Algebraic $K$-theory of quotient stacks

Abstract

We prove some fundamental results like localization, excision, Nisnevich descent, and the regular blow-up formula for the algebraic $K$-theory of certain stack quotients of schemes with affine group scheme actions. We show that the homotopy $K$-theory of such stacks is homotopy invariant. This implies a similar homotopy invariance property of the algebraic $K$-theory with coefficients.