On the extremal Betti numbers of squarefree monomial ideals
aa r X i v : . [ m a t h . A C ] F e b On the extremal Betti numbers of squarefree monomialideals
Luca Amata, Marilena Crupi*
Department of Mathematics and Computer Sciences, Physics and Earth Sci-ences, University of Messina, Viale Ferdinando Stagno d’Alcontres 31, 98166 Messina,Italy
E-mail addresses : [email protected] ; [email protected] Abstract
Let K be a field and S = K [ x , . . . , x n ] be a polynomial ring over K . Wediscuss the behaviour of the extremal Betti numbers of the class of squarefreestrongly stable ideals. More precisely, we give a numerical characterizationof the possible extremal Betti numbers (values as well as positions) of sucha class of squarefree monomial ideals. Let K be a field and S = K [ x , . . . , x n ] be the polynomial ring in n variables withcoefficients in K . A squarefree monomial ideal of S is a monomial ideal generatedby squarefree monomials. Such ideals are also known as Stanley–Reisner ideals,and quotients by them are called Stanley–Reisner rings. The combinatorial na-ture of these algebraic objects comes from their close connections to simplicialtopology. Many authors have studied the class of squarefree monomial idealsfrom the viewpoint of commutative algebra and combinatorics (see, for example[2, 3, 6, 18], and the references therein).Let I be a graded ideal of S . 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 ) [4]. The pair( k, ℓ ) is called a corner of I . If β k i ,k i + ℓ i ( I ) ( i = 1 , . . . , r ) are extremal Bettinumbers of a graded ideal I , then the set Corn( I ) = { ( k , ℓ ) , ( k , ℓ ) , . . . , ( k r , ℓ r ) } will be called the corner sequence of I [7, Definition 4.1]. In the Macaulay orCoCoA Betti diagram of I , the graded Betti number β i,j ( I ) is plotted in column Keywords: graded ideals, squarefree monomial ideals, minimal graded resolutions.
Corresponding author: Marilena Crupi; email: [email protected]. and row j − i . Using such a notation, a graded Betti number β k,k + ℓ ( I ) isextremal if it is the only entry in the quadrant where it is the northwest corner.Projective dimension measures the column index of the easternmost extremalBetti number, whereas regularity measures the row index of the southernmostextremal Betti number. Indeed, the extremal Betti numbers are a generalizationof such meaningful algebraic invariants.For a monomial ideal I of S , let as denote by G ( I ) the unique minimalset of monomial generators of I and for a monomial 1 = u ∈ S , let us definesupp( u ) = { i : x i divides u } . A monomial ideal I of S is strongly stable if for all u ∈ G ( I ) one has ( x j u ) /x i ∈ I for all i ∈ supp( u ) and all j < i [14, 16]; whereasa squarefree monomial ideal I of S is squarefree strongly stable if for all u ∈ G ( I )one has ( x j u ) /x i ∈ I for all i ∈ supp( u ) and all j < i , j / ∈ supp( u ) [2, 16].Assume that the characteristic of the base field K is zero. If I is a gradedideal of S , then the generic initial ideal Gin( I ), with respect to the reverse lex-icographical order on S induced by x > · · · > x n , is a strongly stable ideal of S (see, for instance, [13, 16]). If I is squarefree, then Gin( I ) is not in generalsquarefree. In [3], the authors have introduced a certain operator σ which trans-forms Gin( I ) to a squarefree monomial ideal of S . Such an ideal, denoted byGin( I ) σ , is squarefree strongly stable [3, Lemma 1.2.]. On the other hand, [3,Theorem 2.4.] assures that if I is a squarefree ideal then the extremal Betti num-bers are preserved when we pass from I to Gin( I ) σ . Hence, if one wants to studythe extremal Betti numbers of squarefree monomial ideals in a polynomial ring S = K [ x , . . . , x n ] with char( K ) = 0, it is not restrictive to consider the behaviorof such extremal Betti numbers for the class of squarefree strongly stable ideals.In this paper, we are interested to the study of the extremal Betti numbersof the class of squarefree strongly stable ideals of S .The first result on the behavior of the extremal Betti numbers of such aclass of squarefree monomial ideals can be found in [12, Propostion 4.1]. Moreprecisely, the authors in [12] gave a criterion to determine whether a graded Bettinumber is extremal: let I be a squarefree strongly stable ideal of S . β k, k + ℓ ( I ) isan extremal Betti number if and only if k + ℓ = max { max( u ) : u ∈ G ( I ) ℓ } and max( u ) < k + j , for all j > ℓ and for all u ∈ G ( I ) j (Characterization 2.6); G ( I ) ℓ is the set of monomials u of G ( I ) such that deg u = ℓ . They did not giveany numerical charaterization of the possible extremal Betti numbers of such aclass of ideals. Later, such a criterion was generalized to the class of squarefreestrongly stable submodules of a finitely generated graded free S –module with ahomogeneous basis in [10, Theorem 4.3]. Moreover, a criterion for determiningtheir positions and their number was also given in [10, Section 5]. Such a criterionwill be an important tool for the development of this article.Differently from the non–squarefree case, not much is known about the nu-merical characterization of the possible extremal Betti numbers (values and po-2itions) of the class of squarefree strongly stable ideals. Indeed, many authorshave faced and solved such a question for the class of strongly stable ideals in S ([1, 7, 8, 9, 11, 12, 17]). More precisely, the authors of the previous papers haveexamined the following problem: Problem 1.1
Given two positive integers n, r , ≤ r ≤ n − , r pairs of positiveintegers ( k , ℓ ) , . . . , ( k r , ℓ r ) such that n − ≥ k > k > · · · > k r ≥ and ≤ ℓ < ℓ < · · · < ℓ r and r positive integers a , . . . , a r , under which conditionsdoes there exist a graded ideal I of S = K [ x , . . . , x n ] such that β k ,k + ℓ ( I ) = a , . . . , β k r ,k r + ℓ r ( I ) = a r are its extremal Betti numbers? Positive answers to Problem 1.1 can be found in [7, Propositions 2.5, 3.5,Theorem 3.7], [11, Theorem 3.1] and [17, Theorem 3.7] when K is a field ofcharacteristic 0 (see also [1, Proposition 3.1, Theorem 3.2]). More specifically, inall the previous cited papers, numerical characterizations of the possible extremalBetti numbers of a graded ideal I of initial degree ≥ S (withrespect to the reverse lexicographical order on S ) is strongly stable and since theextremal Betti numbers are preserved by passing from the graded ideal to itsgeneric ideal [4], the problem is equivalent to the characterization of the possibleextremal Betti numbers of a strongly stable ideal of S . Moreover, in [1] a CoCoApackage for computing the smallest strongly stable ideal of S to face Problem1.1 has been developed. In particular, the package is able to determine all thepossible r -tuples of positive integers ( a , . . . , a r ) for which such an ideal doesexist. Finally, a complete answer to such a problem reformulated in terms ofgraded submodules of a finitely generated graded free S –module has been statedin [7, Theorem 4.6], [8, Theorem 4.6] and [9, Theorem 1].The purpose of this paper is to numerically characterize the possible extremalBetti numbers of squarefree monomial ideals of a standard graded polynomialring S over a field of characteristic 0. Our techniques involve overall tools fromenumerative combinatorics.The plan of the paper is as follows. In Section 2, some notions that will be usedthroughout the paper are recalled. In Section 3, firstly we identify the admissiblecorner sequences of a squarefree strongly stable ideal for n = 2 , ,
4. Then, wedetermine the maximal number of corners allowed for a squarefree strongly stableideal I of S with a corner in its initial degree (Propositions 3.7, 3.9). Moreover,given n − ℓ ( n ≥
5) pairs of positive integers ( k , ℓ ) , ( k , ℓ ) , . . . , ( k n − ℓ , ℓ n − ℓ ),with 1 ≤ k n − ℓ < k n − ℓ − < · · · < k ≤ n − ≤ ℓ < ℓ < · · · < ℓ n − ℓ ≤ n −
1, we determine the conditions under which there exists a squarefree lexideal (Definition 2.3) I of K [ x , . . . , x n ] of initial degree ℓ having β k i ,k i + ℓ i ( I ), i = 1 , . . . , r , as extremal Betti numbers (Theorem 3.13). A complete description3f the minimal system of monomial generators of I is given. Squarefree lexideals are a subclass of the class of squarefree strongly stable ideals [2]. Finally,in Section 4, we face the squarefree version of Problem 1.1, i.e. , the followingproblem: Given three positive integers n ≥ , ℓ ≥ and ≤ r ≤ n − ℓ , r pairsof positive integers ( k , ℓ ) , . . . , ( k r , ℓ r ) such that n − ≥ k > k > · · · > k r ≥ and ≤ ℓ < ℓ < · · · < ℓ r , k i + ℓ i ≤ n ( i = 1 , . . . , r ), and r positive integers a , . . . , a r , under which conditions does there exist a squarefree monomial ideal I of S = K [ x , . . . , x n ] such that β k ,k + ℓ ( I ) = a , . . . , β k r ,k r + ℓ r ( I ) = a r are itsextremal Betti numbers? We solve such a problem when char( K ) = 0 (Theorem4.14). In such a case, the question is equivalent to the characterization of thepossible extremal Betti numbers of a squarefree strongly stable ideal of S as wehave pointed out. The idea behind Theorem 4.14 is to establish the bounds for theintegers a i ( i = 1 , . . . , r ), starting with a r and then arriving to a , by computingthe cardinality of suitable sets of monomials. The key result in this Section isTheorem 4.4. Let ( k, ℓ ) be a pair of positive integers and let A s ( k, ℓ ) be the setof all squarefree monomials u of S of degree ℓ and such that max( u ) = k + ℓ , withmax( u ) = max { i : x i divides u } , ordered by the squarefree lex order ≥ slex definedin Section 2. If u ∈ A s ( k, ℓ ), Theorem 4.4 shows a method for determining thecardinality of the set of all squarefree monomials w ∈ A s ( k, ℓ ) such that w ≥ slex u .We provide some examples illustrating the main obstructions to the issue. All theexamples are constructed by means of Macaulay2 packages [15], some of whichwere developed by the authors of this article.
