K-theoretic balancing conditions and the Grothendieck group of a toric variety

We introduce a ring of Z-valued functions on a complete fan delta called Grothendieck weights to describe the ordinary operational K-theory of the associated toric variety X. These functions satisfy a K-theoretic analogue of the balancing condition for Minkowski weights, which is induced by a presentation of the Grothendieck group of X. We explicitly give a combinatorial presentation in low dimensions, and relate Grothendieck weights to other fan-based invariants such as piecewise exponential functions and Minkowski weights. As an application, we give an example of a projective toric surface X such that the forgetful map K???T (X) -> K???(X) is not surjective. (c) 2022 Elsevier Inc. All rights reserved. <comment>Superscript/Subscript Available</comment