JOURNAL OF
ALGEBRAIC
COMBINATORICS

Home > Vol 21, No 4 (2005)

Morten Brun and Tim Römer

FB Mathematik/Informatik Universität Osnabrück 49069 Osnabrück Germany

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.

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