Polytopal linear retractions

Bruns, Winfried; Gubeladze, Joseph

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

Polytopal linear retractions
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by Winfried Bruns and Joseph Gubeladze

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

DOI: https://doi.org/10.1090/S0002-9947-01-02703-9

Published electronically: May 14, 2001

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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).

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