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

Xiaoqi Wei; Yan Gu

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 3, page 753-766
  • ISSN: 0011-4642

Abstract

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Let Δ n , d (resp. Δ n , d ' ) be the simplicial complex and the facet ideal I n , d = ( x 1 x d , x d - k + 1 x 2 d - k , ... , x n - d + 1 x n ) (resp. J n , d = ( x 1 x d , x d - k + 1 x 2 d - k , ... , x n - 2 d + 2 k + 1 x n - d + 2 k , x n - d + k + 1 x n x 1 x k ) ). When d 2 k + 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 1 . When d = 2 k , 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 1 .

How to cite

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Wei, Xiaoqi, and Gu, Yan. "Depth and Stanley depth of the facet ideals of some classes of simplicial complexes." Czechoslovak Mathematical Journal 67.3 (2017): 753-766. <http://eudml.org/doc/294092>.

@article{Wei2017,
abstract = {Let $\Delta _\{n,d\}$ (resp. $\Delta _\{n,d\}^\{\prime \}$) 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. $J_\{n,d\}=(x_\{1\}\cdots x_\{d\},x_\{d-k+1\}\cdots x_\{2d-k\},\ldots ,x_\{n-2d+2k+1\}\cdots x_\{n-d+2k\},x_\{n-d+k+1\}\cdots x_\{n\}x_\{1\}\cdots x_\{k\})$). When $d\ge 2k+1$, 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 $t\ge 1$. 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 $t\ge 1$.},
author = {Wei, Xiaoqi, Gu, Yan},
journal = {Czechoslovak Mathematical Journal},
keywords = {monomial ideal; facet ideal; depth; Stanley depth},
language = {eng},
number = {3},
pages = {753-766},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Depth and Stanley depth of the facet ideals of some classes of simplicial complexes},
url = {http://eudml.org/doc/294092},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Wei, Xiaoqi
AU - Gu, Yan
TI - Depth and Stanley depth of the facet ideals of some classes of simplicial complexes
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 3
SP - 753
EP - 766
AB - Let $\Delta _{n,d}$ (resp. $\Delta _{n,d}^{\prime }$) 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. $J_{n,d}=(x_{1}\cdots x_{d},x_{d-k+1}\cdots x_{2d-k},\ldots ,x_{n-2d+2k+1}\cdots x_{n-d+2k},x_{n-d+k+1}\cdots x_{n}x_{1}\cdots x_{k})$). When $d\ge 2k+1$, 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 $t\ge 1$. 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 $t\ge 1$.
LA - eng
KW - monomial ideal; facet ideal; depth; Stanley depth
UR - http://eudml.org/doc/294092
ER -

References

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  1. Anwar, I., Popescu, D., 10.1016/j.jalgebra.2007.06.005, J. Algebra 318 (2007), 1027-1031. (2007) Zbl1132.13009MR2371984DOI10.1016/j.jalgebra.2007.06.005
  2. Bouchat, R. R., 10.1216/JCA-2010-2-1-1, J. Commut. Algebra 2 (2010), 1-35. (2010) Zbl1238.13028MR2607099DOI10.1216/JCA-2010-2-1-1
  3. Bruns, W., Herzog, J., 10.1017/CBO9780511608681, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1998). (1998) Zbl0909.13005MR1251956DOI10.1017/CBO9780511608681
  4. Cimpoeaş, M., 10.2478/s11533-009-0037-0, Cent. Eur. J. Math. 7 (2009), 629-634. (2009) Zbl1185.13027MR2563437DOI10.2478/s11533-009-0037-0
  5. Cimpoeaş, M., On the Stanley depth of edge ideals of line and cyclic graphs, Rom. J. Math. Comput. Sci. 5 (2015), 70-75. (2015) Zbl06664242MR3371758
  6. Duval, A. M., Goeckner, B., Klivans, C. J., Martin, J. L., 10.1016/j.aim.2016.05.011, Adv. Math. 299 (2016), 381-395. (2016) Zbl1341.05256MR3519473DOI10.1016/j.aim.2016.05.011
  7. Faridi, S., 10.1007/s00229-002-0293-9, Manuscr. Math. 109 (2002), 159-174. (2002) Zbl1005.13006MR1935027DOI10.1007/s00229-002-0293-9
  8. Herzog, J., Vladoiu, M., Zheng, X., 10.1016/j.jalgebra.2008.01.006, J. Algebra 322 (2009), 3151-3169. (2009) Zbl1186.13019MR2567414DOI10.1016/j.jalgebra.2008.01.006
  9. Morey, S., 10.1080/00927870903286900, Commun. Algebra 38 (2010), 4042-4055. (2010) Zbl1210.13020MR2764849DOI10.1080/00927870903286900
  10. Okazaki, R., 10.1216/JCA-2011-3-1-83, J. Commut. Algebra 3 (2011), 83-88. (2011) Zbl1242.13025MR2782700DOI10.1216/JCA-2011-3-1-83
  11. Popescu, D., 10.1016/j.jalgebra.2009.03.009, J. Algebra 321 (2009), 2782-2797. (2009) Zbl1179.13016MR2512626DOI10.1016/j.jalgebra.2009.03.009
  12. Rauf, A., 10.1080/00927870902829056, Commun. Algebra 38 (2010), 773-784. (2010) Zbl1193.13025MR2598911DOI10.1080/00927870902829056
  13. Stanley, R. P., 10.1007/BF01394054, Invent. Math. 68 (1982), 175-193. (1982) Zbl0516.10009MR0666158DOI10.1007/BF01394054
  14. Ştefan, A., Stanley depth of powers of the path ideal, Available at arXiv:1409.6072v1 [math.AC] (2014), 6 pages. (2014) 
  15. Villarreal, R. H., Monomial Algebras, Pure and Applied Mathematics 238, Marcel Dekker, New York (2001). (2001) Zbl1002.13010MR1800904

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