Czechoslovak Mathematical Journal, online first, 12 pp.

The cleanness of (symbolic) powers of Stanley-Reisner ideals

Somayeh Bandari, Ali Soleyman Jahan

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

Somayeh Bandari, Department of Engineering Sciences and Physics, Buein Zahra Technical University, 3451745346, Buein Zahra, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran, e-mail:; Ali Soleyman Jahan, Department of Mathematics, University of Kurdistan, 66177-15175, Sanandaj, Iran, e-mail:

Abstract: Let $\Delta$ be a pure simplicial complex on the vertex set $[n]=\{1,\ldots,n\}$ and $I_\Delta$ its Stanley-Reisner ideal in the polynomial ring $S=K[x_1,\ldots,x_n]$. We show that $\Delta$ is a matroid (complete intersection) if and only if $S/I_\Delta^{(m)}$ ($S/I_\Delta^m$) is clean for all $m\in\mathbb{N}$ and this is equivalent to saying that $S/I_\Delta^{(m)}$ ($S/I_\Delta^m$, respectively) is Cohen-Macaulay for all $m\in\mathbb{N}$. By this result, we show that there exists a monomial ideal $I$ with (pretty) cleanness property while $S/I^m$ or $S/I^{(m)}$ is not (pretty) clean for all integer $m\geq3$. If $\dim(\Delta)=1$, we also prove that $S/I_\Delta^{(2)}$ ($S/I_\Delta^2$) is clean if and only if $S/I_\Delta^{(2)}$ ($S/I_\Delta^2$, respectively) is Cohen-Macaulay.

Keywords: clean; Cohen-Macaulay simplicial complex; complete intersection; matroid; symbolic power

Classification (MSC 2010): 13F20, 05E40, 13F55

DOI: 10.21136/CMJ.2017.0173-16

Full text available as PDF.

[1] R. Achilles, W. Vogel: Über vollständige Durchschnitte in lokalen Ringen. Math. Nachr. 89 (1979), 285-298. (In German.) DOI 10.1002/mana.19790890123 | MR 0546888 | Zbl 0416.13015
[2] S. Bandari, K. Divaani-Aazar, A. S. Jahan: Almost complete intersections and Stanley's conjecture. Kodai Math. J. 37 (2014), 396-404. DOI 10.2996/kmj/1404393894 | MR 3229083 | Zbl 1297.13024
[3] A. Björner, M. L. Wachs: Shellable nonpure complexes and posets. I. Trans. Am. Math. Soc. 348 (1996), 1299-1327. DOI 10.1090/S0002-9947-96-01534-6 | MR 1333388 | Zbl 0857.05102
[4] W. Bruns, J. Herzog: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1993). MR 1251956 | Zbl 0788.13005
[5] A. Dress: A new algebraic criterion for shellability. Beitr. Algebra Geom. 34 (1993), 45-55. MR 1239277 | Zbl 0780.52012
[6] S. Faridi: Monomial ideals via square-free monomial ideals. Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects (A. Corso et al., eds.). Proc. Conf., Sevilla, 2003. Lecture Notes in Pure and Applied Mathematics 244, Chapman & Hall/CRC, Boca Raton (2006), 85-114. DOI 10.1201/9781420028324.ch8 | MR 2184792 | Zbl 1094.13034
[7] J. Herzog, T. Hibi: Monomial Ideals. Graduate Texts in Mathematics 260, Springer, London (2011). DOI 10.1007/978-0-85729-106-6 | MR 2724673 | Zbl 1206.13001
[8] J. Herzog, D. Popescu: Finite filtrations of modules and shellable multicomplexes. Manuscr. Math. 121 (2006), 385-410. DOI 10.1007/s00229-006-0044-4 | MR 2267659 | Zbl 1107.13017
[9] J. Herzog, D. Popescu, M. Vladoiu: On the Ext-modules of ideals of Borel type. Commutative Algebra: Interactions with Algebraic Geometry (L. L. Avramov et al., eds.). Proc. Conf., Grenoble, 2001, Contemp. Math. 331, AMS, Providence (2003), 171-186. DOI 10.1090/conm/331 | MR 2013165 | Zbl 1050.13008
[10] D. T. Hoang, N. C. Minh, T. N. Trung: Combinatorial characterizations of the Cohen-Macaulayness of the second power of edge ideals. J. Comb. Theory, Ser. A 120 (2013), 1073-1086. DOI 10.1016/j.jcta.2013.02.008 | MR 3033662 | Zbl 1277.05174
[11] A. S. Jahan: Prime filtrations of monomial ideals and polarizations. J. Algebra 312 (2007), 1011-1032. DOI 10.1016/j.jalgebra.2006.11.002 | MR 2333198 | Zbl 1142.13022
[12] N. C. Minh, N. V. Trung: Cohen-Macaulayness of powers of two-dimensional squarefree monomial ideals. J. Algebra 322 (2009), 4219-4227. DOI 10.1016/j.jalgebra.2009.09.014 | MR 2558862 | Zbl 1206.13028
[13] N. C. Minh, N. V. Trung: Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley-Reisner ideals. Adv. Math. 226 (2011), 1285-1306; corrigendum ibid. 228 (2011), 2982-2983. DOI 10.1016/j.aim.2010.08.005 | MR 2737785 | Zbl 1204.13015
[14] J. G. Oxley: Matroid Theory. Oxford Graduate Texts in Mathematics 3, Oxford Science Publications, Oxford University Press, Oxford (1992). MR 1207587 | Zbl 0784.05002
[15] R. P. Stanley: Combinatorics and Commutative Algebra. Progress in Mathematics 41, Birkhäuser, Basel (1996). DOI 10.1007/b139094 | MR 1453579 | Zbl 0838.13008
[16] N. Terai, N. V. Trung: Cohen-Macaulayness of large powers of Stanley-Reisner ideals. Adv. Math. 229 (2012), 711-730. DOI 10.1016/j.aim.2011.10.004 | MR 2855076 | Zbl 1246.13032
[17] M. Varbaro: Symbolic powers and matroids. Proc. Am. Math. Soc. 139 (2011), 2357-2366. DOI 10.1090/S0002-9939-2010-10685-8 | MR 2784800 | Zbl 1223.13012
[18] R. H. Villarreal: Monomial Algebras. Pure and Applied Mathematics 238, Marcel Dekker, New York (2001). MR 1800904 | Zbl 1002.13010

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at
Subscribers of Springer need to access the articles on their site, which is

[List of online first articles] [Contents of Czechoslovak Mathematical Journal] [Full text of the older issues of Czechoslovak Mathematical Journal at DML-CZ]