj . Then by (i), C ∪ D = ( F ∩ G ) c = { x , . . . , x n } . By (ii), ( C ∩ D ) c = F ∪ G < Γ ∨ so C ∩ D ∈ Γ . Finallyby (iii) and Equation (1), ˜ H | C |− a − ( Γ C , k ) , H | D |− b − ( Γ D , k ) , (cid:3) heorem 3.7 ( Subadditivity of syzygies of monomial ideals).
If I is a monomial ideal in thepolynomial ring S = k [ x , . . . , x n ] where k is a field, and d is the smallest possible degree of agenerator of I. Suppose i = n − d + , β i , n ( S / I ) , and i = a + b, for some positive integers a andb. Then t i t a + t b .Proof. By polarization, it is enough to consider I a square-free monomial ideal. By Hochster’sformula (Equation (1)), if Γ = N ( I ), then β n − d + , n ( S / I ) = β n − d + , x ··· x n ( S / I ) = dim k ˜ H d − ( Γ , k ) , . If n − d + = a + b , then by Corollary 3.6, there are nonempty subsets C , D ⊆ { x , . . . , x n } suchthat C ∪ D = { x , . . . , x n } and C ∩ D ∈ Γ , and ˜ H | C |− a − ( Γ C , k ) , H | D |− b − ( Γ D , k ) , . By Equation (1), this means that β a , | C | ( S / I ) , β b , | D | ( S / I ) , , so that t a > | C | and t b > | D | . Putting this all together we get t a + t b > | C | + | D | > n = t i , which settles our claim. (cid:3) Discussion 3.8.
Given a square-free monomial ideal I if we are looking for top degree Betti num-bers, by Hochster’s formula (Equation (1)) β n − i − , n ( S / I ) = dim k ˜ H i ( Γ , k ) . Now if d is the smallest possible degree of a generator of I , then all monomials of degree d − I , which means all faces of dimension d − Γ = N ( I ). This means that the smallestindex i with ˜ H i ( Γ , k ) , d −
2, that is˜ H i ( Γ , k ) = i < d − β j , n ( S / I ) = j = n − i − > n − d + . So n − d + n − d + S / I or very close to it.12 xample 3.9. Let I = ( abc , ace , ade , bcd , bde ) be an ideal of S = k [ a , ..., e ] in 5 variables. Here thesmallest degree of a generator of I is d =
3, so n − d + =
3, so we pick a = b =
2. Accordingto Macaulay2 [M2] the Betti table of S / I is 0 1 2 3total : 1 5 5 10 : 1 . . . . . . . . t = , t = , t = = ⇒ t < t + t = . Example 3.10.
In Example 2.4, I = ( ac , bc , ad , bd , ae , be , cde ) is a square-free monomial ideal in5 variables where d = n − d + =
4. According to the Betti table of I , t = t = t = t =
3. Here t < t + t = t < t =
8. Note that we also have β , ( S / I ) , < = n − d + t < t + t = In this section, we consider breaking up special classes of cycles, where we can provide a combina-torial description for the lower-dimensional cycles.
We begin with an example.
Example 4.1.
Let N ( I ) = Γ = h uv , xy , yz , xz i be a simplicial complex on n = xy zu v Here ˜ H ( Γ , k ) , β , uvxyz ( S / I ) ,
0. If 4 = a + b , then using Corollary 3.6 we havethe following two cases to consider.1. a = b =
3. Let C = { u , x } and D = { u , v , y , z } . Then C ∪ D = { u , v , x , y , z } , C ∩ D = { u } ∈ Γ and ˜ H | C |− a − ( Γ C , k ) = ˜ H ( h u , x i , k ) , H | D |− b − ( Γ D , k ) = ˜ H ( h uv , yz i , k ) , . a = b =
2. Let C = { u , x , v } and D = { u , y , z } . Then C ∪ D = { u , v , x , y , z } , C ∩ D = { u } ∈ Γ and˜ H | C |− a − ( Γ C , k ) = ˜ H ( h uv , x i , k ) , H | D |− b − ( Γ D , k ) = ˜ H ( h u , yz i , k ) , .
