A generalization of k-Cohen–Macaulay simplicial complexes

Hassan Haghighi; Siamak Yassemi; Rahim Zaare-Nahandi

doi:10.1007/s11512-010-0136-y

October 2012 A generalization of k-Cohen–Macaulay simplicial complexes

Hassan Haghighi, Siamak Yassemi, Rahim Zaare-Nahandi

Author Affiliations +

Ark. Mat. 50(2): 279-290 (October 2012). DOI: 10.1007/s11512-010-0136-y

Abstract

For a positive integer k and a non-negative integer t, a class of simplicial complexes, to be denoted by k-CM_t, is introduced. This class generalizes two notions for simplicial complexes: being k-Cohen–Macaulay and k-Buchsbaum. In analogy with the Cohen–Macaulay and Buchsbaum complexes, we give some characterizations of CM_t (=1−CM_t) complexes, in terms of vanishing of some homologies of its links, and in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We give a result on the behavior of the CM_t property under the operation of join of two simplicial complexes. We show that a complex is k-CM_t if and only if the links of its non-empty faces are k-CM_t−1. We prove that for an integer s≤d, the (d−s−1)-skeleton of a (d−1)-dimensional k-CM_t complex is (k+s)-CM_t. This result generalizes Hibi’s result for Cohen–Macaulay complexes and Miyazaki’s result for Buchsbaum complexes.

References

1.

Athanasiadis, C. A. and Welker, V., Buchsbaum* complexes, Preprint, 2009. 0909.1931v2Athanasiadis, C. A. and Welker, V., Buchsbaum* complexes, Preprint, 2009. 0909.1931v2

2.

Baclawski, K., Cohen–Macaulay connectivity and geometric lattices, European J. Combin. 3 (1982), 293–305. MR687728 0504.06005Baclawski, K., Cohen–Macaulay connectivity and geometric lattices, European J. Combin. 3 (1982), 293–305. MR687728 0504.06005

3.

Björner, A. and Hibi, T., Betti numbers of Buchsbaum complexes, Math. Scand. 67 (1990), 193–196. MR1096455 0727.55011Björner, A. and Hibi, T., Betti numbers of Buchsbaum complexes, Math. Scand. 67 (1990), 193–196. MR1096455 0727.55011

4.

Björner, A., Wachs, M. and Welker, V., On sequentially Cohen–Macaulay complexes and posets, Israel J. Math. 169 (2009), 295–316. MR2460907 05508742 10.1007/s11856-009-0012-2Björner, A., Wachs, M. and Welker, V., On sequentially Cohen–Macaulay complexes and posets, Israel J. Math. 169 (2009), 295–316. MR2460907 05508742 10.1007/s11856-009-0012-2

5.

Fløystad, G., Enriched homology and cohomology modules of simplicial complexes, J. Algebraic Combin. 25 (2007), 285–307. MR2317335 10.1007/s10801-006-0038-zFløystad, G., Enriched homology and cohomology modules of simplicial complexes, J. Algebraic Combin. 25 (2007), 285–307. MR2317335 10.1007/s10801-006-0038-z

6.

Fröberg, R., A note on the Stanley–Reisner ring of a join and of a suspension, Manuscripta Math. 60 (1988), 89–91. MR920761 0639.13013 10.1007/BF01168149Fröberg, R., A note on the Stanley–Reisner ring of a join and of a suspension, Manuscripta Math. 60 (1988), 89–91. MR920761 0639.13013 10.1007/BF01168149

7.

Hibi, T., Level rings and algebras with straightening laws, J. Algebra 117 (1988), 343–362. MR957445 0652.13010 10.1016/0021-8693(88)90111-1Hibi, T., Level rings and algebras with straightening laws, J. Algebra 117 (1988), 343–362. MR957445 0652.13010 10.1016/0021-8693(88)90111-1

8.

Hibi, T., Buchsbaum complexes with linear resolutions, J. Algebra 179 (1996), 127–136. MR1367844 0839.55012 10.1006/jabr.1996.0006Hibi, T., Buchsbaum complexes with linear resolutions, J. Algebra 179 (1996), 127–136. MR1367844 0839.55012 10.1006/jabr.1996.0006

9.

Hochster, M., Cohen–Macaulay rings, combinatorics, and simplicial complexes, in Ring Theory II, Proc. of the 2nd Oklahoma Conf., Lect. Notes in Pure and Appl. Math. 26, pp. 171–223, Dekker, New York, 1977.Hochster, M., Cohen–Macaulay rings, combinatorics, and simplicial complexes, in Ring Theory II, Proc. of the 2nd Oklahoma Conf., Lect. Notes in Pure and Appl. Math. 26, pp. 171–223, Dekker, New York, 1977.

