On the symbolic powers of binomial edge ideals
aa r X i v : . [ m a t h . A C ] S e p ON THE SYMBOLIC POWERS OF BINOMIAL EDGE IDEALS
VIVIANA ENE, J ¨URGEN HERZOG
Abstract.
We show that under some conditions, if the initial ideal in < ( I ) ofan ideal I in a polynomial ring has the property that its symbolic and ordinarypowers coincide, then the ideal I shares the same property. We apply this resultto prove the equality between symbolic and ordinary powers for binomial edgeideals with quadratic Gr¨obner basis. Introduction
Binomial edge ideals were introduced in [11] and, independently, in [12]. Let S = K [ x , . . . , x n , y , . . . , y n ] be the polynomial ring in 2 n variables over a field K and G a simple graph on the vertex set [ n ] with edge set E ( G ) . The binomial edge ideal of G is generated by the set of 2-minors of the generic matrix X = x x · · · x n y y · · · y n ! indexed by the edges of G . In other words, J G = ( x i y j − x j y i : i < j and { i, j } ∈ E ( G )) . We will often use the notation [ i, j ] for the maximal minor x i y j − x j y i of X. In the last decade, several properties of binomial edge ideals have been studied.In [11], it was shown that, for every graph G, the ideal J G is a radical ideal and theminimal prime ideals are characterized in terms of the combinatorics of the graph.Several articles considered the Cohen-Macaulay property of binomial edge ideals;see, for example, [1, 8, 14, 15, 16]. A significant effort has been done for studyingthe resolution of binomial edge ideals. For relevant results on this topic we refer tothe recent survey [17] and the references therein.In this paper, we consider symbolic powers of binomial edge ideals. The study anduse of symbolic powers have been a reach topic of research in commutative algebrafor more than 40 years. Symbolic powers and ordinary powers do not coincide ingeneral. However, there are classes of homogeneous ideals in polynomial rings forwhich the symbolic and ordinary powers coincide. For example, if I is the edgeideal of a graph, then I k = I ( k ) for all k ≥ I (∆) of a simplicial complex ∆ has the property that I (∆) k = I (∆) ( k ) for all k ≥ I (∆) is normally torsion free) if andonly if ∆ is a Mengerian complex; see [10, Section 10.3.4]. The ideal of the maximalminors of a generic matrix shares the same property, that is, the symbolic andordinary powers coincide [6]. Mathematics Subject Classification.
Key words and phrases. symbolic power, binomial edge ideal, chordal graphs. o the best of our knowledge, the comparison between symbolic and ordinarypowers for binomial edge ideals was considered so far only in [13]. In Section 4 ofthis paper, Ohtani proved that if G is a complete multipartite graph, then J kG = J ( k ) G for all integers k ≥ . In our paper we prove that, for any binomial edge ideal with quadratic Gr¨obnerbasis, the symbolic and ordinary powers of J G coincide. The proof is based on thetransfer of the equality for symbolic and ordinary powers from the initial ideal tothe ideal itself.The structure of the paper is the following. In Section 2 we survey basic resultsneeded in the next section on symbolic powers of ideals in Noetherian rings and onbinomial edge ideals and their primary decomposition.In Section 3 we discuss symbolic powers in connection to initial ideals. Under somespecific conditions on the homogeneous ideal I in a polynomial ring over a field, onemay derive that if in < ( I ) k = in < ( I ) ( k ) for some integer k ≥ , then I k = I ( k ) ; seeLemma 3.1. By using this lemma and the properties of binomial edge ideals, we showin Theorem 3.3 that if in < ( J G ) is a normally torsion-free ideal, then the symbolicand ordinary powers of J G coincide. This is the case, for example, if G is a closedgraph (Corollary 3.4) or the cycle C . However, in general, in < ( J G ) is not a normallytorsion-free ideal. For example, for the binomial edge ideal of the 5–cycle, we have J C = J (2) C , but (in < ( J C )) ( (in < ( J C )) (2) . Preliminaries
In this section we summarize basic facts about symbolic powers of ideals andbinomial edge ideals.2.1.
Symbolic powers of ideals.
