Czechoslovak Mathematical Journal, Vol. 67, No. 3, pp. 753-766, 2017


Depth and Stanley depth of the facet ideals of some classes of simplicial complexes

Xiaoqi Wei, Yan Gu

Received April 7, 2016.   First published August 8, 2017.

Abstract:  Let $\Delta_{n,d}$ (resp. $\Delta_{n,d}'$) be the simplicial complex and the facet ideal $I_{n,d}=(x_1\cdots x_d,x_{d-k+1}\cdots x_{2d-k},\ldots,x_{n-d+1}\cdots x_n)$ (resp. $J_{n,d}=(x_1\cdots x_d,x_{d-k+1}\cdots x_{2d-k},\ldots,x_{n-2d+2k+1}\cdots x_{n-d+2k},x_{n-d+k+1}\cdots x_nx_1\cdots x_k)$). When $d\geq2k+1$, we give the exact formulas to compute the depth and Stanley depth of quotient rings $S/J_{n,d}$ and $S/I_{n,d}^t$ for all $t\geq1$. When $d=2k$, we compute the depth and Stanley depth of quotient rings $S/J_{n,d}$ and $S/I_{n,d}$, and give lower bounds for the depth and Stanley depth of quotient rings $S/I_{n,d}^t$ for all $t\geq1$.
Keywords:  monomial ideal; facet ideal; depth; Stanley depth
Classification MSC:  13C15, 13P10, 13F20, 13F55
DOI:  10.21136/CMJ.2017.0172-16


References:
[1] I. Anwar, D. Popescu: Stanley conjecture in small embedding dimension. J. Algebra 318 (2007), 1027-1031. DOI 10.1016/j.jalgebra.2007.06.005 | MR 2371984 | Zbl 1132.13009
[2] R. R. Bouchat: Free resolutions of some edge ideals of simple graphs. J. Commut. Algebra 2 (2010), 1-35. DOI 10.1216/JCA-2010-2-1-1 | MR 2607099 | Zbl 1238.13028
[3] W. Bruns, J. Herzog: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1998). DOI 10.1017/CBO9780511608681 | MR 1251956 | Zbl 0909.13005
[4] M. Cimpoeaş: Stanley depth of monomial ideals with small number of generators. Cent. Eur. J. Math. 7 (2009), 629-634. DOI 10.2478/s11533-009-0037-0 | MR 2563437 | Zbl 1185.13027
[5] M. Cimpoeaş: On the Stanley depth of edge ideals of line and cyclic graphs. Rom. J. Math. Comput. Sci. 5 (2015), 70-75. MR 3371758 | Zbl 06664242
[6] A. M. Duval, B. Goeckner, C. J. Klivans, J. L. Martin: A non-partitionable Cohen-Macaulay simplicial complex. Adv. Math. 299 (2016), 381-395. DOI 10.1016/j.aim.2016.05.011 | MR 3519473 | Zbl 1341.05256
[7] S. Faridi: The facet ideal of a simplicial complex. Manuscr. Math. 109 (2002), 159-174. DOI 10.1007/s00229-002-0293-9 | MR 1935027 | Zbl 1005.13006
[8] J. Herzog, M. Vladoiu, X. Zheng: How to compute the Stanley depth of a monomial ideal. J. Algebra 322 (2009), 3151-3169. DOI 10.1016/j.jalgebra.2008.01.006 | MR 2567414 | Zbl 1186.13019
[9] S. Morey: Depths of powers of the edge ideal of a tree. Commun. Algebra 38 (2010), 4042-4055. DOI 10.1080/00927870903286900 | MR 2764849 | Zbl 1210.13020
[10] R. Okazaki: A lower bound of Stanley depth of monomial ideals. J. Commut. Algebra 3 (2011), 83-88. DOI 10.1216/JCA-2011-3-1-83 | MR 2782700 | Zbl 1242.13025
[11] D. Popescu: Stanley depth of multigraded modules. J. Algebra 321 (2009), 2782-2797. DOI 10.1016/j.jalgebra.2009.03.009 | MR 2512626 | Zbl 1179.13016
[12] A. Rauf: Depth and Stanley depth of multigraded modules. Commun. Algebra 38 (2010), 773-784. DOI 10.1080/00927870902829056 | MR 2598911 | Zbl 1193.13025
[13] R. P. Stanley: Linear Diophantine equations and local cohomology. Invent. Math. 68 (1982), 175-193. DOI 10.1007/BF01394054 | MR 0666158 | Zbl 0516.10009
[14] A. Ştefan: Stanley depth of powers of the path ideal. Available at arXiv:1409.6072v1 [math.AC] (2014), 6 pages.
[15] R. H. Villarreal: Monomial Algebras. Pure and Applied Mathematics 238, Marcel Dekker, New York (2001). MR 1800904 | Zbl 1002.13010

Affiliations:   Xiaoqi Wei, Yan Gu (corresponding author), School of Mathematical Sciences, Soochow University, No. 1 Shizi Street, Suzhou 215006, Jiangsu, P. R. China, e-mail: weixiaoqi1989@sina.com, guyan@suda.edu.cn

Springer subscribers can access the papers on Springer website.
Access to full texts on this site is restricted to subscribers of Myris Trade. To activate your access, please send an e-mail to myris@myris.cz.
[List of online first articles] [Contents of Czechoslovak Mathematical Journal] [Full text of the older issues of Czechoslovak Mathematical Journal at DML-CZ]

 
PDF available at: