Upper bounds for Extremal Betti Numbers of t-Spread Strongly Stable Ideals
aa r X i v : . [ m a t h . A C ] F e b Upper bounds for Extremal Betti Numbers of t -Spread Strongly Stable Ideals Luca Amata, Antonino Ficarra, Marilena Crupi
Department of Mathematical and Computer Sciences, Physical and Earth SciencesUniversity of MessinaViale Ferdinando Stagno d’Alcontres 31, 98166 Messina, Italye-mail: [email protected], antfi[email protected], [email protected]
February 16, 2021
Abstract
We study the extremal Betti numbers of the class of t –spread strongly stableideals. More precisely, we determine the maximal number of admissible extremalBetti numbers for such ideals, and thereby we generalize the known results for t ∈ { , } . Let S = K [ x , . . . , x n ] be the standard polynomial ring in n variables with coefficientsin a field K . A squarefree monomial ideal I of S is an ideal generated by squarefreemonomials. This class of ideals plays an important role in Commutative Algebra, notonly for its intrinsic value but overall for its strong connections to Combinatorics andTopology. Recently, Ene, Herzog, and Qureshi [13] have generalized the notion of(squarefree) monomial ideal by introducing the class of t –spread monomial ideals.Let t ≥ x i x i · · · x i d , with 1 ≤ i ≤ i ≤ · · · ≤ i d ≤ n ,is called t –spread if i j +1 − i j ≥ t for all j = 1 , . . . , d −
1. A t –spread monomial idealis an ideal generated by t –spread monomials. In the last years, many authors havefocused their attention on such a class of monomial ideals in order to analyze the mainalgebraic invariants that may be associated to a graded ideal in S and thereby theyhave generalized some classical results [3, 4, 5, 13, 11]. Indeed, a 0–spread monomialideal is a monomial ideal, whereas a 1–spread ideal is a squarefree monomial ideal. Keywords: monomial ideals, minimal graded resolution, extremal Betti numbers, t -spread ideals. t –spread monomial ideals.Among all the algebraic invariants of a graded ideal I of S , the role of the gradedBetti numbers is unquestionable. An important subset of the graded Betti numbersof I consists of the extremal Betti numbers (Definition 2.3) introduced in [7] as arefinement of the Castelnuovo-Mumford regularity and the projective dimension of theideal I . Many characterizations of these graded Betti numbers for classes of monomialideals can be found in [16, 8, 9, 10, 1, 2, 3] and in the references therein.Our aim is to solve the following problem. Problem 1
Given two positive integers t and n , let S t,n be the set of all t –spreadstrongly stable ideals in S = K [ x , . . . , x n ]. What is the maximal number of extremalBetti numbers allowed for an ideal in S t,n ?In [2], the authors determined the maximal number of such graded Betti numbersfor a 1–spread strongly stable ideal, whereas in [3], the same authors computed themaximal number of extremal Betti numbers of a 2–spread strongly stable ideal of initialdegree 2, where for the initial degree of a graded ideal I of S , denoted by indeg I , wemean the minimum j such that I j = 0 ( I j is the K -vector space spanned by thehomogeneous elements of I of degree j ). The problem of determining such number for t > t –spread strongly stable ideal of initial degree ≥
2, for all integers t ≥ t –spread monomials of S (Subsections 3.1.1, 3.1.2) which allow us to solve Problem 1.We provide some examples illustrating our techniques. In Section 4, we prove our mainresult (Theorem 4.1). It establishes what is the maximum number of extremal Bettinumbers of a t –spread strongly stable ideal I of S of initial degree 2. As a consequenceof this theorem we obtain the results stated in [2] and [3] for t = 1 ,
2, respectively. InSection 5, an analogous result of Theorem 4.1 is stated (Theorem 5.1). Such result istrue when the initial degree of t –spread strongly stable ideal I of S is greater than 2.Finally, Section 6 contains our conclusions and perspectives. Almost all the examplesin the paper have been verified with specific packages of Macaulay2 [14] some of whichdeveloped by the authors of the paper.
Let S = K [ x , . . . , x n ] be the standard polynomial ring in n variables with coefficientsin K . S is an N –graded ring where deg x i = 1, for all i = 1 , . . . , n . A monomial ideal I of S is an ideal generated by monomials. By G ( I ) we denote the unique minimal set2f monomial generators of I . For a monomial u ∈ S , u = 1, we setsupp( u ) = (cid:8) i : x i divides u (cid:9) , and write max( u ) = max (cid:8) i : i ∈ supp( u ) (cid:9) , min( u ) = min (cid:8) i : i ∈ supp( u ) (cid:9) . Moreover, we set max(1) = min(1) = 0.
