Nagoya Mathematical Journal

Dualizing complex of a toric face ring

Ryota Okazaki and Kohji Yanagawa

Full-text: Open access


A toric face ring, which generalizes both Stanley-Reisner rings and affine semigroup rings, is studied by Bruns, Römer and their coauthors recently. In this paper, under the "normality" assumption, we describe a dualizing complex of a toric face ring $R$ in a very concise way. Since $R$ is not a graded ring in general, the proof is not straightforward. We also develop the squarefree module theory over $R$, and show that the Cohen-Macaulay, Buchsbaum, and Gorenstein* properties of $R$ are topological properties of its associated cell complex.

Article information

Nagoya Math. J., Volume 196 (2009), 87-116.

First available in Project Euclid: 15 January 2010

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13F55: Stanley-Reisner face rings; simplicial complexes [See also 55U10] 13D25


Okazaki, Ryota; Yanagawa, Kohji. Dualizing complex of a toric face ring. Nagoya Math. J. 196 (2009), 87--116.

Export citation


  • M. Brun, W. Bruns and T. Römer, Cohomology of partially ordered sets and local cohomology of section rings, Adv. Math., 208 (2007), 210--235.
  • 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, pp. 1--51.
  • W. Bruns and J. Gubeladze, Polytopes, rings, and $K$-theory, Springer Monographs in Mathematics, Springer, 2009.
  • W. Bruns and J. Herzog, Cohen-Macaulay rings, revised edition, Cambridge University Press, 1998.
  • W. Bruns, R. Koch, and T. Römer, Gröbner bases and Betti numbers of monoidal complexes, Michigan Math. J., 57 (2008), 71--91.
  • Z. Caijun, Cohen-Macaulay section rings, Trans. Amer. Math. Soc., 349 (1997), 4659--4667.
  • R. Hartshorne, Residues and duality, Lecture notes in Mathematics 20, Springer, 1966.
  • B. Ichim and T. Römer, On toric face rings, J. Pure Appl. Algebra, 210 (2007), 249--266.
  • M.-N. Ishida, The local cohomology groups of an affine semigroup ring, Algebraic Geometry and Commutative Algebra, vol. I, Kinokuniya, Tokyo, 1988, pp. 141--153.
  • B. Iversen, Cohomology of sheaves, Springer-Verlag, 1986.
  • R. Y. Sharp, Dualizing complexes for commutative Noetherian rings, Math. Proc. Comb. Phil. Soc., 78 (1975), 369--386.
  • R. P. Stanley, Generalized $H$-vectors, intersection cohomology of toric varieties, and related results, Commutative algebra and combinatorics, Adv. Stud. Pure Math., 11, 1987, 187--213.
  • S. Stuckrad and W. Vogel, Buchsbaum rings and applications, Springer-Verlag, 1986.
  • K. Yanagawa, Sheaves on finite posets and modules over normal semigroup rings, J. Pure Appl. Algebra, 161 (2001), 341--366.
  • K. Yanagawa, Squarefree modules and local cohomology modules at monomial ideals, Local cohomology and its applications, Lecture Notes in Pure and Appl. Math., 226, Dekker, New York, 2002, pp. 207--231.
  • K. Yanagawa, Stanley-Reisner rings, sheaves, and Poincaré-Verdier duality, Math. Res. Lett., 10 (2003), 635--650.
  • K. Yanagawa, Dualizing complex of the incidence algebra of a finite regular cell complex, Illinois J. Math., 49 (2005), 1221--1243.
  • K. Yanagawa, Notes on $C$-graded modules over an affine semigroup ring $K[C]$, Comm. Algebra, 38 (2008), 3122--3146.
  • G. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, Vol. 152, Springer, 1995 (Revised edition 1998).