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Transactions of the American Mathematical Society

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Polytopal linear retractions


Authors: Winfried Bruns and Joseph Gubeladze
Journal: Trans. Amer. Math. Soc. 354 (2002), 179-203
MSC (2000): Primary 13F20, 14M25; Secondary 52C07
DOI: https://doi.org/10.1090/S0002-9947-01-02703-9
Published electronically: May 14, 2001
MathSciNet review: 1859031
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Abstract:

We investigate graded retracts of polytopal algebras (essentially the homogeneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are again polytopal algebras and that codimension $1$ retractions factor through retractions preserving the semigroup structure. We expect that these results hold in general.

This paper is a part of the project started by the authors in 1999, where we investigate the graded automorphism groups of polytopal algebras. Part of the motivation comes from the observation that there is a reasonable `polytopal' generalization of linear algebra (and, subsequently, that of algebraic $K$-theory).


References [Enhancements On Off] (What's this?)

  • [Bor] A. Borel, Linear Algebraic Groups, Second edition, Grad. Texts in Math., Springer-Verlag, 1991. MR 92d:20001
  • [BG1] W. Bruns and J. Gubeladze, Polytopal linear groups, J. Algebra 218 (1999), 715-737. MR 2000g:14059
  • [BG2] W. Bruns and J. Gubeladze, Polyhedral algebras, toric arrangements, and their groups, Proceedings of the Osaka meeting on computational commutative algebra and combinatorics, 1999. Adv. Stud. Pure Math., to appear.
  • [BGT] W. Bruns, J. Gubeladze, and N. V. Trung, Normal polytopes, triangulations, and Koszul algebras, J. Reine Angew. Math. 485 (1997), 123-160. MR 96c:52016
  • [BH] W. Bruns and J. Herzog. Cohen-Macaulay rings (Rev. Ed.), Cambridge University Press, 1998.
  • [Cos] D. Costa, Retracts of polynomial rings, J. Algebra 44 (1977), 492-502. MR 55:2876
  • [ES] D. Eisenbud and B. Sturmfels, Binomial ideals, Duke Math. J. 84 (1996), 1-45. MR 97d:13031
  • [Fo] R. Fossum, The Divisor Class Group of Krull Domain, Springer-Verlag, 1973. MR 52:3139
  • [Fu] W. Fulton, Introduction to Toric Varieties, Princeton University Press, 1993. MR 94g:14028
  • [GKZ] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser Boston, 1994. MR 95e:14045
  • [Gi] R. Gilmer, Commutative Semigroup Rings, Univ. Chicago Press, 1984. MR 85e:20058
  • [Gu] J. Gubeladze, The isomorphism problem for commutative monoid rings, J. Pure Appl. Algebra 129 (1998), 35-65. MR 99h:20094
  • [Ha] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math., Springer-Verlag, 1977. MR 57:3116
  • [Oda] T. Oda, Convex Bodies and Algebraic Geometry (An introduction to the theory of toric varieties), Springer-Verlag, 1988. MR 88m:14038

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Additional Information

Winfried Bruns
Affiliation: Universität Osnabrück, FB Mathematik/Informatik, 49069 Osnabrück, Germany
Email: Winfried.Bruns@mathematik.uni-osnabrueck.de

Joseph Gubeladze
Affiliation: A. Razmadze Mathematical Institute, Alexidze St. 1, 380093 Tbilisi, Georgia
Email: gubel@rmi.acnet.ge

DOI: https://doi.org/10.1090/S0002-9947-01-02703-9
Keywords: Polytopal algebra, retracts, affine semigroup ring, binomial ideal
Received by editor(s): January 10, 2000
Received by editor(s) in revised form: April 10, 2000
Published electronically: May 14, 2001
Additional Notes: The second author was supported by the Max-Planck-Institut für Mathematik in Bonn and INTAS, Grant 93-2618-Ext
Article copyright: © Copyright 2001 American Mathematical Society

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