On the Stanley depth of the path ideal of a cycle graph Mircea Cimpoea¸s Abstract We give tight bounds for the Stanley depth of the quotient ring of the path ideal of a cycle graph. In particular, we prove that it satisfies the Stanley inequality. Keywords: Stanley depth, cycle graph, path ideal. 2010 Mathematics Subject Classification: 13C15, 13P10, 13F20. 6 1 0 Introduction 2 n a Let K be a field and S = K[x ,...,x ] the polynomial ring over K. Let M be a Zn-graded 1 n 3 J S-module. A Stanley decomposition of M is a direct sum D : M = ri=1miK[Zi] as a Zn-graded K-vector space, where m ∈ M is homogeneous with respect to Zn-grading, i L ] Z ⊂ {x ,...,x } such that m K[Z ] = {um : u ∈ K[Z ]} ⊂ M is a free K[Z ]-submodule C i 1 n i i i i i of M. We define sdepth(D) = min |Z | and sdepth(M) = max{sdepth(D)| D is a A i=1,...,r i Stanley decomposition of M}. The number sdepth(M) is called the Stanley depth of M. . h Herzog, Vladoiu and Zheng show in [10] that sdepth(M) can be computed in a finite t a number of steps if M = I/J, where J ⊂ I ⊂ S are monomial ideals. In [13], Rinaldo give a m computer implementation for this algorithm, in the computer algebra system CoCoA [6]. In [ [2], J. Apel restated a conjecture firstly given by Stanley in [14], namely that sdepth(M) ≥ 1 depth(M) for any Zn-graded S-module M. This conjecture proves to be false, in general, v 1 for M = S/I and M = J/I, where 0 6= I ⊂ J ⊂ S are monomial ideals, see [7]. For a 6 friendly introduction in the thematic of Stanley depth, we refer the reader [11]. 2 0 Let ∆ ⊂ 2[n] be a simplicial complex. A face F ∈ ∆ is called a facet, if F is maximal 0 with respect to inclusion. We denote F(∆) the set of facets of ∆. If F ∈ F(∆), we denote . 1 x = x . Then the facet ideal I(∆) associated to ∆ is the squarefree monomial ideal 0 F j∈F j 6 I = (xF : F ∈ F(∆)) of S. The facet ideal was studied by Faridi [8] from the depth Q 1 perspective. : v Theline graph oflenght n,denotedbyL ,isagraphwiththevertex set V = [n] andthe n i X edge set E = {{1,2},{2,3},...,{n−1,n}}. Let ∆ be the simplicial complex with the n,m r set of facets F(∆ ) = {{1,2,...,m},{2,3,...,m+1},...,{n−m+1,n−m+2,...,n}}, a n,m where 1 ≤ m ≤ n. We denote I = (x x ···x ,x x ···x ,...,x x ···x ) n,m 1 2 m 2 3 m+1 n−m+1 n−m+2 n , the associated facet ideal. Note that I is the m-path ideal of the graph L , provided n,m n with the direction given by 1 < 2 < ... < n, see [9] for further details. According to [9, Theorem 1.2], 2(n−d), n ≡ d(mod (m+1)) with 0 ≤ d ≤ m−1, pd(S/I ) = m+1 n,m (2n−m+1, n ≡ m(mod (m+1)). m+1 1The support from grant ID-PCE-2011-1023 of Romanian Ministry of Education, Research and Inno- vation is gratefully acknowledged. 