10.

Miller, E., Novik, I. and Swartz, E., Face rings of simplicial complexes with singularities, to appear in Math. Ann.Miller, E., Novik, I. and Swartz, E., Face rings of simplicial complexes with singularities, to appear in Math. Ann.

11.

Miyazaki, M., On 2-Buchsbaum complexes, J. Math. Kyoto Univ. 30 (1990), 367–392. MR1075292 0723.13002Miyazaki, M., On 2-Buchsbaum complexes, J. Math. Kyoto Univ. 30 (1990), 367–392. MR1075292 0723.13002

12.

Munkres, J., Topological results in combinatorics, Michigan Math. J. 31 (1984), 113–128. MR736476 0585.57014 10.1307/mmj/1029002969Munkres, J., Topological results in combinatorics, Michigan Math. J. 31 (1984), 113–128. MR736476 0585.57014 10.1307/mmj/1029002969

13.

Provan, S. J. and Billera, L. J., Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Oper. Res. 5 (1980), 576–594. MR593648 0457.52005 10.1287/moor.5.4.576Provan, S. J. and Billera, L. J., Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Oper. Res. 5 (1980), 576–594. MR593648 0457.52005 10.1287/moor.5.4.576

14.

Reisner, G. A., Cohen–Macaulay quotients of polynomial rings, Adv. Math. 21 (1976), 30–49. MR407036 0345.13017 10.1016/0001-8708(76)90114-6Reisner, G. A., Cohen–Macaulay quotients of polynomial rings, Adv. Math. 21 (1976), 30–49. MR407036 0345.13017 10.1016/0001-8708(76)90114-6

15.

Sabzrou, H., Tousi, M. and Yassemi, S., Simplicial join via tensor products, Manuscripta Math. 126 (2008), 255–272. MR2403189 1165.13003 10.1007/s00229-008-0175-xSabzrou, H., Tousi, M. and Yassemi, S., Simplicial join via tensor products, Manuscripta Math. 126 (2008), 255–272. MR2403189 1165.13003 10.1007/s00229-008-0175-x

16.

Schenzel, P., On the number of faces of simplicial complexes and the purity of Frobenius, Math. Z. 178 (1981), 125–142. MR627099 0472.13012 10.1007/BF01218376Schenzel, P., On the number of faces of simplicial complexes and the purity of Frobenius, Math. Z. 178 (1981), 125–142. MR627099 0472.13012 10.1007/BF01218376

17.

Stanley, R., Combinatorics and Commutative Algebra, 2nd ed., Birkhäuser, Boston, MA, 1995.Stanley, R., Combinatorics and Commutative Algebra, 2nd ed., Birkhäuser, Boston, MA, 1995.

18.

Stückrad, J. and Vogel, W., Buchsbaum Rings and Applications. An Interaction Between Algebra, Geometry, and Topology, Mathematical Monographs 21, VEB Deutscher Verlag der Wissenschaften, Berlin, 1986.Stückrad, J. and Vogel, W., Buchsbaum Rings and Applications. An Interaction Between Algebra, Geometry, and Topology, Mathematical Monographs 21, VEB Deutscher Verlag der Wissenschaften, Berlin, 1986.

19.

Terai, N. and Hibi, T., Finite free resolutions and 1-skeletons of simplicial complexes, J. Algebraic Combin. 6 (1997), 89–93. MR1431826 0871.55015 10.1023/A:1008648302195Terai, N. and Hibi, T., Finite free resolutions and 1-skeletons of simplicial complexes, J. Algebraic Combin. 6 (1997), 89–93. MR1431826 0871.55015 10.1023/A:1008648302195

20.

Zaare-Nahandi, R. and Zaare-Nahandi, R., The minimal free resolution of a class of square-free monomial ideals, J. Pure Appl. Algebra 189 (2004), 263–278. MR2038574 1058.13008 10.1016/j.jpaa.2003.10.026Zaare-Nahandi, R. and Zaare-Nahandi, R., The minimal free resolution of a class of square-free monomial ideals, J. Pure Appl. Algebra 189 (2004), 263–278. MR2038574 1058.13008 10.1016/j.jpaa.2003.10.026

Citation Download Citation

Hassan Haghighi, Siamak Yassemi, and Rahim Zaare-Nahandi "A generalization of k-Cohen–Macaulay simplicial complexes," Arkiv för Matematik 50(2), 279-290, (October 2012). https://doi.org/10.1007/s11512-010-0136-y

Received: 7 June 2010; Published: October 2012

Access the abstract

JOURNAL ARTICLE
12 PAGES

DOWNLOAD PDF + SAVE TO MY LIBRARY