Let I ⊂ R be an ideal in a Noetherian ring R, and let Min( I ) the set of the minimal prime ideals of I. For an iteger k ≥ , onedefines the k th symbolic power of I as follows: I ( k ) = \ p ∈ Min( I ) ( I k R p ∩ R ) = \ p ∈ Min( I ) ker( R → ( R/I k ) p ) == { a ∈ R : for every p ∈ Min( I ) , there exists w p p with w p a ∈ I k } == { a ∈ R : there exists w [ p ∈ Min( I ) p with wa ∈ I k } . By the definition of the symbolic power, we have I k ⊆ I ( k ) for k ≥ . Symbolicpowers do not, in general, coincide with the ordinary powers. However, if I is acomplete intersection or it is the determinantal ideal generated by the maximalminors of a generic matrix, then it is known that I k = I ( k ) for k ≥
1; see [6] or [2,Corollary 2.3].Let I = Q ∩ · · · ∩ Q m an irredundant primary decomposition of I with √ Q i = p i for all i. If the minimal prime ideals of I are p , . . . p s , then I ( k ) = Q ( k )1 ∩ · · · ∩ Q ( k ) s . n particular, if I ⊂ R = K [ x , . . . , x n ] is a square-free monomial ideal in apolynomial ring over a field K , then I ( k ) = \ p ∈ Min( I ) p k . Moreover, I is normally torsion-free (i.e. Ass( I m ) ⊆ Ass( I ) for m ≥
1) if and onlyif I k = I ( k ) for all k ≥ , if and only if I is the Stanley-Reisner ideal of a Mengeriansimplicial complex; see [10, Theorem 1.4.6, Corollary 10.3.15]. In particular, if G is a bipartite graph, then its monomial edge ideal I ( G ) is normally torsion-free [10,Corollary 10.3.17].In what follows, we will often use the binomial expansion of symbolic powers [9].Let I ⊂ R and J ⊂ R ′ be two homogeneous ideals in the polynomial algebras R, R ′ in disjoint sets of variables over the same field K . We write I, J for the extensionsof these two ideals in R ⊗ K R ′ . Then, the following binomial expansion holds.
Theorem 2.1. [9, Theorem 3.4]
In the above settings, ( I + J ) ( n ) = X i + j = n I ( i ) J ( j ) . Moreover, we have the following criterion for the equality of the symbolic andordinary powers.
Corollary 2.2. [9, Corollary 3.5]
In the above settings, assume that I t = I t +1 and J t = J t +1 for t ≤ n − . Then ( I + J ) ( n ) = ( I + J ) n if and only if I ( t ) = I t and J ( t ) = J t for every t ≤ n. Binomial edge ideals.
Let G be a simple graph on the vertex set [ n ] with edgeset E ( G ) and let S be the polynomial ring K [ x , . . . , x n , y , . . . , y n ] in 2 n variablesover a field K. The binomial edge ideal J G ⊂ S associated with G is J G = ( f ij : i < j, { i, j } ∈ E ( G )) , where f ij = x i y j − x j y i for 1 ≤ i < j ≤ n. Note that f ij are exactly the maximalminors of the 2 × n generic matrix X = x x · · · x n y y · · · y n ! . We will use thenotation [ i, j ] for the 2- minor of X determined by the columns i and j. We consider the polynomial ring S endowed with the lexicographic order inducedby the natural order of the variables, and in < ( J G ) denotes the initial ideal of J G with respect to this monomial order. By [11, Corollary 2.2], J G is a radical ideal.Its minimal prime ideals may be characterized in terms of the combinatorics of thegraph G. We introduce the following notation. Let
S ⊂ [ n ] be a (possible empty)subset of [ n ], and let G , . . . , G c ( S ) be the connected components of G [ n ] \S where G [ n ] \S is the induced subgraph of G on the vertex set [ n ] \ S . For 1 ≤ i ≤ c ( S ) , let˜ G i be the complete graph on the vertex set V ( G i ) . Let P S ( G ) = ( { x i , y i } i ∈S ) + J ˜ G + · · · + J ˜ G c ( S ) . Then P S ( G ) is a prime ideal. Since the symbolic powers of an ideal of maxi-mal minors of a generic matrix coincide with the ordinary powers, and by using orollary 2.2, we get(1) P S ( G ) ( k ) = P S ( G ) k for k ≥ . By [11, Theorem 3.2], J G = T S⊂ [ n ] P S ( G ) . In particular, the minimal primes of J G are among the prime ideals P S ( G ) with S ⊂ [ n ] . The following propositioncharacterizes the sets S for which the prime ideal P S ( G ) is minimal. Proposition 2.3. [11, Corollary 3.9] P S ( G ) is a minimal prime of J G if and onlyif either S = ∅ or S is non-empty and for each i ∈ S , c ( S \ { i } ) < c ( S ) . In combinatorial terminology, for a connected graph G , P S ( G ) is a minimal primeideal of J G if and only if S is empty or S is non-empty and is a cut-point set of G, that is, i is a cut point of the restriction G ([ n ] \S ) ∪{ i } for every i ∈ S . Let C ( G ) be theset of all sets S ⊂ [ n ] such that P S ( G ) ∈ Min( J G ) . Let us also mention that, by [4, Theorem 3.1] and [4, Corollary 2.12], we have(2) in < ( J G ) = \ S∈C ( G ) in < P S ( G ) . Remark 2.4.