Let us consider the polynomial ring S = K [ x , . . . , x n ] as an N -graded ring wheredeg x i = 1, i = 1 , . . . , n . A monomial ideal I of S is an ideal generated bymonomials. If I is a monomial ideal of S , we denote by G ( I ) the unique minimalset of monomial generators of I , by G ( I ) ℓ the set of monomials u of G ( I ) suchthat deg u = ℓ , and by G ( I ) ≥ ℓ the set of monomials u of G ( I ) such that deg u ≥ ℓ .If I = ⊕ j ≥ I j is a graded ideal of the polynomial ring S , we denote by indeg I the initial degree of I , i.e. , the minimum j such that I j = 0.For a monomial 1 = u ∈ S , we setsupp( u ) = { i : x i divides u } , and we writemax( u ) = max { i : i ∈ supp( u ) } , min( u ) = min { i : i ∈ supp( u ) } . moreover, we set max(1) = min(1) = 0. 4 monomial m ∈ S is called a squarefree monomial if m = x i x i · · · x i d with1 ≤ i < i < · · · < i d ≤ n. If T is a subset of S , we denote by Mon d ( T ) the setof all monomials in T and by Mon sd ( T ) the set of all squarefree monomials in T .A monomial ideal I is a squarefree monomial ideal if I is a monomial idealof S generated by squarefree monomials. Definition 2.1
Let I be a squarefree monomial ideal of S . I is called a squarefreestable ideal if for all u ∈ G ( I ) one has ( x j u ) /x max( u ) ∈ I for all j < max( u ) , j / ∈ supp( u ). I is called a squarefree strongly stable ideal if for all u ∈ G ( I ) one has ( x j u ) /x i ∈ I for all i ∈ supp( u ) and all j < i , j / ∈ supp( u ). Remark 2.2
Let T be a set of squarefree monomials in S of degree d . T will becalled a squarefree stable set if for all u ∈ T one has ( x j u ) /x max( u ) ∈ T for all j < max( u ) , j / ∈ supp( u ) . T will be called a squarefree strongly stable set if forall u ∈ T one has ( x j u ) /x i ∈ T for all i ∈ supp( u ) and all j < i , j / ∈ supp( u ) .Hence, a squarefree monomial ideal I of S is squarefree (strongly) stable if Mon sd ( I ) is a squarefree (strongly) stable set, for all d . For every 1 ≤ d ≤ n , we can order Mon sd ( S ) with the squarefree lexicographicorder ≥ slex [2]. More precisely, let u = x i x i · · · x i d , v = x j x j · · · x j d , with 1 ≤ i < i < · · · < i d ≤ n , 1 ≤ j < j < · · · < j d ≤ n , be squarefreemonomials of degree d in S , then u > slex v if i = j , . . . , i s − = j s − and i s < j s , (1)for some 1 ≤ s ≤ d .A nonempty set L ⊆ Mon sd ( S ) is called a squarefree lexsegment set of degree d if for u ∈ L , v ∈ Mon sd ( S ) such that v > slex u , then v ∈ L . Definition 2.3
Let I be a squarefree monomial ideal of S . I is a squarefreelexsegment ideal of S if for all squarefree monomials u ∈ I and all squarefreemonomials v of the same degree with v > slex u , it follows that v ∈ I . Example 2.4
Let S = K [ x , x , x , x , x ] . The ideal I = ( x x x , x x x ,x x x , x x x , x x x x ) is a squarefree lexsegment ideal of S . For any graded ideal I of S , there is a minimal graded free S -resolution [5] F . : 0 → F s → · · · → F → F → I → , where F i = ⊕ j ∈ Z S ( − j ) β i,j . The integers β i,j = β i,j ( I ) = dim K Tor i ( K, I ) j arecalled the graded Betti numbers of I . 5 efinition 2.5 [4] 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 I is a squarefree stable ideal, there exists a formula to compute the gradedBetti numbers of I ([2]): β k, k + ℓ ( I ) = X u ∈ G ( I ) ℓ (cid:18) max( u ) − ℓk (cid:19) . (2)Because of relation (2), next characterization holds true [10, 12]. Characterization 2.6
Let I be a squarefree stable ideal of S . β k, k + ℓ ( I ) is anextremal Betti number if and only if k + ℓ = max { max( u ) : u ∈ G ( I ) ℓ } and max( u ) < k + j , for all j > ℓ and for all u ∈ G ( I ) j . As a consequence of such a characterization, one has that if I is a squarefreestable ideal of S and β k, k + ℓ ( I ) is an extremal Betti number of I , then β k, k + ℓ ( I ) = |{ u ∈ G ( I ) ℓ : max( u ) = k + ℓ }| . (3)Moreover, setting ℓ = max { j : G ( I ) j = ∅} , m = max { max( u ) : u ∈ G ( I ) } , then β m − ℓ, m is the unique extremal Betti number of I if and only if m = max { max( u ) : u ∈ G ( I ) ℓ } , and max( w ) < m , for all w ∈ G ( I ) j with j < ℓ . Remark 2.7 If I is a squarefree stable monomial ideal of S and β k,k + ℓ ( I ) is anextremal Betti number of I , then from Characterization 2.6, we have the followingbound: ≤ β k,k + ℓ ( I ) ≤ (cid:18) k + ℓ − ℓ − (cid:19) . (4) In fact, there exist exactly (cid:0) k + ℓ − ℓ − (cid:1) squarefree monomials of degree ℓ in S with max( u ) = k + ℓ . Now, let ( k , ℓ ) , . . . , ( k r , ℓ r ) ( n − ≥ k > k > · · · > k r ≥
1, 1 ≤ ℓ < ℓ < · · · < ℓ r ) be corners of a graded ideal I , according to [7], the following notionscan be introduced:Corn( I ) = { ( k , ℓ ) , . . . , ( k r , ℓ r ) } , a ( I ) = ( β k ,k + ℓ ( I ) , . . . , β k r ,k r + ℓ r ( I )) . Corn( I ) is called the corner sequence of I , and a ( I ) the corner values sequence of I .If I is a squarefree ideal of S , then k i + ℓ i ≤ n , for all i = 1 , . . . , r .6 xample 2.8 Let S = K [ x , x , x , x , x , x ] and let I = ( 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 ) be a squarefree strongly stable ideal of S . The extremal Betti numbers of I are β , ( I ) = 2 , β , ( I ) = 1 , as the Betti table of I shows: − Hence, the corner sequence and the corner values sequence of I are Corn( I ) = { (3 , , (2 , } , and a ( I ) = (2 , , respectively. We close this Section with some notations from [10, Section 5] that will beuseful in the sequel.Let I be a squarefree stable ideal of S . If I is generated in one degree ℓ , then I has a unique extremal Betti number β m − ℓ, m ( I ), where m = max { max( u ) : u ∈ G ( I ) } .Assume I to be generated in degrees 1 ≤ ℓ < ℓ < · · · < ℓ t ≤ n , and denoteby [ t ] the set { , . . . , t } .Setting m ℓ j = max { max( u ) : u ∈ G ( I ) ℓ j } , for j = 1 , . . . , t , let us consider the following sequence of non negative integersassociated to I : ds ( I ) = ( m ℓ − ℓ , m ℓ − ℓ , . . . , m ℓ t − ℓ t ) . (5)Such a sequence is called the degree-sequence of I .One can observe that, if m ℓ − ℓ > m ℓ − ℓ > · · · > m ℓ t − ℓ t , (6)then, from Characterization 2.6, β m ℓi − ℓ i , m ℓi ( I ) is an extremal Betti number of I , for i = 1 , . . . , t . If (6) does not hold, one can construct a suitable subsequenceof the degree-sequence ds ( I ), say \ ds ( I ) = ( m ℓ i − ℓ i , m ℓ i − ℓ i , . . . , m ℓ iq − ℓ i q ) , (7)with ℓ ≤ ℓ i < ℓ i < · · · < ℓ i q = ℓ t , and such that, for j = 1 , . . . , q , β m ℓij − ℓ ij , m ℓij ( I ) is an extremal Betti number of I .The integer q ≤ t , denoted by dl ( I ), and called the degree-length of I , gives thenumber of the extremal Betti numbers of the squarefree stable ideal I .For more details on this subject see [10].7 Extremal Betti numbers of squarefree strongly sta-ble ideals
In this Section, we examine the extremal Betti numbers of squarefree stronglystable ideals in S = K [ x , . . . , x n ]. More precisely, we identify the admissiblecorner sequence of a squarefree strongly stable ideal in S .From now on, we assume Mon ℓs ( S ) to be endowed with the squarefree lexorder > slex induced by x > x > · · · > x n .At first, we analyze the simple cases occurring when n = 2 , Case 1.
