aa r X i v : . [ m a t h . A C ] A p r STANLEY DEPTH OF EDGE IDEALS
MUHAMMAD ISHAQ AND MUHAMMAD IMRAN QURESHI
Abstract.
We give an upper bound for the Stanley depth of the edge ideal I ofa k -partite complete graph and show that Stanley’s conjecture holds for I . Alsowe give an upper bound for the Stanley depth of the edge ideal of a k -uniformcomplete bipartite hypergraph.Key words : Monomial Ideals, Stanley decompositions, Stanley depth.2000 Mathematics Subject Classification: Primary 13C15, Secondary 13P10, 13F20,05E45, 05C65. Introduction
Let S = K [ x , . . . , x n ] be the polynomial ring in n variables over a field K and M be a finitely generated Z n -graded S -module. If u ∈ M is a homogeneous elementin M and Z ⊂ { x , . . . , x n } then let uK [ Z ] ⊂ M denote the linear K -subspaceof all elements of the form uf , f ∈ K [ Z ]. This space is called a Stanley space ofdimension | Z | if uK [ Z ] is a free K [ Z ]-module. A Stanley decomposition of module M is a presentation of the K -vector space M as a finite direct sum of Stanley spaces D : M = r M i =1 u i K [ Z i ] . The number sdepth( D ) = min {| Z i | : i = 1 , . . . , r } . is called the Stanley depth of decomposition D and the numbersdepth( M ) := max { sdepth( D ) : D is a Stanley decomposition of M } is called the Stanley depth of M . This is a combinatorial invariant which doesnot depend on the characteristic of K . In 1982 Stanley conjectured (see [13]) thatsdepth M ≥ depth M . This conjecture has been proved in several special cases (forexample see [1], [2], [8],[10], [11] and [12]) but it is still open in general. A methodto compute the Stanley depth is given in [5]. Even when it does not provide thevalue of the Stanley depth, this method allows one to obtain fairly good estimationsfor the invariant of interest.The aim of this paper is to bound the Stanley depth of the edge ideal of a complete k -partite graph and an s -uniform complete bipartite hypergraph(see Lemma 2.4,Theorem 3.4). In Corollary 2.8 we showed that Stanley’s conjecture holds for the The authors would like to express their gratitude to ASSMS of GC University Lahore for creatinga very appropriate atmosphere for research work. This research is partially supported by HECPakistan. dge ideal of a complete k -partite graph. Acknowledgement:
Both authors are grateful to Professor D. Popescu for helpfuldiscussions during the preparation of this paper.2.
Stanley depth of edge ideal of k -partite graph Definition 2.1.
Let G ( V, E ) be a graph with vertex set V and edge set E . Then G ( V, E ) is called a complete graph if every e ⊂ V such that | e | = 2 belongs to E . Definition 2.2.
A graph G ( V, E ) with vertex set V and edge set E is called complete k -partite if the vertex set V is partitioned into k disjoint subset V , V , . . . , V k and E = {{ u, v } : u ∈ V i , v ∈ V j , i = j } . Definition 2.3.
Let G be a graph. Then the edge ideal I associated to G is thesquarefree monomial ideal I = ( x i x j : { v i , v j } ∈ E ) of S .Now let G be a complete k -partite graph with vertex set V ( G ) = V ∪ V ∪ · · · ∪ V k with | V i | = r i , where r i ∈ N and 2 ≤ r ≤ · · · ≤ r k . Let r + · · · + r k = n . Let I = ( x , . . . , x r ), I = ( x r +1 , . . . , x r + r ) , . . . , I k = ( x r + ··· + r k − +1 , . . . , x n ) be themonomial ideals in S . Then the edge ideal of G is of the form I = ( X i = j I i ∩ I j ) . We recall the method of Herzog, Vladoiu and Zheng [5] for computing the Stanleydepth of a squarefree monomial ideal I using posets. Let G ( I ) = { u , . . . , v l } be theset of minimal monomial generators of I . The characteristic poset of I with respectto h = (1 , , . . . , P hI is in fact the set P hI = { C ⊂ [ n ] | C contains the supp( u i ) for some i } where supp( u i ) = { j : x j | u i } ⊆ [ n ] := { , . . . , n } . For every A, B ⊂ P hI with