Characterization and Newton Complementary Dual of Quasi f-Ideals
aa r X i v : . [ m a t h . A C ] J a n CHARACTERIZATION AND NEWTON COMPLEMENTARYDUAL OF QUASI f -IDEALS ∗ F. U. REHMAN , H. HASAN , H. MAHMOOD , M. A. BINYAMIN Abstract.
The notion of quasi f -ideals was first presented in [14] which gen-eralize the idea of f -ideals. In this paper, we give the complete characterizationof quasi f -ideals of degree greater or equal to 2. Additionally, we show thatthe property of being quasi f -ideals remain the same after taking the Newtoncomplementary dual of a squarefree monomial ideal I provided that the minimalgenerating set of I is perfect. Key words: f -vector; facet complex; Stanley-Reisner complex; quasi f -ideal; : 13F20, 05E45, 13F55, 13C14. Introduction
Throughout this paper, F is a field and R = F [ x , x , ..., x n ] is a polynomial ringwith n indeterminate. Any squarefree monomial ideal I ⊂ R can be associated totwo different simplicial complexes over the finite set of vertices, denoted by δ F ( I ) and δ N ( I ), called the facet complex of I and the non-face complex (or Stanley-Reisnercomplex) of I respectively. The f -vectors of these two simplicial complexes δ F ( I )and δ N ( I ) have the accompanying prospects :1) f ( δ F ( I )) = f ( δ N ( I )) or2) f ( δ F ( I )) = f ( δ N ( I )) but dim ( δ F ( I )) = dim ( δ N ( I )) or3) f ( δ F ( I )) = f ( δ N ( I )) but dim ( δ F ( I )) = dim ( δ N ( I ))A squarefree monomial ideal I ⊂ R with property that mention in (1) is called an f -ideal of the polynomial ring R . This notion has been studied for various propertiesof f -ideals relevant to combinatorial commutative algebra in the papers [3], [4], [11],[12], and [13]. If we look at (1) and (2) collectively, there is one thing common, i.e, di-mensions of both δ F ( I ) and δ N ( I ) are same. This means that the f -vectors of δ F ( I )and δ N ( I ) can be added or subtracted usually. In this paper we will think just thosesquarefree monomial ideal I in the polynomial ring R = F [ x , x , ..., x n ] with proper-ties that dim ( δ F ( I )) = s = dim ( δ N ( I )) and f ( δ N ( I )) − f ( δ F ( I )) = ( a , a , . . . , a s ),we call it a quasi f -ideal of type ( a , a , . . . , a s ). It is noted that if a i = 0 for all i = 1 , , ..., r , then clearly we have the property (1), and equivalently saying that ∗ The third and fourth authors are supported by the Higher Education Commission of Pakistanfor this research (Grant no. 7515). Government College University Lahore, Pakistan. Government College University Faisalabad,Pakistan.
E-mails : [email protected], [email protected], [email protected], [email protected] . very f -ideal is a quasi f -ideal of type -vector. It is natural to ask: Is it possibleto characterize all the squarefree monomial quasi f -ideals? Mahmood. H. et al,classified all the pure squarefree monomial quasi f -ideals of degree 2 in two differentapproaches in [14]. In this paper, we characterize all quasi f -ideals of degree d ≥ f -ideal regarding its Newton complementarydual. In 2013, during the study of Cremona maps; Costa and Simis [5] introducedthe notion of the Newton complementary dual in general context. After that, Doriaand Simis [6] were examined different properties of Newton complementary dual.Ansaldi, Lin, and Shin [2] investigated the Newton complementary duals of mono-mial ideals.In this paper, we use the following outlines. In section 2, we give essential conceptsthat help in the expected results. Section 3 focuses on the primary study of quasi f -ideals; Theorem 3.2 characterizes all pure squarefree quasi f -ideals of degree d ≥ R modulo squarefree monomial quasi f -ideal I . Thefourth section of this paper is devoted to the Newton complementary duals of quasi f -ideals. We prove that an ideal I is a quasi f -ideal if and only if b I is a quasi f -idealprovided that G ( I ) is perfect, Theorem 4.5.2. Basic Set Up