13n general if Γ is a disconnected complex on n vertices with Stanley-Reisner ideal I , then β n − , n ( S / I ) ,
0, and if n − = a + b for some a , b >
0, then we can always find disconnectedinduced subcomplexes Γ C and Γ D where C = a + D = b +
1, as in the example above. Belowwe demonstrate how this can be done.If Γ is disconnected, then it has the form Γ = Γ ∪ · · · ∪ Γ t where Γ , . . . , Γ t are connected components and t >
1. In this case, | V ( Γ i ) | > i t , V ( Γ ) = V ( Γ ) ∪ · · · ∪ V ( Γ t ) and V ( Γ k ) ∩ V ( Γ l ) = ∅ for all 1 k < l t .Without loss of generality and up to renaming the variables, we can assume the following: • | V ( Γ ) | | V ( Γ ) | · · · | V ( Γ t ) | , • x k ∈ V ( Γ k ) for 1 k t , • V ( Γ ) = { x , x t + , . . . , x t + | V ( Γ ) |− }• V ( Γ k ) = { x k , x ( t + | V ( Γ ) | + ··· + | V ( Γ k − ) |− k + , . . . , x ( t + | V ( Γ ) | + ··· + | V ( Γ k ) |− k ) } for each 1 < k t . Example 4.2.
The simplicial complex
Γ = h uv , xy , yz , xz i from the previous example can be writtenas Γ = Γ ∪ Γ where Γ = h x x i and Γ = h x x , x x , x x i .For each 1 a < n −
1, define C = { x , x , . . . , x a + } and D = { x , x a + , . . . , x n } . Clearly C ∪ D = { x , . . . , x n } , | C | = a + | D | = n − a and C ∩ D = { x } ∈ Γ . Moreover, it is easy to seethat both Γ C and Γ D are disconnected induced subcomplexes of Γ on the subsets { x , x , . . . , x a + } and { x , x a + , . . . , x n } , respectively. Therefore, if b = n − a − H | C |− a − ( Γ C , k ) = ˜ H ( Γ C , k ) , H | D |− b − ( Γ D , k ) = ˜ H ( Γ D , k ) , . Recall that a cycle in a graph G is an ordered list of distinct vertices x , . . . , x n where the edgesare x i − x i for 2 i n and x n x . Graph cycles characterize nontrivial 1-homology in simplicialcomplexes; see for example Theorem 3.2 in [C].Suppose Γ is a simplicial complex on the set { x , . . . , x n } that is the support complex of a face-minimal graph cycle, so that ˜ H ( Γ , k ) ,
0. This means that β n − , n ( S / I ) ,
0. Suppose n − = a + b for some a , b > Γ can be written in the form Γ = h x x , x x , . . . , x n − x n , x n x i . For 1 a < n −
2, define C = { x , x , x , . . . , x a + } and D = { x , x a + , . . . , x n } . C ∪ D = { x , . . . , x n } , | C | = a + | D | = n − a − C ∩ D = ∅ ∈ Γ . Moreover,it is easy to see that both Γ C and Γ D are disconnected induced subcomplexes of Γ on the subsets { x , x , x , . . . , x a + } and { x , x a + , . . . , x n } , respectively. Therefore,˜ H | C |− a − ( Γ C , k ) = ˜ H ( Γ C , k ) , H | D |− b − ( Γ D , k ) = ˜ H ( Γ D , k ) , b = n − a − Example 4.3.
