| Title:
|
Depth and Stanley depth of the facet ideals of some classes of simplicial complexes (English) |
| Author:
|
Wei, Xiaoqi |
| Author:
|
Gu, Yan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
3 |
| Year:
|
2017 |
| Pages:
|
753-766 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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.\ $J_{n,d}=(x_{1}\cdots x_{d},x_{d-k+1}\cdots x_{2d-k},\ldots ,x_{n-2d+2k+1}\cdots x_{n-d+2k},x_{n-d+k+1}\cdots x_{n}x_{1}\cdots x_{k})$). When $d\geq 2k+1$, 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 $t\geq 1$. 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 $t\geq 1$. (English) |
| Keyword:
|
monomial ideal |
| Keyword:
|
facet ideal |
| Keyword:
|
depth |
| Keyword:
|
Stanley depth |
| MSC:
|
13C15 |
| MSC:
|
13F20 |
| MSC:
|
13F55 |
| MSC:
|
13P10 |
| idZBL:
|
Zbl 06770128 |
| idMR:
|
MR3697914 |
| DOI:
|
10.21136/CMJ.2017.0172-16 |
| . |
| Date available:
|
2017-09-01T12:24:02Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146857 |
| . |
| Reference:
|
[1] Anwar, I., Popescu, D.: Stanley conjecture in small embedding dimension.J. Algebra 318 (2007), 1027-1031. Zbl 1132.13009, MR 2371984, 10.1016/j.jalgebra.2007.06.005 |
| Reference:
|
[2] Bouchat, R. R.: Free resolutions of some edge ideals of simple graphs.J. Commut. Algebra 2 (2010), 1-35. Zbl 1238.13028, MR 2607099, 10.1216/JCA-2010-2-1-1 |
| Reference:
|
[3] Bruns, W., Herzog, J.: Cohen-Macaulay Rings.Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1998). Zbl 0909.13005, MR 1251956, 10.1017/CBO9780511608681 |
| Reference:
|
[4] Cimpoeaş, M.: Stanley depth of monomial ideals with small number of generators.Cent. Eur. J. Math. 7 (2009), 629-634. Zbl 1185.13027, MR 2563437, 10.2478/s11533-009-0037-0 |
| Reference:
|
[5] Cimpoeaş, M.: On the Stanley depth of edge ideals of line and cyclic graphs.Rom. J. Math. Comput. Sci. 5 (2015), 70-75. Zbl 06664242, MR 3371758 |
| Reference:
|
[6] Duval, A. M., Goeckner, B., Klivans, C. J., Martin, J. L.: A non-partitionable Cohen-Macaulay simplicial complex.Adv. Math. 299 (2016), 381-395. Zbl 1341.05256, MR 3519473, 10.1016/j.aim.2016.05.011 |
| Reference:
|
[7] Faridi, S.: The facet ideal of a simplicial complex.Manuscr. Math. 109 (2002), 159-174. Zbl 1005.13006, MR 1935027, 10.1007/s00229-002-0293-9 |
| Reference:
|
[8] Herzog, J., Vladoiu, M., Zheng, X.: How to compute the Stanley depth of a monomial ideal.J. Algebra 322 (2009), 3151-3169. Zbl 1186.13019, MR 2567414, 10.1016/j.jalgebra.2008.01.006 |
| Reference:
|
[9] Morey, S.: Depths of powers of the edge ideal of a tree.Commun. Algebra 38 (2010), 4042-4055. Zbl 1210.13020, MR 2764849, 10.1080/00927870903286900 |
| Reference:
|
[10] Okazaki, R.: A lower bound of Stanley depth of monomial ideals.J. Commut. Algebra 3 (2011), 83-88. Zbl 1242.13025, MR 2782700, 10.1216/JCA-2011-3-1-83 |
| Reference:
|
[11] Popescu, D.: Stanley depth of multigraded modules.J. Algebra 321 (2009), 2782-2797. Zbl 1179.13016, MR 2512626, 10.1016/j.jalgebra.2009.03.009 |
| Reference:
|
[12] Rauf, A.: Depth and Stanley depth of multigraded modules.Commun. Algebra 38 (2010), 773-784. Zbl 1193.13025, MR 2598911, 10.1080/00927870902829056 |
| Reference:
|
[13] Stanley, R. P.: Linear Diophantine equations and local cohomology.Invent. Math. 68 (1982), 175-193. Zbl 0516.10009, MR 0666158, 10.1007/BF01394054 |
| Reference:
|
[14] Ştefan, A.: Stanley depth of powers of the path ideal.Available at arXiv:1409.6072v1 [math.AC] (2014), 6 pages. |
| Reference:
|
[15] Villarreal, R. H.: Monomial Algebras.Pure and Applied Mathematics 238, Marcel Dekker, New York (2001). Zbl 1002.13010, MR 1800904 |
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