A ⊂ B , define the interval [ A, B ] to be { C ∈ P hI : A ⊆ C ⊆ B } . Let P : P hI = ∪ ri =1 [ C i , D i ] be a partition of P hI , and for each i , let c ( i ) ∈ { , } n be the n -tuplesuch that supp( x c ( i ) ) = C i . Then there is a Stanley decomposition D ( P ) of I D ( P ) : I = s M i =1 x c ( i ) K [ { x k | k ∈ D i } ] . By [5] we get thatsdepth( I ) = max { sdepth( D ( P )) | P is a partition of P hI } . Lemma 2.4. sdepth( I ) ≤ (cid:0) n (cid:1) − ( k P i =1 (cid:0) r i (cid:1) ) P ≤ i C, D ] in P with | C | = 2we have d − P ≤ i Let us consider I = ( I ∩ I , I ∩ I , I ∩ I , I ∩ I , I ∩ I , I ∩ I ) be a monomial ideal in S = K [ x , . . . , x ], where I = ( x , . . . , x ) , I =( x , . . . , x ) , I = ( x , . . . , x ) , I = ( x , . . . , x ).Applying Lemma 2.4 we get sdepth( I ) ≤ Lemma 2.6. Ass( S/I ) = { P , . . . , P k } where P i = ( x j | x j I i ) , ∀ i = 1 , . . . , k. Proof. We proceed as follows, Let I = ( I : x ) ∩ ( I, x ). We see that ( I : x ) = P .Let J = ( I, x ). Now J = ( J : x ) ∩ ( J , x ) we have ( J : x ) = ( P , x )But we can omit ( J : x ) because P already appear in the primary decomposition.Proceeding in this way up to step r we get I = P ∩ ( I , P ≤ i = j I i ∩ I j )Let J = ( I , P ≤ i = j I i ∩ I j ). Now we take J = ( J : x r +1 ) ∩ ( J , x r +1 ) and we get ( J : x r +1 ) = P . In this way, after r + r steps we get I = P ∩ P ∩ ( I , I , P ≤ i = j I i ∩ I j )and finally I = P ∩ · · · ∩ P k . (cid:3) Definition 2.7. We call the big size of I (see [10]) the minimal number t = t ( I ) < s such that the sum of all possible ( t + 1)-prime ideals of Ass( S/I ) = { P , ..., P k } isthe maximal ideal ( x , . . . , x n ). Corollary 2.8. Let I be the edge ideal of complete k -partite graph then Stanley’sconjecture holds for I . roof. We see that the big size of I is 1 by Lemma 2.6 so by [10, Corollary 1.6] (seealso [6, Theorem 1.2]) Stanley’s conjecture holds. (cid:3) Let I ′ = ( I, x n +1 , . . . , x n + p ) be a monomial ideal in S ′ = S [ x n +1 , . . . , x n + p ]. Letdenote by A the upper bound of sdepth( I ) found by Lemma 2.4. Theorem 2.9. Then sdepth( I ′ ) ≤ (cid:0) n (cid:1) − k P i =1 (cid:0) r i (cid:1) + (cid:0) p (cid:1) + n (cid:0) p (cid:1) + p (cid:0) n (cid:1)P ≤ i Note that I ′ is a squarefree monomial ideal generated by monomials of degree2 and 1. Let d = sdepth( I ′ ). The poset P I ′ has the partition P : P I ′ = S si =1 [ C i , D i ],satisfying sdepth( D ( P )) = d , where D ( P ) is the Stanley decomposition of I ′ withrespect to the partition P . We may choose P such that | D | = d whenever C = D in the interval [ C, D ].For each interval [ C i , D i ] in P with | C i | = 2 when in the corresponding monomialthe variables belong to { x , . . . , x n } we have | D i | − | C i | subsets of cardinality 3 inthis interval. We have P ≤ i 1) in number. Since the partition is disjoint, we subtractthis from total number of C l ’s, so that we have at least( X ≤ i 2) + p (cid:18) d − (cid:19) + (cid:2) np + (cid:18) p (cid:19) − p ( d − (cid:3) ( d − X ≤ i 2) + p (cid:18) d − (cid:19) + (cid:2) np + (cid:18) p (cid:19) − p ( d − (cid:3) ( d − ≤ (cid:18) n (cid:19) − X ≤ i 2) + p (cid:18) d − (cid:19) + (cid:2) np + (cid:18) p (cid:19) − p ( d − (cid:3) ( d − (cid:3) Example 2.10. Let I ′ = ( I, x , . . . , x ) ⊂ S ′ = S [ x , . . . , x ] be a monomialideal, where I is the same ideal as in Example 2.5. Then by [7, Theorem 2.11]sdepth( I ′ ) ≤ 23. We see that n = 30 , k = 4 , p = 10 , A = 13 , r = 7 , r = 7 , r =7 , r = 9 Now by our Theorem 2.9 we have sdepth( I ′ ) ≤ Stanley depth of edge ideal of an s -uniform complete bipartitehypergraph Definition 3.1. Let V = { v , . . . , v t } be a finite set and E = { E , . . . , E r } be acollection of distinct subsets of V . The pair G = G ( V, E ) is said to be a hypergraphif E j = ∅ for all j , where V and