Definition 2.1
Let t ≥ x i x i · · · x i d with 1 ≤ i ≤ i ≤· · · ≤ i d ≤ n is called t –spread, if i j +1 − i j ≥ t , for all j = 1 , . . . , d −
1. A monomial ideal I of S is called a t –spread monomial ideal, if it is generated by t –spread monomials.For instance, x x x ∈ K [ x , x , x , x , x ] is a 2–spread monomial, but not a 3–spread monomial. Every monomial is 0–spread and every monomial ideal is a 0–spreadmonomial ideal. A squarefree monomial is a 1–spread monomial and a squarefreemonomial ideal is a 1–spread monomial ideal. If t ≥
1, then every t –spread monomialis a squarefree monomial. Definition 2.2 A t –spread monomial ideal I of S is called t –spread stable, if for all t –spread monomials u ∈ I and for all i < max( u ) such that x i ( u/x max( u ) ) is a t –spreadmonomial, it follows that x i ( u/x max( u ) ) ∈ I . The ideal I is called t –spread stronglystable, if for all t –spread monomials u ∈ I , all j ∈ supp( u ) and all i < j such that x i ( u/x j ) is t –spread, it follows that x i ( u/x j ) ∈ I .Let u , . . . , u r be t –spread monomials of S . The unique t –spread strongly stableideal containing u , . . . , u r will be denoted by B t ( u , . . . , u r ) [13]. The monomials u , . . . , u r are called t –spread Borel generators, and B t ( u , . . . , u r ) is called the finitelygenerated t –spread Borel ideal.Let us denote by M n,d,t the set of all t –spread monomials of degree d in S = K [ x , . . . , x n ]. From [13, Theorem 2.3], the cardinality of M n,d,t is given by | M n,d,t | = (cid:18) n − ( d − t − d (cid:19) . Let t ≥
1. We endow the set M n,d,t with the squarefree lexicographic order , ≥ slex [6]. More precisely, let u = x i x i · · · x i d and v = x j x j · · · x j d , be t –spread monomialsof degree d , with 1 ≤ i < i < · · · < i d ≤ n , 1 ≤ j < j < · · · < j d ≤ n , then u > slex v if i = j , . . . , i s − = j s − and i s < j s , for some 1 ≤ s ≤ d .It’s easy to verify that if u is a t –spread monomial of S , then for all v ∈ B t ( u ) wehave v ≥ slex u .Note that the existence of a t –spread monomial of degree d in S implies that n ≥ ( d − t + 1. Indeed, the monomial x x t x t · · · x ( d − t +1 is the greatest t –spread monomial of M n,d,t , with respect to > slex .3et T be a not empty subset of M n,d,t . We denote by max T (min T , respectively)the maximal (minimum, respectively) monomial w ∈ T , with respect to > slex .From now on, we assume that M n,d,t ( t ≥
1) is endowed by the squarefree lexico-graphic order .Furthermore, we define the t –shadow of T Shad t ( T ) = n x i w : w ∈ T, i = 1 , . . . , n o ∩ M n,d +1 ,t = (cid:8) x i w : w ∈ T and x i w is t –spread monomial, i = 1 , . . . , n (cid:9) . The set Shad t ( T ) could be empty. We define Shad t ( T ) = Shad t ( T ) and Shad nt ( T ) =Shad t (Shad n − t ( T )) for all n ≥
2, by induction.If I is a t –spread strongly stable ideal, then the graded Betti numbers of I can becomputed by [13, Corollary 1.12] β k,k + ℓ ( I ) = X u ∈ G ( I ) ℓ (cid:18) max( u ) − t ( ℓ − − k (cid:19) . (1)Such a formula returns the Eliahou–Kervaire formula [12] for (strongly) stable idealswhenever t = 0 and the Aramova–Herzog–Hibi formula [6, 15] for squarefree (strongly)stable ideals whenever t = 1. Definition 2.3 ([7]) A graded Betti number β k,k + ℓ ( I ) = 0 is called extremal if β i,i + j ( I ) =0 for all i ≥ k , j ≥ ℓ, ( i, j ) = ( k, ℓ ).The pair ( k, ℓ ) is called a corner of I .If ( k , ℓ ) , . . . , ( k r , ℓ r ) ( n − ≥ k > k > · · · > k r ≥
1, 1 ≤ ℓ < ℓ < · · · < ℓ r ) arethe corners of a graded ideal I of S , the setCorn( I ) = n ( k , ℓ ) , ( k , ℓ ) , . . . , ( k r , ℓ r ) o is called the corner sequence of I [8]; whereas the r -uple a ( I ) = (cid:0) β k ,k + ℓ ( I ) , β k ,k + ℓ ( I ) , . . . , β k r ,k r + ℓ r ( I ) (cid:1) is called the corner values sequence of I [8].We conclude this section by quoting two results from [3]. Characterization 2.4 ([3, Theorem 1])
Let I be a t –spread strongly stable ideal of S .The following conditions are equivalent: (a) β k,k + ℓ ( I ) is extremal; (b) k + t ( ℓ −
1) + 1 = max { max( u ) : u ∈ G ( I ) ℓ } and max( u ) < k + t ( j −
1) + 1 , forall j > ℓ and for all u ∈ G ( I ) j . orollary 2.5 ([3, Corollary 2]) Let I be a t –spread strongly stable ideal of S and let β k,k + ℓ ( I ) be an extremal Betti number of I . Then β k,k + ℓ ( I ) = (cid:12)(cid:12)(cid:12)n u ∈ G ( I ) ℓ : max( u ) = k + t ( ℓ −
1) + 1 o(cid:12)(cid:12)(cid:12) . t –Spread Strongly Stable Ideals of initialdegree In this Section, if S = K [ x , . . . , x n ], we manage some suitable t –spread monomials of S in order to examine the behavior of the corners of a t –spread strongly stable idealof S of initial degree 2.Let us denote by S t,n the set of all t –spread strongly stable ideals in S and by S t,n, the set of all I ∈ S t,n for which the value of every extremal Betti number equals 1, i.e. ,all the entries of the corner values sequence a ( I ) are equal to 1: S t,n, = n I ∈ S t,n : a ( I ) = = (1 , , . . . , o . Our goal is to determine the greatest admissible number of corners for an ideal ofinitial degree 2 in S t,n, .The starting point of our work has been the analysis of several examples ( t = 2, 3,4, 5) using the computer algebra system Macaulay2 . In each of these cases, we havefixed two positive integers n and ℓ , and using techniques similar to those in [2] and [3],we have determined the maximum number of admissible corners of a t –spread stronglystable ideal I of a polynomial ring in n variables and such that indeg I = ℓ . Then,all the data obtained have been collected in some tables to analyze how the maximumnumber of corners varied with respect to the parameters (see for instance, Table 1 andTable 2). t = 2 n ℓ − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − Table 1: Maximum number of corners of 2–spread strongly stable ideals5 = 3 n ℓ − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − Table 2: Maximum number of corners of 3-spread strongly stable idealsGiven ℓ (the initial degree), the output of each table shows that the maximumnumber of admissible corners remains eventually unchanged for t consecutive valuesof n and then increases by 1. For this reason, we have estimated that it may be con-venient to decompose n with respect to t by writing n = d + kt for suitable positiveintegers d and k .For later use, recall that the floor function of a real number x is defined as follows: ⌊ x ⌋ = max { n ∈ Z : n ≤ x } . In particular, if − ≤ x < ⌊ x ⌋ = − ≤ x < ⌊ x ⌋ = 0. Let I ∈ S t,n, such that indeg I = 2. We will verify that if one wants I to have themaximal number of extremal Betti numbers, then- G ( I ) = B t ( x x n ), and- n = d + kt with k ≥ ≤ d ≤ t . Claim.
