Browse issues
no. 5
Higher nerves of simplicial complexes
Dao, Hailong1; Doolittle, Joseph1; Duna, Ken1; Goeckner, Bennet2; Holmes, Brent1; Lyle, Justin1
1 University of Kansas Department of Mathematics 1460 Jayhawk Blvd Lawrence KS 66045, USA
2 Department of Mathematics University of Washington Seattle, WA 98195-4350, USA
Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 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:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.64
Classification: 05E40, 05E45, 13C15, 13D03
Keywords: Nerve Complex, depth, $k$-connectivity, homologies, poset, monomial ideals

Dao, Hailong 1; Doolittle, Joseph 1; Duna, Ken 1; Goeckner, Bennet 2; Holmes, Brent 1; Lyle, Justin 1

1 University of Kansas Department of Mathematics 1460 Jayhawk Blvd Lawrence KS 66045, USA
2 Department of Mathematics University of Washington Seattle, WA 98195-4350, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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/articles/10.5802/alco.64/

[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 | DOI | MR | Zbl

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

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

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

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

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

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

[8] Giblin, Peter Graphs, surfaces and homology, Cambridge University Press, Cambridge, 2010, xx+251 pages | DOI | MR | Zbl

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

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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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

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

[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 | DOI | MR | Zbl

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