Let n = 2 and S = K [ x , x ]. A squarefree strongly stable ideal I of S can have at most one corner. More precisely, Corn( I ) = { (1 , } with a ( I ) = (1), i.e. , I = ( x , x ). Case 2.
Let n = 3 and S = K [ x , x , x ]. Also in such a case, a squarefreestrongly stable ideal I of S can have at most one corner ( k, ℓ ), k + ℓ ≤
3. Indeed,the only situations that may occur are listed in Table 1.
Corners Corner values Squarefree strongly stable ideal
Corn( I ) = { (2 , } a ( I ) = (1) I = ( x , x , x )Corn( I ) = { (1 , } a ( I ) = (1) I = ( x , x )Corn( I ) = { (1 , } a ( I ) = (1) I = ( x x , x x )Corn( I ) = { (1 , } a ( I ) = (2) I = ( x x , x x , x x )Table 1: Corner sequences for n = 3.Such easy cases allow us to yield the next result. Proposition 3.1
Let S = K [ x , . . . , x n ] , n ≥ . If I is a squarefree stronglystable ideal of S with ( k, ∈ Corn( I ) , then | Corn( I ) | = 1 . More precisely, I = ( x , x , . . . , x k +1 ) .Proof. First of all one can observe that G ( I ) = { x , . . . , x k +1 } . If G ( I ) ≥ = ∅ ,then there exists a monomial u ∈ G ( I ) of degree ℓ ≥ u ) ≥ k + 2.A contradiction, since ( k,
1) is a corner of I . (cid:3) Now, let us consider the case n = 4. Case 3.
Let n = 4 and S = K [ x , x , x , x ]. Assume I to be a squarefreestrongly stable ideal S of initial degree ≥ k, ℓ ) ∈ Corn( I ) must satisfy the inequality k + ℓ ≤
4, the situations that canoccur in such a case are described in Table 2.8 orners Corner values Squarefree strongly stable ideal
Corn( I ) = { (2 , , (1 , } a ( I ) =(1,1) I = ( x x , x x , x x , x x x )Corn( I ) = { (1 , } a ( I ) = (1) I = ( x x , x x )Corn( I ) = { (1 , } a ( I ) = (2) I = ( x x , x x , x x )Corn( I ) = { (2 , } a ( I ) = (1) I = ( x x , x x , x x )Corn( I ) = { (2 , } a ( I ) = (2) I = ( x x , x x , x x , x x , x x )Corn( I ) = { (2 , } a ( I ) = (3) I = ( x x , x x , x x , x x , x x , x x )Corn( I ) = { (1 , } a ( I ) = (1) I = ( x x x , x x x )Corn( I ) = { (1 , } a ( I ) = (2) I = ( x x x , x x x , x x x )Corn( I ) = { (1 , } a ( I ) = (3) I = ( x x x , x x x , x x x , x x x ) Table 2: Corner sequences for n = 4. Remark 3.2
All the squarefree strongly stable ideals described in Tables 1 and2 are the smallest strongly stable ideals with the given data, with respect to theinclusion relation.
Let T be a subset of Mon sd ( S ), d < n . The set of squarefree monomials ofdegree d + 1 of S Shad( T ) = { x i u : u ∈ T, i / ∈ supp( u ) , i = 1 , . . . , n } is called the squarefree shadow of T . Moreover, we define the i -th squarefreeshadow recursively by Shad i ( T ) = Shad(Shad i − ( T )), i ≥
1, with Shad ( T ) = T . Next notion will be crucial for the further developments in this paper. Definition 3.3
Let u = x i · · · x i q be a squarefree monomial of S of degree q < n .We say that u has a j -gap if i j +1 − i j > ≤ j < q . The positiveinteger i j +1 − i j − j -gap.The j -gap of a squarefree monomial u = x i · · · x i q ∈ S will be denoted by j -gap( u ), whereas its width will be denoted by wd( j -gap( u )). Moreover, wedefine Gap( u ) := { j ∈ [ q ] : there exists a j -gap( u ) } . Definition 3.4
A squarefree monomial u = x i · · · x i q of S will be said gap–freeif Gap( u ) = ∅ . Example 3.5
Let S = K [ x , . . . , x ] . The monomial u = x x x x x ∈ S has three gaps. Indeed, Gap( u ) = { , , } , u ) , u ) have both widthequal to and u ) has width equal to ; on the contrary, the monomial v = x x x x x ∈ S is gap–free. emma 3.6 Let u = x i · · · x i q be a squarefree monomial of degree q < n − of S . Assume u has a gap whose width is ≥ , or u has at least two gaps.Then there exist at least two squarefree monomials v, w ∈ S of degree q + 1 with v > slex w , max( v ) = max( w ) = n and such that (i) v is a multiple of u ; (ii) w is not a multiple of u .Proof. If max( u ) < n , we can choose v = ux n = x i · · · x i q x n . Setting t =max Gap( v ), the greatest squarefree monomial following v in the squarefree lexorder is ˜ v = x i · · · x i t − x i t +1 · · · x i t + q − t +2 . If i t + q − t + 2 = n , we choose w = ˜ v , otherwise, if i t + q − t + 2 < n , we choose w = x i · · · x i t − x i t +1 · · · x i t + q − t +1 x n . Finally, v > slex w , u | v and u ∤ w . Notethat t ≤ q .Now, assume max( u ) = n . If t = max Gap( u ), let v = x i · · · x i t x i t +1 − x i t +1 · · · x i q − x i q = x i · · · x i t x i t +1 − x i t +1 · · · x i q − x n . Furthermore, if p = max Gap( v ), then the greatest squarefree monomial following v in the squarefree lex order is˜ v = x i · · · x i p − x i p +1 · · · x i p + q − p +2 . Hence, if i p + q − p + 2 = n , we choose w = ˜ v , otherwise, if i p + q − p + 2 < n ,we choose w = x i · · · x i p − x i p +1 · · · x i p + q − p +1 x n .Note that the assumption on the gaps of the squarefree monomial u assuresus that we can construct both the monomials v and w . (cid:3) Next results easily follow.