Set ω = x x n and let G ( I ) = B t ( x x n ). There exist t –spread monomials ω , . . . , ω k + ⌊ d − t ⌋ − such that ω j := max n u ∈ M n,j +2 ,t : u / ∈ j − [ i =0 Shad j − it ( B t ( ω i )) and max( u ) = n o , for j = 1 , . . . , k + (cid:4) d − t (cid:5) − ω j , it is sufficient to find the minimum v of Shad t ( B t ( ω j − )). Then ω j will be the largest t –spread monomial of degree j + 2with max( ω j ) = n following v in the squarefree lexicographic order (see also [2]).In order to prove the Claim , we need the next crucial lemma.
Lemma 3.1
Let n, t be two positive integers. Let u = x i x i · · · x i d , ≤ i < i < · · · < i d ≤ n be a t –spread monomial of S = K [ x , . . . , x n ] such that max( u ) = n . If j +1 − i j = t , for all j = 1 , . . . , d − then B t ( u ) is the t –spread Veronese ideal of degree d . Otherwise, if p = max { j : i j +1 − i j > t } , then the largest t –spread monomial v ofdegree d of S with max( v ) = n following u , with respect to > slex , is v = x i · · · x i p − x i p +1 x i p +1+ t · · · x i p +1+ t ( d − p − x n . Proof. If i j +1 − i j = t , for all j = 1 , . . . , d −
1, then u = x n − t ( d − · · · x n − t x n . Hence, u is the smallest t –spread monomial of degree d with max( u ) = n and B t ( u ) is generatedby all t –spread monomials of degree d of S , i.e. , B t ( u ) is the t –spread Veronese idealof degree d [13].Now, suppose i j +1 − i j > t for some j and let p = max { j : i j +1 − i j > t } . If w = x s x s · · · x s d is a t –spread monomial of degree d with max( w ) = n and u > slex w ,then i = s , . . . , i j − = s j − and i j < s j for some index j .It is j ≤ p . Indeed, if j > p , then i p +2 − i p +1 = · · · = i d − i d − = t . Hence, i d − i j = t ( d − j ), i d = s d = n , s d − s j ≥ t ( d − j ), and i j < s j . Thus, t ( d − j ) ≤ s d − s j = i d − s j < i d − i j = t ( d − j ) , and so t ( d − j ) < t ( d − j ). This is absurd. Hence, j ≤ p .Therefore, setting v = x i · · · x i p − x i p +1 x i p +1+ t · · · x i p +1+ t ( d − p − x n , (2)one has u > slex v. Moreover, it is easy to verify that v is the monomial we are looking for. (cid:3) Lemma 3.1 will play a key role in getting the ω i ’s in the claim.In order to simplify the notation, we setΩ j := n u ∈ M n,j +2 ,t : u / ∈ j − [ i =0 Shad j − it ( B t ( ω i )) and max( u ) = n o and so ω j := max Ω j , for all j .Next remark will be pivotal for the rest. Remark 3.2
Let t ≥ n = d + kt , 1 ≤ d ≤ t and k ∈ { , , } .(i) If k = 0, then there is no t –spread monomial of degree two. Indeed, in such acase n = d < t + 1.(ii) If k = 1, then n = d + t . Hence, ω = x x d + t and Shad t ( B t ( ω )) = ∅ .7iii) Let k = 2, then ω = x x n = x x d +2 t . If d = 1, then Shad t ( B t ( ω )) = { x x t x t } = M n, ,t , so Ω = ∅ . If 2 ≤ d ≤ t , then Shad t ( B t ( ω )) = B t ( x x d + t x d +2 t ). Hence, min Shad t ( B t ( ω )) = x x d + t x d +2 t , and by Lemma 3.1, ω = max Ω = x x t x d +2 t is the largest t –spread monomial u with max( u ) = n following x x d + t x d +2 t , with respect to > slex .Remark 3.2 points out that the decomposition n = d + kt, ≤ d ≤ t does not work well in the sense of the Claim , whenever k ∈ { , , } . In this Subsection, if S = K [ x , . . . , x n ], n = d + kt ( k ≥
3, 1 ≤ d ≤ t ), setting, ω = x x n , we construct a set of monomials ω , . . . , ω q of S , q ≤ k + (cid:4) d − t (cid:5) −
1, of thetype described in the
Claim . Such monomials will be called basic monomials of thefirst type .Let k ≥
3. For the sake of clarity, we distinguish two cases.
Case 1.
Let k = 3, n = d + 3 t .Set ω = x x n . The minimum of Shad t ( B t ( ω )) is u = x x n − t x n = x x d +2 t x d +3 t .By Lemma 3.1, the largest t –spread monomial v of degree 3 with max( v ) = n thatfollows u in the squarefree lexicographic order is v = x x t x d +3 t . It is clear that v = ω = max Ω .Let us discuss the “distance” between the last two variables of ω = x x t x d +3 t .We need two consider some cases.If d = 1, then 1 + 3 t − (2 + t ) = 2 t −
1, Shad t ( B t ( ω )) = { x x t x t x t } = M t, ,t and ω does not exist. Hence, we have k − i.e. , ω , ω .If d = 2, then 2 + 3 t − (2 + t ) = 2 t , Shad t ( B t ( ω )) = B t ( x x t x t x t ) = M n, ,t and we cannot construct ω . Hence, also in such a case, we have k − ≤ d ≤ t . In such a case Shad t ( B t ( ω )) = B t ( x x t x d +2 t x d +3 t ).Since min Shad t ( B t ( ω )) = x x t x d +2 t x d +3 t , then, by Lemma 3.1, one has that ω = x x t x t x d +3 t . On the other hand, since d + 4 − t ≤
4, we have | M n, ,t | = | M d +3 t, ,t | = (cid:18) d + 3 t − t + 45 (cid:19) = (cid:18) d + 4 − t (cid:19) = 0 . Thus, M n, ,t = ∅ and ω does not exist. Hence, in such a case, we have constructed k = 3 monomials. Case 2.
