Simplicial complexes of small codimension
HTML articles powered by AMS MathViewer
- by Matteo Varbaro and Rahim Zaare-Nahandi PDF
- Proc. Amer. Math. Soc. 147 (2019), 3347-3355 Request permission
Abstract:
We show that $\textrm {CM}_t$ simplicial complexes, a notion generalizing Buchsbaum-ness, of small codimension must have large depth, proving more precise results in the codimension 2 case. In the paper, we show that the $\textrm {CM}_t$ property is a topological invariant of a simplicial complex.References
- U. Brehm and W. Kühnel, Combinatorial manifolds with few vertices, Topology 26 (1987), no. 4, 465–473. MR 919730, DOI 10.1016/0040-9383(87)90042-5
- A. Conca and M. Varbaro, Square-free Gröbner degenerations, arXiv:1805.11923, 2018.
- Hailong Dao, Craig Huneke, and Jay Schweig, Bounds on the regularity and projective dimension of ideals associated to graphs, J. Algebraic Combin. 38 (2013), no. 1, 37–55. MR 3070118, DOI 10.1007/s10801-012-0391-z
- John A. Eagon and Victor Reiner, Resolutions of Stanley-Reisner rings and Alexander duality, J. Pure Appl. Algebra 130 (1998), no. 3, 265–275. MR 1633767, DOI 10.1016/S0022-4049(97)00097-2
- David Eisenbud, Mark Green, Klaus Hulek, and Sorin Popescu, Restricting linear syzygies: algebra and geometry, Compos. Math. 141 (2005), no. 6, 1460–1478. MR 2188445, DOI 10.1112/S0010437X05001776
- E. Graham Evans and Phillip Griffith, The syzygy problem, Ann. of Math. (2) 114 (1981), no. 2, 323–333. MR 632842, DOI 10.2307/1971296
- Ralf Fröberg, A note on the Stanley-Reisner ring of a join and of a suspension, Manuscripta Math. 60 (1988), no. 1, 89–91. MR 920761, DOI 10.1007/BF01168149
- Hassan Haghighi, Seyed Amin Seyed Fakhari, Siamak Yassemi, and Rahim Zaare-Nahandi, A generalization of Eagon-Reiner’s theorem and a characterization of bi-$\textrm {CM}_t$ bipartite and chordal graphs, Comm. Algebra 46 (2018), no. 9, 3889–3898. MR 3820603, DOI 10.1080/00927872.2018.1427244
- Hassan Haghighi, Siamak Yassemi, and Rahim Zaare-Nahandi, A generalization of $k$-Cohen–Macaulay simplicial complexes, Ark. Mat. 50 (2012), no. 2, 279–290. MR 2961323, DOI 10.1007/s11512-010-0136-y
- Hassan Haghighi, Siamak Yassemi, and Rahim Zaare Nahandi, Cohen-Macaulay bipartite graphs in arbitrary codimension, Proc. Amer. Math. Soc. 143 (2015), no. 5, 1981–1989. MR 3314108, DOI 10.1090/S0002-9939-2015-12433-1
- Robin Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017–1032. MR 384816, DOI 10.1090/S0002-9904-1974-13612-8
- Jürgen Herzog and Takayuki Hibi, Monomial ideals, Graduate Texts in Mathematics, vol. 260, Springer-Verlag London, Ltd., London, 2011. MR 2724673, DOI 10.1007/978-0-85729-106-6
- Jürgen Herzog and Hema Srinivasan, On the subadditivity problem for maximal shifts in free resolutions, Commutative algebra and noncommutative algebraic geometry. Vol. II, Math. Sci. Res. Inst. Publ., vol. 68, Cambridge Univ. Press, New York, 2015, pp. 245–249. MR 3496869
- Ezra Miller, Isabella Novik, and Ed Swartz, Face rings of simplicial complexes with singularities, Math. Ann. 351 (2011), no. 4, 857–875. MR 2854116, DOI 10.1007/s00208-010-0620-5
- Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
- Satoshi Murai and Naoki Terai, $h$-vectors of simplicial complexes with Serre’s conditions, Math. Res. Lett. 16 (2009), no. 6, 1015–1028. MR 2576690, DOI 10.4310/MRL.2009.v16.n6.a10
- Isabella Novik and Ed Swartz, Socles of Buchsbaum modules, complexes and posets, Adv. Math. 222 (2009), no. 6, 2059–2084. MR 2562774, DOI 10.1016/j.aim.2009.07.001
- Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579
- Maria-Laura Torrente and Matteo Varbaro, Computing the Betti table of a monomial ideal: a reduction algorithm, J. Symbolic Comput. 87 (2018), 87–98. MR 3744341, DOI 10.1016/j.jsc.2017.06.001
- Kohji Yanagawa, Alexander duality for Stanley-Reisner rings and squarefree $\mathbf N^n$-graded modules, J. Algebra 225 (2000), no. 2, 630–645. MR 1741555, DOI 10.1006/jabr.1999.8130
- Kohji Yanagawa, Dualizing complex of the face ring of a simplicial poset, J. Pure Appl. Algebra 215 (2011), no. 9, 2231–2241. MR 2786613, DOI 10.1016/j.jpaa.2011.02.009
- Ali Akbar Yazdan Pour, Candidates for nonzero Betti numbers of monomial ideals, Comm. Algebra 45 (2017), no. 4, 1483–1492. MR 3576671, DOI 10.1080/00927872.2016.1177828
Additional Information
- Matteo Varbaro
- Affiliation: Dipartimento di Matematica, Universita’ di Genova, Via Dodecaneso 35, Genova 16146, Italy
- MR Author ID: 873871
- Email: varbaro@dima.unige.it
- Rahim Zaare-Nahandi
- Affiliation: School of Mathematics, Statistics & Computer Science, University of Tehran, Tehran, Iran
- MR Author ID: 211459
- ORCID: 0000-0002-9257-6554
- Email: rahimzn@ut.ac.ir
- Received by editor(s): July 23, 2018
- Received by editor(s) in revised form: December 7, 2018
- Published electronically: April 8, 2019
- Additional Notes: The second author was supported in part by a grant from the University of Tehran
- Communicated by: Claudia Polini
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 3347-3355
- MSC (2010): Primary 13H10, 13F55
- DOI: https://doi.org/10.1090/proc/14510
- MathSciNet review: 3981113