Mathematische Annalen

, Volume 351, Issue 4, pp 857–875 | Cite as

Face rings of simplicial complexes with singularities

  • Ezra Miller
  • Isabella Novik
  • Ed Swartz


The face ring of a simplicial complex modulo m generic linear forms is shown to have finite local cohomology if and only if the link of every face of dimension m or more is nonsingular, i.e., has the homology of a wedge of spheres of the expected dimension. This is derived from an enumerative result for local cohomology of face rings modulo generic linear forms, as compared with local cohomology of the face ring itself. The enumerative result is generalized to squarefree modules. A concept of Cohen–Macaulay in codimension c is defined and characterized for arbitrary finitely generated modules and coherent sheaves. For the face ring of an r-dimensional complex Δ, it is equivalent to nonsingularity of Δ in dimension rc; for a coherent sheaf on projective space, this condition is shown to be equivalent to the same condition on any single generic hyperplane section. The characterization of nonsingularity in dimension m via finite local cohomology thus generalizes from face rings to arbitrary graded modules.

Mathematics Subject Classification (2010)

13F55 05E40 05E45 13H10 14M05 13D45 13C14 


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  1. 1.
    Bruns, W., Herzog, J.: Cohen–Macaulay rings. In: Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)Google Scholar
  2. 2.
    Cuong N.T., Schenzel P., Trung N.V.: Verallgemeinerte Cohen–Macaulay-Moduln. Math. Nachr. 85, 57–73 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Eisenbud D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag, New York (1995)zbMATHGoogle Scholar
  4. 4.
    Flenner H.: Die Sätze von Bertini für lokale Ringe. Math. Ann. 229(2), 97–111 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Goto S., Takayama Y.: Stanley–Reisner ideals whose powers have finite length cohomologies. Proc. Am. Math. Soc. 135, 2355–2364 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Gräbe H.-G.: The canonical module of a Stanley–Reisner ring. J. Algebra 86, 272–281 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Haghighi, H., Yassemi, S., Zaare-Nahandi, R.: A generalization of k-Cohen–Macaulay complexes. Preprint. arXiv:math.AC/0912.4097Google Scholar
  8. 8.
    Haghighi, H., Terai, N., Yassemi, S., Zaare-Nahandi, R.: Sequentially S r simplicial complexes and sequentially S 2 graphs. arXiv:math.AC/1004.3376Google Scholar
  9. 9.
    Miller E.: The Alexander duality functors and local duality with monomial support. J. Algebra 231, 180–234 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Miller, E.: Topological Cohen–Macaulay criteria for monomial ideals. In: Ene, V., Miller, E. (eds.) Combinatorial Aspects of Commutative Algebra. Contemporary Mathematics, vol. 502, pp. 137–156. American Mathematical Society, Providence (2009). arXiv:math.AC/0809.1458Google Scholar
  11. 11.
    Miller, E., Sturmfels, B.: Combinatorial commutative algebra. In: Graduate Texts in Mathematics, vol. 227. Springer–Verlag, New York (2005)Google Scholar
  12. 12.
    Murai S., Terai N.: h-Vectors of simplicial complexes with Serre’s conditions. Math. Res. Lett. 16(6), 1015–1028 (2009)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Reisner G.: Cohen–Macaulay quotients of polynomial rings. Adv. Math. 21, 30–49 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Schäfer U., Schenzel P.: Dualizing complexes of affine semigroup rings. Trans. Am. Math. Soc. 322(2), 561–582 (1990)zbMATHCrossRefGoogle Scholar
  15. 15.
    Schenzel P.: On the number of faces of simplicial complexes and the purity of Frobenius. Math. Z. 178, 125–142 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Spreafico M.L.: Axiomatic theory for transversality and Bertini type theorems. Arch. Math. (Basel) 70(5), 407–424 (1998)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Stanley R.: Cohen–Macaulay complexes. In: Aigner, M. (ed.) Higher Combinatorics, pp. 51–62. Reidel, Dordrecht (1977)Google Scholar
  18. 18.
    Stanley R.: Combinatorics and Commutative Algebra. Birkhäuser, Boston (1996)zbMATHGoogle Scholar
  19. 19.
    Stückrad J., Vogel W.: Buchsbaum Rings and Applications. Springer-Verlag, Berlin (1986)Google Scholar
  20. 20.
    Trung N.V.: Toward a theory of generalized Cohen–Macaulay modules. Nagoya Math. J. 102, 1–49 (1986)MathSciNetGoogle Scholar
  21. 21.
    Takayama Y.: Combinatorial characterizations of generalized Cohen–Macaulay ideals. Bull. Math. Soc. Sci. Math. Roumanie 48(96), 327–344 (2005)MathSciNetGoogle Scholar
  22. 22.
    Yanagawa K.: Alexander duality for Stanley-Reisner rings and squarefree \({\mathbb{N}^n}\)-graded modules. J. Algebra 225(2), 630–645 (2000)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Duke UniversityDurhamUSA
  2. 2.University of WashingtonSeattleUSA
  3. 3.Cornell UniversityIthacaUSA

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