1 ByAuslander-Buchsbaum formula(see[15]),itfollowsthatdepth(S/I ) = n−pd(S/I ) n,m n,m and, by a straightforward computation, we can see depth(S/I ) = n + 1 − n+1 − n,m m+1 n+1 =: ϕ(n,m). We proved in [5] that sdepth(S/I ) = ϕ(n,m). m+1 n,m (cid:4) (cid:5) The cycle graph of lenght n, denoted by C , is a graph with the vertex set V = n (cid:6) (cid:7) [n] and the edge set E = {{1,2},{2,3},...,{n − 1,n},{n,1}}. Let ∆¯ be the sim- n,m ¯ plicial complex with the set of facets F(∆ ) = {{1,2,...,m},{2,3,...,m + 1},··· , n,m {n − m + 1,n − m + 2,...,n},{n − m + 2,...,n,1},...,{n,1,...,m − 1}}. We denote J = (x x ···x ,x x ···x ,...,x x ···x ,...,x x ···x ), the associ- n,m 1 2 m 2 3 m+1 n−m+1 n−m+2 n n 1 m−1 ated facet ideal. Note that J is the m-path ideal of the graph C . n,m n Let p = n and d = n−(m+1)p. According to [1, Corollary 5.5], m+1 (cid:4) (cid:5) 2p+1, d 6= 0, pd(S/J ) = n,m (2p, d = 0. By Auslander-Buchsbaum formula, it follows that depth(S/J ) = n − pd(S/J ) = n,m n,m n − n − n =: ψ(n,m). Note that ψ(n,m) = ϕ(n − 1,m). Our main result is m+1 m+1 Theorem 1.4, in which we prove that ϕ(n,m) ≥ sdepth(S/J ) ≥ ψ(n,m). We also prove n,m (cid:4) (cid:5) (cid:6) (cid:7) that, sdepth(J /I ) = depth(J /I ) = ψ(n,m)+m−1, see Proposition 1.6. These n,m n,m n,m n,m results generalize [4, Theorem 1.9] and [4, Proposition 1.10]. 1 Main results First, we recall the well known Depth Lemma, see for instance [15, Lemma 1.3.9]. Lemma 1.1. (Depth Lemma) If 0 → U → M → N → 0 is a short exact sequence of modules over a local ring S, or a Noetherian graded ring with S local, then 0 a) depthM ≥ min{depthN,depthU}. b) depthU ≥ min{depthM,depthN +1}. c) depthN ≥ min{depthU −1,depthM}. In [12], Asia Rauf proved the analog of Lemma 1.1(a) for sdepth: Lemma 1.2. Let 0 → U → M → N → 0 be a short exact sequence of Zn-graded S- modules. Then: sdepth(M) ≥ min{sdepth(U),sdepth(N)}. The following result is well known. However, we present an original proof. Lemma 1.3. Let I ⊂ S be a nonzero proper monomial ideal. Then, I is principal if and only if sdepth(S/I) = n−1. Proof. Assume sdepth(S/I) = n−1 and let S/I = r u K[Z ] be a Stanley decomposi- i=1 i i tion with |Z | = n−1 for all i, and u ∈ S monomials. Since 1 ∈/ I, we may assume that i i L u = 1. Let x be the variable which is not in Z . If x ∈ I, since S/(x ) = K[Z ] and 1 j1 1 j1 j1 1 K[Z ] ⊂ S/I, then I = (x ). Otherwise, we may assume that u = x . 