Mayer–Vietoris sequences and equivariant $K$-theory rings of toric varieties

Abstract

We apply a Mayer-Vietoris sequence argument to identify the Atiyah-Segal equivariant complex K-theory rings of certain toric varieties with rings of integral piecewise Laurent polynomials on the associated fans. We provide necessary and sufficient conditions for this identification to hold for toric varieties of complex dimension 2, including smooth and singular cases. We prove that it always holds for smooth toric varieties, regardless of whether or not the fan is polytopal or complete. Finally, we introduce the notion of fans with "distant singular cones," and prove that the identification holds for them. The identification has already been made by Hararda, Holm, Ray and Williams in the case of divisive weighted projective spaces; in addition to enlarging the class of toric varieties for which the identification holds, this work provides an example in which the identification fails. We make every effort to ensure that our work is rich in examples.