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Dualizing Complex of a Toric Face Ring

Published online by Cambridge University Press:  11 January 2016

Ryota Okazaki  and
Kohji Yanagawa
Ryota Okazaki
Affiliation:
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology Osaka University Toyonaka, Osaka, 560-0043, Japan, [email protected]
Kohji Yanagawa
Affiliation:
Department of Mathematics, Faculty of Engineering Science Kansai University Suita, 564-8680, Japan, [email protected]
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Abstract

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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 square-free module theory over R, and show that the Cohen-Macaulay, Buchsbaum, and Gorenstein* properties of R are topological properties of its associated cell complex.

Keywords

13F5513D25
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 196 , 2009 , pp. 87 - 116
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2009

References

[1] Brun, M., Bruns, W. and Römer, T., Cohomology of partially ordered sets and local cohomology of section rings, Adv. Math., 208 (2007), 210–235.Google Scholar
[2] Bruns, W. and Gubeladze, J., Polyhedral algebras, arrangements of toric varieties, and their groups, Computational commutative algebra and combinatorics, Adv. Stud. Pure Math., 33, 2001, pp. 1–51.Google Scholar
[3] Bruns, W. and Gubeladze, J., Polytopes, rings, and K-theory, Springer Monographs in Mathematics, Springer, 2009.Google Scholar
[4] Bruns, W. and Herzog, J., Cohen-Macaulay rings, revised edition, Cambridge University Press, 1998.Google Scholar
[5] Bruns, W., Koch, R., and Römer, T., Gröbner bases and Betti numbers of monoidal complexes, Michigan Math. J., 57 (2008), 71–91.CrossRefGoogle Scholar
[6] Caijun, Z., Cohen-Macaulay section rings, Trans. Amer. Math. Soc., 349 (1997), 4659–4667.Google Scholar
[7] Hartshorne, R., Residues and duality, Lecture notes in Mathematics 20, Springer, 1966.Google Scholar
[8] Ichim, B. and Römer, T., On toric face rings, J. Pure Appl. Algebra, 210 (2007), 249–266.Google Scholar
[9] Ishida, M.-N., The local cohomology groups of an affine semigroup ring, Algebraic Geometry and Commutative Algebra, vol. I, Kinokuniya, Tokyo, 1988, pp. 141–153.CrossRefGoogle Scholar
[10] Iversen, B., Cohomology of sheaves, Springer-Verlag, 1986.Google Scholar
[11] Sharp, R. Y., Dualizing complexes for commutative Noetherian rings, Math. Proc. Comb. Phil. Soc., 78 (1975), 369–386.Google Scholar
[12] Stanley, R. P., Generalized H-vectors, intersection cohomology of toric varieties, and related results, Commutative algebra and combinatorics, Adv. Stud. Pure Math., 11, 1987, 187–213.CrossRefGoogle Scholar
[13] Stuckrad, S. and Vogel, W., Buchsbaum rings and applications, Springer-Verlag, 1986.Google Scholar
[14] Yanagawa, K., Sheaves on finite posets and modules over normal semigroup rings, J. Pure Appl. Algebra, 161 (2001), 341–366.Google Scholar
[15] Yanagawa, K., 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.Google Scholar
[16] Yanagawa, K., Stanley-Reisner rings, sheaves, and Poincaré-Verdier duality, Math. Res. Lett., 10 (2003), 635–650.Google Scholar
[17] Yanagawa, K., Dualizing complex of the incidence algebra of a finite regular cell complex, Illinois J. Math., 49 (2005), 1221–1243.Google Scholar
[18] Yanagawa, K., Notes on C-graded modules over an affine semigroup ring K[C], Comm. Algebra, 38 (2008), 3122–3146.Google Scholar
[19] Ziegler, G., Lectures on Polytopes, Graduate Texts in Mathematics, Vol. 152, Springer, 1995 (Revised edition 1998).Google Scholar