Stanley decompositions and partitionable simplicial complexes

Jrgen Herzog 0 1 Ali Soleyman Jahan 0 1 Siamak Yassemi 0 1 0 S. Yassemi Institute for Theoretical Physics and Mathematics (IPM) , Tehran, Iran 1 S. Yassemi Department of Mathematics, University of Tehran , P.O. Box 13145448, Tehran, Iran We study Stanley decompositions and show that Stanley's conjecture on Stanley decompositions implies his conjecture on partitionable Cohen-Macaulay simplicial complexes. We also prove these conjectures for all Cohen-Macaulay monomial ideals of codimension 2 and all Gorenstein monomial ideals of codimension 3. In this paper we discuss the conjecture of Stanley [19] concerning a combinatorial upper bound for the depth of a Zn-graded module. Here we consider his conjecture only for S/I , where I is a monomial ideal. Let K be a field, S = K [x1, . . . , xn] the polynomial ring in n variables. Let u S be a monomial and Z a subset of {x1, . . . , xn}. We denote by uK [Z] the K -subspace Dedicated to Takayuki Hibi on the occasion of his fiftieth birthday. - of S whose basis consists of all monomials uv, where v is a monomial in K[Z]. The K -subspace uK[Z] S is called a Stanley space of dimension |Z|. Let I S be a monomial ideal, and denote by I c S the K -linear subspace of S spanned by all monomials which do not belong to I . Then S = I c I as a K -vector space, and the residues of the monomials in I c form a K -basis of S/I . A decomposition D of I c as a finite direct sum of Stanley spaces is called a Stanley decomposition of S/I . The minimal dimension of a Stanley space in the decomposition D is called the Stanley depth of D, denoted sdepth(D). We set sdepth(S/I ) = max{sdepth(D) : D is a Stanley decomposition of S/I } and call this number the Stanley depth of S/I . In [17, Conjecture 5.1] Stanley conjectured the inequality sdepth(S/I ) depth(S/I ). We say that I is a Stanley ideal if Stanleys conjecture holds for S/I . Not many classes of Stanley ideals are known. Apel [3, Corollary 3] showed that all monomial ideals I with dim S/I 1 are Stanley ideals. He also showed [3, Theorem 3 and Theorem 5] that all generic monomial ideals and all cogeneric Cohen Macaulay monomial ideals are Stanley ideals, and Soleyman Jahan [15, Proposition 2.1] proved that all monomial ideals in a polynomial ring in n variables of dimension less than or equal to 1 are Stanley ideals. The above facts imply in particular a result of Apel which says that all monomial ideals in the polynomial ring in three variables are Stanley ideals. The same result for four variables has been recently obtained in [2]. Moreover, Stanleys conjecture for small dimensions is also discussed in [1]. In [13] the authors attach to each monomial ideal a multi-complex and introduce the concept of shellable multi-complexes. In case I is a squarefree monomial ideal, this concept of shellability coincides with the nonpure shellability introduced by Bjrner and Wachs [4]. It is shown in [13, Theorem 10.5] that if I is pretty clean (see the definition in Sect. 3), then the multi-complex attached to I is shellable and I is a Stanley ideal. The concept of pretty clean modules is a generalization of clean modules introduced by Dress [8]. He showed that a simplicial complex is shellable if and only if its StanleyReisner ideal is clean. We use these results to prove that any CohenMacaulay monomial ideal of codimension 2 and any Gorenstein monomial ideal of codimension 3 is a Stanley ideal, see Proposition 2.4 and Theorem 3.1. For the proof of Proposition 2.4, we observe that the polarization of a perfect codimension 2 ideal is shellable and show this by using Alexander duality and the result of [11] that any monomial ideal with 2-linear resolution has linear quotients. The proof of Theorem 3.1 is based on the structure theorem for Gorenstein monomial ideals given in [5]. It also uses the result, proved in Proposition 3.3, that a pretty clean monomial ideal remains pretty clean after applying a substitution replacing the variables by a regular sequence of monomials. In the last section of this paper we introduce squarefree Stanley spaces and show in Proposition 4.2 that for a squarefree monomial ideal I , the Stanley decompositions of S/I into squarefree Stanley spaces correspond bijectively to partitions into intervals of the simplicial complex whose StanleyReisner ideal is the ideal I . Stanley calls a simplicial complex partitionable if there exists a partition = ir=1[Fi , Gi ] of such that for all intervals [Fi , Gi ] = {F : Fi F Gi } one has that Gi is a facet of . We show in Corollary 4.5 that the StanleyReisner ideal I of a Cohen Macaulay simplicial complex is a Stanley ideal if and only if is partitionable. In other words, Stanleys conjecture on Stanley decompositions implies his conjecture on partitionable simplicial complexes. 2 Stanley decompositions Let S = K[x1, . . . , xn] be a polynomial ring and I S a monomial ideal. Note that I and I c as well as all Stanley spaces are K -linear subspaces (...truncated)