Equivariant Algebraic K-Theory and Derived completions II: the case of Equivariant Homotopy K-Theory and Equivariant K-Theory

In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K-Theory similar to the well-known Atiyah-Segal completion theorem for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem in the sense of Atiyah and Segal for equivariant Algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorems proven by him, and also appearing in the original Atiyah-Segal theorem. In an earlier work by the first two authors, we solved this conjecture by providing a derived completion theorem for equivariant G-theory. In the present paper, we provide a similar derived completion theorem for the homotopy Algebraic K-theory of equivariant perfect complexes, on schemes that need not be regular. Our solution is broad enough to allow actions by all linear algebraic groups, irrespective of whether they are connected or not, and acting on any normal quasi-projective scheme of finite type over a field, irrespective of whether they are regular or projective. This allows us therefore to consider the Equivariant Homotopy Algebraic K-Theory of large classes of varieties like all toric varieties (for the action of a torus) and all spherical varieties (for the action of a reductive group). With finite coefficients invertible in the base fields, we are also able to obtain such derived completion theorems for equivariant algebraic K-theory but with respect to actions of diagonalizable group schemes. These enable us to obtain a wide range of applications, several of which are also explored.