K-theory and Absolute Cohomology for Algebraic Stacks

In this paper we consider the K-theory of smooth algebraic stacks, establish λ and Adams operations and show that the higher K-theory of such stacks is always a pre-λ-ring and is a λ-ring if every coherent sheaf is the quotient of a vector bundle. As a consequence we are able to define Adams operations and absolute cohomology for smooth algebraic stacks satisfying this hypothesis. We also define a Riemann-Roch transformation and prove a Riemann-Roch theorem for strongly projective morphisms between smooth stacks. When the stack is a scheme, all these are shown to reduce to the corresponding results for schemes.