Let k ≥
4. Firstly, we consider an example.
Example 3.3
Let n = 9 and t = 2, we can write n = d + kt , with d = 1 , k = 4. Then,8 = x x and ω = x x x . Observe that in such a caseΩ := n u ∈ M , , : u / ∈ Shad ( B ( ω )) ∪ Shad ( B ( ω )) and max( u ) = 9 o = ∅ . Indeed, | M , , | = (cid:0) (cid:1) = 15 and the monomials u ∈ M , , with max( u ) = 9 are thefollowing ones x x x x , x x x x , x x x x , x x x x , x x x x , x x x x ,x x x x , x x x x , x x x x . Hence, Ω = { x x x x } and ω = x x x x .Note that ω = x x t +1 x t +1 x n = x x t x t x n , as in the case k = 3 for 3 ≤ d ≤ t .Assume k ≥
4. For j ≥
1, let us define the following monomials of S of degree j + 2 ω j := (cid:18) j − Y i =0 x i + it (cid:19) x ( j +1)+ jt x d + kt = x x t x t · · · x ( j +1)+( j − t x ( j +1)+ jt x d + kt . (3)For j = 1 ,
2, one has: ω = x x t x d + kt ,ω = x x t x t x d + kt . It is clear that ω = max Ω and ω = max Ω .The monomials ω j are t –spread if j satisfies the inequality ( j + 1) + jt ≤ n − t .Let us determine the greatest such an integer, i.e. , j max = max (cid:8) j : ( j + 1) + jt ≤ n − t (cid:9) . For every j ∈ (cid:8) j : ( j + 1) + jt ≤ n − t (cid:9) one has j (1 + t ) ≤ n − t −
1. Therefore, j ≤ n − (1 + t )1 + t = n t − , and j max = (cid:22) n t (cid:23) − . Now, we want to verify that ω j = max Ω i , for j = 1 , . . . , j max .Assume max Ω j − = ω j − . Since,min Shad t (cid:0) B t ( ω j − ) (cid:1) = ω j − x n − t = (cid:18) j − Y i =0 x i + it (cid:19) x j +( j − t x d +( k − t x d + kt ,
9y Lemma 3.1, one hasmax Ω j = (cid:18) j − Y i =0 x i + it (cid:19) x ( j +1)+( j − t x ( j +1)+ jt x n = ω j . Note that (3) describes also the ω j ’s of the case k = 3.In the sequel, the monomials ω j ( j = 1 , . . . , j max ) will be called basic monomials ofthe first type , or also basic forward monomials , because each monomial ω j is obtainedby changing the penultimate variable of the preceding monomial ( ω j − ) of the list, asnext example illustrates. Example 3.4
Let n = 46 and t = 3, we can write n = 1 + 15 t . We determine j max . j max = (cid:22) n t (cid:23) − (cid:22) (cid:23) − . Firstly, we set ω = x x n = x x . Then, we have the following j max “further mono-mials”: ω j := (cid:18) j − Y i =0 x i + it (cid:19) x ( j +1)+ jt x d + kt = x x t · · · x ( j +1)+( j − t x ( j +1)+ jt x d + kt , for all j = 1 , . . . ,
10. More in details, ω = x x x , ω = x x x x x x x x ,ω = x x x x , ω = x x x x x x x x x ,ω = x x x x x , ω = x x x x x x x x x x ,ω = x x x x x x , ω = x x x x x x x x x x x ,ω = x x x x x x x , ω = x x x x x x x x x x x x . Note that every monomial ω i of the list can be obtained by changing the second tolast variable of the previous monomial ω i − = x q · · · x q r of the list by adding 1 to theindex of such a variable and inserting a new variable indexed by q r − + 1 + t .For example, ω = x x x x and ω = x x x x x .Observe, that in this case we can construct another monomial of the kind describedin the Claim . Indeed Ω is not empty and it is easy to verify that ω j max +1 = ω = max Ω = x x x x x x x x x x x x x . .1.2 Basic monomials of the second type Example 3.4 suggests us the construction of further t –spread monomials of S whichwill be fundamental for our aim. Such monomials will be called basic monomials ofthe second type .Let us consider the t –spread monomial of degree j max + 2 ω j max = x i x i · · · x i ( j max+2) = (cid:18) j max − Y i =0 x i + it (cid:19) x ( j max +1)+ j max t x d + kt . We observe that i m +1 − i m = t + 1 for all m = 1 , . . . , j max −
1, and i j max +1 − i j max = t .Moreover, d + kt − [( j max + 1) + j max t ] = n − [( j max + 1) + j max t ] ≤ t. Indeed, if n − [( j max + 1) + j max t ] > t , then n − t > ( j max + 1) + j max t + t . Hence, n − t ≥ ( j max + 2) + ( j max + 1) t and j max + 1 would be an integer greater than j max which belongs to the set (cid:8) j : ( j + 1) + jt ≤ n − t (cid:9) . It is an absurd. Finally, n − [( j max + 1) + j max t ] ≤ t .Now, let us examine the integer s = 2 t − h n − [( j max + 1) + j max t ] i = 2 t − n + j max (1 + t ) + 1 . We need to distinguish two cases: j max − − s ≥ j max − − s < j max − − s ≥
1. In such a case, v = min Shad t ( B t ( ω j max )) = x i x i · · · x i ( j max − − s ) (cid:18) k Y i = k − − s x d + it (cid:19) . By Lemma 3.1, ω j max +1 does exist and it is the largest t –spread monomial u of degree j max + 3, with max( u ) = n following v in the squarefree lexicographic order: ω j max +1 = x i ( j max − − s )+1 ( v/x i ( j max − − s ) ) . Now we focus on the variable we are going to “change” in v in order to obtain themonomial ω j max +1 : i ( j max − − s ) + 1 = 2 + j max − − s + ( j max − − s ) t + 1= 2 + j max − − t + n − j max (1 + t ) − j max − − s ) t + 1= d + ( k − − s ) t. j max +1 = Ω ⌊ n t ⌋ is ω j max +1 = max Ω j max +1 = x i x i · · · x i ( j max − − s ) (cid:18) k Y i = k − − s x d + it (cid:19) . We call such monomial the critic monomial , since from now on the next monomials weare going to construct are no longer obtained by changing the penultimate variable.
