Hostname: page-component-cb9f654ff-5kfdg Total loading time: 0 Render date: 2025-08-19T21:48:45.322Z Has data issue: false hasContentIssue false
  • English
  • Français

On Canonical Modules of Toric Face Rings

Published online by Cambridge University Press:  11 January 2016

Bogdan Ichim  and
Tim Römer
Bogdan Ichim
Affiliation:
Universität Osnabrück, FB Mathematik/Informatik, 49069 Osnabrück, Germany Institute of Mathematics, C.P. 1-764, 70700 Bucharest, Romania, bogdan.ichim@math.uos.de, bogdan.ichim@imar.ro
Tim Römer
Affiliation:
Universität Osnabrück, FB Mathematik/Informatik, 49069 Osnabrück, Germany, troemer@uos.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Generalizing the concepts of Stanley-Reisner and affine monoid algebras, one can associate to a rational pointed fan Σ in ℝd the ℤd -graded toric face ring K[Σ]. Assuming that K[Σ] is Cohen-Macaulay, the main result of this paper is to characterize the situation when its canonical module is isomorphic to a ℤd -graded ideal of K[Σ]. From this result several algebraic and combinatorial consequences are deduced. As an application, we give a relation between the cleanness of K[Σ] and the shellability of Σ.

Keywords

13C1413D4505E99

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 194 , 2009 , pp. 69 - 90
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2009

References

[1] Björner, A. and Wachs, M. L., Shellable nonpure complexes and posets. I, Trans. Am. Math. Soc., 348 (1996), no. 4, 1299–1327.Google Scholar
[2] Björner, A. and Wachs, M. L., Shellable nonpure complexes and posets. II, Trans. Am. Math. Soc., 349 (1997), no. 10, 3945–3975.Google Scholar
[3] Björner, A., Las Vergnas, M., Sturmfels, B., White, N. and Ziegler, G., Oriented ma-troids. 2nd ed., Encyclopedia of Mathematics and Its Applications 46, Cambridge University Press, 1999.Google Scholar
[4] Brun, M., Bruns, W. and Römer, T., Cohomology of partially ordered sets and local cohomology of section rings, Adv. Math., 208 (2007), no. 1, 210–235.Google Scholar
[5] Brun, M. and Römer, T., Subdivisions of toric complexes, J. Algebr. Comb., 21 (2005), no. 4, 423–448.Google Scholar
[6] Brun, M. and Römer, T., On algebras associated to partially ordered sets, to appear in Math. Scand.Google Scholar
[7] Bruns, W. and Gubeladze, J., Polyhedral algebras, arrangements of toric varieties, and their groups, Adv. Stud. Pure Math., 33 (2001), 1–51.Google Scholar
[8] Bruns, W. and Gubeladze, J., Polytopes, rings, and K-theory, preprint (2008).Google Scholar
[9] Bruns, W. and Herzog, J., Cohen-Macaulay rings, Rev. ed., Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, 1998.Google Scholar
[10] Danaraj, G. and Klee, V., Shellings of spheres and polytopes, Duke Math. J., 41 (1974), 443–451.Google Scholar
[11] Dress, A., A new algebraic criterion for shellability, Beitr. Alg. Geo., 34 (1993), 45–55.Google Scholar
[12] Fløystad, G., Cohen-Macaulay cell complexes, Algebraic and geometric combinatorics (C. A. Athanasiadis et al., eds.), Contemporary Mathematics 423, American Mathematical Society, 2006, pp. 205–220.Google Scholar
[13] Gräbe, H. G., The canonical module of a Stanley-Reisner ring, J. Algebra, 86 (1984), 272–281.Google Scholar
[14] Gräbe, H. G., A dualizing complex for Stanley-Reisner rings, Math. Proc. Camb. Philos. Soc., 96 (1984), 203–212.Google Scholar
[15] Ichim, B. and Römer, T., On toric face rings, J. Pure Appl. Algebra, 210 (2007), no. 1, 249–266.Google Scholar
[16] McMullen, P. and Shephard, G. C., Convex polytopes and the upper bound conjecture, London Mathematical Society Lecture Note Series 3, Cambridge University Press, 1971.Google Scholar
[17] Miller, E., Cohen-Macaulay quotients of normal semigroup rings via irreducible resolutions, Math. Research Letters, 9 (2002), 117–128.Google Scholar
[18] Munkres, J., Topological results in combinatorics, Michigan Math. J., 31 (1984), 113–128.Google Scholar
[19] Reisner, G. A., Cohen-Macaulay quotients of polynomial rings, Adv. Math., 21 (1976), 30–49.Google Scholar
[20] 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.Google Scholar
[21] Stanley, R. P., Enumerative combinatorics. Vol. 1, Paperback ed., Cambridge Studies in Advanced Mathematics 49, Cambridge, Cambridge University Press, 1999.Google Scholar
[22] Stanley, R. P., Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics 41, Birkhäuser, 2005.Google Scholar
[23] Yanagawa, K., Notes on C-graded modules over an affine semigroup ring K[C], preprint math.AC/0506457.Google Scholar
[24] Ziegler, G. M., Lectures on polytopes, Springer, 1994.Google Scholar