Operational $K$-theory

Abstract

We study the operational bivariant theory associated to the covariant theory of Grothendieck groups of coherent sheaves, and prove that it has many geometric properties analogous to those of operational Chow theory. This operational $K$-theory agrees with Grothendieck groups of vector bundles on smooth varieties, admits a natural map from the Grothen­dieck group of perfect complexes on general varieties, satisfies descent for Chow envelopes, and is $A^1$-homotopy invariant.

Furthermore, we show that the operational $K$-theory of a complete linear variety is dual to the Grothendieck group of coherent sheaves. As an application, we show that the $K$-theory of perfect complexes on any complete toric threefold surjects onto this group. Finally we identify the equivariant operational $K$-theory of an arbitrary toric variety with the ring of integral piecewise exponential functions on the associated fan.