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.

38 pages; v2: new exampes in Sections 5 and 7, and an new application (Theorem 1.4), showing that the natural map from K-theory of perfect complexes to the dual of the Grothendieck group of coherent sheaves is surjective for complete toric threefolds; v3: final version published in Documenta Math

References in corpus (1)

• The equivariant $K$-theory and cobordism rings of divisive weighted projective spaces

Cited by in corpus (9)

• On the $K$-theoretic Hall algebra of a surface
• Precobordism and cobordism
• Equivariant Grothendieck-Riemann-Roch and localization in operational K-theory
• Localization in equivariant operational K-theory and the Chang-Skjelbred property
• The equivariant $K$-theory and cobordism rings of divisive weighted projective spaces
• The Wheel Conditions and K-theoretic Hall Algebras
• Vector bundles on proper toric 3-folds and certain other schemes
• Algebraic rational cells and equivariant intersection theory
• Formal Group Rings of Toric Varieties