The linear syzygy graph of a monomial ideal and linear resolutions

Erfan Manouchehri; Ali Soleyman Jahan

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 3, page 785-802
  • ISSN: 0011-4642

Abstract

top
For each squarefree monomial ideal I ⊂ S = k [ x 1 , ... , 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 x u i = y u 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 Δ ≅ G I Δ ∨ is a cycle or a tree.

How to cite

top

Manouchehri, Erfan, and Soleyman Jahan, Ali. "The linear syzygy graph of a monomial ideal and linear resolutions." Czechoslovak Mathematical Journal 71.3 (2021): 785-802. <http://eudml.org/doc/297484>.

@article{Manouchehri2021,
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.},
author = {Manouchehri, Erfan, Soleyman Jahan, Ali},
journal = {Czechoslovak Mathematical Journal},
keywords = {monomial ideal; linear resolution; linear quotient; variable-decomposability; Cohen-Macaulay simplicial complex},
language = {eng},
number = {3},
pages = {785-802},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The linear syzygy graph of a monomial ideal and linear resolutions},
url = {http://eudml.org/doc/297484},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Manouchehri, Erfan
AU - Soleyman Jahan, Ali
TI - The linear syzygy graph of a monomial ideal and linear resolutions
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 3
SP - 785
EP - 802
AB - 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.
LA - eng
KW - monomial ideal; linear resolution; linear quotient; variable-decomposability; Cohen-Macaulay simplicial complex
UR - http://eudml.org/doc/297484
ER -

References

top
  1. Abbott, J., Bigatti, A. M., Lagorio, G., CoCoA-5: A system for doing Computations in Commutative Algebra, Available at http://cocoa.dima.unige.it/. 
  2. Ajdani, S. M., Jahan, A. S., 10.1007/s40840-015-0129-x, Bull. Malays. Math. Sci. Soc. (2) 39 (2016), 609-617. (2016) Zbl1333.13031MR3471335DOI10.1007/s40840-015-0129-x
  3. Bigdeli, M., Herzog, J., Zaare-Nahandi, R., 10.1080/00927872.2017.1339058, Commun. Algebra 46 (2018), 1080-1095. (2018) Zbl1428.13032MR3780221DOI10.1080/00927872.2017.1339058
  4. Conca, A., Negri, E. De, 10.1006/jabr.1998.7740, J. Algebra 211 (1999), 599-624. (1999) Zbl0924.13012MR1666661DOI10.1006/jabr.1998.7740
  5. Conca, A., Herzog, J., Castelnuovo-Mumford regularity of product of ideals, Collect. Math. 54 (2003), 137-152. (2003) Zbl1074.13004MR1995137
  6. Eagon, J. A., Reiner, V., 10.1016/S0022-4049(97)00097-2, J. Pure Appl. Algebra 130 (1998), 265-275. (1998) Zbl0941.13016MR1633767DOI10.1016/S0022-4049(97)00097-2
  7. Emtander, E., 10.7146/math.scand.a-15124, Math. Scand. 106 (2010), 50-66. (2010) Zbl1183.05053MR2603461DOI10.7146/math.scand.a-15124
  8. Francisco, C. A., Tuyl, A. Van, 10.1090/S0002-9939-07-08841-7, Proc. Am. Math. Soc. 135 (2007), 2327-2337. (2007) Zbl1128.13013MR2302553DOI10.1090/S0002-9939-07-08841-7
  9. Fröberg, R., 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. (1990) Zbl0741.13006MR1171260
  10. Herzog, J., Hibi, T., 10.1007/978-0-85729-106-6, Graduate Texts in Mathematics 260. Springer, London (2011). (2011) Zbl1206.13001MR2724673DOI10.1007/978-0-85729-106-6
  11. Herzog, J., Hibi, T., Zheng, X., 10.7146/math.scand.a-14446, Math. Scand. 95 (2004), 23-32. (2004) Zbl1091.13013MR2091479DOI10.7146/math.scand.a-14446
  12. Herzog, J., Takayama, Y., 10.4310/HHA.2002.v4.n2.a13, Homology Homotopy Appl. 4 (2002), 277-294. (2002) Zbl1028.13008MR1918513DOI10.4310/HHA.2002.v4.n2.a13
  13. Morales, M., 10.1016/j.jalgebra.2010.08.025, J. Algebra 324 (2010), 3431-3456. (2010) Zbl1217.13007MR2735392DOI10.1016/j.jalgebra.2010.08.025
  14. Rahmati-Asghar, R., Yassemi, S., 10.1142/S1005386715000656, Algebra Colloq. 22 (2015), 745-756. (2015) Zbl1332.13019MR3420707DOI10.1142/S1005386715000656
  15. Madani, S. Saeedi, Kiani, D., Terai, N., Sequentially Cohen-Macaulay path ideals of cycles, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 54 (2011), 353-363. (2011) Zbl1265.13016MR2917856
  16. Woodroofe, R., Chordal and sequentially Cohen-Macaulay clutters, Electron. J. Comb. 18 (2011), Article ID P208, 20 pages. (2011) Zbl1236.05213MR2853065

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.