The cited results of [4] require that K is algebraically closed. How-ever, in our case, we may remove this condition on the field K. Indeed, neither theGr¨obner basis of J G nor the primary decomposition of J G depend on the field K, thus we may extend the field K to its algebraic closure ¯ K. When we study symbolic powers of binomial edge ideals, we may reduce to con-nected graphs. Let G = G ∪· · ·∪ G c where G , . . . , G c are the connected componentsof G and J G ⊂ S the binomial edge ideal of G. Then we may write J G = J G + · · · + J G c where J G i ⊂ S i = K [ x j , y j : j ∈ V ( G i )] for 1 ≤ i ≤ c. In the above equality, we usedthe notation J G i for the extension of J G i in S as well. Proposition 2.5.
In the above settings, we have J kG = J ( k ) G for every k ≥ if andonly if J kG i = J ( k ) G i for every k ≥ . Proof.
The equivalence is a direct consequence of Corollary 2.2. (cid:3) Symbolic powers and initial ideals
In this section we discuss the transfer of the equality between symbolic and ordi-nary powers from the initial ideal to the ideal itself.Let R = K [ x , . . . , x n ] be the polynomial ring over the field K and I ⊂ R ahomogeneous ideal. We assume that there exists a monomial order < on R suchthat in < ( I ) is a square-free monomial ideal. In particular, it follows that I is aradical ideal. Let Min( I ) = { p , . . . , p s } . Then I = T si =1 p i . Lemma 3.1.
In the above settings, we assume that the following conditions arefulfilled: (i) in < ( I ) = T si =1 in < ( p i ); ii) For an integer t ≥ we have: (a) p ( t ) i = p ti for ≤ i ≤ s ;(b) in < ( p ti ) = (in < ( p i )) t for ≤ i ≤ s ;(c) (in < ( I )) ( t ) = (in < ( I )) t . Then I ( t ) = I t . Proof.
In our hypothesis, we obtain:in < ( I t ) ⊇ (in < ( I )) t = (in < ( I )) ( t ) = s \ i =1 (in < ( p i )) ( t ) ⊇ s \ i =1 (in < ( p i )) t = s \ i =1 in < ( p ti ) ⊇⊇ in < ( s \ i =1 p ti ) = in < ( s \ i =1 p ( t ) i ) = in < ( I ( t ) ) ⊇ in < ( I t ) . Therefore, it follows that in < ( I ( t ) ) = in < ( I t ) . Since I t ⊆ I ( t ) , we get I t = I ( t ) . (cid:3) We now investigate whether one may use the above lemma for studying symbolicpowers of binomial edge ideals. Note that, by (2), the first condition in Lemma 3.1holds for any binomial edge ideal J G . In addition, as we have seen in (1), condition(a) in Lemma 3.1 holds for any prime ideal P S ( G ) and any integer t ≥ . Lemma 3.2.
Let
S ⊂ [ n ] . Then in < ( P S ( G ) t ) = (in < ( P S ( G ))) t , for every t ≥ . Proof.
