Czechoslovak Mathematical Journal, online first, 14 pp.

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.

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

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 2010): 13C15, 13P10, 13F20, 13F55

DOI: 10.21136/CMJ.2017.0172-16

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