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Journal of Algebraic Combinatorics

, Volume 21, Issue 4, pp 423–448 | Cite as

Subdivisions of Toric Complexes

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Abstract

We introduce toric complexes as polyhedral complexes consisting of rational cones together with a set of integral generators for each cone, and we define their associated face rings. Abstract simplicial complexes and rational fans can be considered as toric complexes, and the face ring for toric complexes extends Stanley and Reisner’s face ring for abstract simplicial complexes [20] and Stanley’s face ring for rational fans [21]. Given a toric complex with defining ideal I for the face ring we give a geometrical interpretation of the initial ideals of I with respect to weight orders in terms of subdivisions of the toric complex generalizing a theorem of Sturmfels in [23]. We apply our results to study edgewise subdivisions of abstract simplicial complexes.

Keywords

initial ideal toric ideal polyhedral complex regular subdivision edgewise subdivision face ring 
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References

  1. 1.
    J. Backelin and R. Fröberg, “{Koszul algebras, Veronese subrings and rings with linear resolutions},” Rev. Roum. Math. Pures Appl. 30 (1985), 85–97.Google Scholar
  2. 2.
    M. Bökstedt, W.C. Hsiang, and I. Madsen, “The cyclotomic trace and algebraic K-theory of spaces,” Invent. Math. 111 (1993), 465–539Google Scholar
  3. 3.
    L.A. Borisov, L. Chen, and G.G. Smith, “{The orbifold Chow ring of toric Deligne-Mumford stacks},” J. Am. Math. Soc. 18 (2005), 193–215Google Scholar
  4. 4.
    W. Bruns and A. Conca, “{Groebner bases, initial ideals and initial algebras}.” In L.L. Avramov et al. (Hrsg.), Homological methods in commutative algebra, IPM Proceedings, Teheran 2004.Google Scholar
  5. 5.
    W. Bruns and J. Gubeladze, “Polyhedral algebras, arrangements of toric varieties, and their groups. Computational commutative algebra and combinatorics,” Adv. Stud. Pure Math. 33 (2001), 1–51Google Scholar
  6. 6.
    W. Bruns and J. Herzog, Cohen-Macaulay Rings. Rev. ed., Cambridge Studies in Advanced Mathematics 39, Cambridge University Press (1998)Google Scholar
  7. 7.
    D.A. Cox, “The homogeneous coordinate ring of a toric variety,” J. Algebr. Geom. 4 (1995), 17–50Google Scholar
  8. 8.
    H. Edelsbrunner and D.R. Grayson, “Edgewise subdivision of a simplex,” Discrete Comput. Geom. 24 (2000), 707–719Google Scholar
  9. 9.
    D. Eisenbud, A. Reeves, and B. Totaro, “Initial ideals, Veronese subrings, and rates of algebras,” Adv. Math. 109 (1994), 168–187Google Scholar
  10. 10.
    D. Eisenbud and B. Sturmfels, “Binomial ideals,” Duke Math. J. 84 (1996), 1–45Google Scholar
  11. 11.
    H. Freudenthal, “Simplizialzerlegungen von beschränkter Flachheit,” Ann. Math. 43 (1942), 580–582Google Scholar
  12. 12.
    W. Fulton, “Introduction to Toric Varieties,” Annals of Mathematics Studies, Princeton University Press, 1993, vol. 131Google Scholar
  13. 13.
    D. Grayson, “Exterior power operations on higher K-theory,” K-Theory 3 (1989), 247–260Google Scholar
  14. 14.
    S. Hosten, D. MacLagan, and B. Sturmfels, “{Supernormal vector configurations},” J. Algebr. Comb. 19 (2004), 297–313Google Scholar
  15. 15.
    G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal Embeddings. I. Lecture Notes in Mathematics, Springer, 1973, vol. 339Google Scholar
  16. 16.
    D. Notbohm and N. Ray, “{On Davis-Januszkiewicz homotopy types I; formality and rationalisation},” Algebr. Geom. Topol. 5 (2005), 31–51Google Scholar
  17. 17.
    H. Ohsugi and T. Hibi, “Compressed polytopes, initial ideals and complete multipartite graphs,” Ill. J. Math. 44 (2000), 391–406Google Scholar
  18. 18.
    A. Schrijver, Theory of Linear and Integer Programming, Wiley, 1998Google Scholar
  19. 19.
    E. O’Shea and R.R. Thomas, “{Toric initial ideals of Δ-normal configurations: Cohen-Macaulayness and degree bounds},” J. Algebr. Comb. 21 (2005), 247–268Google Scholar
  20. 20.
    R.P. Stanley, Combinatorics and Commutative Algebra. 2nd ed., Progress in Mathematics Birkhäuser, 1996, vol. 41Google Scholar
  21. 21.
    R. Stanley, “Generalized h-vectors, intersection cohomology of toric varieties, and related results. Commutative algebra and combinatorics,” Adv. Stud. Pure Math. 11 (1987), 187–213Google Scholar
  22. 22.
    B. Sturmfels, “Gröbner bases of toric varieties,” Tôhoku Math. J. 43 (1991), 249–261Google Scholar
  23. 23.
    B. Sturmfels, “Gröbner bases and convex polytopes,” Univ. Lecture Series 8, AMS, 1996Google Scholar
  24. 24.
    G.M. Ziegler, Lectures on polytopes. Graduate Texts in Mathematics, Springer, 1995, vol. 152Google Scholar

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© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.FB Mathematik/InformatikUniversität OsnabrückOsnabrückGermany
  2. 2.FB Mathematik/InformatikUniversität OsnabrückOsnabrückGermany

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