Proposition 3.7
Let I be a squarefree strongly stable ideal of S = K [ x , . . . , x n ] , n ≥ , with initial degree and with a corner in degree . Then (1) I has at most n − corners for n = 4 ; (2) I has at most n − corners for n ≥ .Proof. (1). It follows from Case 3.(2). Let n ≥
5. An admissible degree–sequence of I is the following one ds ( I ) = ( n − , n − , · · · , n − ( n −
2) = 2) . Indeed, setting w = x x n , since 1 -gap( w ) has width n −
2, then Lemma 3.6assures that there exist at least n − w , . . . , w n − in S
10f degrees 3 , . . . , n −
2, respectively, with max( w i ) = n , and n − v , . . . , v n − of degrees 3 , . . . , n −
2, respectively, with max( v i ) = n and such that v i > slex w i , w i − | v i , v i ∤ w i , for i = 2 , . . . , n −
3. Using the sametechniques as in Lemma 3.6, one can easily verify that w i ∤ w i +1 ( i = 1 , . . . , n − w v w v w v w v w v w v . . . The monomials w i ( i = 1 , . . . , n −
3) will be called basic monomials .Next tables list the basic monomials for n = 5 , . . . ,
9. For n ≥
10, theconstruction of such elements proceeds smoothly. n = v i w i x x x x x x x x x x x x − n = v i w i x x x x x x x x x x x x x x x x x x x x x − n = v i w i 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 n = v i w i 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 n = v i w i 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 Note that the construction of the basic elements ends up as soon as one getsa gap–free monomial. (cid:3) xample 3.8 Let S = K [ x , x , x , x , x , x , x , x ] , and let I = ( 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 ) be a squarefree strongly stable ideal of S . The degree-sequence of I is ds ( I ) = ( m − , m − , m − , m − , m −
6) = (6 , , , , .I has initial degree and dl ( I ) = 5 . The extremal Betti numbers of I are β − , ( I ) = β − , ( I ) = β − , ( I ) = β − , ( I ) = β − , ( I ) = 1 , as the Betti tableof I shows: − − − − − − − − − − Proposition 3.9
Let n ≥ and let I be a squarefree strongly stable ideal of S = K [ x , . . . , x n ] with initial degree ℓ ≥ and with a corner in degree ℓ . Then I has at most n − ℓ corners.Proof. Using the same reasoning as in Proposition 3.7, an admissible degree–sequence of I is the following one: ds ( I ) = ( n − ℓ, n − ( ℓ + 1) , · · · , n − ( n −
1) = 1) , with dl ( I ) = n − ℓ .Next tables show the basic monomials for n = 5 , . . . , ℓ = 3. For n ≥ ℓ = 3), the construction of such elements proceeds smoothly. n = v i w i x x x x x x x x x x x x x x x x − n = v i w i 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 − Also in this case, the construction of the basic elements ends up as soon asone gets a gap–free monomial. (cid:3) n = v i w i 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 n = v i w i 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 Example 3.10
Let S = K [ x , x , x , x , x , x , x , x ] and let I = ( 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 x x ) be a squarefree strongly stable ideal of S initial degree . The degree-sequence of I is ds ( I ) = ( m − , m − , m − , m − , m −
7) = (5 , , , , . The extremal Betti numbers of I are β − , ( I ) = β − , ( I ) = β − , ( I ) = β − , ( I )= β − , ( I ) = 1 , as the Betti table of I shows − − − − − − − − − − The next example considers a squarefree monomial ideal I of S without acorner in its initial degree, and shows the construction of a squarefree monomialideal J of S with a corner in its initial degree and with the same extremal Bettinumbers (positions and values) of I . Example 3.11
Consider the following monomial ideal I of S = K [ x , . . . , x ] : I = ( x x , x x x , x x x , x x x x ) .I is squarefree strongly stable of initial degree and with Corn( I ) = { (2 , , (1 , } . From the Betti table of I , one can note that there is no corner in itsinitial degree:
13 1 22 : 1 − − − Figure 1: Betti Table of I Furthermore, we can construct a squarefree strongly stable ideal J in S withinitial degree and Corn( J ) = { (2 , , (1 , } . It is J = ( x x x , x x x , x x x , x x x x ) . Note that J is the smallest squarefree strongly stable ideal of S with cornersequence { (2 , , (1 , } : − Figure 2: Betti Table of J Remark 3.12
It is worthy to point out that a squarefree strongly stable ideal I of S = K [ x , . . . , x n ] ( n ≥ ) of initial degree ℓ ≥ with a corner in degree ℓ andsuch that ds ( I ) = ( n − , n − , . . . , , for ℓ = 2 , ds ( I ) = ( n − ℓ, n − ℓ − , . . . , , for ℓ ≥ is a squarefree lex ideal of S .Hence, one can observe that a squarefree lex ideal of the polynomial ring S ofinitial degree ℓ ≥ and with a corner in degree ℓ can have at most n − ℓ cornersunlike the non–squarefree case. Indeed, a lex ideal I of a polynomial ring canhave at most corners [11, Theorem 3.2] (see also [12, Proposition 2.1]). For u, v ∈ Mon sd ( S ), u ≥ slex v , let us define the following set of squarefreemonomials: L ( u, v ) = { z ∈ Mon sd ( S ) : u ≥ slex z ≥ slex v } . Theorem 3.13
Let n ≥ and ℓ ≥ two integers. Given n − ℓ pairs of positiveintegers ( k , ℓ ) , ( k , ℓ ) , . . . , ( k n − ℓ , ℓ n − ℓ ) , (8)14 ith ≤ k n − ℓ < k n − ℓ − < · · · < k ≤ n − and ≤ ℓ < ℓ < · · · < ℓ n − ℓ ≤ n − , then there exists a squarefree lex ideal I of S of initial degree ℓ and withthe pairs in (8) as corners if and only if k i + ℓ i = n , for i = 1 , . . . , n − ℓ .Proof. Set S = K [ x , . . . , x n ]. If there exists a squarefree lex ideal I of S ofinitial degree ℓ and with the pairs in (8) as corners, then Proposition 3.9 forcesthat k i + ℓ i = n , for i = 1 , . . . , n − ℓ .Conversely, assume there exist n − ℓ pairs of positive integers( k , ℓ ) , ( k , ℓ ) , . . . , ( k n − ℓ , ℓ n − ℓ ) , (9)with 1 ≤ k n − ℓ < k n − ℓ − < · · · < k ≤ n −
3, 3 ≤ ℓ < ℓ < · · · < ℓ n − ℓ ≤ n − k i + ℓ i = n , for i = 1 , . . . , n − ℓ .We prove that there exists a squarefree lex ideal I of S generated in degrees ℓ , ℓ , . . . , ℓ n − ℓ with Corn( I ) = { ( k , ℓ ) , . . . , ( k n − ℓ , ℓ n − ℓ ) } .Setting s = max { i : ℓ + 2 i − ≤ n − } , the required monomial ideal I canbe constructed as follows. Step 1.
For i = 1 , . . . , s , let- G ( I ) ℓ = L ( u , v ), with u = x x · · · x ℓ and v = x x · · · x ℓ − x n ;- G ( I ) ℓ i = G ( I ) ℓ + i − = L ( u i , v i ), with u i = x x · · · x ℓ − i − Y j =0 x ℓ +2 j x ℓ +2( i − x ℓ +2( i − = x x · · · x ℓ − i − Y j =0 x ℓ +2 j x ℓ +2 i − x ℓ +2 i − and v i = x · · · x ℓ − i − Y j =0 x ℓ +2 j x ℓ +2( i − x n = x · · · x ℓ − i − Y j =0 x ℓ +2 j x ℓ +2 i − x n . Step 2.
Let us consider the squarefree monomial v s = x x · · · x ℓ − s − Y j =0 x ℓ +2 j x ℓ +2 s − x n . Since, ℓ + 2 s − ≤ n −
2, the smallest monomial belonging to the Shad( G ( I ) ℓ s )is w s +1 = x x · · · x ℓ − s − Y j =0 x ℓ +2 j x ℓ +2 s − x n − x n .