Let us review some fundamental ideas to get familiar with simplicial complex andsquarefree monomial ideals. Let V = { v , v , . . . , v n } be a vertex set. A subset ∆ of P ( V ) is said to be a simplicial complex on V if and only if each one point subset of V lies in ∆ and, if E is any subset of F ∈ ∆, then E ∈ ∆. Note that, each elementof ∆ is known as face and the maximal faces under ⊆ are known as facets. For anyface F of ∆, the dimension of F is given by dim ( F ) = | F | −
1. Note that ∅ ∈ ∆and dim ( ∅ ) = −
1. A simplicial complex ∆ is said to be pure if all of its facets havethe same dimension. The dimension of ∆ is the maximum of the dimension of allfacets of ∆. If ∆ is a d -dimensional simplicial complex on V , then the f -vector of∆ is the d + 2 tuple f (∆) = ( f − (∆) , f (∆) , f (∆) , ..., f d (∆))where f − (∆) = 1 and f i (∆) = |{ F ∈ ∆ : dim( F ) = i }| , 0 ≤ i ≤ d ,. Note thatin d -dimensional simplicial complex ∆, we may assume if needed f i (∆) = 0 for i > d . This means that the f -vector of simplicial complex ∆ can be written as f (∆) = ( f − (∆) , f (∆) , f (∆) , ..., f d (∆) , , , ..., I = h u , u , ..., u r i is a squarefree monomial ideal of a polynomialring R = F [ x , x , . . . , x n ] with supp ( I ) = { x , x , . . . , x n } . We use G ( I ) to denotethe unique set of minimal generators of I . We can associate to I two simplicialcomplexes. The facet complex is a simplicial complex denoted by δ F ( I ) and isdefined as δ F ( I ) = h F , F , ..., F r i where F i = { v j : x j divide u i } is the facet onthe vertices v , v , . . . , v n , and i = 1 , , , ..., r . The non-face complex (or Stanley-Reisner complex) is a simplicial complex on V = { v , v , . . . v n } such that a subset v i , v i , ..., v i k } of V is a face of this non-face complex if and only if the correspondingmonomial x i x i · · · x i k does not belong to I . We denote it by δ N ( I ). In this paper,we are interested in the following family of squarefree monomial ideals: Definition 2.1.
A squarefree monomial ideal I in the polynomial ring R with thefield F is said to be a quasi f -ideal of type ( a , a , . . . , a r ) ∈ Z r if and only if f i ( δ N ( I )) − f i ( δ F ( I )) = a i for all i ∈ { , , , ..., r } . Remark 2.2.
It is noted that, if a i = 0 for all i ∈ { , , , ..., r } , then obviously I is an f -ideal. This means that every f -ideal is a quasi f -ideal whose type is azero vector, and obviously any quasi f -ideal with type some non-zero vector can notbe an f -ideal. It is impotent to mention that the class of quasi f -ideals is muchmore bigger than the class of f -ideals; moreover, unlike f -ideals, examples of quasi f -ideals can be found in R = F [ x , x , ..., x n ], for any n .We would like to recall the [12, Definition 2.1] of perfect sets of R . We use sm ( R )and sm ( R ) d to denote the set of all squarefree monomials and the set of all squarefreemonomials of degree d in R respectively; Definition 2.3.
Let R = F [ x , x , ..., x n ], and let A ⊆ sm ( R ). We define(i) ⊔ ( A ) = { gx i | g ∈ A, x i does not divide g, ≤ i ≤ n } and(ii) ⊓ ( A ) = { h | h = g/x i for some g ∈ A and some x i with x i | g } A is called lower perfect if ⊓ ( A ) = sm ( R ) d − holds. Dually, A is called upper perfectif ⊔ ( A ) = sm ( R ) d +1 holds. If A is both lower perfect and upper perfect, then A iscalled a perfect set.3. Characterization of Quasi f -Ideals The purpose of this section is to characterize quasi f -ideals of degree d ≥ R = F [ x , x , ..., x n ].The following lemma is key to understanding how to count the number of facesfor particular dimensions of non-face complex of a pure squarefree monomial idealfor any degree. Lemma 3.1.
