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On Rational Equivalence in Tropical Geometry

Published online by Cambridge University Press:  20 November 2018

Lars Allermann ,
Simon Hampe  and
Johannes Rau
Lars Allermann
Affiliation:
Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany e-mail: allerman@mathematik.uni-kl.de
Simon Hampe
Affiliation:
Fachrichtung Mathematik, Universität der Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany e-mail: simon.hampe@math.uni-sb.de johannes.rau@math.uni-sb.de
Johannes Rau
Affiliation:
Fachrichtung Mathematik, Universität der Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany e-mail: simon.hampe@math.uni-sb.de johannes.rau@math.uni-sb.de
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Abstract

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This article discusses the concept of rational equivalence in tropical geometry (and replaces an older, imperfect version). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the “bounded” Chow groups of ${{\mathbf{R}}^{n}}$ by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest. We show that every tropical cycle in ${{\mathbf{R}}^{n}}$ is a sum of (translated) fan cycles. This also proves that the intersection ring of tropical cycles is generated in codimension 1 (by hypersurfaces).

Keywords

14T05tropical geometryrational equivalence
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 68 , Issue 2 , 01 April 2016 , pp. 241 - 257
Copyright
Copyright © Canadian Mathematical Society 2016

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