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

Full text available as PDF.

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