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In the mid 1980s, while working on establishing completion theorems forequivariant Algebraic K- Theory similar to the well-known Atiyah-Segalcompletion theorem for equivariant topological K-theory, the late RobertThomason found the strong finiteness conditions that are required in suchtheorems to be too restrictive. Then he made a conjecture on the existence of acompletion theorem in the sense of Atiyah and Segal for equivariant AlgebraicG-theory, for actions of linear algebraic groups on schemes that holds withoutany of the strong finiteness conditions that are required in such theoremsproven by him, and also appearing in the original Atiyah-Segal theorem. Themain goal of the present paper is to provide a proof of this conjecture in asbroad a context as possible, making use of the technique of derived completion,and to consider several of the applications.  Our solution is broad enough to allow actions by all linear algebraic groups,irrespective of whether they are connected or not, and acting on anyquasi-projective scheme of finite type over a field, irrespective of whetherthey are regular or projective. This allows us therefore to consider theEquivariant Algebraic G-Theory of large classes of varieties like all Toricvarieties (for the action of a torus) and all Spherical varieties (for theaction of a reductive group). Restricting to actions by split tori, we are alsoable to consider actions on Algebraic Spaces. These enable us to obtain a widerange of applications, some of which are briefly sketched and which we plan toexplore in detail in the future. A comparison of our results with previouslyknown results, none of which made use of derived completions, shows thatwithout the use of derived completions, one can only obtain results which areindeed very restrictive.

Atiyah-Segal Derived Completions for Equivariant Algebraic G-Theory and K-Theory