Toric rings attached to simplicial complexes

• Herzog, Jürgen ^[1] ; Moradi, Somayeh ^[2] ; Qureshi, Ayesha Asloob ^[3]
  1. [1] University of Duisburg-Essen

     University of Duisburg-Essen

     Kreisfreie Stadt Essen, Alemania

     
  2. [2] Ilam University

     Ilam University

     Irán

     
  3. [3] Sabancı University

     Sabancı University

     Turquía

     

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• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 77, Fasc. 3, 2026, págs. 827-841
• Idioma: inglés
• DOI: 10.1007/s13348-025-00493-9
• Texto completo no disponible (Saber más ...)
• Resumen

  • We consider standard graded toric rings R_{\Delta } whose generators correspond to the faces of a simplicial complex \Delta. When R_{\Delta } is normal, it is shown that its divisor class group is free. For a flag complex \Delta which is the clique complex of a perfect graph, a nice description for the class group and the canonical module of R_{\Delta } in terms of the minimal vertex covers of the graph is given. Moreover, for a quasi-forest simplicial complex a quadratic Gröbner basis for the defining ideal of R_{\Delta } is presented. Using this fact we give combinatorial descriptions for the a-invariant and the Gorenstein property of R_{\Delta }.

• Referencias bibliográficas
  • Bruns, W., Gubeladze, J.: Polytopes, rings, and K-theory. Springer Monographs in Mathematics, Springer, Dordrecht (2009)
  • Bruns, W., Herzog, J.: Cohen–Macaulay rings, revised ed., Cambridge Stud. Adv. Math., vol. 39, Cambridge University Press, Cambridge (1998)
  • Bruns, W., Ichim, B., Römer, T., Söger, C.: Normaliz, Algorithms for rational cones and affine monoids, Available at http://www.math.uos.de/normaliz
  • Chouinard, L.G., II.: Krull semigroups and divisor class groups. Canadian J. Math. 33(6), 1459–1468 (1981)
  • Chvátal, V.: On certain polytopes associated with graphs. J. Combin. Theory Ser. B 18, 138–154 (1975)
  • Engström, A., Norén, P.: Ideals of graph homomorphisms. Ann. Comb. 17, 71–103 (2013)
  • Ficarra, A., Moradi, S.: Sortable simplicial complexes and their associated toric rings, arXiv preprint (2025). https://arxiv.org/abs/2412.10113
  • Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry http://www.math.uiuc.edu/Macaulay2/
  • Herzog, J., Hibi, T.: Monomial ideals. Graduate Texts in Mathematics, Springer, New York (2018)
  • Herzog, J., Hibi, T., Moradi, S.: Toric rings arising from vertex cover ideals. Adv. Appl. Math. 152, 102596 (2024)
  • Herzog, J., Hibi, T., Ohsugi, H.: Binomial ideals. Graduate Texts in Mathematics, Springer, New York (2018)
  • Herzog, J., Hibi, T., Zheng, X.: Dirac’s theorem on chordal graphs and alexander duality. Eur. J. Combin. 25(7), 949–960 (2004)
  • Herzog, J., Hibi, T., Zheng, X.: Cohen-macaulay chordal graphs. J. Combin. Theory Ser. A 113(5), 911–916 (2006)
  • Higashitani, A., Matsushita, K.: Three families of toric rings arising from posets or graphs with small class groups. J. Pure Appl. Algebra...
  • Hochster, M.: Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes. Ann. of Math. 96, 318–337 (1972)
  • Matsushita, K.: Torsionfreeness for divisor class groups of toric rings of integral polytopes (2022) arXiv:2202.10686
  • Ohsugi, H., Herzog, J., Hibi, T.: Combinatorial pure subrings. Osaka J. Math. 37, 745–757 (2000)
  • Ohsugi, H., Hibi, T.: Normal polytopes arising from finite graphs. J. Algebra 207, 409–426 (1998)
  • Ohsugi, H., Hibi, T.: Convex polytopes all of whose reverse lexicographic initial ideals are squarefree. Proc. Amer. Math. Soc. 129(9), 2541–2546...
  • Ohsugi, H., Hibi, T.: Special simplices and gorenstein toric rings. J. Combin. Theory Ser. A 113, 718–725 (2006)
  • Sturmfels, B.: Gröbner Bases and Convex Polytopes. American Mathematical Society, Providence (1996)
  • Villarreal, R.H.: Monomial algebras. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL (2015)