On the Stanley depth of squarefree Veronese ideals
Mitchel T. Keller
, Yi-Huang Shen
, Noah Streib
and Stephen J. Young
DOI: 10.1007/s10801-010-0249-1
Abstract
Let K be a field and S= K[ x 1,\cdots , x n ]. In 1982, Stanley defined what is now called the Stanley depth of an S-module M, denoted sdepth\thinspace ( M), and conjectured that depth\thinspace ( M)\leq sdepth\thinspace ( M) for all finitely generated S-modules M. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when M= I/ J with J\subset I being monomial S-ideals. Specifically, their method associates M with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze squarefree Veronese ideals in S. In particular, if I n, d is the squarefree Veronese ideal generated by all squarefree monomials of degree d, we show that if 1\leq d\leq n<5 d+4, then sdepth\thinspace ( I n, d )=\lfloor ( n - d)/( d+1)\rfloor + d, and if d\geq 1 and n\geq 5 d+4, then d+3\leq sdepth\thinspace ( I n, d )\leq \lfloor ( n - d)/( d+1)\rfloor + d.
Pages: 313–324
Keywords: keywords Stanley depth; squarefree monomial ideal; interval partition; squarefree Veronese ideal
Full Text: PDF
References
1. Alcántar, A.: Rees algebras of square-free Veronese ideals and their a-invariants. Discrete Math. 302, 7-21 (2005)
2. Apel, J.: On a conjecture of R.P. Stanley. I. Monomial ideals. J. Algebr. Comb. 17(1), 39-56 (2003)
3. Apel, J.: On a conjecture of R.P. Stanley. II. Quotients modulo monomial ideals. J. Algebr. Comb. 17(1), 57-74 (2003)
4. Biró, Cs., Howard, D.M., Keller, M.T., Trotter, W.T., Young, S.J.: Interval partitions and Stanley depth. J. Comb. Theory Ser. A. 117(4), 475-482 (2010)
5. Cimpoea\?s, M.: Stanley depth of complete intersection monomial ideals. Bull. Math. Soc. Sci. Math. Roum. (N.S.) 51(99)(3), 205-211 (2008)
6. Cimpoea\?s, M.: A note on Stanley's conjecture for monomial ideals. [math.AC] (2009)
7. Cimpoea\?s, M.: Stanley depth of square free Veronese ideals. [math.AC] (2009)
8. Dress, A.: A new algebraic criterion for shellability. Beitr. Algebra Geom. 34(1), 45-55 (1993)
9. Herzog, J., Hibi, T.: Cohen-Macaulay polymatroidal ideals. Eur. J. Comb. 27, 513-517 (2006)
10. Herzog, J., Jahan, A.S., Yassemi, S.: Stanley decompositions and partitionable simplicial complexes. J. Algebr. Comb. 27, 113-125 (2008)
11. Herzog, J., Vladoiu, M., Zheng, X.: How to compute the Stanley depth of a monomial ideal. J. Algebra 322(9), 3151-3169 (2009)
12. Keller, M.T., Young, S.J.: Stanley depth of squarefree monomial ideals. J. Algebra 322(10), 3789- 3792 (2009)
13. Nasir, S.: Stanley decompositions and localization. Bull. Math. Soc. Sci. Math. Roum. (N.S.) 51(99)(2), 151-158 (2008)
14. Okazaki, R.: A lower bound of Stanley depth of monomial ideals. J. Commut. Algebra (2009, to appear)
15. Popescu, D.: Stanley depth of multigraded modules. J. Algebra 321(10), 2782-2797 (2009)
16. Shen, Y.H.: Stanley depth of complete intersection monomial ideals and upper-discrete partitions.
2. Apel, J.: On a conjecture of R.P. Stanley. I. Monomial ideals. J. Algebr. Comb. 17(1), 39-56 (2003)
3. Apel, J.: On a conjecture of R.P. Stanley. II. Quotients modulo monomial ideals. J. Algebr. Comb. 17(1), 57-74 (2003)
4. Biró, Cs., Howard, D.M., Keller, M.T., Trotter, W.T., Young, S.J.: Interval partitions and Stanley depth. J. Comb. Theory Ser. A. 117(4), 475-482 (2010)
5. Cimpoea\?s, M.: Stanley depth of complete intersection monomial ideals. Bull. Math. Soc. Sci. Math. Roum. (N.S.) 51(99)(3), 205-211 (2008)
6. Cimpoea\?s, M.: A note on Stanley's conjecture for monomial ideals. [math.AC] (2009)
7. Cimpoea\?s, M.: Stanley depth of square free Veronese ideals. [math.AC] (2009)
8. Dress, A.: A new algebraic criterion for shellability. Beitr. Algebra Geom. 34(1), 45-55 (1993)
9. Herzog, J., Hibi, T.: Cohen-Macaulay polymatroidal ideals. Eur. J. Comb. 27, 513-517 (2006)
10. Herzog, J., Jahan, A.S., Yassemi, S.: Stanley decompositions and partitionable simplicial complexes. J. Algebr. Comb. 27, 113-125 (2008)
11. Herzog, J., Vladoiu, M., Zheng, X.: How to compute the Stanley depth of a monomial ideal. J. Algebra 322(9), 3151-3169 (2009)
12. Keller, M.T., Young, S.J.: Stanley depth of squarefree monomial ideals. J. Algebra 322(10), 3789- 3792 (2009)
13. Nasir, S.: Stanley decompositions and localization. Bull. Math. Soc. Sci. Math. Roum. (N.S.) 51(99)(2), 151-158 (2008)
14. Okazaki, R.: A lower bound of Stanley depth of monomial ideals. J. Commut. Algebra (2009, to appear)
15. Popescu, D.: Stanley depth of multigraded modules. J. Algebra 321(10), 2782-2797 (2009)
16. Shen, Y.H.: Stanley depth of complete intersection monomial ideals and upper-discrete partitions.
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