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

Published electronically:
May 14, 2001

MathSciNet review:
1859031

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Abstract | References | Similar Articles | Additional Information

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 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 -theory).

**[Bor]**Armand Borel,*Linear algebraic groups*, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR**1102012****[BG1]**Winfried Bruns and Joseph Gubeladze,*Polytopal linear groups*, J. Algebra**218**(1999), no. 2, 715–737. MR**1705750**, 10.1006/jabr.1998.7831**[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]**Antoine Deza, Komei Fukuda, and Vera Rosta,*Wagner’s theorem and combinatorial enumeration of 3-polytopes*, Sūrikaisekikenkyūsho Kōkyūroku**872**(1994), 30–34. Computational geometry and discrete geometry (Japanese) (Kyoto, 1993). MR**1330480****[BH]**W. Bruns and J. Herzog.*Cohen-Macaulay rings*(Rev. Ed.), Cambridge University Press, 1998.**[Cos]**Douglas L. Costa,*Retracts of polynomial rings*, J. Algebra**44**(1977), no. 2, 492–502. MR**0429866****[ES]**David Eisenbud and Bernd Sturmfels,*Binomial ideals*, Duke Math. J.**84**(1996), no. 1, 1–45. MR**1394747**, 10.1215/S0012-7094-96-08401-X**[Fo]**Robert M. Fossum,*The divisor class group of a Krull domain*, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74. MR**0382254****[Fu]**William Fulton,*Introduction to toric varieties*, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR**1234037****[GKZ]**I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky,*Discriminants, resultants, and multidimensional determinants*, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR**1264417****[Gi]**Robert Gilmer,*Commutative semigroup rings*, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1984. MR**741678****[Gu]**Joseph Gubeladze,*The isomorphism problem for commutative monoid rings*, J. Pure Appl. Algebra**129**(1998), no. 1, 35–65. MR**1626643**, 10.1016/S0022-4049(97)00063-7**[Ha]**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR**0463157****[Oda]**Tadao Oda,*Convex bodies and algebraic geometry*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR**922894**

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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:
http://dx.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