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Simplicial complexes satisfying Serre’s condition: a survey with some new results. (English) Zbl 1345.13014

Summary: The problem of finding a characterization of Cohen-Macaulay simplicial complexes has been studied intensively by many authors. There are several attempts at this problem available for some special classes of simplicial complexes satisfying some technical conditions. This paper is a survey, with some new results, of some of these developments. The new results about simplicial complexes with Serre’s condition are an analogue of the known results for Cohen-Macaulay simplicial complexes.

MSC:

13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
05E99 Algebraic combinatorics
13C13 Other special types of modules and ideals in commutative rings
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References:

[1] M. Barile, A note on the edge ideals of Ferrers graphs .v2. · Zbl 1166.13024
[2] A. Björner and M.L. Wachs, Shellable nonpure complexes and posets , I, Trans. Amer. Math. Soc. 348 (1996), 1299-1327. · Zbl 0857.05102 · doi:10.1090/S0002-9947-96-01534-6
[3] W. Bruns and J. Herzog, Cohen-Macaulay rings , Cambridge Stud. Adv. Math. 39 , Cambridge University Press, 1993. · Zbl 0788.13005
[4] A.M. Duval, Algebraic shifting and sequentially Cohen-Macaulay simplicial complexes , Electron. J. Combin. 3 (1996), Research Paper 21. · Zbl 0883.06003
[5] J.A. Eagon and V. Reiner, Resolutions of Stanley-Reisner rings and Alexander duality , J. Pure Appl. Alg. 130 (1998), 265-275. · Zbl 0941.13016 · doi:10.1016/S0022-4049(97)00097-2
[6] D. Eisenbud, M. Green, K. Hulek and S. Popescu, Restricting linear syzygies: algebra and geometry , Compos. Math. 141 (2005), 1460-1478. · Zbl 1086.14044 · doi:10.1112/S0010437X05001776
[7] M. Estrada and R.H. Villarreal, Cohen-Macaulay bipartite graphs , Arch. Math. (Basel) 68 (1997), 124-128. · Zbl 0869.13003 · doi:10.1007/s000130050040
[8] C.A. Francisco and H.T. Há, Whiskers and sequentially Cohen-Macaulay graphs , J. Combin. Theor. 115 (2008), 304-316. · Zbl 1142.13021 · doi:10.1016/j.jcta.2007.06.004
[9] C.A. Francisco and A. Van Tuyl, Sequentially Cohen-Macaulay edge ideals , Proc. Amer. Math. Soc. 135 (2007), 2327-2337. · Zbl 1128.13013 · doi:10.1090/S0002-9939-07-08841-7
[10] A. Goodarzi, M.R. Pournaki, S.A. Seyed Fakhari and S. Yassemi, On the \(h\)-vector of a simplicial complex with Serre’s condition , J. Pure Appl. Alg. 216 (2012), 91-94. · Zbl 1237.13042 · doi:10.1016/j.jpaa.2011.05.005
[11] H. Haghighi, N. Terai, S. Yassemi and R. Zaare-Nahandi, Sequentially \(S_r\) simplicial complexes and sequentially \(S_2\) graphs , Proc. Amer. Math. Soc. 139 (2011), 1993-2005. · Zbl 1220.13014 · doi:10.1090/S0002-9939-2010-10646-9
[12] H. Haghighi, S. Yassemi and R. Zaare-Nahandi, Bipartite \(S_2\) graphs are Cohen-Macaulay , Bull. Math. Soc. Sci. Math. Roum. 53 (2010), 125-132.
[13] R. Hartshorne, Complete intersections in characteristic \(p>0\) , Amer. J. Math. 101 (1979), 380-383. · Zbl 0418.14027 · doi:10.2307/2373984
[14] J. Herzog and T. Hibi, Componentwise linear ideals , Nagoya Math. J. 153 (1999), 141-153. · Zbl 0930.13018 · doi:10.1017/S0027763000006930
[15] —-, Distributive lattices, bipartite graphs and Alexander duality , J. Alg. Combin. 22 (2005), 289-302. · Zbl 1090.13017 · doi:10.1007/s10801-005-4528-1
[16] —-, Monomial ideals , Springer-Verlag, London, Ltd., London, 2011.
[17] J. Herzog, T. Hibi and X. Zheng, Dirac’s theorem on chordal graphs and Alexander duality , Europ. J. Combin. 25 (2004), 949-960. · Zbl 1062.05075 · doi:10.1016/j.ejc.2003.12.008
[18] —-, Monomial ideals whose powers have a linear resolution , Math. Scand. 95 (2004), 23-32. · Zbl 1091.13013 · doi:10.7146/math.scand.a-14446
[19] J. Herzog, T. Hibi and X. Zheng, Cohen-Macaulay chordal graphs , J. Combin. Theor. 113 (2006), 911-916. · Zbl 1172.13307 · doi:10.1016/j.jcta.2005.08.007