1 j1 2 j1 2 Let x be the variable which is not in Z . If x x ∈ I, then, one can easily see that j2 2 j1 j2 I = (x x ). If x x ∈/ I, then we may assume u = x x and so on. Thus, we have j1 j2 j1 j2 3 j1 j2 u = x ···x , for all 1 ≤ i ≤ r+1, where x is the variable which is not in Z . Moreover, i j1 ji−1 ji i I = (u ), and therefore I is principal. r+1 In order to prove theother implication, assume that I = (u)andwrite u = r x . We i=1 ji letu = i−1 x andZ = {x ,...,x }\{x },forall1 ≤ i ≤ r.Then,S/I = r u K[Z ] i k=1 jk i 1 n ji Qi=1 i i is a Stanley decomposition with |Z | = n−1 for all i. Therefore sdepth(S/I) = n−1. i Q L Our main result, is the following Theorem. Theorem 1.4. ϕ(n,m) ≥ sdepth(S/J ) ≥ depth(S/J ) = ψ(n,m). n,m n,m Proof. If n = m, then J = (x ...x ) is a principal ideal, and, according to Lemma 1.3 n,n 1 n we are done. Also, if m = 1, then J = (x ,...,x ) and so there is nothing to prove, since n,1 1 n S/J = K. The case m = 2 follows from [4, Proposition 1.8] and [4, Theorem 1.9]. n,1 Assume n > m ≥ 3. If n = m + 1, then we consider the short exact sequence 0 → S/(J : x ) → S/J → S/(J ,x ) → 0. Note that (J : x ) = n,n−1 n n,n−1 n,n−1 n n,n−1 n ∼ (x ···x ,x ···x ,x ···x x ,··· ,x x ···x ) = J S. Therefore, by in- 1 n−2 2 n−1 3 n−1 1 n−1 1 n−3 n−1,n−2 duction hypothesis and [10, Lemma 3.6], sdepth(S/(J : x )) = depth(S/(J : x )) = 1+ψ(n−1,n−2) = n−2. n,n−1 n n,n−1 n ∼ Also,(J ,x ) = (x ···x ,x )andthusS/(J ,x ) = K[x ,...,x ]/(x ···x ). n,n−1 n 1 n−1 n n,n−1 n 1 n−1 1 n−1 Therefore, by Lemma 1.3, we have sdepth(S/(J ,x )) = n−2 = depth(S/(J ,x )). n,n−1 n n,n−1 n Now, assume n > m + 1 > 3. We consider the ideals L = J , L = (L : x ) 0 n,m k+1 k n−k and U = (L ,x ), for 0 ≤ k ≤ m − 2. Note that L = (J : x ···x ) = k k n−k m−1 n,m n−m+2 n (x ,x ···x ,...,x ···x ,x ). 1 2 m+1 n−2m+1 n−m n−m+1 Ifn−2m ≤ 2,thenL = (x ,x )andthussdepth(S/L ) = depth(S/L ) = m−1 1 n−m+1 m−1 m−1 n − 2 = ϕ(n,m), since n+1 = 1 and n+1 = 2. If n − 2m > 2, then S/L ∼= m+1 m+1 m−1 K[x ,...,x ,x ,...,x ]/(x ···x ,...,x ···x ) and therefore, by [10, 2 n−m n−m+2 n 2 m+1 n−2m+1 n−m (cid:4) (cid:5) (cid:6) (cid:7) Lemma 3.6] and [5, Theorem 1.3], we have sdepth(S/L ) = depth(S/L ) = n − m−1 m−1 1 − n−m − n−m = ϕ(n,m). On the other hand, for example by [3, Proposition 2.7], m+1 m+1 sdepth(S/L ) ≥ sdepth(S/J ). Thus, sdepth(S/J ) ≤ ϕ(n,m). m−1 n,m n,m (cid:4) (cid:5) (cid:6) (cid:7) For any 0 < k < m, we have L = (x ···x ,x ···x ,...,x ···x , k 1 m−k 2 m+1 n−m−k n−k−1 x ···x ,x ···x x ,...,x x ···x ). Therefore, U = (x ···x , n−m+1 n−k n−m+2 n−k 1 n−k 1 m−k−1 k 1 m−k x ···x ,...,x ···x ,x ), for k ≤ m−2. We consider two cases: 2 m+1 n−m−k n−k−1 n−k (i) If n−m−k < 2 and 0 ≤ k ≤ m−2, then U = (x ···x ,x ) and therefore k 1 m−k n−k sdepth(S/U ) = depth(S/U ) = n−2 = ϕ(n,m), since n+1 = 1 and n+1 = 2. k k m+1 m+1 (ii) If n − m − k ≥ 2, then, for any 0 ≤ j ≤ k ≤ m − 2, we consider the ideals (cid:4) (cid:5) (cid:6) (cid:7) V := (x ···x ,x ···x ,...,x ···x ) in S := K[x ,...,n ]. Note k,j 1 m−j 2 m+1 n−m−k