Let I be a pure squarefree monomial ideal of degree d + 1 in R = F [ x , x , ..., x n ]. Then f i ( δ N ( I )) = (cid:0) ni +1 (cid:1) , where 0 ≤ i < d. Proof. If n i is the number of i -dimensional non-faces of facet complex of I where0 ≤ i < d , then it means that f i ( δ F ( I )) = (cid:0) ni +1 (cid:1) − n i and δ N ( I ) contains at least n i number of i -dimensional faces. The [3, Lemma 3.6] tells us that the face ofdimension less than d of δ F ( I ) is the face of δ N ( I ). Therefore, we have f i ( δ N ( I )) = (cid:0) ni +1 (cid:1) − n i + n i = (cid:0) ni +1 (cid:1) Theorem 3.2. (Characterization)
Let I be an equigenerated squarefree monomial ideal of R of degree d and let G ( I ) = { u , u , . . . , u r } be minimal set of generator of I . Then I is quasi f -ideal of type( a , a , a , ..., a d − ) ∈ Z d if and only if the following conditions hold true:(1) ht ( I ) = n − d ; (cid:0) nd (cid:1) ≡ ( mod
2) if a d − is even1 ( mod
2) if a d − is odd , and r = | G ( I ) | = ( (cid:0) nd (cid:1) − a d − ). ;(3) a i is the number of i -dimensional non-faces of δ F ( I ), where i = 0 , , , ..., d − Proof. If I is a quasi f -ideal of degree d with type ( a , a , a , ..., a d − ), then obviously,dimensions of both facet complex and non-face complex of I are same. By [1, Lemma3.4] the height of ideal I must be equal to n − d . As I is a squarefree monomialideal of degree d , using the [1, Lemma 3.2], we have that(1) f d − ( δ F ( I )) = (cid:18) nd (cid:19) − f d − ( δ N ( I ))It is noted that if I is a quasi f -ideal of type ( a , a , a , ..., a d − ), then(2) f d − ( δ N ( I )) = f d − ( δ F ( I )) + a d − From equation (1) and equation (2), we have that 2 f d − ( δ F ( I )) = (cid:0) nd (cid:1) − a d − . Itmeans that the parity of (cid:0) nd (cid:1) is same as the parity of a d − , and | G ( I ) | = f d − ( δ F ( I )) = ( (cid:0) nd (cid:1) − a d − ). Now we want to show that a i is the number of i -dimensional non-facesof δ F ( I ), where i = 0 , , , ..., d −
2. Note that for any squarefree monomial ideal I of degree d the Lemma 3.1, tells us that the number of i -dimensional faces of δ N ( I )will be equal to (cid:0) ni +1 (cid:1) for all 0 ≤ i < d −
1. As I is a quasi f -ideal of degree d with type ( a , a , a , ..., a d − ). Therefore, f i ( δ N ( I )) − f i ( δ F ( I )) = a i which implies (cid:0) ni +1 (cid:1) − f i ( δ F ( I )) = a i for all 0 ≤ i < d − I , [15, Proposition 5.3.10] gives dim ( δ N ( I )) = n − ht ( I ) − I is n − d , so we have dim ( δ N ( I )) = dim ( δ F ( I )). Now, we will showthat f ( δ N ( I )) − f ( δ N ( I )) = ( a , a , a , ..., a d − ) . Since I is a squarefree monomialideal of degree d , f d − ( δ N ( I )) = (cid:0) nd (cid:1) − f d − ( δ F ( I )). This implies(3) f d − ( δ N ( I )) − f d − ( δ F ( I )) = (cid:18) nd (cid:19) − f d − ( δ F ( I ))As f d − ( δ F ( I )) = ( (cid:0) nd (cid:1) − a d − ), this implies (cid:0) nd (cid:1) − f d − ( δ F ( I )) = a d − . Therefore,equation (3) becomes f d − ( δ N ( I )) − f d − ( δ F ( I )) = a d − . Since a i is the number of i -dimensional non-faces of δ F ( I ) where i = 0 , , , ..., d −
2, this means that (cid:0) ni +1 (cid:1) − f i ( δ F ( I )) = a i . From Lemma 3 .
