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

Bandari, Somayeh ; Jahan, Ali Soleyman

Czechoslovak Mathematical Journal, Tome 67 (2017) no. 3, pp. 767-778

Abstract (VO)

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 $m\geq 3$. 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.

Détail

DOI : 10.21136/CMJ.2017.0173-16

Classification : 05E40, 13F20, 13F55
Keywords: clean; Cohen-Macaulay simplicial complex; complete intersection; matroid; symbolic power

Bibliographie

[1] Achilles, R., Vogel, W.: Über vollständige Durchschnitte in lokalen Ringen. Math. Nachr. 89 (1979), 285-298 German. | DOI | MR | JFM

[2] Bandari, S., Divaani-Aazar, K., Jahan, A. S.: Almost complete intersections and Stanley's conjecture. Kodai Math. J. 37 (2014), 396-404. | DOI | MR | JFM

[3] Björner, A., Wachs, M. L.: Shellable nonpure complexes and posets. I. Trans. Am. Math. Soc. 348 (1996), 1299-1327. | DOI | MR | JFM

[4] Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1993). | MR | JFM

[5] Dress, A.: A new algebraic criterion for shellability. Beitr. Algebra Geom. 34 (1993), 45-55. | MR | JFM

[6] Faridi, S.: Monomial ideals via square-free monomial ideals. Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects A. Corso et al. Proc. Conf., Sevilla, 2003. Lecture Notes in Pure and Applied Mathematics 244, Chapman & Hall/CRC, Boca Raton (2006), 85-114. | DOI | MR | JFM

[7] Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics 260, Springer, London (2011). | DOI | MR | JFM

[8] Herzog, J., Popescu, D.: Finite filtrations of modules and shellable multicomplexes. Manuscr. Math. 121 (2006), 385-410. | DOI | MR | JFM

[9] Herzog, J., Popescu, D., Vladoiu, M.: On the Ext-modules of ideals of Borel type. Commutative Algebra: Interactions with Algebraic Geometry L. L. Avramov et al. Proc. Conf., Grenoble, 2001, Contemp. Math. 331, AMS, Providence (2003), 171-186. | DOI | MR | JFM

[10] Hoang, D. T., Minh, N. C., Trung, T. N.: Combinatorial characterizations of the Cohen-Macaulayness of the second power of edge ideals. J. Comb. Theory, Ser. A 120 (2013), 1073-1086. | DOI | MR | JFM

[11] Jahan, A. S.: Prime filtrations of monomial ideals and polarizations. J. Algebra 312 (2007), 1011-1032. | DOI | MR | JFM

[12] Minh, N. C., Trung, N. V.: Cohen-Macaulayness of powers of two-dimensional squarefree monomial ideals. J. Algebra 322 (2009), 4219-4227. | DOI | MR | JFM

[13] Minh, N. C., Trung, N. V.: Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley-Reisner ideals. Adv. Math. 226 (2011), 1285-1306 corrigendum ibid. 228 2982-2983 2011. | DOI | MR | JFM

[14] Oxley, J. G.: Matroid Theory. Oxford Graduate Texts in Mathematics 3, Oxford Science Publications, Oxford University Press, Oxford (1992). | MR | JFM

[15] Stanley, R. P.: Combinatorics and Commutative Algebra. Progress in Mathematics 41, Birkhäuser, Basel (1996). | DOI | MR | JFM

[16] Terai, N., Trung, N. V.: Cohen-Macaulayness of large powers of Stanley-Reisner ideals. Adv. Math. 229 (2012), 711-730. | DOI | MR | JFM

[17] Varbaro, M.: Symbolic powers and matroids. Proc. Am. Math. Soc. 139 (2011), 2357-2366. | DOI | MR | JFM

[18] Villarreal, R. H.: Monomial Algebras. Pure and Applied Mathematics 238, Marcel Dekker, New York (2001). | MR | JFM