Higher nerves of simplicial complexes
Algebraic Combinatorics, Volume 2 (2019) no. 5, p. 803-813

We investigate generalized notions of the nerve complex for the facets of a simplicial complex. We show that the homologies of these higher nerve complexes determine the depth of the Stanley-Reisner ring k[Δ] as well as the f-vector and h-vector of Δ. We present, as an application, a formula for computing regularity of monomial ideals.

Received : 2018-04-18
Revised : 2019-01-29
Accepted : 2019-01-29
Published online : 2019-10-08
DOI : https://doi.org/10.5802/alco.64
Classification:  05E40,  05E45,  13C15,  13D03
Keywords: Nerve Complex, depth, k-connectivity, homologies, poset, monomial ideals
@article{ALCO_2019__2_5_803_0,
     author = {Dao, Hailong and Doolittle, Joseph and Duna, Ken and Goeckner, Bennet and Holmes, Brent and Lyle, Justin},
     title = {Higher nerves of simplicial complexes},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {2},
     number = {5},
     year = {2019},
     pages = {803-813},
     doi = {10.5802/alco.64},
     language = {en},
     url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_5_803_0}
}
Dao, Hailong; Doolittle, Joseph; Duna, Ken; Goeckner, Bennet; Holmes, Brent; Lyle, Justin. Higher nerves of simplicial complexes. Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 803-813. doi : 10.5802/alco.64. https://alco.centre-mersenne.org/item/ALCO_2019__2_5_803_0/

[1] Basu, Saugata Different bounds on the different Betti numbers of semi-algebraic sets, Discrete and Computational Geometry, Volume 30 (2003) no. 1, pp. 65-85 | MR 1991587 | Zbl 1073.14556

[2] Björner, Anders Topological methods, Handbook of combinatorics, Vol. 2, Elsevier Sci. B. V., Amsterdam (1995), pp. 1819-1872 | MR 1373690 | Zbl 0851.52016

[3] Björner, Anders Nerves, fibers and homotopy groups, Journal of Combinatorial Theory, Series A, Volume 102 (2003) no. 1, pp. 88-93 | MR 1970978 | Zbl 1030.55006

[4] Borsuk, Karol On the imbedding of systems of compacta in simplicial complexes, Fundamenta Mathematicae, Volume 35 (1948) no. 1, pp. 217-234 | MR 28019 | Zbl 0032.12303

[5] Bruns, Winfried; Herzog, Jürgen Cohen-Macaulay rings, Cambridge University Press (1998), xii+403 pages | MR 1251956 | Zbl 0909.13005

[6] Cavanna, Nicholas J; Jahanseir, Mahmoodreza; Sheehy, Donald R A geometric perspective on sparse filtrations (2015) (https://arxiv.org/abs/1506.03797 ) | Zbl 1378.68163

[7] Gasharov, Vesselin; Peeva, Irena; Welker, Volkmar The lcm-lattice in monomial resolutions, Mathematical Research Letters, Volume 6 (1999) no. 5, pp. 521-532 | Article | MR 1739211 | Zbl 0970.13004

[8] Giblin, Peter Graphs, surfaces and homology, Cambridge University Press, Cambridge (2010), xx+251 pages | Article | MR 2722281 | Zbl 1201.55001

[9] Grünbaum, Branko Nerves of simplicial complexes, aequationes mathematicae, Volume 4 (1970) no. 1-2, pp. 63-73 | Article | MR 264648 | Zbl 0193.52903

[10] Hartshorne, Robin Complete intersections and connectedness, American Journal of Mathematics, Volume 84 (1962) no. 3, pp. 497-508 | Article | MR 142547 | Zbl 0108.16602

[11] Hibi, Takayuki Quotient algebras of Stanley-Reisner rings and local cohomology, J. Algebra, Volume 140 (1991) no. 2, pp. 336-343 | MR 1120426 | Zbl 0761.55015

[12] Holmes, Brent; Lyle, Justin Rank Selection and Depth Conditions for Balanced Simplicial Complexes (2018) (https://arxiv.org/abs/1802.03129 )

[13] Kalai, Gil; Meshulam, Roy A topological colorful Helly theorem, Adv. Math., Volume 191 (2005) no. 2, pp. 305-311 | Article | MR 2103215 | Zbl 1064.52008

[14] Katzman, Mordechai; Lyubeznik, Gennady; Zhang, Wenliang An extension of a theorem of Hartshorne, Proc. Amer. Math. Soc., Volume 144 (2016) no. 3, pp. 955-962 | Article | MR 3447649 | Zbl 06549098

[15] Lipsky, David; Skraba, Primoz; Vejdemo-Johansson, Mikael A spectral sequence for parallelized persistence (2011) (https://arxiv.org/abs/1112.1245 )

[16] Lyubeznik, Gennady On some local cohomology modules, Adv. Math., Volume 213 (2007) no. 2, pp. 621-643 | Article | MR 2332604 | Zbl 1121.13016

[17] Miller, Ezra; Sturmfels, Bernd Combinatorial commutative algebra, Springer-Verlag, New York, Graduate Texts in Mathematics, Volume 227 (2005), xiv+417 pages | MR 2110098 | Zbl 1090.13001

[18] Munkres, James R. Topological results in combinatorics, Michigan Math. J., Volume 31 (1984) no. 1, pp. 113-128 | MR 736476 | Zbl 0585.57014

[19] Panov, Taras; Ustinovskiy, Yury; Verbitsky, Misha Complex geometry of moment-angle manifolds, Mathematische Zeitschrift, Volume 284 (2016) no. 1-2, pp. 309-333 | Article | MR 3545497 | Zbl 1352.32008

[20] Peeva, Irena Graded syzygies, Springer-Verlag London, Ltd., London, Algebra and Applications, Volume 14 (2011), xii+302 pages | MR 2560561 | Zbl 1213.13002

[21] Quillen, Daniel Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. Math., Volume 28 (1978) no. 2, pp. 101-128 | Article | MR 493916 | Zbl 0388.55007