1, the number of i -dimensional faces of δ N ( I )) are (cid:0) ni +1 (cid:1) therefore, we have f i ( δ N ( I )) − f i ( δ F ( I )) = a i for all i ∈ { , , , ..., d − } . Thus I is quasi f -ideal of type ( a , a , a , ..., a d − ) ∈ Z d . (cid:3) It is worth noting that [3, Theorem 3.9] and [14, Theorem 4.1] can be deduceby setting a i = 0 for all i ∈ { , , ..., d − } in Theorem 3.2 such as the followingcorollary: Corollary 3.3.
Let I = ( u , u , . . . , u r ) be an equigenerated squarefree monomialideal of R of degree d. Then I is quasi f -ideal of type = (0 , , ..., ∈ Z d if andonly if the following conditions hold true: ht ( I ) = n − d ;(2) (cid:0) nd (cid:1) ≡ mod , and r = | G ( I ) | = (cid:0) nd (cid:1) ;(3) f d − ( δ F ( I )) = (cid:0) nd − (cid:1) Proof.
It is clear from above Theorem. Note that if a d − = 0, then r = | G ( I ) | = (cid:0) nd (cid:1) . From Lemma 3.1 and [1, Lemma 3.2], together implies f d − ( δ F ( I )) = (cid:0) nd − (cid:1) . (cid:3) Corollary 3.4.
Let I be an equigenerated squarefree monomial ideal of R of degree2 and let G ( I ) = { u , u , . . . , u r } be minimal set of generator of I . Then I is quasi f -ideal of type ( a , a ) ∈ Z if and only if the following conditions hold true:(1) ht ( I ) = n − (cid:0) n (cid:1) ≡ ( mod
2) if a is even1 ( mod
2) if a is odd , and r = | G ( I ) | = ( (cid:0) n (cid:1) − a ). ;(3) a is the number of i -dimensional non-faces of δ F ( I ). Example 3.5.
Let I = ( x x x , x x x , x x x , x x x , x x x ) be a pure squarefreemonomial ideal of degree 3 in the polynomial ring F [ x , x , x , x , x ] . Then theprimary decomposition of I is ( x , x ) T ( x , x ) T ( x , x ) T ( x , x ) T ( x , x ) . Thefacet complex and the non-face complex of I are δ F ( I ) = h{ , , } , { , , } , { , , } , { , , } , { , , }i and δ N ( I ) = h{ , , } , { , , } , { , , } , { , , } , { , , }i . As { , } is non-face of δ F ( I ), ht ( I ) = 2, (cid:0) (cid:1) ≡ mod
2) and | G ( I ) | = 5 = ( (cid:0) (cid:1) − . Thus I is a quasi f -ideal with type (0 , , , . Example 3.6.
Let I = ( x x x , x x x , x x x , x x x ) be a pure squarefree mono-mial ideal of degree 3 in the polynomial ring F [ x , x , x , x , x ] . Then the primarydecomposition of I is ( x , x ) T ( x , x ) T ( x , x ) T ( x , x ) T ( x , x ) T ( x , x ) . Thefacet and non-face complexes of I are δ F ( I ) = h{ , , } , { , , } , { , , } , { , , }i and δ N ( I ) = h{ , , } , { , , } , { , , } , { , , } , { , , } , { , , }i . Then f ( δ F ( I )) = (1 , , ,
4) and f ( δ N ( I )) = (1 , , , . Thus I is a quasi f -idealwith type (0 , , , . In the following theorem, we give the formulae to compute the Hilbert functionand Hilbert series of polynomial ring R modulo a squarefree monomial quasi f -idealI. Theorem 3.7.
