## Journal of Commutative Algebra

### SURVEY ARTICLE: Simplicial complexes satisfying Serre's condition: A survey with some new results

#### Abstract

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.

#### Article information

Source
J. Commut. Algebra, Volume 6, Number 4 (2014), 455-483.

Dates
First available in Project Euclid: 5 January 2015

https://projecteuclid.org/euclid.jca/1420466340

Digital Object Identifier
doi:10.1216/JCA-2014-6-4-455

Mathematical Reviews number (MathSciNet)
MR3294858

Zentralblatt MATH identifier
1345.13014

#### Citation

Pournaki, M.R.; Fakhari, S.A. Seyed; Terai, N.; Yassemi, S. SURVEY ARTICLE: Simplicial complexes satisfying Serre's condition: A survey with some new results. J. Commut. Algebra 6 (2014), no. 4, 455--483. doi:10.1216/JCA-2014-6-4-455. https://projecteuclid.org/euclid.jca/1420466340

#### References

• M. Barile, A note on the edge ideals of Ferrers graphs.v2.
• A. Björner and M.L. Wachs, Shellable nonpure complexes and posets, I, Trans. Amer. Math. Soc. 348 (1996), 1299–1327.
• W. Bruns and J. Herzog, Cohen–Macaulay rings, Cambridge Stud. Adv. Math. 39, Cambridge University Press, 1993.
• A.M. Duval, Algebraic shifting and sequentially Cohen–Macaulay simplicial complexes, Electron. J. Combin. 3 (1996), Research Paper 21.
• J.A. Eagon and V. Reiner, Resolutions of Stanley–Reisner rings and Alexander duality, J. Pure Appl. Alg. 130 (1998), 265–275.
• D. Eisenbud, M. Green, K. Hulek and S. Popescu, Restricting linear syzygies: algebra and geometry, Compos. Math. 141 (2005), 1460–1478.
• M. Estrada and R.H. Villarreal, Cohen–Macaulay bipartite graphs, Arch. Math. (Basel) 68 (1997), 124–128.
• C.A. Francisco and H.T. Há, Whiskers and sequentially Cohen–Macaulay graphs, J. Combin. Theor. 115 (2008), 304–316.
• C.A. Francisco and A. Van Tuyl, Sequentially Cohen–Macaulay edge ideals, Proc. Amer. Math. Soc. 135 (2007), 2327–2337.
• 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.
• 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.
• 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.
• R. Hartshorne, Complete intersections in characteristic $p>0$, Amer. J. Math. 101 (1979), 380–383.
• J. Herzog and T. Hibi, Componentwise linear ideals, Nagoya Math. J. 153 (1999), 141–153.
• ––––, Distributive lattices, bipartite graphs and Alexander duality, J. Alg. Combin. 22 (2005), 289–302.
• ––––, Monomial ideals, Springer-Verlag, London, Ltd., London, 2011.
• J. Herzog, T. Hibi and X. Zheng, Dirac's theorem on chordal graphs and Alexander duality, Europ. J. Combin. 25 (2004), 949–960.
• ––––, Monomial ideals whose powers have a linear resolution, Math. Scand. 95 (2004), 23–32.
• J. Herzog, T. Hibi and X. Zheng, Cohen–Macaulay chordal graphs, J. Combin. Theor. 113 (2006), 911–916.
• J. Herzog, Y. Takayama and N. Terai, On the radical of a monomial ideal, Arch. Math. 85 (2005), 397–408.
• G. Kalai, Algebraic shifting, Adv. Stud. Pure Math. 33 (2001), 121–163.
• D. König, Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre, Math. Ann. 77 (1916), 453–465.
• ––––, Theory of finite and infinite graphs, Birkhauser Boston, Inc., Boston, 1990.
• G. Lyubeznik, On the arithmetical rank of monomial ideals, J. Alg. 112 (1988), 86–89.
• E. Miller and B. Sturmfels, Combinatorial commutative algebra, Springer-Verlag, New York, 2005.
• 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.
• J.R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984.
• S. Murai and N. Terai, $h$-Vectors of simplicial complexes with Serre's conditions, Math. Res. Lett. 16 (2009), 1015–1028.
• 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.
• G.A. Reisner, Cohen–Macaulay quotients of polynomial rings, Adv. Math. 21 (1976), 30–49.
• G. Rinaldo, N. Terai and K.I. Yoshida, Cohen–Macaulayness for symbolic power ideals of edge ideals, J. Alg. 347 (2011), 1–22.
• ––––, On the second powers of Stanley–Reisner ideals, J. Comm. Alg. 3 (2011), 405–430.
• P. Schenzel, Dualisierende Komplexe in der lokalen Algebra und Buchsbaum–Ringe, Lect. Notes Math. 907, Springer-Verlag, New York, 1982.
• A. Simis, On the Jacobian module associated to a graph, Proc. Amer. Math. Soc. 126 (1998), 989–997.
• A. Simis, W.V. Vasconcelos and R.H. Villarreal, On the ideal theory of graphs, J. Alg. 167 (1994), 389–416.
• R.P. Stanley, Combinatorics and commutative algebra, Second Edition, Progr. Math. 41, Birkhauser Boston, Inc., Boston, 1996.
• A. Taylor, The inverse Gröbner basis problem in codimension two, J. Symbol. Comp. 33 (2002), 221–238.
• N. Terai, Alexander duality in Stanley–Reisner rings, Affine algebraic geometry, Osaka University Press, Osaka, 2007.
• N. Terai and N.V. Trung, Cohen–Macaulayness of large powers of Stanley–Reisner ideals, Adv. Math. 229 (2012), 711–730.
• 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.
• A. Van Tuyl, Sequentially Cohen–Macaulay bipartite graphs: Vertex decomposability and regularity, Arch. Math. 93 (2009), 451–459.
• R.H. Villarreal, Cohen–Macaulay graphs, Manuscr. Math. 66 (1990), 277–293.
• R. Woodroofe, Vertex decomposable graphs and obstructions to shellability, Proc. Amer. Math. Soc. 137 (2009), 3235–3246.
• K. Yanagawa, Alexander duality for Stanley–Reisner rings and squarefree $\mathbb{N}^n$-graded modules, J. Alg. 225 (2000), 630–645.