E are the sets of vertices and edges of G respectively.A hypergraph is said to be s -uniform hypergraph, if | E j | = s for all j . Definition 3.2. Associate to each vertex v j of a hypergraph G a variable x j of apolynomial ring S = K [ x , . . . , x t ] then the edge ideal of G is defined as E ( G ) = (cid:0) { Y v i ∈ E j x i : E j ∈ E } (cid:1) ⊂ S Let I ⊂ S be the edge ideal of a complete bipartite graph over n vertices with n ≥ I ) ≤ n + 22 . Now our aim is to give an upper bound for the Stanley depth of an edge ideal of ahypergraph which is a kind of generalization to the complete bipartite graph.We say that G s ( V, E ) is an s -uniform complete bipartite hypergraph if the follow-ing conditions holds(1) The vertex set V is partitioned into 2 disjoint subsets V and V .(2) For all hyperedges E i , E i ∩ V j = E i , j = 1 , s -subset of V such that F ∩ V j = F for j = 1 , E .If s = 2 then the hypergraph G ( V, E ) is just a complete bipartite graph. Example 3.3. Let G ( V ∪ V , E ) be a 3-uniform bipartite hypergraph with | V | = 3 and | V | = 3 . Then the edge ideal of G ( V ∪ V , E ) is 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 ) ⊂ K [ x , . . . , x ] . et I s ⊂ K [ x , x , . . . , x | V | ] denote the monomial edge ideal of the hypergraph G s ( V, E ). Then Theorem 3.4. s ≤ sdepth( I s ) ≤ s + (cid:0) | V | s +1 (cid:1) − (cid:0) | V | s +1 (cid:1) − (cid:0) | V | s +1 (cid:1)(cid:0) | V | s (cid:1) − (cid:0) | V | s (cid:1) − (cid:0) | V | s (cid:1) . Proof. Note that I s is a squarefree monomial ideal generated by squarefree mono-mials of degree s . By [9, Lemma 2.1] s ≤ sdepth( I s ). Now we count the number ofmonomials of degree s in I s . To count the number of monomials of degree s in I wehave to count the number of hyperedges of cardinality s in G s ( V, E ). The hypergraph G s ( V, E ) contains all the edges E i of cardinality s such that E i ∩ V j = E i , j = 1 , C of cardinality s does not belongs to G s ( V, E ) if C ⊂ V j for some j . Now let N denotes the number of hyperedges which belongsto G s ( V, E ). Then N = (cid:0) | V | s (cid:1) − (cid:0) | V | s (cid:1) − (cid:0) | V | s (cid:1) and if s > | V j | for some j then wetake (cid:0) | V j | s (cid:1) = 0. Similarly to count the number of squarefree monomials of degree s + 1 in I , we have to count the number of hyperedges of the hypergraph G s +1 ( V, E ).Let M be the number of hyperedges of the hypergraph G s +1 ( V, E ) then as before M = (cid:0) | V | s +1 (cid:1) − (cid:0) | V | s +1 (cid:1) − (cid:0) | V | s +1 (cid:1) and if s + 1 > | V j | for some j then we take (cid:0) | V j | s +1 (cid:1) = 0.By repeating the proof of Lemma 2.4 for I s we have sdepth( I s ) ≤ s + MN and therequired result follows. (cid:3) Example 3.5. Let I ⊂ S = K [ x , . . . , x ] be the edge ideal of the hypergraph G ( V, E ) with | V | = 15, | V | = 7 and | V | = 8. Then by Theorem 3.4 we have5 ≤ sdepth( I ) ≤ Remark 3.6. Let I be the edge ideal of the hypergraph G s ( V, E ), if s > | V | , | V | then I is the squarefree Veronese ideal I | V | , s . References [1] J. Apel, On a conjecture of R.P. Stanley, Part I — Monomial ideals, J. Algebraic Combin. (2003) 39–56.[2] I. Anwar, D. 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Popescu, Stanley depth of multigraded modules , J. Algebra (2009) 2782–2797.[12] D. Popescu, M. I. Qureshi, Computing the Stanley depth , J. Algebra (2010) 2943–2959.[13] R. Stanley, Linear Diophantine equations and local cohomology , Invent. Math. (1982) 175–193. Muhammad Ishaq, Abdus Salam School of Mathematical Sciences, GC University,Lahore, 68-B New Muslim town Lahore, Pakistan E-mail address : ishaq [email protected] Muhammad Imran Qureshi, Abdus Salam School of Mathematical Sciences, GCUniversity, Lahore, 68-B New Muslim town Lahore, Pakistan E-mail address : [email protected]@gmail.com