Example 3.5
Let us consider again Example 3.4. In such a case, it is s = 2 t − h n − [( j max + 1) + j max t ] i = 6 − [46 −
41] = 1 . Since j max − − s ≥
1, then ω j max +1 exists. Setting ω j max = ω = x i x i · · · x i , then i ( j max − − s ) + 1 = i − − + 1 = i + 1 = 31 and the critic monomial ω j max +1 = ω isthe following one ω j max +1 = x i x i · · · x i ( j max − − s ) (cid:18) k Y i = k − − s x d + it (cid:19) = x x x x x x x x x x x x x . Observe that ω j max +2 exists and ω j max +2 = x x x x x x x x x x x x x x . Now, our question is:
How may admissible t –spread monomials can we constructstarting from ω j max +1 ? Consider the critic monomial ω j max +1 = x i x i · · · x i ( j max − − s ) (cid:18) k Y i = k − − s x d + it (cid:19) . Since i m +1 − i m = t + 1 for all m = 1 , . . . , j max − − s , from Lemma 3.1 (see (2)), wehave max Ω j max +2 = ω j max +2 = x i x i · · · x i ( j max − − s − t − (cid:18) k Y i = k − − s − t − x d + it (cid:19) . Proceeding in such a way, we can get the further ω j max +1+ ν monomials, ω j max +1+ ν = x i x i · · · x i ( j max − − s − νt ) (cid:18) k Y i = k − − s − ν (1+ t ) x d + it (cid:19) , as long as i ( j max − − s − νt ) ≥ i , i.e. j max − − s − νt ≥ ν max = max (cid:8) ν : j max − − s − νt ≥ (cid:9) . If ν is such that j max − − s − νt ≥
1, then νt ≤ j max − − s . Hence νt ≤ j max − − s = j max − − t + n − j max (1 + t ) − j max − − t + d + kt − j max − j max t − d − k − − j max ) t. Thus, we have ν max = $ d − k − − j max ) tt % = (cid:22) d − t (cid:23) + k − − j max . So we can construct other ν max t –spread monomials ω j max +2 , . . . , ω j max + ν max +1 .Finally, we have constructed the following t –spread monomials of S :- ω = x x n (one monomial);- ω , . . . , ω j max ( j max basic monomials of the first type);- ω j max +1 , ω j max +2 , . . . , ω j max +1+ ν max ( ν max + 1 monomials)which satisfy the Claim . Their total number is1 + j max + 1 + ν max = 1 + j max + 1 + (cid:22) d − t (cid:23) + k − − j max = k + (cid:22) d − t (cid:23) . The monomials ω j max +1 , ω j max +2 , . . . , ω j max +1+ ν max will be called basic monomialsof second type , or also basic backward monomials , because each of these monomials isobtained, for all ν = 0 , . . . , ν max , by changing the ( j max − − s − νt + 1) − th variableof the preceding monomial of the list.Recall that we are considering t ≥
2. We observe that k + (cid:22) d − t (cid:23) = k − d = 1 , ,k if 3 ≤ d ≤ t. Let us show that ω j max + ν max +1 is the last monomial which satisfies the Claim , i.e. Ω j max + ν max +2 = ∅ .We need to examine some cases.If d = 1 or d = 2, then if one may construct another monomial of the type describedin the Claim , its degree would be j max + ν max + 4 = k + ⌊ d − t ⌋ + 2 = k + 1 and | M n,k +1 ,t | = (cid:18) n − ( k + 1 − t − k + 1 (cid:19) = (cid:18) d + kt − kt + kk + 1 (cid:19) = (cid:18) d + kk + 1 (cid:19) . d = 1, one has | M n,k +1 ,t | = 1 and M n,k +1 ,t = { x x t · · · x kt } . Since, x x t · · · x kt ∈ Shad k − t ( B t ( ω )) = Shad k − t ( B t ( x x kt )), then we have Ω j max + ν max +2 = ∅ ; whereas, if d = 2, then | M n,k +1 ,t | = k + 1. Moreover, in such a case, max M n,k +1 ,t = x x t · · · x kt and min M n,k +1 ,t = x x t · · · x kt . Let z ∈ M n,k +1 ,t with max( z ) =2+ kt . If min( z ) = 1, then z ∈ Shad k − t (cid:0) B t ( ω ) (cid:1) ; if min( z ) = 2, then z ∈ Shad k − t (cid:0) B t ( ω ) (cid:1) .Therefore, Ω j max + ν max +2 = ∅ .Now, let 3 ≤ d ≤ t . If one could construct another monomial of the type describedin the Claim , then its degree would be equal to j max + ν max + 4 = k + ⌊ d − t ⌋ + 2 = k + 2and | M n,k +2 ,t | = (cid:18) n − ( k + 2 − t − k + 2 (cid:19) = (cid:18) k + 1 + d − tk + 2 (cid:19) = 0 . In fact k +1+ d − t < k +2. Hence M n,j max + ν max +4 ,t = M n,k +2 ,t = ∅ and Ω j max + ν max +2 = ∅ . Hence, in every admissible case, Ω j max + ν max +2 = ∅ and consequently ω j max + ν max +1 isthe last t –spread monomial of the type described in the Claim that one may construct.
Example 3.6
We consider again Example 3.4. In such case, ν max = (cid:4) d − t (cid:5) + k − − j max = − − −
10 = 2 . There are ν max = 2 monomials of the second type todetermine. We set ω j max +1 = x i x i · · · x i ( j max − − s ) (cid:18) k Y i = k − − s x d + it (cid:19) = x x x x x x x x x x x x x . We determine ω j max +2 by shifting backward by t = 3, i.e. ω = x x x x x x x x x x x x x x x x , y ω = x x x x x x x x x x x x x x . It remains to determine ω j max + ν max +1 = ω j max +3 = ω . Shifting backward by t = 3again, we have ω = x x x x x x x x x x x x x x x x x , y ω = x x x x x x x x x x x x x x x . Hence, we have obtained all the monomials we need.It may happen that ω j max +1 does not exist, as next example shows.14 xample 3.7 Let n = 32 and t = 5, we can write n = 2+6 t . Then j max = j n t k − (cid:4) (cid:5) − ω j max = ω = x x x x x x . Observe that s = 2 t − h n − [( j max + 1) + j max t ] i = 10 − [32 −
25] = 3, and j max − − s =0 <
1, therefore ω j max +1 does not exist. Since ν max = (cid:22) d − t (cid:23) + k − − j max = − , then the total number of monomials constructed is j max + 1 = 5. On the other hand,we can note that j max + ν max + 2 = k + (cid:22) d − t (cid:23) = 6 − . It is important to underline that in Example 3.7, even though ω j max +1 does notexist, the formula k + (cid:4) d − t (cid:5) works well. Such a situation has forced us to analyze thecase above.Let j max − − s < ω j max is the last monomial of the type described in the Claim thatwe can construct, and consequently we get j max + 1 = j n t k monomials.We show that in this case k + (cid:4) d − t (cid:5) = j max + 1.In fact, j max < s = 3 + 2 t − n + j max + j max t and so j max t > n − − t =( d −
3) + ( k − t . Hence j max > (cid:4) d − t (cid:5) + k −
2, so j max ≥ (cid:4) d − t (cid:5) + k −
1. Moreover, ν max = (cid:22) d − t (cid:23) + k − − j max ≤ (cid:22) d − t (cid:23) + k − − (cid:18)(cid:22) d − t (cid:23) + k − (cid:19) = − . If we show that ν max = −
1, then we will have k + (cid:22) d − t (cid:23) = j max + ν max + 2 = j max + 1 = (cid:22) n t (cid:23) , as desired. Indeed, if ν max ≤ −
2, then (cid:4) d − t (cid:5) + k − j max ≤ i.e. k ≤ j max − (cid:22) d − t (cid:23) = (cid:22) n t (cid:23) − − (cid:22) d − t (cid:23) . (4)Now, we need to consider two possible cases.If d = 1 or d = 2, then (cid:4) d − t (cid:5) = −
1, and k ≤ j n t k − − (cid:4) d − t (cid:5) = j n t k ≤ n t .Hence, k (1 + t ) ≤ n = d + kt and consequently k ≤ d ; this is absurd since k ≥ d ≤
2. 15f 3 ≤ d ≤ t , then (cid:4) d − t (cid:5) = 0 and k ≤ j n t k − − (cid:4) d − t (cid:5) = j n t k − ≤ n t −
1. Itfollows that k (1 + t ) ≤ n − − t = d + kt − − t. Hence k ≤ d − − t . But d ≤ t , so k ≤ t − − t = −
1. This is an absurd. Indeed, k ≥ ν max = −
1, as desired.