To shorten the notation, we write P instead of P S ( G ), c instead of c ( S ) , and J i instead of J ˜ G i for 1 ≤ i ≤ c. Let R ( P ) , respectively R (in < ( P )) be the Reesalgebras of P, respectively in < ( P ) . Then, as the sets of variables { x j , y j : j ∈ V ( ˜ G i ) } are pairwise disjoint, we get(3) R ( P ) = R (( { x i , y i } i ∈S )) ⊗ K ( ⊗ ci =1 R ( J i )) . On the other hand, since in < ( P ) = ( { x i , y i } i ∈S ) + in < ( J ) + · · · + in < ( J c ) , due to thefact that J , . . . , J c are ideals in disjoint sets of variables different from { x i , y i } i ∈S (see [11]), we obtain R (in < P ) = R (( { x i , y i } i ∈S )) ⊗ K ( ⊗ ci =1 R (in < J i )) =(4) = R (( { x i , y i } i ∈S )) ⊗ K ( ⊗ ci =1 in < R ( J i )) . For the last equality we used the equality in < ( J ti ) = (in < J i ) t for all t ≥ R (in < J i ) = in < R ( J i ) due to [5,Theorem 2.7]. We know that R ( P ) and in < ( R ( P )) have the same Hilbert function.On the other hand, equalities (3) and (4) show that R ( P ) and R (in < P ) have thesame Hilbert function since R ( J i ) and in < R ( J i ) have the same Hilbert function forevery 1 ≤ i ≤ s. Therefore, R (in < P ) and in < R ( P ) have the same Hilbert function.As R (in < P ) ⊆ in < ( R ( P )), we have R (in < P ) = in < ( R ( P )), which implies by [5,Theorem 2.7] that in < ( P t ) = (in < P ) t for all t. (cid:3) Theorem 3.3.
Let G be a connected graph on the vertex set [ n ] . If in < ( J G ) is anormally torsion-free ideal, then J ( k ) G = J kG for k ≥ . Proof.
The proof is a consequence of Lemma 3.2 combined with relations (2) and(1). (cid:3) here are binomial edge ideals whose initial ideal with respect to the lexicographicorder are normally torsion-free. For example, the binomial edge ideals which havea quadratic Gr¨obner basis have normally torsion-free initial ideals. They were char-acterized in [11, Theorem 1.1] and correspond to the so-called closed graphs. Thegraph G is closed if there exists a labeling of its vertices such that for any edge { i, k } with i < k and for every i < j < k , we have { i, j } , { j, k } ∈ E ( G ) . If G isclosed with respect to its labeling, then, with respect to the lexicographic order < on S induced by the natural ordering of the indeterminates, the initial ideal of J G is in < ( J G ) = ( x i y j : i < j and { i, j } ∈ E ( G )) . This implies that in < ( J G ) is the edgeideal of a bipartite graph, hence it is normally torsion-free. Therefore, we get thefollowing. Corollary 3.4.
Let G be a closed graph on the vertex set [ n ] . Then J ( k ) G = J kG for k ≥ . Let C be the 4-cycle with edges { , } , { , } , { , } , { , } . Let < be the lexico-graphic order on K [ x , . . . , x , y , . . . , y ] induced by x > x > x > x > y > y >y > y . With respect to this monomial order, we havein < ( J C ) = ( x x y , x y , x y , x y y , x y , x y ) . Let ∆ be the simplicial complex whose facet ideal I (∆) = in < ( J C ) . It is easilyseen that ∆ has no special odd cycle, therefore, by [10, Theorem 10.3.16], it followsthat I (∆) is normally torsion-free. Note that the 4-cycle is a complete bipartitegraph, thus the equality J kC = J ( k ) C for all k ≥ C be the 5-cycle with edges { , } , { , } , { , } , { , } , { , } and I = in < ( J C ) the initial idealof J C with respect to the lexicographic order on K [ x . . . , x , y , . . . , y ] . By using
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Viviana Ene, Faculty of Mathematics and Computer Science, Ovidius University,Bd. Mamaia 124, 900527 Constanta, Romania
E-mail address : [email protected] J¨urgen Herzog, Fachbereich Mathematik, Universit¨at Duisburg-Essen, CampusEssen, 45117 Essen, Germany
E-mail address : [email protected]@uni-essen.de