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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