Let I be a quasi f -ideal in a polynomial ring R of type ( a − , a , a ,..., a d ) . Then(1) The Hilbert series of
R/I is given by F ( R/I, k ) = d X i = − f i ( δ F ( I )) + a i (1 − k ) i +1 k i +1
2) The Hilbert function of
R/I is given by H ( R/I, k ) = d X i =0 (cid:18) k − i (cid:19) ( f i ( δ F ( I )) + a i )where k ≥ H ( R/I,
0) = 1.
Proof. As I is a quasi f -ideal in a polynomial R of type ( a − , a , a , ..., a d ). Therefore, f i ( δ N ( I )) = f i ( δ F ( I )) + a i , this means that the f -vector of f ( δ N ( I )) is ( f − ( δ F ( I )) + a − , f ( δ F ( I )) + a , ..., f d ( δ F ( I )) + a d ) . Since I is non face ideal of the simplicialcomplex δ N ( I )). Now using [15, Theorem 6.7.2] and [15, Theorem 6.7.3], we havethat F ( R/I, k ) = d X i = − f i ( δ F ( I )) + a i (1 − k ) i +1 k i +1 and H ( R/I, k ) = d X i =0 (cid:18) k − i (cid:19) ( f i ( δ F ( I )) + a i )as desired. (cid:3) Example 3.8.
We return to the ideal I in Example 3.6. This ideal I is a quasi f -ideal with type (0 , , ,
2) (i.e. a − = 0 , a = 0 , a = 2 and a = 2) and the facetcomplex of I is δ F ( I ) = h{ , , } , { , , } , { , , } , { , , }i clearly, f ( δ F ( I )) = (1 , , ,
4) (i.e. f − = 1 , f = 5 , f = 8 and f = 4. Thus theHilbert function and Hilbert series of R [ x , x , x , x , x ] /I are H ( R/I, k ) = (cid:18) k − (cid:19) (5 + 0) + (cid:18) k − (cid:19) (8 + 2) + (cid:18) k − (cid:19) (4 + 2)and F ( R/I, k ) = 1 + 0(1 − k ) − k − + 5 + 0(1 − k ) k + 8 + 2(1 − k ) k + 4 + 2(1 − k ) k respectively, where R = F [ x , x , x , x , x ] . Newton complementary dual of Quasi f -Ideal In [4], Budd and Van Tuyl examined that the property of being f -ideals remainthe same after taking the Newton complementary dual of a squarefree monomialideal I . As quasi f -ideal is the generalization of f -ideal. Therefore, it is natural toask the following question:When is Newton’s complementary dual of I a quasi f -ideal?In this section, we have tended to this inquiry. Firstly, we recall the [4, Definition3.1] of Newton complementary dual of a squarefree monomial ideal. efinition 4.1. Let I be a monomial ideal of R with G ( I ) = { u , u , . . . , u r } isa minimal set of generators of I . The Newton complementary dual of I is simplydenoted by b I and is defined as b I = h n Q i =1 x i u | u ∈ G ( I ) i Example 4.2.
The Newton complementary dual of an ideal I in Example 3.5 is b I = ( x x , x x , x x , x x , x x ) . It is easy to understand that b I = ( x x , x x , x x , x x , x x ) is not a quasi f -ideal. Because the minimal generating set G ( b I ) of b I is not upper perfect since x x x / ∈ ⊔ G ( b I ). But keeping in mind that I is a quasi f -ideal. It means that theNewton complementary dual of a quasi f -ideal need not be quasi f -ideal. The fol-lowing remark helps us to overcome the problem of duality of a squarefree monomialquasi f -ideal. Remark 4.3.
For a squarefree monomial ideal I of degree d in a polynomial ring R = F [ x , x , ..., x n ], G ( I ) is lower (upper) perfect set if and only if G ( b I ) is upper(lower) perfect. Indeed, If G ( I ) is lower perfect and suppose that there is a squarefreemonomial m / ∈ b I of degree n − d + 1. Then we must have a squarefree monomial u / ∈ I of degree d − G ( I ) is lower perfectset. This means that G ( I ) is perfect set if and only if G ( b I ) is perfect set. Theperfectness of G ( I ) guaranties that the dimensions of both, the facet complex of b I and the non-face complex of b I , coincide.Now, we recall [4, Corollary 3.6]. Corollary 4.4.