By the materials in Section 3.1, we are able to state the main result in the paper.
Theorem 4.1
Let n, t, k be three positive integers such that n, t ≥ and k ≥ .Assume n = d + kt, ≤ d ≤ t. Then, every ideal I ∈ S t,n, of initial degree two and with a corner in degree two canhave at most k + (cid:22) d − t (cid:23) = k − if d = 1 , ,k if ≤ d ≤ t, corners. Proof.
Let us consider the k + (cid:4) d − t (cid:5) monomials of S = K [ x , . . . , x n ] defined in the Claim and consider the t –strongly stable ideal I = B t (cid:0) ω , ω , . . . , ω j max , ω j max +1 , ω j max +2 , . . . , ω j max +1+ ν max (cid:1) . The construction of the monomials ω j , together with Characterization 2.4, guaranteesthat I is an ideal of S t,n, with a corner in degree two and such that | Corn( I ) | = k + (cid:22) d − t (cid:23) = k − d = 1 , ,k if 3 ≤ d ≤ t. More in details,
Corn( I ) = (cid:26) ( k i , ℓ i ) : k i = n − t ( ℓ i − − , ℓ i = 2 + ( i − , i = 1 , . . . , k + (cid:22) d − t (cid:23) (cid:27) = (cid:26) ( n − t − , , ( n − t − , , . . . , (cid:18) n − (cid:16) k + (cid:22) d − t (cid:23) (cid:17) t − , k + (cid:22) d − t (cid:23) + 1 (cid:19) (cid:27) . It is clear that | Corn( I ) | is the maximum number of corners for a t –spread stronglystable ideal of S . (cid:3) The results obtained in [3] are now consequences of Theorem 4.1.16 orollary 4.2 ([3, Theorem 2])
Let n ≥ be odd. A –spread strongly stable ideal I of S of initial degree two and with a corner in degree two can have at most n − corners. Proof.
It is sufficient to write n = d + kt = 1 + 2 k , with d = 1 , t = 2 and k ≥ (cid:3) Corollary 4.3 ([3, Theorem 4]).
Let n ≥ be even. A –spread strongly stable ideal I of S of initial degree two and with a corner in degree two can have at most n − corners. Proof.
It is sufficient to write n = d + kt = 2 + 2 k , with d = 2 , t = 2 and k ≥ (cid:3) From Characterization 2.4 and Theorem 4.1, next result follows.
Theorem 4.4
Let n = d + kt be a positive integer, with t ≥ , ≤ d ≤ t and k ≥ .Set ℓ = 2 . Given r = k + (cid:4) d − t (cid:5) pairs of positive integers ( k , ℓ ) , ( k , ℓ ) , . . . , ( k r , ℓ r ) , (5) with ≤ k r < k r − < · · · < k ≤ n − t − and ℓ < ℓ < · · · < ℓ r ≤ k + (cid:4) d − t (cid:5) + 1 ,then there exists a t –spread strongly stable ideal of S = K [ x , . . . , x n ] of initial degree ℓ = 2 and with the pairs in (5) as corners if and only if k j + t ( ℓ j −
1) + 1 = n , for all j = 1 , . . . , k + (cid:4) d − t (cid:5) . We finish this Section with an example which illustrates our methods.
Example 4.5
Let n = 14 and t = 3, we can write n = 2 + 4 t . We determine j max and ν max . j max = (cid:22) n t (cid:23) − (cid:22) (cid:23) − ,ν max = (cid:22) d − t (cid:23) + k − − j max = − . Since ν max = −
1, then the critic monomial does not exist. Setting, ω = x x n = x x ,then, we have two forward monomials ω = x x x ,ω = x x x x . Hence I = B (cid:0) x x , x x x , x x x x (cid:1) = (cid:0) x x , x x , x x , x x , x x , x x , x x , x x , x x , x x , x x ,x x x , x x x , x x x , x x x , x x x , x x x , x x x ,x x x x , x x x x , x x x x (cid:1)
17s the 3–spread strongly stable ideal we are looking for. The highlithed monomials arethe 3–spread Borel generators of I . The Betti diagram of I is0 1 2 3 4 5 6 7 8 9 102 : 11 55 165 330 462 462 330 165 55 11 13 : 7 28 56 70 56 28 8 1 - - -4 : 3 9 10 5 1 - - - - - - Theorem 4.1 gives the maximal number of corners allowed for a t –spread stronglystable ideal whenever the initial degree of the ideal is two.Nevertheless, it is worthy to see how this number changes with respect to the initialdegree of the given t –spread strongly stable ideal I .In this Section, if I is t –spread strongly stable ideal, we focus on indeg I = ℓ ≥ ℓ = 2 (Theorem 4.1). Theorem 5.1
Let n, t, k be three positive integers such that n, t ≥ and k ≥ .Assume n = d + kt, ≤ d ≤ t. Then, every ideal I ∈ S t,n, of initial degree ℓ , ≤ ℓ ≤ k + (cid:4) d − t (cid:5) + 1 , and with acorner in degree ℓ can have at most k + (cid:22) d − t (cid:23) − ( ℓ − corners. Proof.