n−k−1 k 1 n−k−1 ∼ that S/U = (S /V )[x ,...,x ] and thus, by [10, Lemma 3.6], depth(S/U ) = k k k,k n−k+1 n k depth(S /V )+k and sdepth(S/U ) = sdepth(S /V )+k. k k,k k k k,k For any 0 ≤ j < k ≤ m−2, we claim that V /V is isomorphic to k,j k,j+1 (K[x ,...,x ]/(x ···x ,...,x ···x ))[x ,...,x ]. m−j+2 n−k−1 m−j+2 2m−j+1 n−m−k n−k−1 1 m−j 3 Indeed, if u ∈ V \ V is a monomial, then x ···x |u and x ∤ u. Also, k,j k,j+1 1 m−j m−j+1 x ···x ∤ u, ..., x ···x ∤ u. Denoting v = u/(x ···x ), we can m−j+2 2m−j+1 n−m−k n−k−1 1 m−j writev = v′v′′,withv′ ∈ K[x ,...,x ]\(x ···x ,...,x ···x ) m−j+2 n−k−1 m−j+2 2m−j+1 n−m−k n−k−1 and v′′ ∈ K[x ,...,x ]. 1 m−j By [10, Lemma 3.6] and [5, Theorem 1.3], sdepth(V /V ) = depth(V /V ) = k,j k,j+1 k,j k,j+1 m − j + ϕ(n − k − m + j − 2,m) = n − k − 1 − n−m−1−k+j − n−m−1−k+j = n − m+1 m+1 k + 1 − n−k+j − n−k+j ≥ ϕ(n,m) − k. On the other hand, V = I for any m+1 m+1 (cid:4) (cid:5) k,0(cid:6) n−k−1,m(cid:7) 0 ≤ k ≤ m − 2 and therefore, by [5, Theorem 1.3], sdepth(S/V ) = depth(S/V ) = k,0 k,0 (cid:4) (cid:5) (cid:6) (cid:7) ϕ(n − k − 1,m) = n − k − n−k − n−k ≥ ϕ(n,m) − k, for any k ≥ 1. From the m+1 m+1 short exact sequences 0 → V /V → S/V → S/V → 0, 0 ≤ j < k, Lemma 1.1 k,j k,j+1 k,j+1 k,j (cid:4) (cid:5) (cid:6) (cid:7) and Lemma 1.2, it follows that sdepth(S/V ) ≥ depth(S/V ) = ϕ(n,m)−k, for all k,j+1 k,j+1 0 ≤ j < k ≤ m−2. Thus sdepth(S/U ) ≥ depth(S/U ) ≥ ϕ(n,m),forall0 < k ≤ m−2.On k k the other hand, sdepth(S/V ) = depth(S/V ) = ϕ(n−1,m) = ψ(n,m) −m, and thus 0,0 0,0 sdepth(S/U ) = depth(S/U ) = ψ(n,m). 0 0 Now, we consider short exact sequences 0 → S/L → S/L → S/U → 0. for 0 ≤ k < m. k+1 k k By Lemma 1.1 and Lemma 1.2 we get sdepth(S/L ) ≥ depth(S/L ) = ϕ(n,m), for any k k 0 < k ≤ m−2, and sdepth(S/L ) ≥ depth(S/L ) = ψ(n,m). 0 0 Corollary 1.5. If n+1 = n and n+1 = n , then m+1 m+1 m+1 m+1 (cid:4) sd(cid:5)epth(cid:4)(S/J(cid:5)n,m) =(cid:6)dept(cid:7)h(S/(cid:6)Jn,m(cid:7)) = ϕ(n,m). Proposition 1.6. sdepth(J /I ) ≥ depth(J /I ) = ψ(n,m)+m−1. n,m n,m n,m n,m Proof. We claim that J /I is isomorphic to n,m n,m xn−m+2···xnx1(K[x2,...,xn−m]/(x2···xm,x3···xm+2...,xn−2m+1···xn−m))[xn−m+2,...,xn,x1]⊕ ⊕xn−m+3···xnx1x2(K[x3,...,xn−m+1]/(x3···xm,x4···xm+3,...,xn−2m+2···xn−m+1))[xn−m+3,...,xn,x1,x2]⊕ ···⊕xnx1···xm−1(K[xm,...,xn−2]/(xm,xm+1···x2m,...,xn−m−1···xn−2))[xn,x1...,xm−1]. Indeed, let u ∈ J \ I be a monomial. If x ···x x |u, then x ∤ u and n,m n,m n−m+2 n 1 n−m+1 x ···x ∤ u. It follows that: 2 m u∈xn−m+2···xnx1(K[x2,...,xn−m]/(x2···xm,x3···xm+2...,xn−2m+1···xn−m))[xn−m+2,...,xn,x1]. If x ···x x ∤ u and x ···x x x |u then x ∤ u and x ···x ∤ u. Thus: n−m+2 n 1 n−m+3 n 1 2 n−m+2 3 m u∈xn−m+3···xnx1x2(K[x3,...,xn−m+1]/(x3···xm,x4···xm+3,...,xn−2m+2···xn−m+1))[xn−m+3,...,xn,x1,x2]. Finally,ifx ···x x ∤ u,..., x x x ···x ∤ uandx x ···x |u,thenitfollows n−m+2 n 1 n−1 n 1 m−2 n 1 m−1 that x ∤ u and x ∤ u. Therefore: n−1 m u∈/ xnx1···xm−1(K[xm,...,xn−2]/(xm,xm+1···x2m,...,xn−m−1···xn−2))[xn,x1...,xm−1]. AsintheproofofTheorem3.1(seethecomputationsforV ’s),byapplyingLemma1.1and k,j Lemma 1.2, it follows that sdepth(J /I ) ≥ depth(J /I ) = ϕ(n−m−2,m)+m = n,m n,m n,m n,m ψ(n,m)+m−1, as required. Inspiredby[4,Conjecture1.12]andcomputerexperiments [6],weproposethefollowing: Conjecture 1.7. For any n ≥ 3(m+1)+1, we have sdepth(S/J ) = ϕ(n,m). n,m 4 References [1] A. Alilooee, S. Faridi, On the resolution of path ideals of cycles, Communications in Algebra, Vol. 43, Issue 12, (2015), 5413-5433 . [2] J. Apel, On a conjecture of R. P. Stanley; Part II - Quotients Modulo Monomial Ideals, J. of Alg. Comb. 17, (2003), 57-74. [3] M.Cimpoeas,Several inequalities regarding Stanley depth,RomanianJournalofMath. and Computer Science 2(1), (2012), 28-40. [4] M. Cimpoeas, On the Stanley depth of edge ideals of line and cyclic graphs, Romanian Journal of Math. and Computer Science 5(1), (2015), 70-75. [5] M. Cimpoeas, Stanley depth of the path ideal associated to a line graph,http://arxiv.org/pdf/1508.07540v2.pdf, to appear in Mathematical Reports. [6] CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra, Available at http://cocoa.dima.unige.it [7] A. M. Duval, B. Goeckneker, C. J. Klivans, J. L. Martine, A non-partitionable Cohen- Macaulay simplicial complex, http://arxiv.org/pdf/1504.04279 [8] S. Faridi, The facet ideal of a simplicial complex, Manuscripta Mat., 109 (2002), 159- 174. [9] Jing He, Adam Van Tuyl, Algebraic properties of the path ideal of a tree, Comm. Algebra 38 (2010), no. 5, 1725-742. [10] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, Journal of Algebra 322(9), (2009), 3151-3169. [11] J. Herzog, A survey on Stanley depth, In Monomial Ideals, Computations and Appli- cations, Springer, (2013), 3-45. [12] A. Rauf, Depth and sdepth of multigraded module, Communications in Algebra, Vol. 38, Issue 2, (2010), 773-784. [13] G.Rinaldo,An algorithm to compute the Stanley depth of monomial ideals,LeMatem- atiche, Vol. LXIII (ii), (2008), 243-256. [14] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68, (1982), 175-193. [15] R. H. Villarreal, Monomial algebras. Monographs and Textbooks in Pure and Applied Mathematics, 238, Marcel Dekker, Inc., New York, (2001). Mircea Cimpoea¸s, Simion Stoilow Institute of Mathematics, Research unit 5, P.O.Box 1-764, Bucharest 014700,Romania, E-mail: mircea.cimpoeas@imar.ro 5

On the Stanley depth of the path ideal of a cycle graph

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