Polytopal linear retractions

Bruns, Winfried; Gubeladze, Joseph

doi:10.1090/S0002-9947-01-02703-9

Polytopal linear retractions
HTML articles powered by AMS MathViewer

by Winfried Bruns and Joseph Gubeladze PDF

Trans. Amer. Math. Soc. 354 (2002), 179-203 Request permission

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

Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
Winfried Bruns and Joseph Gubeladze, Polytopal linear groups, J. Algebra 218 (1999), no. 2, 715–737. MR 1705750, DOI 10.1006/jabr.1998.7831
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.
Antoine Deza, Komei Fukuda, and Vera Rosta, Wagner’s theorem and combinatorial enumeration of $3$-polytopes, Sūrikaisekikenkyūsho K\B{o}kyūroku 872 (1994), 30–34. Computational geometry and discrete geometry (Japanese) (Kyoto, 1993). MR 1330480
W. Bruns and J. Herzog. Cohen–Macaulay rings (Rev. Ed.), Cambridge University Press, 1998.
Douglas L. Costa, Retracts of polynomial rings, J. Algebra 44 (1977), no. 2, 492–502. MR 429866, DOI 10.1016/0021-8693(77)90197-1
David Eisenbud and Bernd Sturmfels, Binomial ideals, Duke Math. J. 84 (1996), no. 1, 1–45. MR 1394747, DOI 10.1215/S0012-7094-96-08401-X
Robert M. Fossum, The divisor class group of a Krull domain, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74, Springer-Verlag, New York-Heidelberg, 1973. MR 0382254, DOI 10.1007/978-3-642-88405-4
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, DOI 10.1515/9781400882526
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, DOI 10.1007/978-0-8176-4771-1
Robert Gilmer, Commutative semigroup rings, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1984. MR 741678
Joseph Gubeladze, The isomorphism problem for commutative monoid rings, J. Pure Appl. Algebra 129 (1998), no. 1, 35–65. MR 1626643, DOI 10.1016/S0022-4049(97)00063-7
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
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