Czechoslovak Mathematical Journal, first online, pp. 1-18


The linear syzygy graph of a monomial ideal and linear resolutions

Erfan Manouchehri, Ali Soleyman Jahan

Received March 4, 2020.   Published online November 18, 2020.

Abstract:  For each squarefree monomial ideal $I\subset S = k[x_1,\ldots, x_n] $, we associate a simple finite graph $G_I$ by using the first linear syzygies of $I$. The nodes of $G_I$ are the generators of $I$, and two vertices $u_i$ and $u_j$ are adjacent if there exist variables $x, y$ such that $xu_i = yu_j$. In the cases, where $G_I$ is a cycle or a tree, we show that $I$ has a linear resolution if and only if $I$ has linear quotients and if and only if $ I $ is variable-decomposable. In addition, with the same assumption on $G_I$, we characterize all squarefree monomial ideals with a linear resolution. Using our results, we characterize all Cohen-Macaulay codimension $2$ monomial ideals with a linear resolution. As another application of our results, we also characterize all Cohen-Macaulay simplicial complexes in the case, where $G_{\Delta}\cong G_{I_{\Delta^{\vee}}}$ is a cycle or a tree.
Keywords:  monomial ideal; linear resolution, linear quotient; variable-decomposability; Cohen-Macaulay simplicial complex
Classification MSC:  13D02, 13F55, 13F20
DOI:  10.21136/CMJ.2020.0099-20

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References:
[1] J. Abbott, A. M. Bigatti, G. Lagorio: CoCoA-5: A system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it/. SW 00143
[2] S. M. Ajdani, A. S. Jahan: Vertex decomposability of 2-CM and Gorenstein simplicial complexes of codimension 3. Bull. Malays. Math. Sci. Soc. (2) 39 (2016), 609-617. DOI 10.1007/s40840-015-0129-x | MR 3471335 | Zbl 1333.13031
[3] M. Bigdeli, J. Herzog, R. Zaare-Nahandi: On the index of powers of edge ideals. Commun. Algebra 46 (2018), 1080-1095. DOI 10.1080/00927872.2017.1339058 | MR 3780221 | Zbl 1428.13032
[4] A. Conca, E. De Negri: $M$-sequences, graph ideals, and ladder ideals of linear type. J. Algebra 211 (1999), 599-624. DOI 10.1006/jabr.1998.7740 | MR 1666661 | Zbl 0924.13012
[5] A. Conca, J. Herzog: Castelnuovo-Mumford regularity of product of ideals. Collect. Math. 54 (2003), 137-152. MR 1995137 | Zbl 1074.13004
[6] J. A. Eagon, V. Reiner: Resolutions of Stanley-Reisner rings and Alexander duality. J. Pure Appl. Algebra 130 (1998), 265-275. DOI 10.1016/S0022-4049(97)00097-2 | MR 1633767 | Zbl 0941.13016
[7] E. Emtander: A class of hypergraphs that generalizes chordal graphs. Math. Scand. 106 (2010), 50-66. DOI 10.7146/math.scand.a-15124 | MR 2603461 | Zbl 1183.05053
[8] C. A. Francisco, A. Van Tuyl: Sequentially Cohen-Macaulay edge ideals. Proc. Am. Math. Soc. 135 (2007), 2327-2337. DOI 10.1090/S0002-9939-07-08841-7 | MR 2302553 | Zbl 1128.13013
[9] R. Fröberg: On Stanley-Reisner rings. Topics in Algebra. Part 2. Commutative Rings and Algebraic Groups. Banach Center Publications 26. Polish Academy of Sciences, Institute of Mathematics, Warszaw (1990), 57-70. MR 1171260 | Zbl 0741.13006
[10] 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
[11] J. Herzog, T. Hibi, X. Zheng: Monomial ideals whose powers have a linear resolution. Math. Scand. 95 (2004), 23-32. DOI 10.7146/math.scand.a-14446 | MR 2091479 | Zbl 1091.13013
[12] J. Herzog, Y. Takayama: Resolutions by mapping cones. Homology Homotopy Appl. 4 (2002), 277-294. DOI 10.4310/HHA.2002.v4.n2.a13 | MR 1918513 | Zbl 1028.13008
[13] M. Morales: Simplicial ideals, 2-linear ideals and arithmetical rank. J. Algebra 324 (2010), 3431-3456. DOI 10.1016/j.jalgebra.2010.08.025 | MR 2735392 | Zbl 1217.13007
[14] R. Rahmati-Asghar, S. Yassemi: $k$-decomposable monomial ideals. Algebra Colloq. 22 (2015), 745-756. DOI 10.1142/S1005386715000656 | MR 3420707 | Zbl 1332.13019
[15] S. Saeedi Madani, D. Kiani, N. Terai: Sequentially Cohen-Macaulay path ideals of cycles. Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 54 (2011), 353-363. MR 2917856 | Zbl 1265.13016
[16] R. Woodroofe: Chordal and sequentially Cohen-Macaulay clutters. Electron. J. Comb. 18 (2011), Article ID P208, 20 pages. MR 2853065 | Zbl 1236.05213

Affiliations:   Erfan Manouchehri (corresponding author), Ali Soleyman Jahan, Department of Mathematics, University of Kurdistan, P.O.Box 66177-15175, Sanadaj, Iran, e-mail: erfanm6790@yahoo.com, solymanjahan@gmail.com


 
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