15e distinguish two cases: ℓ + 2 s − n −
2, and ℓ + 2 s − < n − Claim 1 . If ℓ + 2 s − < n −
2, then ℓ + 2 s − n − s , ℓ + 2( s + 1) − ≥ n −
1. Hence, ℓ + 2 s − ≥ n − n − ≤ ℓ + 2 s − < n − ℓ + 2 s − n −
3. The claim follows.Let us consider ℓ + 2 s − ℓ + 2( s −
2) + 1 = n −
2. In such a case, w s +1 = x · · · x ℓ − s − Y j =0 x ℓ +2 j x ℓ +2( s − x n − x n = x · · · x ℓ − s − Y j =0 x ℓ +2 j x n − x n − x n − x n . Hence, the greatest squarefree monomial of S following w s +1 is u s +1 = x x · · · x ℓ − s − Y j =0 x ℓ +2 j x ℓ +2( s − x ℓ +2( s − · · · x ℓ +2( s − . Note that max( u s +1 ) = ℓ + 2( s −
3) + 5 = ℓ + 2 s − n − n ,whereupon we choose G ( I ) ℓ s +1 = { u s +1 } . The smallest squarefree monomial belonging to Shad( G ( I ) ℓ s +1 ) is w s +2 = x x · · · x ℓ − s − Y j =0 x ℓ +2 j x ℓ +2( s − x ℓ +2( s − x ℓ +2( s − · · · x ℓ +2( s − = x x · · · x ℓ − s − Y j =0 x ℓ +2 j x n − x n − x n − x n − x n − x n . Therefore, the greatest squarefree monomial of S following w s +2 is u s +2 = x x · · · x ℓ − s − Y j =0 x ℓ +2 j x ℓ +2( s − x ℓ +2( s − · · · x ℓ +2( s − . Note that max( u s +2 ) = ℓ + 2( s −
4) + 7 = ℓ + 2 s − n − n . Thus,we choose G ( I ) ℓ s +2 = { u s +2 } , and so on. In general, G ( I ) ℓ s + q = { u s + q } , u s + q = x x · · · x ℓ − s − − ( q +1) Y j =0 x ℓ +2 j x ℓ +2( s − − q )+1 x ℓ +2( s − − q )+2 · · · x ℓ +2( s − − q )+2 q +3 , for q = 1 , . . . , t , where t is the positive integer such that s − − ( t + 1) = 0. It iseasy to verify that max( u s + q ) = n . Claim 2. s + t = n − ℓ − u s + t ) = n , and t + 1 = s − t = s − n = ℓ + 2( s − − t ) + 2 t + 3 = ℓ + 2( t + 1 − t ) + 2 t + 3 = ℓ + 2 t + 5 . Hence, n − ℓ − ℓ + 2 t + 5 − ℓ − t + 3 = 2 s − s + t. The claim follows.Finally, we choose G ( I ) ℓ n − ℓ − = G ( I ) s + t +1 = { u s + t +1 } = { x x · · · x ℓ − x ℓ +1 · · · x n } ,G ( I ) ℓ n − ℓ = G ( I ) s + t +2 = { u s + t +2 } = { x x · · · x ℓ − x ℓ − x ℓ · · · x n } . Now, let us consider the case ℓ + 2 s − n −
3. In such a case, the smallestmonomial belonging to Shad( G ( I ) ℓ s ) is w s +1 = x x · · · x ℓ − s − Y j =0 x ℓ +2 j x ℓ +2( s − x n − x n = x x · · · x ℓ − s − Y j =0 x ℓ +2 j x n − x n − x n . Therefore, the greatest squarefree monomial of S following w s +1 is u s +1 = x x · · · x ℓ − s − Y j =0 x ℓ +2 j x n − x n − x n . Since max( u s +1 ) = n , we choose G ( I ) ℓ s +1 = { u s +1 } .
17y hypothesis, ℓ + 2( s −
2) = n −
4, so that the smallest squarefree monomialbelonging to Shad( G ( I ) ℓ s +1 ) is w s +2 = x · · · x ℓ − s − Y j =0 x ℓ +2 j x n − x n − x n − x n = x · · · x ℓ − s − Y j =0 x ℓ +2 j x n − x n − x n − x n − x n . Consequently, the greatest squarefree monomial of S following w s +2 is u s +2 = x x · · · x ℓ − s − Y j =0 x ℓ +2 j x ℓ +2( s − x ℓ +2( s − · · · x ℓ +2( s − . Note that max( u s +2 ) = ℓ + 2( s −
3) + 6 = ℓ + 2 s = n , whence we choose G ( I ) ℓ s +2 = { u s +2 } . In general, G ( I ) ℓ s + q = { u s + q } , with u s + q = x x · · · x ℓ − s − − q Y j =0 x ℓ +2 j x ℓ +2( s − − ( q − · · · x ℓ +2( s − − ( q − q +2 , for q = 1 , . . . , t , where t is the positive integer such that s − − t = 0 ( t = s − u s + q ) = n .Also in such a case we can verify that s + t = n − ℓ −
2. Indeed, sincemax( u s + t ) = n , and t = s −
2, then n = ℓ + 2( s − − ( t − t + 2 = ℓ + 2 t + 4 , and n − ℓ − ℓ + 2 t + 4 − ℓ − t + 2 = 2( s −
2) + 2 = s + t. Finally, as in the previous case, we can choose G ( I ) ℓ n − ℓ − = G ( I ) s + t +1 = { x x · · · x ℓ − x ℓ +1 · · · x n } , and G ( I ) ℓ n − ℓ = G ( I ) s + t +2 = { x x · · · x ℓ − x ℓ − x ℓ · · · x n } . It is worthy observing that I is the smallest squarefree lex ideal of S withCorn( I ) = { ( k , ℓ ) , ( k , ℓ ) , . . . , ( k r , ℓ r ) } and such that β k i , k i + ℓ i ( I ) = 1, for all i , i.e. , a ( I ) = (1 , . . . , (cid:3) A numerical characterization of extremal Betti num-bers
In this Section, we face the following problem.
Problem 4.1
Given three positive integers n ≥ , ℓ ≥ and ≤ r ≤ n − ℓ , r pairs of positive integers ( k , ℓ ) , . . . , ( k r , ℓ r ) such that n − ≥ k > k > · · · >k r ≥ and ≤ ℓ < ℓ < · · · < ℓ r , k i + ℓ i ≤ n ( i = 1 , . . . , r ), and r positiveintegers a , . . . , a r , under which conditions does there exist a squarefree monomialideal I of S = K [ x , . . . , x n ] such that β k ,k + ℓ ( I ) = a , . . . , β k r ,k r + ℓ r ( I ) = a r are its extremal Betti numbers? For a pair of positive integers ( k, ℓ ) such that k + ℓ ≤ n , we define the followingset: A s ( k, ℓ ) = { u ∈ Mon sℓ ( S ) : max( u ) = k + ℓ } . Setting A s ( k, ℓ ) = { u , . . . , u q } , we can suppose, possibly after a permutationof the indices, that u > slex u > slex · · · > slex u q . (10)For the i -th monomial u of degree ℓ with max( u ) = k + ℓ , we mean the monomialof A s ( k, ℓ ) that appears in the i -th position of (10), for 1 ≤ i ≤ q . Note that u = x x · · · x ℓ − x k + ℓ , u q = x k +1 · · · x k + ℓ , and q = | A s ( k, ℓ ) | = (cid:0) k + ℓ − ℓ − (cid:1) .Furthermore, if u i , u j , i < j , are two monomials in (10), we define the follow-ing subsets of A s ( k, ℓ ):[ u i , u j ] = { w ∈ A s ( k, ℓ ) : u i ≥ slex w ≥ slex u j } , [ u i , u j ) = { w ∈ A s ( k, ℓ ) : u i ≥ slex w > slex u j } ;[ u i , u j ] will be called the segment of A s ( k, ℓ ) of initial element u i and final element u j , whereas [ u i , u j ) will be called the left segment of A s ( k, ℓ ) of initial element u i and final element u j . If i = j , we set [ u i , u j ] = { u i } . Remark 4.2
From (3), if ( k, ℓ ) is a corner of a squarefree stable ideal I and β k,k + ℓ ( I ) = a , then there exists a segment [ v , v a ] of A s ( k, ℓ ) such that a = | [ v , v a ] | . Next lemma will be crucial in the sequel. It can be easily proved by inductionon n . Lemma 4.3
Let n and q ≥ be two positive integers such that n ≥ q . Then (cid:18) nq (cid:19) = (cid:18) n − q − (cid:19) + (cid:18) n − q − (cid:19) + · · · + (cid:18) q − q − (cid:19) . u ∈ A s ( k, ℓ ), the next proposition shows a method, in-volving Lemma 4.3, to count the number of monomials v ∈ A s ( k, ℓ ) such that v ≥ slex u . Theorem 4.4
Let ( k, ℓ ) be a pair of positive integers with ℓ ≥ and let u = x i x i · · · x i ℓ − x i ℓ be a monomial of A s ( k, ℓ ) . Setting ˜ u = x i x i · · · x i ℓ − , then | [ x x · · · x ℓ − x k + ℓ , u ] | is a sum of t suitable binomial coefficients, where t = i , if Gap(˜ u ) = ∅ ,i + P ps =1 wd( g s -gap(˜ u )) , if Gap(˜ u ) = { g , . . . , g p } 6 = ∅ . Proof.
