logo AFST
Monomial ideals with 3-linear resolutions
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 4, pp. 877-891.

Dans cet article nous étudions la régularité de Castelnuovo-Mumford des idéaux engendrés par des monômes libres de carré et de degré trois. Nous définissons des opérations sur l’ensemble des clutters associés à ces idéaux et démontrons que la régularité de Castelnuovo-Mumford est conservée par ces opérations. Ces opérations nous permettent d’introduire certaines classes d’idéaux ayant une résolution linéaire. En particulier nous démontrons qu’aucun clutter correspondant à une triangulation de la sphère a une résolution linéaire, mais par contre que tout subclutter propre a une résolution linéaire.

In this paper, we study the Castelnuovo-Mumford regularity of square-free monomial ideals generated in degree 3. We define some operations on the clutters associated to such ideals and prove that the regularity is preserved under these operations. We apply these operations to introduce some classes of ideals with linear resolutions and also show that any clutter corresponding to a triangulation of the sphere does not have linear resolution while any proper subclutter of it has a linear resolution.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/afst.1428
@article{AFST_2014_6_23_4_877_0,
     author = {Marcel Morales and Abbas Nasrollah Nejad and Ali Akbar Yazdan Pour and Rashid Zaare-Nahandi},
     title = {Monomial ideals with 3-linear resolutions},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {877--891},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 23},
     number = {4},
     year = {2014},
     doi = {10.5802/afst.1428},
     zbl = {06374892},
     mrnumber = {3270427},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1428/}
}
TY  - JOUR
AU  - Marcel Morales
AU  - Abbas Nasrollah Nejad
AU  - Ali Akbar Yazdan Pour
AU  - Rashid Zaare-Nahandi
TI  - Monomial ideals with 3-linear resolutions
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2014
DA  - 2014///
SP  - 877
EP  - 891
VL  - Ser. 6, 23
IS  - 4
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1428/
UR  - https://zbmath.org/?q=an%3A06374892
UR  - https://www.ams.org/mathscinet-getitem?mr=3270427
UR  - https://doi.org/10.5802/afst.1428
DO  - 10.5802/afst.1428
LA  - en
ID  - AFST_2014_6_23_4_877_0
ER  - 
Marcel Morales; Abbas Nasrollah Nejad; Ali Akbar Yazdan Pour; Rashid Zaare-Nahandi. Monomial ideals with 3-linear resolutions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 4, pp. 877-891. doi : 10.5802/afst.1428. https://afst.centre-mersenne.org/articles/10.5802/afst.1428/

[1] Bruns (W.), Herzog (J.).— Cohen-Macaulay Rings, Revised Edition, Cambridge University Press, Cambridge (1996). | MR 1251956 | Zbl 0909.13005

[2] CoCoATeam: CoCoA.— A System for Doing Computations in Commutative Algebra, available at http://cocoa.dima.unige.it.

[3] Connon (E.), Faridi (S.).— A criterion for a monomial ideal to have a linear resolution in characteristic 2, arXiv:1306.2857 [math.AC]. | MR 3336577

[4] Eagon (J. A.), Reiner (V.).— Resolutions of Stanley-Reisner rings and Alexander duality, J. Pure and Applied Algebra 130, p. 265-275 (1998). | MR 1633767 | Zbl 0941.13016

[5] Emtander (E.).— Betti numbers of hypergraphs. Commun. Algebra 37, No. 5, p. 1545-1571 (2009). | MR 2526320 | Zbl 1191.13015

[6] Emtander (E.).— A class of hypergraphs that generalizes chordal graphs, Math. Scand. 106, no. 1, p. 50-66 (2010). | MR 2603461 | Zbl 1183.05053

[7] Fröberg (R.).— On Stanley-Reisner rings, Topics in algebra, Part 2 (Warsaw, 1988), Banach Center Publ., vol. 26, PWN, Warsaw, p. 57-70 (1990). | MR 1171260 | Zbl 0741.13006

[8] Morales (M.).— Simplicial ideals, 2-linear ideals and arithmetical rank, J. Algebra 324, no. 12, p. 3431-3456 (2010). | MR 2735392 | Zbl 1217.13007

[9] Morales (M.), Yazdan Pour (A.-A.), Zaare-Nahandi (R.).— The regularity of edge ideals of graphs. J. Pure Appl. Algebra 216, No. 12, p. 2714-2719 (2012). | MR 2943752 | Zbl pre06141857

[10] Decker (W.), Greuel (G.-M.), G. Pfister (G.), H. Schönemann (H.).— Singular 3-1-3 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2011). | Zbl 0902.14040

[11] Stanley (R.).— Combinatorics and Commutative Algebra, second ed., Progress in Mathematics, vol. 41, Birkhäuser Boston Inc., Boston, MA, (1996). | MR 1453579 | Zbl 0838.13008

[12] Terai (N.).— Generalization of Eagon-Reiner theorem and h-vectors of graded rings, preprint (2000).

[13] Woodroofe (R.).— Chordal and sequentially Cohen-Macaulay clutters, Electron. J. Combin. 18 (2011), no. 1, Paper 208, 20 pages, arXiv:0911.4697. | MR 2853065 | Zbl 1236.05213

Cité par Sources :