Let I be a squarefree monomial ideal of R .(1) If f ( δ F ( I )) = ( f − , f , ..., f d ), then f ( δ N ( b I )) = ( (cid:18) n (cid:19) − f n − , ..., (cid:18) ni (cid:19) − f n − i − ) , ..., (cid:18) nn − (cid:19) − f , (cid:18) nn (cid:19) − f − )(2) If f ( δ N ( I )) = ( f − , f , ..., f d ), then f ( δ F ( b I )) = ( (cid:18) n (cid:19) − f n − , ..., (cid:18) ni (cid:19) − f n − i − ) , ..., (cid:18) nn − (cid:19) − f , (cid:18) nn (cid:19) − f − )In both cases, f i = 0 if i > d . Theorem 4.5.
Let I be a squarefree monomial ideal of degree d in a polynomialring R = F [ x , x , ..., x n ] and let the minimal generating set G ( I ) of I be perfectset. Then I is a quasi f -ideal with type ( a − , a , a , ..., a d − ) ∈ Z d +1 , where a i = 0for all i ≥ d if and only if b I is a quasi f -ideal of type ( a n − , a n − , a n − , ..., a − ) . Proof. If I is a quasi f -ideal with type ( a − , a , a , ..., a d − ) ∈ Z d +1 , then f i ( δ F ( I )) = f i ( δ N ( I )) − a i for all i ∈ {− . , , ..., d − } . This means that the f -vector of facet omplex of I will be ( f − ( δ N ( I )) − a − , ( f ( δ N ( I )) − a , ..., ( f d − ( δ N ( I )) − a d − ). Now,using [4, Corollary 3.6], we have(4) f j ( δ N ( b I )) = (cid:18) nj + 1 (cid:19) − f n − j − ( δ N ( I )) + a n − j − where j ∈ {− . , , ..., n − } . It is noted that if ( f − , f , ..., f d − ) is f -vector of δ N ( I ), then from [4, Corollary 3.6], we get(5) f j ( δ F ( b I )) = (cid:18) nj + 1 (cid:19) − f n − j − ( δ N ( I ))where j ∈ {− . , , ..., n − } . From equation (3) and equation (4), we have(6) f j ( δ N ( b I )) = f j ( δ F ( b I )) + a n − j − This implies f j ( δ N ( b I )) − f j ( δ F ( b I )) = a n − j − for all j ∈ {− . , , ..., n − } . Thisshows that b I is a quasi f -ideal of type ( a n − , a n − , a n − , ..., a − ) . Similarly for reveresimplication, we simply replace I with b I . Corollary 4.6.
Let I be a squarefree monomial ideal of a polynomial ring R = F [ x , x , ..., x n ]. Then I is a quasi f -ideal of type -vector if and only if b I is a quasi f -ideal of type -vector. Proof.
It is obvious. Since, a quasi f -ideal I of type -vector is an f -ideal. It impliesthat the minimal generating set G ( I ) of I is perfect. Hence, we have needed to showfollows from the above theorem. Example 4.7.
Let us consider the ideal I in the polynomial ring F [ x , x , x , x , x , x ]given by I = ( x x , x x , x x , x x , x x , x x , x x , x x ). It is easy to see that I is a quasi f -ideal of type (0 , , −
1) ( or it can be written as (0 , , − , , , , , here a − = 0, a = 0, a = − a = 0, a = 0, a = 0 and a = 0).Clearly the Newton complementary dual b I of I is generated by the monomials { x x x x , x x x x , x x x x , x x x x , x x x x , x x x x , x x x x , x x x x } . Theprimary decomposition of b I is ( x , x ) T ( x , x ) T ( x , x ) T ( x , x ) T ( x , x ) T ( x , x ) T ( x , x ) T ( x , x , x ) T ( x , x , x ). It is easy to calculate that f ( δ F ( b I )) =(1 , , , ,
8) and f ( δ N ( b I )) = (1 , , , , b I is a quasi f -ideal with type(0 , , , , −
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