  In the mid 1980s, while working on establishing completion theorems for
equivariant Algebraic K-Theory similar to the well-known completion theorems
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 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. 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.