Set m = | [ x x · · · x ℓ − x k + ℓ , u ] | . m is the number of all monomials w ∈ A s ( k, ℓ ) such that w ≥ slex u . By Lemma 4.3, the binomial coefficient (cid:0) k + ℓ − ℓ − (cid:1) = | A s ( k, ℓ ) | can be decomposed as a sum of k + 1 binomial coefficients, as follows: (cid:18) k + ℓ − ℓ − (cid:19) = k +1 X j =1 (cid:18) k + ℓ − − jℓ − (cid:19) = (cid:18) k + ℓ − ℓ − (cid:19) + (cid:18) k + ℓ − ℓ − (cid:19) + · · · + (cid:18) ℓ − ℓ − (cid:19) . (11)One can observe that (cid:0) k + ℓ − ℓ − (cid:1) counts the monomials w ∈ A s ( k, ℓ ) such thatmin( w ) = 1, the binomial coefficient (cid:0) k + ℓ − ℓ − (cid:1) counts the monomials w ∈ A s ( k, ℓ )such that min( w ) = 2. In general, the binomial coefficient (cid:0) k + ℓ − iℓ − (cid:1) counts themonomials w ∈ A s ( k, ℓ ) such that min( w ) = i −
1, for i = 4 , . . . , k + 2. Notethat (cid:0) ℓ − ℓ − (cid:1) = (cid:0) k + ℓ − ( k +2) ℓ − (cid:1) counts the monomials w ∈ A s ( k, ℓ ) with min( w ) = k + 1. Indeed, there exists only a monomial w of such a type. It is w = x k +1 x k +2 · · · x k + ℓ = min A s ( k, ℓ ). It is clear that all monomials w ∈ A s ( k, ℓ )with min( w ) < i = min(˜ u ) = min( u ) are greater than u . Hence, the first i − m .We need to distinguish two cases: Gap(˜ u ) = ∅ , Gap(˜ u ) = ∅ .Note that Gap(˜ u ) = Gap( u ), or Gap(˜ u ) = Gap( u ) − Case 1.
Let Gap(˜ u ) = ∅ . In such a case, u is the greatest monomial of A s ( k, ℓ )with min( u ) = i . More precisely, the following sum of binomial coefficients i − X j =1 (cid:18) k + ℓ − − jℓ − (cid:19) (12)gives the number of all monomials w ∈ A s ( k, ℓ ) greater than u . Since i , i , . . . , i ℓ are consecutive integers, then other monomials greater than u which are differentfrom the w ’s counted by (12) do not exist. Hence, m = | [ x x · · · x ℓ − x k + ℓ , u ] | = i − X j =1 (cid:18) k + ℓ − − jℓ − (cid:19) + 1 .
20n the other hand, 1 = (cid:0) (cid:1) , and consequently m is the sum of t = i − i = min(˜ u ) = min( u ) binomial coefficients. Case 2.
Let Gap(˜ u ) = { g , . . . , g p } , p ≥
1. It is worthy to point out thatthe existence of the gaps g j ( j = 1 , . . . , p ) implies that i g j +1 − i g j − > i.e. , supp(˜ u ) ∩ { q : i g j < q < i g j +1 } = ∅ , for all j ∈ [ p ]. Thus, all monomials w ∈ A s ( k, ℓ ) of the type x i x i · · · x i gj z , where z is a monomial of degree ℓ − g j and max( z ) = k + ℓ such that supp( z ) ∩ { q : i g j < q < i g j +1 } 6 = ∅ , are greaterthan u .It is clear that all these monomials make up the left segment [ x x · · · x ℓ − x k + ℓ , u ).Let us consider the i –th binomial in (11): (cid:18) k + ℓ − − i ℓ − (cid:19) = k +1 X j =1 (cid:18) k + ℓ − − i − jℓ − (cid:19) . (13)In order to compute all monomials w of the type x i x i · · · x i g z , we need toevaluate g successive binomial decompositions until the next one: (cid:18) k + ℓ − i g − ℓ + i − i g − (cid:19) = k − i +1 X j =1 (cid:18) k + ℓ − i g − − jℓ + i − i g − (cid:19) . (14)The sum of the first wd( g -gap(˜ u )) = i g +1 − i g − w ∈ A s ( k, ℓ ) we are looking for.In order to compute all monomials w ∈ A s ( k, ℓ ) of the type x i x i · · · x i g z , weconsider the (wd( g -gap(˜ u )) − (cid:18) k + ℓ − i g − − wd( g -gap(˜ u )) − ℓ + i − i g − (cid:19) = (cid:18) k + ℓ − i g +1 − ℓ + i − i g − (cid:19) == k − i + i g − i g +2 X j =1 (cid:18) k + ℓ − i g +1 − − jℓ + i − i g − (cid:19) . Hence, evaluating the i g − i g +1 successive binomial decompositions until (cid:18) k + ℓ − i g − ℓ + i − i g − i g + i g +1 − (cid:19) = k − i + i g − i g +2 X j =1 (cid:18) k + ℓ − i g − − jℓ + i − i g − i g + i g +1 − (cid:19) , (15)the number of all required monomials w ∈ A s ( k, ℓ ) will be given by the sum ofthe first wd( g -gap(˜ u )) = i g +1 − i g − g j ∈ Gap(˜ u ), j ≥ | [ x x · · · x ℓ − x k + ℓ , u ) | = i − P ps =1 wd( g s -gap(˜ u )). Hence, inorder to get | [ x x · · · x ℓ − x k + ℓ , u ] | , we must take into account the binomial (cid:0) (cid:1) which counts the monomial u : t = i − p X s =1 wd( g s -gap(˜ u )) + 1 = i + p X s =1 wd( g s -gap(˜ u )) . (cid:3) Remark 4.5
Our choice to focus on the monomial ˜ u = x i x i · · · x i ℓ − , insteadof u , in Theorem 4.4 is due to the fact that if i ℓ − < k + ℓ − , i.e. , Gap(˜ u ) =Gap( u ) − , then all monomials z ∈ A s ( k, ℓ ) such that k + ℓ − ∈ supp( z ) aresmaller than u , with respect to ≥ slex . Next example illustrates Theorem 4.4.
Example 4.6
Let S = K [ x , . . . , x ] and consider the monomial u = x x x x .Set ˜ u = x x x . From Remark 2.7, | A s (4 , | = (cid:0) (cid:1) = 35 . In order to compute m = | [ x x x x , u ] | , we consider the following binomial decomposition: (cid:18) (cid:19) = (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) . Since, min( u ) = 2 , then all monomials w ∈ A s (4 , with min( w ) = 1 are greaterthan u , so we must take into account the binomial coefficient (cid:0) (cid:1) = 15 for thecomputation of m .Now, let us consider the following binomial decomposition: (cid:18) (cid:19) = (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) . Since
Gap(˜ u ) = { , } and wd(1 -gap(˜ u )) = 2 , the sum (cid:0) (cid:1) + (cid:0) (cid:1) = 7 gives thenumber of all monomials of the type x z ∈ A s (4 , , with z squarefree monomialof degree and max( z ) = 8 such that supp( z ) ∩ { q : 2 < q < } 6 = ∅ .At this stage, we have
15 + 7 = 22 monomials.The next decomposition we need to consider is (cid:18) (cid:19) = (cid:18) (cid:19) + (cid:18) (cid:19) . Since ∈ Gap(˜ u ) , and wd(2 -gap(˜ u )) = 1 , we must take into account (cid:0) (cid:1) = 1 .Finally, we have obtained
22 + 1 = 23 monomials of A s (4 , greater than u , and so m = | [ x x x x , u ] | = 23 + 1 = 24 .The following scheme summarizes the previous calculations. (cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) . ow, consider the monomial v = x x x x and let ˜ v = x x x . Proceedingas before, since Gap(˜ v ) = { } , then | [ x x x x , u ] | = 27 + 1 , where is givenby the sum of the highlighted binomial coefficients in the next scheme: (cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) . Here is the list of all monomials which come into play for u and v : 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 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 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 Now, let u , . . . u r be squarefree monomials of degree q of S . We denote by B ( u , . . . , u r ) the smallest squarefree strongly stable set of Mon sq ( S ) containingthe monomials u , . . . , u r .It is well known that if q < n , Shad( B ( u , . . . , u r )) is a squarefree strongly sta-ble set of monomials of degree q + 1 of S , and consequently Shad i ( B ( u , . . . , u r ))is a squarefree strongly stable set of degree q + i , for 1 ≤ i ≤ n − q .Now, let ( k , ℓ ) and ( k , ℓ ) be two pairs of positive integers such that k >k , ℓ < ℓ , k i + ℓ i ≤ n ( i = 1 , u , . . . , u r ∈ Mon sℓ ( S ) are squarefreemonomials of S such that max( u j ) = k + ℓ , j = 1 , . . . , r , we define the followingset:BShad( u , . . . , u r ) ( k ,ℓ ) = { v ∈ Shad ℓ − ℓ ( B ( u , . . . , u r )) : max( v ) ≤ k + ℓ } . u , . . . , u r ) ( k ,ℓ ) is a squarefree stronglystable set of degree ℓ of S . Remark 4.7