[20] J. Herzog, Y. Takayama and N. Terai, On the radical of a monomial ideal , Arch. Math. 85 (2005), 397-408. · Zbl 1112.13003 · doi:10.1007/s00013-005-1385-z
[21] G. Kalai, Algebraic shifting , Adv. Stud. Pure Math. 33 (2001), 121-163. · Zbl 1034.57021
[22] D. König, Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre , Math. Ann. 77 (1916), 453-465. · JFM 46.0146.03 · doi:10.1007/BF01456961
[23] —-, Theory of finite and infinite graphs , Birkhauser Boston, Inc., Boston, 1990.
[24] G. Lyubeznik, On the arithmetical rank of monomial ideals , J. Alg. 112 (1988), 86-89. · Zbl 0639.13015 · doi:10.1016/0021-8693(88)90133-0
[25] E. Miller and B. Sturmfels, Combinatorial commutative algebra , Springer-Verlag, New York, 2005. · Zbl 1090.13001
[26] N.C. Minh and N.V. Trung, Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley-Reisner ideals , Adv. Math. 226 (2011), 1285-1306. · Zbl 1204.13015 · doi:10.1016/j.aim.2010.08.005
[27] J.R. Munkres, Elements of algebraic topology , Addison-Wesley Publishing Company, Menlo Park, CA, 1984. · Zbl 0673.55001
[28] S. Murai and N. Terai, \(h\)-Vectors of simplicial complexes with Serre’s conditions , Math. Res. Lett. 16 (2009), 1015-1028. · Zbl 1200.13035 · doi:10.4310/MRL.2009.v16.n6.a10
[29] M.R. Pournaki, S.A. Seyed Fakhari and S. Yassemi, On the \(h\)-triangles of sequentially \((S_r)\) simplicial complexes via algebraic shifting , Ark. Mat. 51 (2013), 185-196. · Zbl 1263.13022 · doi:10.1007/s11512-011-0160-6
[30] G.A. Reisner, Cohen-Macaulay quotients of polynomial rings , Adv. Math. 21 (1976), 30-49. · Zbl 0345.13017 · doi:10.1016/0001-8708(76)90114-6
[31] G. Rinaldo, N. Terai and K.I. Yoshida, Cohen-Macaulayness for symbolic power ideals of edge ideals , J. Alg. 347 (2011), 1-22. · Zbl 1239.13030 · doi:10.1016/j.jalgebra.2011.09.007
[32] —-, On the second powers of Stanley-Reisner ideals , J. Comm. Alg. 3 (2011), 405-430. · Zbl 1237.13045 · doi:10.1216/JCA-2011-3-3-405
[33] P. Schenzel, Dualisierende Komplexe in der lokalen Algebra und Buchsbaum-Ringe , Lect. Notes Math. 907 , Springer-Verlag, New York, 1982. · Zbl 0484.13016 · doi:10.1007/BFb0094123
[34] A. Simis, On the Jacobian module associated to a graph , Proc. Amer. Math. Soc. 126 (1998), 989-997. · Zbl 0887.13014 · doi:10.1090/S0002-9939-98-04180-X
[35] A. Simis, W.V. Vasconcelos and R.H. Villarreal, On the ideal theory of graphs , J. Alg. 167 (1994), 389-416. · Zbl 0816.13003 · doi:10.1006/jabr.1994.1192
[36] R.P. Stanley, Combinatorics and commutative algebra , Second Edition, Progr. Math. 41 , Birkhauser Boston, Inc., Boston, 1996. · Zbl 0838.13008
[37] A. Taylor, The inverse Gröbner basis problem in codimension two , J. Symbol. Comp. 33 (2002), 221-238. · Zbl 1048.13020 · doi:10.1006/jsco.2001.0511
[38] N. Terai, Alexander duality in Stanley-Reisner rings , Affine algebraic geometry , Osaka University Press, Osaka, 2007. · Zbl 1128.13014
[39] N. Terai and N.V. Trung, Cohen-Macaulayness of large powers of Stanley-Reisner ideals , Adv. Math. 229 (2012), 711-730. · Zbl 1246.13032 · doi:10.1016/j.aim.2011.10.004
[40] N. Terai and K.I. Yoshida, A note on Cohen-Macaulayness of Stanley-Reisner rings with Serre’s condition \((S_2)\) , Comm. Alg. 36 (2008), 464-477. · Zbl 1137.13013 · doi:10.1080/00927870701716124
[41] A. Van Tuyl, Sequentially Cohen-Macaulay bipartite graphs : Vertex decomposability and regularity , Arch. Math. 93 (2009), 451-459. · Zbl 1184.13062 · doi:10.1007/s00013-009-0049-9
[42] R.H. Villarreal, Cohen-Macaulay graphs , Manuscr. Math. 66 (1990), 277-293. · Zbl 0737.13003 · doi:10.1007/BF02568497
[43] R. Woodroofe, Vertex decomposable graphs and obstructions to shellability , Proc. Amer. Math. Soc. 137 (2009), 3235-3246. · Zbl 1180.13031 · doi:10.1090/S0002-9939-09-09981-X
[44] K. Yanagawa, Alexander duality for Stanley-Reisner rings and squarefree \(\mathbb{N}^n\)-graded modules , J. Alg. 225 (2000), 630-645. · Zbl 0981.13011 · doi:10.1006/jabr.1999.8130
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