zbMATH — the first resource for mathematics

Almost Buchsbaumness of some rings arising from complexes with isolated singularities. (English) Zbl 1452.13029
Author’s abstract: We study properties of the Stanley-Reisner rings of simplicial complexes with isolated singularities modulo two generic linear forms. Miller, Novik, and Swartz [E. Miller et al., Math. Ann. 351, No. 4, 857–875 (2011; Zbl 1236.13017)] proved that if a complex has homologically isolated singularities, then its Stanley-Reisner ring modulo one generic linear form is Buchsbaum. Here we examine the case of nonhomologically isolated singularities, providing many examples in which the Stanley-Reisner ring modulo two generic linear forms is a quasi-Buchsbaum but not Buchsbaum ring.
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D45 Local cohomology and commutative rings
Full Text: DOI arXiv Euclid
[1] M. Goresky and R. MacPherson, “Intersection homology theory”, Topology 19:2 (1980), 135-162. · Zbl 0448.55004
[2] S. Goto, “A note on quasi-Buchsbaum rings”, Proc. Amer. Math. Soc. 90:4 (1984), 511-516. · Zbl 0542.13015
[3] S. Goto and N. Suzuki, “Index of reducibility of parameter ideals in a local ring”, J. Algebra 87:1 (1984), 53-88. · Zbl 0538.13003
[4] H.-G. Gräbe, “The canonical module of a Stanley-Reisner ring”, J. Algebra 86:1 (1984), 272-281. · Zbl 0533.13003
[5] A. Hatcher, Algebraic topology, Cambridge University Press, 2002. · Zbl 1044.55001
[6] J. Herzog and T. Hibi, Monomial ideals, Graduate Texts in Mathematics 260, Springer, 2011. · Zbl 1206.13001
[7] T. Hibi, “Quotient algebras of Stanley-Reisner rings and local cohomology”, J. Algebra 140:2 (1991), 336-343. · Zbl 0761.55015
[8] S. B. Iyengar, G. J. Leuschke, A. Leykin, C. Miller, E. Miller, A. K. Singh, and U. Walther, Twenty-four hours of local cohomology, Graduate Studies in Mathematics 87, Amer. Math. Soc., 2007. · Zbl 1129.13001
[9] S. Klee and I. Novik, “Face enumeration on simplicial complexes”, pp. 653-686 in Recent trends in combinatorics, edited by A. Beveridge et al., IMA Vol. Math. Appl. 159, Springer, 2016. · Zbl 1354.05152
[10] E. Miller, I. Novik, and E. Swartz, “Face rings of simplicial complexes with singularities”, Math. Ann. 351:4 (2011), 857-875. · Zbl 1236.13017
[11] M. Miyazaki, “Characterizations of Buchsbaum complexes”, Manuscripta Math. 63:2 (1989), 245-254. · Zbl 0671.13014
[12] M. Miyazaki, “On the canonical map to the local cohomology of a Stanley-Reisner ring”, Bull. Kyoto Univ. Ed. Ser. B 79 (1991), 1-8. · Zbl 0741.55013
[13] J. R. Munkres, “Topological results in combinatorics”, Michigan Math. J. 31:1 (1984), 113-128. · Zbl 0585.57014
[14] I. Novik and E. Swartz, “Face numbers of pseudomanifolds with isolated singularities”, Math. Scand. 110:2 (2012), 198-222. · Zbl 1255.13014
[15] G. A. Reisner, “Cohen-Macaulay quotients of polynomial rings”, Advances in Math. 21:1 (1976), 30-49. · Zbl 0345.13017
[16] P. Schenzel, “On the number of faces of simplicial complexes and the purity of Frobenius”, Math. Z. 178:1 (1981), 125-142. · Zbl 0472.13012
[17] R. P. Stanley, “The upper bound conjecture and Cohen-Macaulay rings”, Studies in Appl. Math. 54:2 (1975), 135-142. · Zbl 0308.52009
[18] R. P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics 41, Birkhäuser, 1996. · Zbl 0838.13008
[19] J. Stückrad and W. Vogel, Buchsbaum rings and applications: an interaction between algebra, geometry, and topology, Mathematische Monographien 21, VEB Deutscher Verlag der Wissenschaften, 1986. · Zbl 0606.13017
[20] N. Suzuki, “On quasi-Buchsbaum modules: an application of theory of FLC-modules”, pp. 215-243 in Commutative algebra and combinatorics (Kyoto, 1985), edited by M. Nagata and H. Matsumura, Adv. Stud. Pure Math. 11, North-Holland, 1987. · Zbl 0652.13014
[21] W. Vogel, “A non-zero-divisor characterization of Buchsbaum modules”, Michigan Math. J. 28:2 (1981), 147-152. · Zbl 0441.13014
[22] K. · Zbl 1185.13009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.