Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
clean; Cohen-Macaulay simplicial complex; complete intersection; matroid; symbolic power
clean; Cohen-Macaulay simplicial complex; complete intersection; matroid; symbolic power
Summary:
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.
References:
[1] Achilles, R., Vogel, W.: Über vollständige Durchschnitte in lokalen Ringen. Math. Nachr. 89 (1979), 285-298 German. DOI 10.1002/mana.19790890123 | MR 0546888 | Zbl 0416.13015
[2] Bandari, S., Divaani-Aazar, K., Jahan, A. S.: 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] Björner, A., Wachs, M. L.: 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] Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1993). MR 1251956 | Zbl 0788.13005
[5] Dress, A.: A new algebraic criterion for shellability. Beitr. Algebra Geom. 34 (1993), 45-55. MR 1239277 | Zbl 0780.52012
[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 10.1201/9781420028324.ch8 | MR 2184792 | Zbl 1094.13034
[7] Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics 260, Springer, London (2011). DOI 10.1007/978-0-85729-106-6 | MR 2724673 | Zbl 1206.13001
[8] Herzog, J., Popescu, D.: 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] 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 10.1090/conm/331 | MR 2013165 | Zbl 1050.13008
[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 10.1016/j.jcta.2013.02.008 | MR 3033662 | Zbl 1277.05174
[11] Jahan, A. S.: 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] Minh, N. C., Trung, N. V.: 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] 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 10.1016/j.aim.2010.08.005 | MR 2737785 | Zbl 1204.13015
[14] Oxley, J. G.: Matroid Theory. Oxford Graduate Texts in Mathematics 3, Oxford Science Publications, Oxford University Press, Oxford (1992). MR 1207587 | Zbl 0784.05002
[15] Stanley, R. P.: Combinatorics and Commutative Algebra. Progress in Mathematics 41, Birkhäuser, Basel (1996). DOI 10.1007/b139094 | MR 1453579 | Zbl 0838.13008
[16] Terai, N., Trung, N. V.: 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] Varbaro, M.: 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] Villarreal, R. H.: Monomial Algebras. Pure and Applied Mathematics 238, Marcel Dekker, New York (2001). MR 1800904 | Zbl 1002.13010
Search
Partner of