The proof is very similar to that of Theorem 4.1. We prove the existenceof a t –spread strongly stable ideal I ∈ S t,n, of initial degree ℓ generated in degrees ℓ , ℓ + 1 , . . . , k + (cid:4) d − t (cid:5) − ( ℓ −
2) + ℓ − k + (cid:4) d − t (cid:5) + 1 and such that | Corn( I ) | = k + (cid:22) d − t (cid:23) − ( ℓ − . Firstly, we set ω = x x t · · · x ℓ − t x n and G ( I ) = B t ( ω ).We claim that for all j = 1 , , . . . , k + (cid:4) d − t (cid:5) + 1, there exist t –spread monomials ω , ω , . . . , ω k + ⌊ d − t ⌋ − ( ℓ − − such that ω j = max > slex n u ∈ M n,j + ℓ ,t : u / ∈ j − [ i =0 Shad j − it ( B t ( ω i )) and max( u ) = n o . j := n u ∈ M n,j + ℓ ,t : u / ∈ j − [ i =0 Shad j − it ( B t ( ω i )) and max( u ) = n o . Let us consider the case k = 3.If d = 1, then n = 1 + 3 t and ℓ = 3. Then, we set ω = x x t x t . Since M t, ,t = { x x t x t x t } and x x t x t x t ∈ Shad t ( B t ( ω )), we cannotconstruct ω and, in such a case, a t –spread strongly stable ideal with ℓ = 3 can haveat most one corner.If d ≥
2, then n = d + 3 t .In particular, if d = 2, then 3 ≤ ℓ ≤ k + (cid:4) d − t (cid:5) + 1 = 3. Since ℓ = 3, we set ω = x x t x d +3 t and ω = x x t x t x d +3 t . Hence, we get two corners.If d ≥
3, then k + (cid:4) d − t (cid:5) + 1 = 4 and ℓ ∈ { , } . If ℓ = 3, we set ω = x x t x d +3 t ,and ω = x x t x t x d +3 t . Since | M n, ,t | = (cid:0) d +4 − t (cid:1) = 0, we cannot construct ω andwe can have at most two corners. If ℓ = 4, setting ω = x x t x t x d +3 t , since M n, ,t = ∅ , we can have at most one corner.Let k ≥
4. For j ≥
1, we consider the monomials ω j := x x t · · · x ℓ − t (cid:18) j − Y i =0 x i +( ℓ − i ) t (cid:19) x ( j +1)+( ℓ − j ) t x d + kt = x x t · · · x ℓ − t x ℓ − t · · · x ( j +1)+( ℓ − j ) t x d + kt . (6)The monomials ω j are t –spread as long as j is such that ( j +1)+( ℓ − j ) t ≤ n − t .We determine the greatest such an integer. We have j max = max (cid:8) j : ( j + 1) + ( ℓ − j ) t ≤ n − t (cid:9) . Proceeding as in the initial degree two case, we have that j max = (cid:22) n − ( ℓ − t t (cid:23) − , and ω j = max Ω j , for all j = 1 , . . . , j max . Now, let ω j max = x i x i · · · x i ( j max+ ℓ = x x t · · · x ℓ − t (cid:18) j max − Y i =0 x i +( ℓ − i ) t (cid:19) x ( j max +1)+( ℓ − j max ) t x d + kt . We have d + kt − [( j max + 1) + ( ℓ − j max ) t ] = n − [( j max + 1) + ( ℓ − j max ) t ] ≤ t s = 2 t − h n − [( j max + 1) + ( ℓ − j max ) t ] i = 2 t − n + j max (1 + t ) + 1 + ( ℓ − t. Hence, it follows that i ( j max + ℓ − − s ) + 1 = 2 + j max − − s + ( ℓ − j max − − s ) t + 1 = d + ( k − − s ) t. Now, we distinguish two cases: j max + ℓ − − s ≥ ℓ − j max + ℓ − − s < ℓ − j max + ℓ − − s ≥ ℓ −
2. As in Theorem 4.1,max Ω j max +1 = ω j max +1 = x i x i · · · x i ( j max+ ℓ − − s ) (cid:18) k Y i = k − − s x d + it (cid:19) . (7)Observe that i ℓ − = 1 + ( ℓ − t and i ℓ − = 2 + ( ℓ − t . Since i m +1 − i m = t, for m = 1 , . . . , ℓ − ,t + 1 , for m = ℓ − , . . . , j max + ℓ − − s, then max Ω j max +2 = ω j max +2 = x i x i · · · x i ( j max+ ℓ − − s − t ) (cid:18) k Y i = k − − s − t − x d + it (cid:19) . Finally, we can construct the monomials ω j max +1+ ν = x i x i · · · x i ( j max+ ℓ − − s − νt ) (cid:18) k Y i = k − − s − ν (1+ t ) x d + it (cid:19) , as long as i ( j max + ℓ − − s − νt ) ≥ i ℓ − , i.e. j max + ℓ − − s − νt ≥ ℓ − i.e. j max − − s − νt ≥
0. Now, let us determine ν max = max (cid:8) ν : j max − − s − νt ≥ (cid:9) . If ν is such that j max − − s − νt ≥
0, then νt ≤ j max − − s . Hence νt ≤ j max − − s = j max − − t + n − j max (1 + t ) − − ( ℓ − t = j max − − t + d + kt − j max − j max t − − ( ℓ − t = d − (cid:0) k − − j max − ( ℓ − (cid:1) t.