Let N ( I ) = Γ = h x x , x x , x x , x x , x x i . x x x x x Then ˜ H ( Γ , k ) , β , x ··· x ( S / I ) ,
0. Taking a = b =
2, set C = { x , x } and D = { x , x , x } . Then˜ H | C |− a − ( Γ C , k ) = ˜ H ( h x , x i , k ) , H | D |− b − ( Γ D , k ) = ˜ H ( h x , x x i , k ) , Question 2.1, Question 2.2, Question 2.3 and Question 2.5 are all equivalent, though their di ff er-ent settings allow the application of di ff erent (inductive) tools. All of them are open in their fullgenerality as far as we know, though each can be answered positively for certain classes of idealsor combinatorial objects. A positive answer to either would settle the subadditivity question formonomial ideals in a polynomial ring. References [A] A. Abedelfatah,
Some results on the subadditivity condition of syzygies ,arXiv:2001.01136v1 [math.AC] (2020).[ACI] L. Avramov, A. Conca and S. Iyengar,
Subadditivity of syzygies of Koszul algebras , Math.Ann. 361, no. 1-2, 511 – 534 (2015).[AN] A. Abedelfatah and E. Nevo,
On vanishing patterns in j-strands of edge ideals , J. AlgebraicCombin. 46, no. 2, 287295 (2017).[B] K. Baclawski,
Galois connections and the Leray spectral sequence , Advances in Math. 25,no. 3, 191 – 215 (1977). 15BH] M. Bigdeli and J. Herzog,
Betti diagrams with special shape , Homological and Computa-tional Methods in Commutative Algebra, 33 – 52 (2017).[C] E. Connon,
On d-dimensional cycles and the vanishing of simplicial homology ,arXiv:1211.7087 [math.AC] (2013).[DS] H. Dao and J. Schweig,
Bounding the projective dimension of a squarefree monomial idealvia domination in clutters , Proceedings of the American Mathematical Society, Volume 143,Number 2. 555 – 565 (2015).[EF1] N. Erey and S. Faridi,
Multigraded Betti numbers of simplicial forests , J. Pure Appl. Algebra218, 1800 – 1805 (2014).[EF2] N. Erey and S. Faridi,
Betti numbers of monomial ideals via facet covers , J. Pure Appl.Algebra 220, no. 5, 1990 – 2000 (2016).[EHU] D. Eisenbud, C. Huneke, and B. Ulrich,
The regularity of Tor and graded Betti numbers ,Amer. J. Math. 128, no. 3, 573 – 605 (2006).[ES] S. El Khoury and H. Srinivasan,
A note on the subadditivity of Syzygies , Journal of Algebraand its Applications, vol.16, no.9, 1750177 (2017).[F1] S. Faridi,
Lattice complements and the subadditivity of syzygies of simplicial forests , Journalof Commutative Algebra, Volume 11, Number 4, 535 – 546 (2019).[F2] S. Faridi,
Simplicial trees are sequentially Cohen-Macaulay , J. Pure Appl. Algebra 190, no.1 – 3, 121 – 136 (2004).[FG] O. Fern´andez-Ramos and P. Gimenez,
Regularity 3 in edge ideals associated to bipartitegraphs , J. Algebraic Combin. 39 (2014).[Fr] R. Fr¨oberg,
On Stanley-Reisner rings , in: Topics in algebra, Banach Center Publications, 26Part 2, 57 70 (1990).[GPW] V. Gasharov, I. Peeva, V. Welker,
The lcm-lattice in monomial resolutions , MathematicalResearch Letters 6, 521 – 532 (1999).[HH] J. Herzog and T. Hibi
Monomial Ideals , Graduate Texts in Mathematics, vol. 260, Springer-Verlag, London (2011). http: // dx.doi.org / / On the subadditivity problem for maximal shifts in free resolu-tions , Commutative Algebra and Noncommutative Algebraic Geometry, II MSRI Publica-tions Volume 68, (2015).[M2] D. Grayson and M. Stillman, Macaulay2, a software system for research in algebraic geom-etry. Available at http: // / Macaulay2 / .16P] I. Peeva, Graded syzygies , Algebra and Applications, 14. Springer-Verlag London, Ltd.,London (2011).[RW] V. Reiner and V. Welker,
Linear syzygies of Stanley-Reisner ideals , Mathematica Scandi-navica, 89(1), 117 – 132 (2001).[T] D. Taylor,