It is worthy to underline that if one wants to compute the minimumof
BShad( u , . . . , u r ) ( k ,ℓ ) , it is sufficient to determine min BShad( u r ) ( k ,ℓ ) . Fur-thermore, in order to obtain such a monomial, one can suitably manage the in-tegers in supp( u r ) , as we will see in a while. Definition 4.8
Let u be a squarefree monomial of degree q of S , q < n . Let p ≤ n a positive integer such that [ p ] \ supp( u ) = ∅ and { j , . . . , j t } a subset of[ p ] \ supp( u ), with j < j < · · · < j t , q + t ≤ n . The monomial x j · · · x j t u ∈ Mon sq + t ( S ) is called the joint of u with the variables x j , . . . , x j t . Example 4.9
Let u = x x x x ∈ K [ x , . . . , x ] . Let p = 7 and consider theset { , , } ⊂ [7] \ { , , , } . The joint of u with x , x , x is the squarefreemonomial x x x x x x x ∈ Mon s ( S ) . With the same notations as before, we give the construction of the monomialmin BShad( u ) ( k ,ℓ ) for a given squarefree monomial u ∈ A s ( k , ℓ ). Construction 4.10
Let ( k , ℓ ) and ( k , ℓ ) be two pairs of positive integers suchthat k > k , ≤ ℓ < ℓ and k i + ℓ i ≤ n , for i = 1 , . Let u = x i · · · x i ℓ be asquarefree monomial of A s ( k , ℓ ) . Assume i t to be the greatest integer belongingto supp( u ) such that i t < k + ℓ , and write u = x i · · · x i t · · · x i ℓ . Let us consider the monomial u = x i · · · x i t and let j , . . . , j ℓ − t be the greatestintegers belonging to [ k + ℓ ] \ supp( u ) . Then, min BShad( u ) ( k ,ℓ ) = x j · · · x j ℓ − t u ∈ A s ( k , ℓ ) . Construction 4.10 assures the correctness of the next algorithm.24 lgorithm 1:
Computation of min BShad( u ) ( k,ℓ ) Input:
Polynomial ring S , monomial u , positive integer k , positiveinteger ℓ Output: monomial v begin j ← k + ℓ ; t ←| { i ∈ supp( u ) : i < j } | ; v ← the first t variables of u ; q ← ℓ − t ; while q > doif j / ∈ supp( v ) thenif j > then v ← v ∗ S j ; elseerror no monomial ; end q ← q − end j ← j − endreturn v ; end Lemma 4.11
Take two pairs of positive integers ( k , ℓ ) and ( k , ℓ ) such that k > k , ≤ ℓ < ℓ with k i + ℓ i ≤ n , for i = 1 , . Let u be a squarefreemonomial of degree ℓ with max( u ) = k + ℓ and let v = min BShad( u ) ( k ,ℓ ) . If Gap( v ) = ∅ , then there exists a monomial w ∈ A s ( k , ℓ ) \ BShad( u ) ( k ,ℓ ) .Proof. Let v = min BShad( u ) ( k ,ℓ ) = x r · · · x r ℓ . One has max( v ) = k + ℓ . Assume p = max Gap( v ), then the greatest square-free monomial following v in the squarefree lex order is x r · · · x r p − x r p +1 · · · x r p + ℓ − p +1 , with r p + ℓ − p + 1 ≤ k + ℓ . Hence, if r p + ℓ − p + 1 = k + ℓ , wechoose w = x r · · · x r p − x r p +1 · · · x r p + ℓ − p +1 . Otherwise, if r p + ℓ − p + 1 < k + ℓ , let w = x r · · · x r p − x r p +1 · · · x r p + ℓ − p x k + ℓ . (cid:3) Next pseudocode describes the procedure in Lemma 4.11.25 lgorithm 2:
Computation of the next monomial smaller than a given u in A s ( k, ℓ ) Input:
Polynomial ring S , monomial u Output: monomial w begin m ← max supp( u ); ℓ ← deg( u ); if Gap( u ) = ∅ then t ← max Gap( u ); w ← the first t − u ; j ← index of variable of u at position t ; foreach i ∈ { . . ℓ − t } do j ← j + 1 ; w ← w ∗ S j ; end w ← w ∗ S m ; elseerror no monomial ; endreturn w ; end The discussion below is significant for solving Problem 4.1.
Discussion k , ℓ ) and ( k , ℓ ) be two pairs of positive integers suchthat k > k , 2 ≤ ℓ < ℓ with k i + ℓ i ≤ n ( i = 1 ,
2) and let a , a be two positiveintegers.Let T be a segment of A s ( k , ℓ ) of cardinality a < (cid:0) k + ℓ − ℓ − (cid:1) . We want todetermine the admissible values for a ≤ (cid:0) k + ℓ − ℓ − (cid:1) so that there exists a segment[ u , u a ] of A s ( k , ℓ ) of cardinality a and such that BShad([ u , u a ]) ( k ,ℓ ) + T .It is clear that it should be a < (cid:0) k + ℓ − ℓ − (cid:1) .Now, set T = [ z , z a ], and assume T * BShad([ u , u a ]) ( k ,ℓ ) . Let v ∈ A s ( k , ℓ ) be the smallest monomial such that z / ∈ BShad( v ) ( k ,ℓ ) . Such amonomial allows us to determine the bound on a for which there exists thesegment T .Indeed, we can compute the following cardinalities (Theorem 4.4): n = |{ u ∈ A s ( k , ℓ ) : u ≥ v }| = | [ x x · · · x ℓ − x k + ℓ , v ] | ,p = |{ v ∈ A s ( k , ℓ ) : v > u }| = | [ x x · · · x ℓ − x k + ℓ , u ) | . Hence, since [ u , u a ] ⊆ [ x x · · · x ℓ − x k + ℓ , v ], we get the following coarsebound for a : a ≤ n ;26hen, we can refine such a bound via p as follows: a ≤ n − p . One can notice, that if u = max A s ( k , ℓ ), then p = 0. Example 4.13
Given S = K [ x , . . . , x ] , let us consider the pairs of positiveintegers (5 , and (2 , , the positive integers a = 8 and a = 6 , and the followingsegment of A s (5 , of cardinality a = 8 : [ 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 } . We want to verify if there exists a segment of A s (2 , of cardinality a = 6 not contained in BShad([ x x x x , x x x x ]) (2 , .First, from Remark 2.7, we know that a ≤ (cid:0) (cid:1) = 56 and a ≤ (cid:0) (cid:1) = 21 .In order to determine p = |{ v ∈ A s (5 ,
4) : v > x x x x }| = | [ x x x x , x x x x ) | ,we need to consider a suitable sequence of binomial decompositions. The first bi-nomial decomposition that we have to examine is (cid:18) (cid:19) = (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) . Then, applying the procedure described in Theorem 4.4 (see also Example 4.6),we obtain the following sequence of binomial decompositions, (cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) , whereupon p = 6 .In order to compute n , we consider the set A consisting of the smallest a = 6 monomials of A s (2 , : A = { 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 } . These monomials can be found using the “reversal” of Algorithm 2.The smallest monomial z of A s (5 , such that max A = x x x x x x / ∈ BShad( z ) (2 , is z = x x x x . The number of all monomials w ∈ A s (5 , greater than or equal to z is determined by the following binomial sequences: (cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) . ence, we have n = (6 + 5 + 4 + 3 + 2) + 1 = 21 monomials. Finally, wehave a ≤ n − p = 21 − .For a = 15 , then a segment of A s (2 , of length a = 6 is A = [ x x x x x x , x x x x x x ] . Discussion 4.12 yields the following result.
