K-rings of smooth toric varieties via piecewise-exponential functions

TL;DR

This paper provides an explicit algebraic description of the K-ring of smooth toric varieties using piecewise-exponential functions, paralleling the known Chow ring presentations, thus advancing the algebraic understanding of these geometric objects.

Contribution

It introduces a new presentation of the K-ring of smooth toric varieties via piecewise-exponential functions, extending the Stanley-Reisner framework.

Findings

Explicit presentation of K-rings as quotients of Stanley-Reisner rings.

Parallel between K-ring and Chow ring presentations.

Enhanced algebraic tools for studying smooth toric varieties.

Abstract

We describe an explicit presentation of the ring of integral piecewise-exponential functions on a unimodular fan as a quotient of the Stanley-Reisner ring of the fan. This gives rise to a presentation of K-rings of smooth toric varieties that is parallel to the well-known presentation of integral Chow rings as quotients of Stanley-Reisner rings.