20e have ν max = $ d − (cid:0) k − − j max − ( ℓ − (cid:1) tt % = (cid:22) d − t (cid:23) + k − − j max − ( ℓ − , and we can construct further ν max t –spread monomials ω j max +2 , . . . , ω j max + ν max +1 . Finally, we have constructed the 1 + j max + 1 + ν max monomials ω , ω , . . . , ω j max , ω j max +1 , ω j max +2 , . . . , ω j max +1+ ν max which satisfy our claim. Note that1 + j max + 1 + ν max = 1 + j max + 1 + (cid:22) d − t (cid:23) + k − − j max − ( ℓ − k + (cid:22) d − t (cid:23) − ( ℓ − . As in the initial degree two case, ω j max + ν max +1 is the last monomial of the type requiredin the claim, that we can construct.Now, suppose j max + ℓ − − s < ℓ − j max ≥ (cid:4) d − t (cid:5) + k − − j max − ( ℓ −
2) and ν max ≤ − ν max = − j max + 1 monomials ( ω , ω , . . . , ω j max ). Moreover, j max + 1 = j max + 1 + 1 + ν max = k + (cid:22) d − t (cid:23) − ( ℓ − . It is important to underline that in both cases we determine k + (cid:4) d − t (cid:5) − ( ℓ − ω j max +1 does not exist.Setting I = B t (cid:0) ω , ω , . . . , ω j max , ω j max +1 , ω j max +2 , . . . , ω j max +1+ ν max (cid:1) , the existence of the monomials ω j , together with Characterization 2.4, guarantees thatthe ideal I is an ideal of S t,n, with a corner in degree ℓ in S and such that | Corn( I ) | = k + (cid:22) d − t (cid:23) − ( ℓ − . (cid:3) The next result (analogous to Theorem 4.4) covers the case t = 1 in [2] and thecases t ≥ Theorem 5.2
Let n = d + kt be a positive integer, with t ≥ , ≤ d ≤ t and k ≥ . et ≤ r ≤ k + (cid:4) d − t (cid:5) be an integer. Given r pairs of positive integers ( k , ℓ ) , ( k , ℓ ) , . . . , ( k r , ℓ r ) , (8) with ≤ k r < k r − < · · · < k ≤ n − t − and ≤ ℓ < ℓ < · · · < ℓ r ≤ k + (cid:4) d − t (cid:5) + 1 ,then there exists a t –spread strongly stable ideal of S = K [ x , . . . , x n ] of initial degree ℓ and with the pairs in (8) as corners if and only if k j + t ( ℓ j −
1) + 1 = n , for all j = 1 , . . . , r . Example 5.3
Let n = 138 and t = 11, we can write n = 6 + 12 t . Let ℓ = 5. Wehave: j max = (cid:22) n − ( ℓ − t t (cid:23) − (cid:22) (cid:23) − ,ν max = (cid:22) d − t (cid:23) + k − − j max − ( ℓ −
2) = 0 . Firstly, we set ω = x x t · · · x ℓ − t x n = x x x x x . Then we determine the j max = 7 monomials given by (6). More precisely, ω = x x x x x x ,ω = x x x x x x x , ω = x x x x x x x x x x ,ω = x x x x x x x x , ω = x x x x x x x x x x x ,ω = x x x x x x x x x , ω = x x x x x x x x x x x x . Following Theorem 5.1, we consider the integer s = 2 t − h n − [( j max + 1) + ( ℓ − j max ) t ] i = 22 − [138 − . Since j max + ℓ − − s ≥ ℓ −
2, then ω j max +1 exists. Let ω j max = ω = x i x i · · · x i j max+ ℓ = x i x i · · · x i , then i ( j max + ℓ − − s ) + 1 = i − − + 1 = i + 1 = 72 and ω j max +1 = ω is given by (7), i.e. ω = x i x i · · · x i ( j max+ ℓ − − s ) (cid:18) k Y i = k − − s x d + it (cid:19) = x x x x x x x x x x x x x . This is the last monomial that we can determine, since ν max = 0. Finally, I = B ( ω , . . . , ω ) is the desired t –spread strongly stable ideal. In this paper, following the approach used in [2] and [3], we have discussed the extremalBetti numbers of t –spread strongly stable ideals and we have determined the maximal22umber of admissible corners of a t –spread strongly stable ideal given its initial degree.As in [3], it is important to “decompose” the integer n with respect to t . In a certainway, we have divided n by t forcing the rest of the division to lie in the set { , . . . , t } .In [2], a numerical characterization of the possible extremal Betti numbers (values aswell as positions) of the class of squarefree strongly stable ideals was given. Theorem5.2 characterizes the positions of the extremal Betti numbers of the class of t –spreadstrongly stable ideals in the Betti diagram. Nothing is known about the possible valuesof the extremal Betti numbers of such a class of ideals. This question is currently underinvestigation by the authors of this paper. References [1] L. Amata, M. Crupi. Computation of graded ideals with given extremal Betti numbersin a polynomial ring. J. Symbolic Computation (2019), 120–132.[2] L. Amata, M. Crupi. On the extremal Betti numbers of squarefree monomial ideals. IEJA,to appear[3] L. Amata, M. Crupi. Extremal Betti Numbers of t-Spread Strongly Stable Ideals. Math-ematics (2019), 695.[4] C. Andrei, V. Ene, B. Lajmiri. Powers of t-spread principal Borel ideals. Archiv derMathematik (2018), 1–11.[5] C. Andrei-Ciobanu, Kruskal–Katona Theorem for t –spread strongly stable ideals. Bull.Math. Soc. Sci. Math. Roum. 2019, 62, 107–122.[6] A. Aramova, J. Herzog, T. Hibi. Squarefree lexsegment ideals. Math.Z. (1998), 353–378.[7] D. Bayer, H. Charalambous, S. Popescu. Extremal Betti numbers and Applications toMonomial Ideals. J. Algebra (1999), 497–512.[8] M. Crupi. Extremal Betti numbers of graded modules, J. Pure Appl. Algebra , (2016),2277–2288.[9] M. Crupi. Computing general strongly stable modules with given extremal Betti numbers.J. Com.Alg., (1) (2020), 53–70[10] M. Crupi, C. Ferr`o. Squarefree monomial modules and extremal Betti numbers, AlgebraColloq. (2016), 519-530.[11] R. Dinu. Gorenstein T -spread Veronese algebras, Osaka J. Math., (4) (2020), 935–947.[12] S. Eliahou, M. Kervaire. Minimal resolutions of some monomial ideals, J. Algebra, (1990), 1–25.[13] V. Ene, J. Herzog, A. A. Qureshi. t-spread strongly stable monomial ideals. Com. Algebra, (12)(2019).[14] D. R. Grayson, M. E. Stillman, Macaulay2, a software system for research in algebraicgeometry . Available at .
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