Theorem 4.14
Consider three positive integers n ≥ , ℓ ≥ and ≤ r ≤ n − ℓ , r pairs of positive integers ( k , ℓ ) , . . . , ( k r , ℓ r ) such that n − ≥ k >k > · · · > k r ≥ and ≤ ℓ < ℓ < · · · < ℓ r , k i + ℓ i ≤ n ( i = 1 , . . . , r ), and r positive integers a , . . . , a r . Let K be a field of characteristic zero. The followingconditions are equivalent: (1) There exists a squarefree graded ideal J of S = K [ x , . . . , x n ] with β k ,k + ℓ ( J ) = a , . . . , β k r ,k r + ℓ r ( J ) = a r as extremal Betti numbers. (2) There exists a squarefree strongly stable ideal I of S = K [ x , . . . , x n ] with β k ,k + ℓ ( I ) = a , . . . , β k r ,k r + ℓ r ( I ) = a r as extremal Betti numbers. (3) Setting (i) v r = x k r +1 · · · x k r + ℓ r , A r = [ w r , v r ] , with w r ∈ A s ( k r , ℓ r ) and such that | A r | = a r ; (ii) for i = 1 , . . . , r − , v r − i = min { u ∈ A s ( k r − i , ℓ r − i ) : max A r − i +1 / ∈ BShad( u ) ( k r − i +1 ,ℓ r − i +1 ) } , A r − i = [ w r − i , v r − i ] , with w r − i ∈ A s ( k r − i , ℓ r − i ) and such that | A r − i | = a r − i ; (iii) for i = 1 , . . . , r , n i = |{ u ∈ A s ( k i , ℓ i ) : u ≥ v i }| , then the integers a i satisfy the following conditions: a i ≤ n i . If a i = | [ u i, , u i,a i ] | , u i,j ∈ A s ( k i , ℓ i ) ( j = 1 , . . . , a i ) and p i = |{ v ∈ A s ( k i , ℓ i ) : v > u i, }| , then a i ≤ n i − p i , for i = 1 , . . . , r .Proof. (1) ⇔ (2). See [3] and the introduction in this paper.(2) ⇒ (3). It follows applying iteratively Discussion 4.12, for i = 1 , . . . , r . Notethat v r = min A s ( k r , ℓ r ), and consequently n r = (cid:0) k r + ℓ r − ℓ r − (cid:1) ; whereas p = 0.(3) ⇒ (2). We construct a squarefree strongly stable ideal I of S generated indegrees ℓ , . . . , ℓ r as follows:- G ( I ) ℓ = B ( u , , . . . , u ,a ); 28 G ( I ) ℓ = B ( u , , . . . , u ,a ) \ BShad ℓ − ℓ ( G ( I ) ℓ ) ( k ,ℓ ) ;- G ( I ) ℓ i = B ( u i, , . . . , u i,a i ) \ BShad ℓ i − ℓ i − (Mon s ( I ℓ i − )) ( k i ,ℓ i ) , for i = 3 , . . . , r ,where Mon s ( I ℓ i − ) is the set of all squarefree monomials of degree ℓ i − belonging to I ℓ i − .The monomials u i, , . . . , u i,a i , for i = 1 , . . . , r , are the basic monomials of I . (cid:3) Remark 4.15
A similar statement can be formulated in the case ℓ = 2 and n ≥ . Next example illustrates Theorem 4.14.
Example 4.16
Let n = 11 , r = 4 , C = { (8 , , (4 , , (3 , , (2 , } and a =( a , a , a , a ) = (7 , , , . We want to construct a squarefree strongly sta-ble ideal I of S = K [ x , . . . , x ] generated in degrees 3,5,6,9 and such that Corn( I ) = C , a ( I ) = a .With the same notations as in Theorem 4.14, before starting the constructionof the ideal, we verify if the coarse bounds are satisfied for each a i , i = 1 , . . . , .First of all, v = x x x x x x x x x and n = | [ x x x x x x x x x , v ] | = (cid:0) (cid:1) = 45 . Hence, a = 2 ≤ n .Moreover, A = { x x x x x x x x x , x x x x x x x x x } , v = x x x x x x , and from the binomial decompositions (cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) we obtain a = 2 ≤ n = | [ x x x x x x , v ] | = 35 + (6 + 3) + 1 = 45 .Furthermore, A = { x x x x x x , x x x x x x } and v = x x x x x .From the binomial decompositions (cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) one has a = 5 ≤ n = | [ x x x x x , v ] | = (35 + 4) + 1 = 40 . inally, A = [ 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 } and v = x x x . The binomial decom-positions (cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) imply a = 7 ≤ n = | [ x x x , v ] | = 8 + 1 = 9 .Now, we proceed with the construction of the ideal I we are looking for, andso doing we refine the previous bounds for the a i ’s.- The greatest monomial of A s (8 , is x x x . Since p must be equal to and a = 7 ≤ n − p = 9 , one can consider the greatest a = 7 monomialsof A s (8 , . Such monomials can be obtained by Algorithm 2. Hence, weset G ( I ) = B ( x x x , x x x , x x x , x x x , x x x , x x x , x x x ) . - Let us consider the corner (4 , . By Algorithm 1, we compute the smallestmonomial of BShad ( G ( I ) ) (4 , , i.e. , the monomial x x x x x ; whereas,by Algorithm 2, we determine the greatest monomial of A s (4 , \ BShad ( G ( I ) ) (4 , , i.e. , x x x x x . Finally, from the binomial decomposition (cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) it follows that p = | [ x x x x x , x x x x x ) | = 35 . Hence, n − p =40 −
35 = 5 monomials are available. Therefore, since a = 5 , we set G ( I ) = B ( 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 ) . - Let us consider the corner (3 , . One has min BShad( G ( I ) ) (3 , = x x x x x x and max( A s (3 , \ BShad( G ( I ) ) (3 , = x x x x x x , andfrom (cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) e have p = | [ x x x x x x , x x x x x x ) | = 43 . Hence, n − p =45 −
43 = 2 .Since a = 2 , we set G ( I ) = B ( x x x x x x , x x x x x x ) . - If one considers the corner (2 , , since min BShad ( G ( I ) ) (2 , = x x x x x x x x x max( A s (2 , \ BShad ( G ( I ) )) (2 , = x x x x x x x x x , from (cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) it follows p = | [ x x x x x x x x x , x x x x x x x x x ) | = 43 .So n − p = (cid:0) (cid:1) − p = 45 −
43 = 2 . Hence, since a = 2 , we can set G ( I ) = B ( x x x x x x x x x , x x x x x x x x x ) . The Betti table of the squarefree strongly stable I just constructed is the fol-lowing one: − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − One can observe that Theorem 4.14 assures the correctness of the next Algo-rithm 3.We close the Section with an example that illustrates a situation where theconstruction of a squarefree strongly stable ideal is not possible.31 lgorithm 3:
Computation of the basic monomials for the given data
Input:
Polynomial ring S , list of corners { ( k i , ℓ i ) } , list of values ( a i ) Output: list of monomials mons begin hyp ← logical conditions required as hypotheses of the Theorem4.14; if hyp then m ← k + ℓ ; w ← S ∗ . . . ∗ S ℓ − ∗ S m ; // first corner mons ← { w } ; foreach j ∈ { . . a } do w ← next monomial of w ; // calling Algorithm 2 if no monomial thenerror no ideal ; else mons ← mons ∪ { w } ; endend r ← number of corners; // successive corners foreach i ∈ { . . r } do w ← min BShad( mons ) ( k i − ,ℓ i − ) ; // calling Algorithm 1 foreach j ∈ { . . a i } do w ← next monomial of w ; // calling Algorithm 2 if no monomial thenerror no ideal ; else mons ← mons ∪ { w } ; endendendendreturn mons ; end xample 4.17 Let n = 10 , r = 3 , C = { (6 , , (5 , , (3 , } and a = ( a , a , a ) =(2 , , . We have | A s (3 , | = (cid:0) (cid:1) = 84 , so it is possible to manage a = 4 ≤ monomials.Let us consider the set A consisting of the smallest four monomials in A s (3 , : A = { 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 } , and let us try to get the smallest monomial z ∈ A s (5 , such that x x x x x x x / ∈ BShad( z ) (3 , . It is z = x x x x . Now, we compute | [ x x x x , z ] | as boundfor a : (cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) + (cid:0) (cid:1) . We have n = 21 + (5 + 4 + 3 + 2) + 1 = 36 monomials greater than z and so a = 1 ≤ .Note that if z does not exist, then it is clear that we can not go on.Now, we try to verify the bound for a taking into account the previous results.Consider the monomial z ∈ A s (5 , , and take the greatest monomial w of A s (6 , such that z / ∈ BShad( w ) (3 , . It is w = x x . We can note that w is the smallestmonomial of A s (6 , , i.e. , | [ x x , w ] | = 1 .Hence, we have that a ≤ . For this reason the requested value for a = 2 is not admissible and there does not exist any squarefree monomial ideal I of K [ x , . . . , x ] such that Corn( I ) = C and a ( I ) = a .Nevertheless, there exists a squarefree monomial ideal J of S such that Corn( J ) = C and a ( J ) = (1 , , . References [1] L. Amata and M. Crupi,
Computation of graded ideals with given extremalBetti numbers in a polynomial ring , J. Symbolic Computation, 93 (2019),120–132.[2] A. Aramova, J. Herzog and T. Hibi,
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