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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 43, No. 2, pp. 479-500 (2002)
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Polytopal Linear Algebra

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Winfried Bruns, Joseph Gubeladze

FB Mathematik/Informatik, Universität Osnabrück, 49069 Osnabrück, Germany, e-mail: Winfried.Bruns@mathematik.uni-osnabrueck.de; A. Razmadze Mathematical Institute, Alexidze St. 1, 380093 Tbilisi, Georgia, e-mail: gubel@rmi.acnet.ge

**Abstract:** We investigate similarities between the category of vector spaces and that of polytopal algebras, containing the former as a full subcategory. In Section 2 we introduce the notion of a polytopal Picard group and show that it is trivial for fields. The coincidence of this group with the ordinary Picard group for general rings remains an open question. In Section 3 we survey some of the previous results on the automorphism groups and retractions. These results support a general conjecture proposed in Section 4 about the nature of arbitrary homomorphisms of polytopal algebras. Thereafter a further confirmation of this conjecture is presented by homomorphisms defined on Veronese singularities.

This is a continuation of the project started in [BG1], [BG2], [BG3]. The higher $K$-theoretic aspects of polytopal linear objects will be treated in [BG4], [BG5].

[BG1] Bruns, W.; Gubeladze, J.: * Polytopal linear groups*, J. Algebra ** 218** (1999), 715-737.

[BG2] Bruns, W.; Gubeladze, J.: * Polytopal linear retractions.* Trans. Amer. Math. Soc. ** 354**(1) (2002), 179-203.

[BG3] Bruns, W.; Gubeladze, J.: * Polyhedral algebras, arrangements of toric varieties, and their groups.* Adv. Stud. Pure Math., to appear.

[BG4] Bruns, W.; Gubeladze, J.: * Polyhedral $K_2$.* Preprint.

[BG5] Bruns, W.; Gubeladze, J.: * Higher polyhedral $K$-groups.* Preprint.

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